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Knowledge Spaces Mathematica Package - Technical Report

1. And EqualSetsQ kx K Al Union A a Check whether a partial intepretation kx already exists kx0 Select KXs 1 a amp If kx0 No simply add new interpretation KXs1 CPAInterpretationsInterpretationAdd K Xs kx Yes join old partial and new partial interpretation kx1 kxO 1 1 Union kx0 1 2 kx 2 KXs1 CPAInterpretationsInterpretationReplace K Xs kx 1 I 95 KSMP Techical Report Andrej Zaluski Return Mapping A1 PsK KXs1 CPAInterpretationsDeleteltemInterpretations K Xs q returns the list of interpratations where all the interpretations on the item q were removed from the original list KXs of interpretations CPAInterpretationsDeleteltemInterpretations KXs0_ q_ Module KXs KXs Select KXs0 1 q amp Return SetDifference K Xs0 K Xs CPAInterpretationFunctionConformTo k returns the interpretation function obtained from the given interpretation function k by reducing its set of items to a competence based item set i e by eliminating non interpreted and non informative interpretations CPAInterpretationFunctionConformTo Mapping A0_ PK_ KXs0_ Module K Apply Union PK KXs A K Apply Union PK select interpretations not conforming to the interpretation function definition KXs Select K Xs0 SetEmptyQ 2 Or EqualSetsQ 2 K amp create the list containing only conforming assertions KXs Se
2. VERBOSITY If OptionVerbosity gt 10 Print Contradiction found for the assertion Aq J Therefore this Aq and its inferences are not open question but rather an missed negative inference therefore the inferences are also missed VeryNewPosAss VeryNewNegAss QEKCollectInferencesForNegativeAswer Q PosAss NegAss Aq tmp Union VeryNewPosAss PosAss PosAss tmp tmp Union VeryNewNegAss NegAss NegAss tmp tmp SetDifference OpenQ Union VeryNewPosAss VeryNewNegAss OpenQ tmp Break I i 1 Length OpenQ VERBOSITY If OptionVerbosity gt 10 Print Open questions after missing inferences added n OpenQ J End of main While J Calculates structure obtained by assertions and returns it in appropriate format assertions or space strucure If OptionOutputStructure Assertions tmp PosAss NegAss tmp QEK Output Q PosAss NegAss OutputStructure gt SpaceStructure I VERBOSITY If OptionVerbosity gt 5 Print Established structure by querying n tmp SONDE Write data to the querying session sonde If OptionQSRReport Report Or OptionQSRReport Both QSR QSRSondeQuestionsAskedReplace QSR NofQuestions l VERBOSITY SONDE If OptionVerbosity gt 10 amp amp OptionQSRReport Report Or OptionQSRReport Both Print Final querying session sonde status QSR l What to return Switch OptionQSRRepor
3. Writeltems usage Writeltems FN Items stores the description on the items Items into the file FN MakeSkill usage MakeSkill Code Name Desc returns the data structure representing a skill consisting of the unique identifier Code the name Name and the description Desc SkillCode usage SkillCode Skill returns the code of the given skill Skill SkillName usage SkillName Skill returns the name of the given skill Skill SkillDescription usage SkillDescription Skill returns the description of the given Skill SkillGet usage SkillGet Skills Code returns the first skill described in Skills whose code matches Code ReadSkills usage ReadSkills FN KSMP_SkillsFile reads the description of skills stored in the file FN using the file format KSMP_SkillsFile n n ReadSkills FN assumes the file format KSMP_SkillsFile WriteSkills usage WriteSkills FN Skills stores the description on the skills Skills into the file FN MakeExpert usage MakeExpert MK CE returns the data structure representing an expert consisting of domain s metaknowledge MK and the uncertainty CE of making a careless error during a querying session ExpertMetaKnowledge usage ExpertMetaKnowledge E returns a domain metaknowledge of the expert E ExpertCarelessErrors usage ExpertCarelessErrors E returns the uncertainty that the expert E makes a careless error during a querying session MakeMe
4. Dol B QEKAssertionPremise AIIBs i Bstar QEKStar PosAss B If SubsetQ A Bstar Aminus Union Aminus SetDifference Q Bstar 1 1 Length AIIBs VERBOSITY If OptionVerbosity gt 10 Print Calculating negative inferences If OptionVerbosity gt 15 Print Old NewNegAss NewNegAss Add new negative inferences to the existing ones tmp Union NewNegAss Map A amp Aminus NewNegAss tmp VERBOSITY If OptionVerbosity gt 10 Print New negative inferences n NewNegAss Next A from the new block or end of Scan i e kBlockAs Add NewPosAss NewNegAss in PosAss and NegAss and update OpenQ tmp SetDifference OpenQ Union NewPosAss NewNegAss OpenQ tmp tmp Union NewPosAss PosAss PosAss tmp tmp Union NewNegAss NegAss NegAss tmp VERBOSITY If OptionVerbosity gt 10 Print Open questions after incomplete inference rules n OpenQ J Find missing negative inferences of incomplete inference rules by finding contradictions in pseudoobservations within the new block ContradictionFound True While ContradictionFound ContradictionFound False Do Aq OpenQI i VeryNewPosAss VeryNewNegAss QEKCollectInferencesForPositiveAnswer Q PosAss NegAss Aq Any contradiction 119 KSMP Techical Report Andrej Zaluski If Intersection VeryNewPosAss NegAss Intersection VeryNewPosAss NegAss ContradictionFound True
5. EntailRelationFromKnowledgeSpace SF returns the entail relation from the given knowledge space SF EntailRelationFromKnowledgeSpace Structure S_ F_ Module Relation S Select PQxPQ S Function AB ForAllQ F Intersection AB 1 Implies Intersection AB 2 4 amp I EntailRelationToKnowledgeSpace ESR returns the knowledge space from the given entail relation ESR EntailRelationToKnowledgeSpace Relation S_ R_ Module Structure S Select PowerSet S Function K 86 KSMP Techical Report Andrej Zaluski ForAllQ R Function AB Intersection K AB 1 LImplies Intersection K AB 2 J EntailmentFromKnowledgeSpace SF returns the entailment from the given knowledge space SF EntailmentFromKnowledgeSpace Structure S_ F_ Module Relation S Select PQxQ S Function AB ForAllQ F Intersection AB 1 LImplies SetElementQ AB 2 amp l EntailmentToKnowledgeSpace ESR returns the knowledge space from the given entail relation ESR EntailmentToKnowledgeSpace Relation S_ R_ Module Structure S Select PowerSet S Function K ForAllQ R Function AB Intersection K AB 1 LImplies SetElementQ AB 2 K HI ShowsInterestingAssertions Ent returns only non redundants assertions of the given entailment Ent ShowInterestingAssertions Entailment_ Module Select Entailment 2
6. QueryText StringJoin Text1 Text2 Text3 nE nAssertion 1 Block Textl QEOSQueryTextsSkills all_minus_one introduction Text2 tmp QEOSQueryTextsSkills all_minus_one question Text3 Apply StringJoin tmp SkillName SkillGet Skills b QueryText StringJoin Text1 Text2 Text3 l True GENERIC Block Textl QEOSQueryTextsSkills generic introduction tmp Map SkillName SkillGet Skills amp A Ntmp Length tmp Text2 Apply StringJoin Table tmp i lt gt i 1 Ntmp 1 Text2 Apply StringJoin Text2 tmp Ntmp tmp QEOSQueryTextsSkills generic question Text3 Apply StringJoin tmp SkillName SkillGet Skills b QueryText StringJoin Text1 Text2 Text3 Text4 You may assume that chance factors such as careless errors and lucky guesses play no rule in the student s performance Return StringJoin QueryText Text4 QEOSQueryExpertOnCompetenceSpace E SkOrExp options queryies a human SkOrExp represents skills description or a simulated expert SkOrExp represents a simulated expert is queried on the set E of elementary competencies and a simulated expert SkOrExp represents the expert and a 123 KSMP Techical Report Andrej Zaluski structure either space or assertions is returned The valid option is QueryingAlgorithm Koppen default For additional valid options see QEK QueryExpertByKoppen QEOSQueryE
7. States FX QK CAnswers NCAnswers If SetCardinality States gt 1 q KADSelectNextQuestionAuxFirst Structure DeleteCases Q Union CAnswers NCAnswers States Return q KAAskStudent Stud_ q_ Simulation Module CF 1 LV3 False tmp 135 KSMP Techical Report Andrej Zaluski If SubsetQ q StudentKnowledge Stud LV3 True tmp StudentMakeAnswer q LV3 CF Return tmp KAAskStudent Items_ q_ Human Module answer CF 1 LV answer Input K A FormulateQuestion Items q If answer ItemCorrectAnswer ItemGet Items q LV True _LV False Return StudentMakeAnswer q LV CF KADSelectNextQuestion Structure S_ F_ Module SF Structure S F KADSelectNextQuestion SF KADSelectNextQuestion Structure S_ F_ CNC_List Module SF Structure S F q C CNC 1 NC CNC 2 If Union C NC q KADSelectNextQuestionAuxFirst SF q KADSelectNextQuestionAuxNonFirst SF C NC l Return q KADAssessKnowledge Structure Q K ItemsStudent_ opts__ Module OpenQ OpenK q Qy Qn Items Student Hist OptionAssessmentResult OptionExaminee OptionVerbosity ans tmp OptionVerbosity Verbosity opts Options KADAssessKnowledge OptionExaminee Examinee opts Options KADAssessKnowledge OptionAssessmentResult AssessmentResult opts Options KADAssessKnowledge VERBOSITY If OptionVerbosity gt 1 Print KSMP v1 0 Deterministic Knowled
8. 2 R And SubsetQ 2 1 5 RD implies 1 2 amp RelationTransitiveQ SR returns True if the relation SR is transitive RelationTransitiveQ Relation SO_ RO_ Module S S0 R RO ForAlITriplesQ S S S SubsetQl 1 2 R And SubsetQ 2 3 R Implies SubsetQl 1 31 RIE RelationConnectedQ SR returns True iff the relation SR is a connected relation RelationConnectedQ Relation SO_ RO_ Module S S0 R RO ForAlIPairsQ S SubsetQl 01 2 R Or SubsetQl 2 ALB RD amp RelationEquivalenceQ SR returns True is the relation SR is an equivalence relation RelationEquivalenceQ SR_Relation Module 78 KSMP Techical Report Andrej Zaluski RelationReflexiveQ SR And RelationSymmetricQ SR And RelationTransitiveQ SR RelationQuasiOrderQ SR returns True if the relation SR is a quasi order RelationQuasiOrderQ SR_Relation Module RelationReflexiveQ SR And RelationTransitiveQ SR RelationPartialOrderQ SR returns True if the relation SR is a partial order RelationPartialOrderQ SR_Relation Module RelationQuasiOrderQ SR And RelationAntiSymmetricQ SR RelationLinearOrderQ SR returns True if the relation SR is a linear order RelationLinearOrderQ SR_Relation Module RelationPartialOrderQ SR And RelationConnectedQ SR NA RowBox KSAxiom1 SF returns
9. Answer J Draw the inferences PosAnsInf NegAnsInf Add both the answer and inferences update the list of judged interpretation assertions If ExpertAnswerLogicalValue Answer PosAnsInf QEOSInterpretationFunctionCollectInferencesForPositiveAnswer OpenQ ExpertAnswerAssertion Answer tmp Join PosIAss PosAnsInf PosIAss tmp NegAnsInf QEOSInterpretationFunctionCollectInferencesForNegativeAnswer OpenQ 129 KSMP Techical Report Andrej Zaluski ExpertAnswerAssertion Answer J Update the list of open questions tmp SetDifference OpenQ Union NegAnsInf PosAnsInf OpenQ tmp VERBOSITY If OptionVerbosity gt 5 Print The answer and its inferences added to the known interpretation assertions I VERBOSITY If OptionVerbosity gt 10 Print Positive answer inferences n PosAnsInf Print Negative answer inferences n NegAnsInf J VERBOSITY If OptionVerbosity gt 5 Print New positively judged interpretation assertions n PosAnsInf Print Open questions left n OpenQ End of While Form the established interpretation function k CPA InterpretationFunctionUpgrade CPAInterpretationFunctionInitialize A K QEOSInterpretationsFromJAssertions PosIAss VERBOSITY If OptionVerbosity gt 10 Print Established interpretations by querying n FunctionMapping k Keep only the interpretations for compet
10. Begin Private MakeItem Code Desc QText CorrAns LofDist returns data object representing an item consisting of unique identifier Code description Desc text of the question QText correct Answer CorrAns and the list of distractors LofDist Makeltem code_ description_ text_ CorrectAnswer_ LofDistractors_List Module code description text CorrectAnswer LofDistractors ItemCode Item returns the code of the given Item ItemCode item_ Module item 1 ItemDescription Item returns the description of the given Item ItemDescription item_ Module item 2 ItemText Item returns the question text of the given Item ItemText item_ Module item 3 ItemCorrectAnswer item_ Module item 4 ItemAnswers item_ Module item 5 74 KSMP Techical Report Andrej Zaluski ItemGet Items code returns the first item its code matches Code ItemGet Items_ code_ Module found found Select Items Function item ItemCode item code If found found 1 ReadItems FN KSMP_ItemsFile reads the information about the collection of items stored within the file FN according to the fileformat KSMP_ItemsFile ReadItems FN_String KSMP_ItemsFile Module concat LinesPerltem 5 Nofltems itemsfile i itemsfile DataFileCommentDrop DataFileRead FN Nofltems Length itemsfile LinesPerltem For i 0 i lt LinesPerltem Nofltems AppendTo
11. InfoTypes CommentSign 2 Head 3 InfoType InfoType Select InfoTypes 1 InfoType0 amp 1 1 2 Return Select FileFormatData 1 FileFormat amp 1 InfoType DataFileHeadExtract SFL FileFormat extracts the head from non commented data list SFL representing the datafile object in accordance with the datafile format FileFormat For the supported file formats see FileFormatsInfo DataFileHeadExtract SFL_ DataFileFormat_String Module Return SFL FileFormatsInfo DataFileFormat Head DataFileBodyExtract SFL FileFormat extracts the body from non commented data list SFL representing the datafile object in accordance with the file format FilFormat For the supported file formats see FileFormatsInfo DataFileBodyExtract SFL_ DataFileFormat_String Module SFL Complement Range Length SFL FileFormatsInfo DataFileFormat Head J DataFileHeadInterpret H FF interprets the extracted head H from the data file object of the FF format DataFileHeadInterpret H_List FF_String Module Switch FF KST_SpaceFile DFHI H KST_SpaceFile KST_PatternFile DFHI H KST_SpaceFile Hidden converts a list of strings to the list of numbers DFHI H_ KST_SpaceFile Map ToExpression amp H DataFileBodyInterpret B FF interprets the extracted body B from the data file object of the FF format DataFileBodyInterpret B_List FF_String
12. QEOSInterpretationFunctionCountInferences QEOSInterpretationFunctionCountInferences IASSes IAss returns the data structure Neg Pos containing the number Neg of inferences drawn if a negative answer on the interpretation assertion Ass has been collected and the number Pos of inferences drawn in case a positive answer has been collected The list ASSes of open interpretation assertions is given in advance QEOSInterpretationFunctionSelectionRuleMaximin QEOSInterpretationFunctionSelectionRuleMaximin OpenQ returns the interpretation assertion which implies a maximal number of inferences according to the maximin selection rule The list OpenQ of open interpretation assertions is given QEOSInterpretationFunctionSelectionRuleWeightedSums QEOSInterpretationFunctionSelectionRuleWeightedSums OpenQ returns the interpretation assertion which implies a maximal number of inferences according to the weighted sums selection rule The list OpenQ of open interpretation assertions is given QEOSInterpretationsFrom Assertions QEOSInterpretationsFromIJAssertions IASSes returns the list of interpretations derived from the list ASSes of interpretation assertions QEOSInterpretationsOnItem QEOSInterpretationsOnItem IASSes q returns all interpretation assertions given the item q and the list ASSes of interpretation assertions QEOSQueryExpertOnCompetenceSpace QEOS QueryExpertOnCompetenceSpace E SkOrExp options queries a human SkOrExp
13. Select M 1 x amp 1 1 2 SetElementQ x FunctionMappingOriginal Mapping D0 C0 M0 FunctionValue functions x applies a list of functions from right to left to value of x FunctionValue f_Mapping x_ FunctionValue f x FunctionValue functions_List x_ FunctionValue First functions FunctionValue Rest functions x FunctionValue f D1 returns a list of function values of the elements of the set D1 FunctionValue f_Mapping D0_List Module D D0 Map FunctionValue f amp D SubsetQ D0 FunctionDomain f NA RowBox RowBox RowBox FunctionValue functions D1 applies a list of functions from right to the left on the subset of the first domain DIR SIN IndentingNewLine FunctionValue functions List DO_List Module D DO IndentingNewLine Map FunctionValue functions amp D IndentingNewLine SubsetQ D0 FunctionDomain Take functions 1 WC NGV IF FunctionSubsetImage f D1 returns the image of the subset D1 under the function f FunctionSubsetImage f_ Mapping D0_List Module D Union D0 Union Map FunctionValue f amp D SubsetQ D0 FunctionDomain f 59 KSMP Techical Report Andrej Zaluski FunctionInjectionQ f tests whether the function fis an injection F
14. SkillDescription Skill returns the description of the given skill Skill SkillGet SkillGet Skills Code returns the first skill described in Skills whose code matches Code SkillIName SkillName Skill returns the name of the given skill Skill SSAttributionQ SSAttributionQ LofCl returns True iff a non empty collection of clauses is assigned to each item within the mapping LofCL of items to clauses SSAxiom1Q SSAxiom1Q LofCl returns True iff the mapping LofCl of items to clauses is an attribution SSAxiom2atItemQ SSAxiom2atItemQ q Clauses returns True iff the item q is an element of all clauses that are assigned to it in the collection Clauses of clauses 36 KSMP Techical Report Andrej Zaluski SSAxiom2Q SSAxiom2Q LofCl returns True iff each item in the mapping LofCL is member of all clauses assigned to it SSAxiom2Q Q LofCl returns True iff each item from Q is member of all clauses assigned to it by the mapping LofCL of items to clauses SSAxiom3Q SSAxiom3Q LofCl returns True iff for each clause C of any item q given by the mapping LofCl there exist a clause C1 assigned to each item q1 of the clause C that is a subset of the clause C SSAxiom4atClausesQ SSAxiom4atClausesQ clauses returns True iff any two clauses for the same item are incomparable SSAxiom4Q SSAxiom4Q LofCl returns True iff any two clauses of any item within the mapping LofCl are incomparable SSClausesOflItem
15. Block flagtmp1 flagtmp2 flagtmp1 flag flagtmp2 SubsetQ Intersection subfamiliescollection LeftDoubleBracket j RightDoubleBracket F flag flagtmp amp amp flagtmp2 If LInvisibleSpace flag Break G If LInvisibleSpace flag Break G flag GridBaseline gt Baseline 1 1 Column Widths gt 0 999 ColumnAlignments gt Left s GridBaseline gt Baseline 1 1 ColumnAlignments gt Left Definition KSAxiom3C Editable gt False FXFilterOrderDown SF f returns all the subsets F1 from the structure SF that are subsets of the subset f FXFilterOrderDown Structure S_ F_ f_ Module SF Structure S F Select F SubsetQ f amp FXFilterOrderUp SF f returns all the subsets F1 from the structure SF that are supersets of the subset f FXFilterOrderUp Structure S_ F_ f_ Module SF Structure S F Select F SubsetQ f amp GenerateFamilyByTakingUnions F returns the family generated by taking unions on the elements from a family F GenerateFamilyByTakingUnions F_List FamilyQ Module j myset F Flatten Table Map Apply Union amp KSubsets myset j j 1 Length myset 11 AtomsAt SF x returns the list of atoms of the structure SF at the element x 82 KSMP Techical Report Andrej Zaluski AtomsAt Structure S_ F_ x0_ Module SF Structu
16. KSMP Techical Report Andrej Zaluski Installation Software and hardware requirements portability and versioning The package has been developed and tested on a Pentium 233 MHz computer equipped with 96 MB of RAM running Windows 2000 Professional operating system Mathematica version 4 0 was used No effort has been taken to check backwards compatibility with 3 x versions of Mathematica In general any hardware operating system configuration with installed at least version 4 x Mathematica may be regarded as sufficient for running the package However an applicant has to be aware that the amount of available memory will greatly influence usability of the package for applications In order to ensure compatibility with the existing knowledge space tools to facilitate exchange of data with other software products e g SPSS and to allow easy migration of the data between different hardware operating systems it was decided to use pure ascii as the format for the data files It was decided so despite expected decrements in the package performance occurring at some points caused by such decision The version 1 0 is the first publicly available version of the package Installing the package The directory KnowledgeSpaces contains all package files including documentation This directory should be on the search path The simplest way would be to place the directory in one of the standard locations for packages for example in ZopDirectory AddOns
17. KSMPListToBinary KSMPListToBinary L List Module max b i max Max I b Table 0 max Dof b max 1 1 i 51 i 1 Length 1 Return b Apply And Map IntegerQ amp amp Positive amp 1 KSMPBinaryToList KSMPBinaryToList BL_List Module n Length BL i L Do If BL i 1 AppendTo L n 1 i i n 1 1 Return L Apply And Map I 0 amp BL KSMPBinaryToWarp BL_List Module Return FromDigits BL 2 4 Apply And Map 1 0 amp BL KSMPListToWarp 0 KSMPListToWarp L_ List Module Return FromDigits KSMPListToBinary L 2 Apply And Map IntegerQ amp amp Positive amp L KSMPWarpToList 0 KSMPWarpToList W_ WarpQ Module KSMPBinaryToList IntegerDigits W 2 J KSMPWarpToBinary W_ WarpQ IntegerDigits W 2 End EndPackage Print KSMP v1 0 Warp Core OK Print Knowledge Spaces Mathematica Package Print C 2001 Andrej Zaluski Free software see attached license 44 KSMP Techical Report Andrej Zaluski KSMP layer one Title Knowledge Spaces Mathematica Package Layer One Context KnowledgeSpaces L1 Author Andrej Zaluski Summary The package KnowledgeSpaces LI represents the abstract data types layer one of the KnowledgeSpaces Mathematica Package KSMP The aim of this package is to be an interface between basic Mathematica da
18. Module Switch FF KST_SpaceFile DFBI B KST_SpaceFile KST_PatternFile DFBI B KST_SpaceFile 65 KSMP Techical Report Andrej Zaluski HIDDEN Rewrites the body of the KST_Space file data file object to a matrix of numbers DFBI B_ KST_SpaceFile Module j Table RowString2List B j j 1 Length B RowList2String R rewrites a list R of numbers as a string RowList2String Row_List Module StringJoin Map ToString amp Row RowString2List R rewrites a string of 1 digits as the list of numbers RowString2List Row_String Module i Table ToExpression StringTake Row i i i 1 StringLength Row Matrix2DataFileBody M_List KST_SpaceFile creates the datafile object body according to KST_SpaceFile format from a matrix M Matrix2DataFileBody M_List KST_SpaceFile Module i Table RowList2String M i 1 1 Length M Matrix2DataFileHead M KST_SpaceFile produces the head part of datafile object according to KST_SpaceFile format from a matrix M Matrix2DataFileHead M_List KST_SpaceFile Module Map ToString amp Length M 1 Length M DataFileWrite FN M KST_SpaceFile writes a matrix M together with the derived head in the datafile FN respecting the KST_SpaceFile format DataFileWrite FileName_String M List KST_SpaceFile Module Export FileName Join Matrix2DataFileHead M KST_SpaceF
19. OptionQueryOn QueryOn opts Options QEK QueryExpertByKoppen Whether to query human or simulated expert 114 KSMP Techical Report Andrej Zaluski OptionHumOrSim Expert opts Options QEK QueryExpertByKoppen The kind of output structure OptionOutputStructure OutputStructure opts Options QEK QueryExpertByKoppen What the function should return as its output OptionQSRReport QSRReport opts Options QEK QueryExpertByKoppen Verbal report on querying or not OptionVerbosity Verbosity opts Options QEK QueryExpertByK oppen If OptionVerbosity gt 1 Print QEOS v1 0 Querying Session Report nKoppen 1993 block by block algorithm used for querying expert OptionHumOrSim on OptionQueryOn Print Function returns Result Report or Both OptionQSRReport Print Resulting structure OptionOutputStructure 1 VERBOSITY If OptionHumOrSim Human amp amp OptionQueryOn KnowledgeSpace amp amp OptionVerbosity gt 5 Print Description of used items n ItemsExpert J If OptionHumOrSim Human amp amp OptionQueryOn CompetenceSpace amp amp OptionVerbosity gt 5 Print Description of used skills n ltemsExpert J If OptionHumOrSim Simulation amp amp OptionVerbosity gt 5 Print Simulated expert n ItemsExpert J Construct the 1st block and initialize it premise size k 1 OpenQ Pos
20. 1 i 2 Length candidates Return MaxAss 1 QEKSelectNextQuestion Q Pos Neg OpenQ options returns the data structure NextAss NegAnsInf PosAnsInf where NextAss is a next assertion for querying accompanied with its both positive and negative inferences NegAnsInf in case of a negative answer and PosAnsInf in case of a positive answer Querying is conducted on set Q of items assuming previously established positive inferences Pos negative inferences Neg open questions OpenQ and it is based on the selection rule defined by the option SelectionRule 113 KSMP Techical Report Andrej Zaluski that accepts the following values Maximin default WeightedSums QEKSelectNextQuestion Q PosAss_ NegAss OpenQ opts Module rule NextAss tule SelectionRule opts Options QEKSelectNextQuestion Switch rule WeightedSums NextAss QEKSelectionRuleWeightedSums Q PosAss NegAss OpenQ SelectionRule gt WeightedSums _ NextAss QEK SelectionRuleMaximin Q PosAss NegAss OpenQ J Return NextAss QEKCollectInferencesForNegativeAnswer Q PosAss NextAss QEKCollectInferencesForPositiveAnswer Q PosAss NegAss NextAss QEKOutput Q PosAss NegAss options returns the output of the querying proces either as knowledge space or as assertions dependingly on the OutputStructure option The OutputStructure option accepts the following values SpaceStructure default As
21. Functions Structures DataFiles Report and Miscellaneous component The aim of this layer is to be an interface between basic Mathematica data structures and functions at one side and functions implementing concepts and procedures from the knowledge space theory at the other The aim of the component DataFiles is to ensure reading and writing of data files both in KSMP native formats as well as in formats used by existing knowledge spaces tools especially KST and SRBT tools The aim of Report component is to equip both the functions from the layer two and the modules with facilities for easy producing uniform reports of interest to an applicant The aim of the components Sets Relations and Functions are to represent selected concepts from the set theory theory of relations and theory of functions in a way suitable for basing the layer two and the modules on them having the aims of the package as the ultimate goal The component Structures represents the concept of structure as it is defined and used within the knowledge space theory The component Miscellaneous is a place for putting all public function that might be of interest mostly for a developer which have not found their place in the other components The list of implemented functions is available in the glossary of KSMP functions KSMP Techical Report Andrej Zaluski KSMP Warp core Warp core is an additional module to the layer one Its purpose is to provide data structures and
22. SSAxiom4Q LofCl returns true iff any two clauses of any item within the mapping LofCl are incomparable SSAxiom4Q LofCl_ ForAlQ LofCl SSAxiom4atClausesQ amp SurmiseFunctionQ f returns true iff the function f represents a surmise function SurmiseFunctionQ Mapping Q_ Null LofCl_ Module If SetNonEmptyQ Q And EqualSetsQ Q Transpose LofCl 1 And SSAxiom1 Q LofCl And SSAxiom2Q LofCl And SSAxiom3Q LofCl And SSAxiom4Q LofCl True False SurmiseSystemQ QS returns true iff the Mathematica object QS represents a surmise system SurmiseSystemQ Q0_ Mapping Q_ Null LofCl_ Module Sigma Mapping Q Null LofCl Apply And Q0 Q SurmiseFunctionQ Sigma MakeSurmiseFunction Q LofCl forms the Mathematica object representing the surmise function based on the set Q and the mapping LofCl of clauses if these objects fulfill necessary requirements MakeSurmiseFunction Q_ LofCl_ Module If SurmiseFunctionQ Mapping Q Null LofCl Mapping Q Null LofCl 84 KSMP Techical Report Andrej Zaluski MakeSurmiseSystem Q LofCl forms the Mathematica object representing the surmise system based on the set Q and the mapping LofCL of clauses if these objects fulfill necessary requirements MakeSurmiseSystem Q_ LofCl_ Module If SurmiseFunctionQ Mapping Q Null LofCl Q Mapping Q Null LofC1 SSSurmiseSet SS returns the set forming the surmi
23. True iff the structure SF contains both the empty set and the whole set S n InterpretationBox GridBox GridBox KSAxiom1 Structure S_List F_List SubsetQ S FI GridBaseline gt Baseline 1 1 ColumnWidths gt 0 999 ColumnAlignments gt Left s GridBaseline gt Baseline 1 1 ColumnAlignments gt Left Definition KSAxiom1 Editable gt False NA RowBox KSAxiom2S SF returns True iff the structure SF is stable under union n InterpretationBox GridBox GridBox KSAxiom2S Structure S_List F_List ForAllPairsQ F Function pair Block f pair LeftDoubleBracket 1 RightDoubleBracket g pair LeftDoubleBracket 2 RightDoubleBracket SubsetQ f Union g F s 79 KSMP Techical Report Andrej Zaluski GridBaseline gt Baseline 1 1 Column Widths gt 0 999 ColumnAlignments gt Left s GridBaseline gt Baseline 1 1 ColumnAlignments gt Left Definition KSAxiom2S Editable gt False Na RowBox KSAxiom3S SF returns True iff the structure SF is stable under intersection n InterpretationBox GridBox GridBox KSAxiom3 Structure S_List F_List ForAllPairsQ F Function pair Block f pair LeftDoubleBracket 1 RightDoubleBracket g pair LeftDoubleBracket 2 RightDoubleBracket SubsetQ f
24. make the package appropriate both for applicants and developers The package is conceived as consisting of a core various modules and applications that form the four abstract layers of the package The core s purpose is to ensure availability of basic data structures and functions representing the basic knowledge space theory concepts to the modules as well as to ensure both flexible and uniform way of working with data files and to provide reporting facilities The modules are conceived to have functionalities corresponding to the different topics and approaches within the knowledge space theory The modules are forming the layer three of the package The layer four is reserved for applications that resides mostly on using the modules The layers and modules are consisting of components that mostly correspond to different concepts topics and approaches in the knowledge space theory In this package version the core of the package consists of the two layers KSMP layer one and KSMP layer two with an additional module KSMP warp core added to the layer one The layer three consist of the following modules skills module querying module and assessment module The first application planed for the layer four is QEOS whose aim is to provide an user friendly interface to an applicant interested in establishing the structures by different methods of querying experts KSMP Layer one The layer one consists of the following components Sets Relations
25. reads the structure stored in the file FN taking KST_SpaceFile as default data file format ReadStructure FN KST_SpaceFile reads the structure stored in the file FN taking KST_SpaceFile as the data file format ReadStructure FN SC reads the structure stored in the file FN and rewrites it using the carrier set SC ReadStructure FN KST_SpaceFile SC reads the structure stored in the file FN and rewrites it using the carrier set SC ReadStructure FN_String ReadStructure FN KST_SpaceFile Normalized ReadStructure FN_String SC_List ReadStructure FN KST_SpaceFile SC ReadStructure FN_String KST_SpaceFile Normalized StructureFromMatrix DataFileBodyInterpret DataFileBodyExtract DataFileCommentDrop DataFileRead FN KST_SpaceFile KST_SpaceFile Normalized ReadStructure FN_String KST_SpaceFile SC_List StructureFromMatrix DataFileBodyInterpret DataFileBodyExtract DataFileCommentDrop DataFileRead FN KST_SpaceFile KST_SpaceFile SC WriteStructure FN SF writes the structure SF in the file FN with KST_SpaceFile as default datafile format WriteStructure FN SF KST_SpaceFile writes the structure SF in the file FN according to KST_SpaceFile datafile format WriteStructure Filename_String SF_ Structure WriteStructure Filename SF KST_SpaceFile WriteStructure Filename_String SF Structure KST_SpaceFile 63 KSMP Techical Report Andrej Zalu
26. returns the datafile object body from the matrix M using KST_SpaceFile format Matrix2DataFileHead usage Matrix2DataFileHead M KST_SpaceFile returns the head of a datafile object from the matrix M usingKST_SpaceFile format DataFileWrite usage DataFileWrite FN M KST_SpaceFile writes the matrix M together with the derived head in the datafile FN using the KST_SpaceFile format n nDataFileWrite FN Items KSMP_ItemsFile writes the description Items into the file FN using the format KSMP_ItemsFile n nDataFileWrite FN Skills KSMP_SkillsFile writes the description Skills into the file FN using the format KSMP_SkillsFile ForAllQ usage ForAllQ X pf returns True iff the property pf holds for all elements of the set X ExistOneQ usage ExistOneQ X pf returns True iff the property pf holds for at least one element of the set X ForAllPairsQ usage ForAl PairsQ X pf returns True iff the predicate pf gives True for all ordered pairs from X n nForAllPairsQ X Y pf returns True iff the predicate pf gives True for all pairs from X x Y ForAllTriplesQ usage ForAllTriplesQ X pf returns True iff the predicate pf gives True for all ordered triples from X ForAllTriplesQ X Y Z pf returns True iff the predicate pf returns True for all ordered triples from X x Y x Z Verbosity usage Verbosity is the default of the option Verbosity Verbosity 0 global default Verbosity u
27. Ass is positively judged APx and added to the previous positive assertions PosAss QEKInferences14b QEK Inferences 14b PosAss NegAss Ass returns the negative inferences contributed by the inference rule APx amp AxPB amp ANy gt BNy when the assertion APx is added to the previous positive PosAss and negative NegAss assertions QEKInferences14c QEK Inferences 14c Q PosAss NegAss Ass returns the negative inferences contributed by the inference rule APX amp ByPA amp BNx gt BNy when the assertion APx is added to the previous positive PosAss and negative NegAss assertions given the set Q of items QEKInferences14d QEKInferences14d Q PosAss Ass returns the negative inferences contributed by the inference rule ANx amp APB amp ByPx gt BNy if ANx is added to the previous negative assertions NegAss given the set Q of items QEK InitializeFirstBlock QEKInitializeFirstBlock Q returns the data structure OQ P N representing the first block in Koppen s algorithm and consisting of open questions OQ positive assertions P and negative assertions N QEK MakeAssertions QEKMakeAssertions Q Premise returns all possible assertions given the premise Premise and the set Q of items assuming that only one element in the consequence of the assertion is present QEKMakeFullBlock QEK MakeFullBlock Q k returns the block list of the assertions of the subsets by items table defined by the given premise size k and the set Q of items Q
28. B SetQ A And SetQ B SetComplement U S returns the complement of the set S in the universe U SetComplement U_ S_ SetDifference U S SubsetQ S U SetQ X_ If Head X List Head X Integer True False SetEmptyQ S_ WarpQ If S 0 True False SetNonEmptyQ S_ WarpQ Not SetEmptyQ S SetCardinality S_ WarpQ If S 0 Apply Plus KSMPWarpToBinary S 0 SetIntersection A_ WarpQ B_ WarpQ BitAnd A B SetUnion A_ WarpQ B_ WarpQ BitOr A B SubsetQ W1_ WarpQ W2_ WarpQ SetIntersection W1 W2 W1 EqualSetsQ W1_ WarpQ W2_ WarpQ W1 W2 SubsetProperQ A_ WarpQ B_ WarpQ SubsetQ A B amp amp Not EqualSetsQ A B SetElementQ e_Integer Positive B_ WarpQ SubsetQ KSMPListToWarp e B 54 KSMP Techical Report Andrej Zaluski SetDifference A_ WarpQ B_ WarpQ Module BLA KSMPWarpToBinary A BLB KSMPWarpToBinary B nA nB n i nA Length BLA nB Length BLB n Max nA nB Do PrependTo BLA 0 i nA n 1 Do PrependTo BLB 0 i nB n 1 KSMPBinaryToWarp BLA BLB 1 gt 0 SetComplement U_ WarpQ A_ WarpQ SetDifference U A MakeRelation S R returns the data object Relation for the given set S and the sebset R of the Cartesian product SxS MakeRelation Set_List Pairs_List Relation Set Pairs MakeRelation S1 S2 R returns the data object representing the relation from the set S1 to the set S2 given by the subset R of the Cartesian Product S1xS2
29. ExtraPackages at system level or alternatively at user s level in PreferencesDirectory AddOns Applications If from any reason it would be desirable that the package resides in a specific non standard location e g MyPackagesDir this could be for example accomplished by issuing the command PrependTo Path MyPackagesDir The package can be autoloaded by issuing the command lt lt KnowledgeSpaces which can also be put in init m file Autoloading the package is advised because this approach eliminates unpleasant situation that might occur when one of its functions is called before the package has been actually loaded Alternatively all packages modules can be loaded at any time using Needs KnowledgeSpaces Querying Needs KnowledgeSpaces Assessment KSMP Techical Report Andrej Zaluski Package documentation The package documentation consists of the technical report user s manual and on line documentation based on these two documents The on line documentation is accessible via the Help Browser under the Add ons Knowledge Spaces node After installing the package it is necessary to rebuild the help index in order to make the package on line documentation visible This can be achieved by choosing Rebuild Help Index from the Help menu Additionally standard ways of getting assistance for built in functions are present for all the package functions as well Using the command one can see the description of a func
30. Intersection g F s GridBaseline gt Baseline 1 1 Column Widths gt 0 999 ColumnAlignments gt Left s GridBaseline gt Baseline 1 1 ColumnAlignments gt Left Definition KSAxiom3S Editable gt False NA RowBox KnowledgeStructureQ SF tests whether the structure SF satisfies formal requirements of a knowledge structure n InterpretationBox GridBox GridBox KnowledgeStructureQ Structure S_List SetNonEmptyQ F_List KSAxiom1 Structure S F GridBaseline gt Baseline 1 1 Column Widths gt 0 999 ColumnAlignments gt Left s GridBaseline gt Baseline 1 1 ColumnAlignments gt Left Definition KnowledgeStructureQ Editable gt False Na 80 KSMP Techical Report Andrej Zaluski RowBox KnowledgeSpaceQ SF tests whether the structure SF satisfies formal requirements of a knowledge space n InterpretationBox GridBox GridBox KnowledgeSpaceQ Structure S_List SetNonEmptyQ F_List KSAxiom1 Structure S F amp amp KSAxiom2C Structure S F GridBaseline gt Baseline 1 1 Column Widths gt 0 999 ColumnAlignments gt Left GridBaseline gt Baseline 1 1 ColumnAlignments gt Left Definition KnowledgeSpaceQ Editable gt False KnowledgeSpaceQuasiOrdinalQ SF tests whether the structure SF satisfies
31. LofCl SurmiseSystemFromKnowledgeSpace QK returns the surmise system representation of the given knowledge space QK SurmiseSystemFromKnowledgeSpace Structure Q_ K_ Module QK Structure Q K SSSF 85 KSMP Techical Report Andrej Zaluski SSSF SurmiseFunctionFromKnowledgeSpace QK Return Q SSSF SurmiseSystemFromBasis B returns the surmise system equivalent to the knowledge space represented by the basis B SurmiseSystemFromBasis B_ Module SurmiseSystemFromKnowledgeSpace BasisToSpace B SurmiseFunctionFromBasis B returns the surmise function of the knowledge space represented by the basis B SurmiseFunctionFromBasis B_ SSSurmiseFunction SurmiseSystemFromBasis B PQxQ Q returns the cartesian product betwen the set Q and its powerset from which the empty set was substracted PQxQ X_ Module SetCartesianProduct SetDifference PowerSet X x SetNonEmptyQ X PQXPQ Q returns the cartesian product on the powerset of the set Q from which the empty set was substracted PQxQ X_ Module SetCartesianProduct SetDifference PowerSet X SetDifference PowerSet X 4 SetNonEmptyQ X RelationRan SR a returns the set of all elements for which holds that the element a is in the relation SR with them RelationRan Relation S_ R_ a_ Map 2 amp Select R 1 a amp SubsetQ a S
32. MakeRelation S1_List S2_List Pairs_List Relation S1 S2 Pairs RelationQ SR returns True is SR is a relation RelationQ Relation S_ R_ SubsetQ R SetCartesianProduct S S RelationQ Relation S1_ S2_ R_ SubsetQ R SetCartesianProduct S 1 S2 Defining primitive functions RelationSet SR returns S from the relation S R RelationSet Relation S_ R_ S RelationSets SR returns the sets S1 S2 forming the relation S1 S2 R RelationSets Relation S1_ S2_ R_ S1 S2 RelationPairs SR returns the pairs R from the relation S R or S1 S2 R RelationPairs Relation S_ R_ R RelationPairs Relation S1_ S2_ R_ R HIDDEN Within S R replaces S by S1 RelationChangeSet Relation S_ R_ S1_ Module SR Relation S R Relation S1 RelationPairs SR HIDDEN Wihtin S1 S2 R replaces S1 by S11 S2 by S22 RelationChangeSets Relation S1_ S2_ R_ S11_ S22_ Module SR Relation S1 S2 R Relation S11 S22 RelationPairs SR HIDDEN Within S R replaces R by R1 RelationChangePairs Relation S R_ R1_ Module SR Relation S R Relation RelationSet SR R1 HIDDEN Within S1 S2 R replaces R by R1 RelationChangePairs Relation SA_ SB_ R_ R1_ Module SR Relation SA SB R Relation RelationSets SR 1 RelationSets SR 2 R1 55 KSMP Techical Report Andrej Zaluski RelationSetCardinality SR returns the cardinality of the set forming the r
33. QEK AssertionConsequence Ass Ax Ys B Ax Union A x B Union QEKStar PosAss Ax Ys Union QEKStar NegAss A If B NotEqual And Ys NotEqual Map B amp Ys QEKInferences 14c Q PosAss NegAss 111 KSMP Techical Report Andrej Zaluski Ass returns all the negative inferences made by the inference rule APx amp ByPA amp BNx gt BNy if the assertion Apx would be added to the previous positive PosAss and negative NegAss assertions given the set Q of items QEKiInferences14c Q_ PosAss_ NegAss_ Ass_ Module A QEKAssertionPremise Ass x QEKAssertionConsequence Ass Bs BNys tmp tmp1 Bs Map QEK AssertionPremise amp Select NegAss QEK AssertionConsequence x amp BNys Dol B Bs i tmp Select Map Union B amp Q EqualSetsQ QEKStar PosAss 1 A amp tmp1 Union BNys tmp BNys tmp1 i 1 Length Bs Return BNys QEKInferences14d Q PosAss Ass returns the negative inferences contributed by the inference rule ANx amp KAPB amp ByPx gt BNy if ANx would be added to the previous negative assertions NegAss given the set Q of items QEKInferences14d Q_ PosAss_ Ass_ Module A QEKAssertionPremise Ass x QEKAssertionConsequence Ass B B QEKStar PosAss A Select Map Union B amp Q SetElementQ 1 x PosAss amp QEKCollectInferencesForNegativeAnswer Q PosAss Ass returns the d
34. RelationChangeCarriers Relation S01_ S02_ RO_ Items01_List Items02_List Module Items1 Items2 S1 S2 R Changes1 Changes2 tmp1 tmp2 Items 1 Union Items0 1 Items2 Union Items02 S1 S01 S2 S02 R RO tmp1 tmp2 Changes 1 Table S 1 i gt Items1 i 1 1 Length S1 Changes2 Table S2 i gt Items2 i 1 1 Length S2 S1 S1 Changes1 S2 S2 Changes2 tmp1 Transpose R 1 Changes1 tmp2 Transpose R 2 Changes2 R Transpose tmp1 tmp2 Return Relation S1 S2 R 56 KSMP Techical Report Andrej Zaluski Length Items01 Length S01 amp amp Length Items02 Length S02 amp amp Length Items01 SetCardinality Items01 amp amp Length Items02 SetCardinality Items02 The changing mappings are bijective functions ElementsInRelationQ SR a b tests whether elements a and b are in the relation SR ElementsInRelationQ Relation S_ R_ a_ b_ Module SetElementQ a b R SubsetQ a b S ElementsInRelationQ Relation S1_ S2_ R_ a_ b_ Module SetElementQ a b R SubsetQ a S1 And SubsetQ b S2 RelationMaximalQ SR Alpha tests whether the element Alpha is the maximal with respect to the relation SR RelationMaximalQ Relation S_ R_ Alpha _ Module SR Relation S R Apply And Map ElementsInRelationQ SR Alpha Implies Alpha Equal amp RelationSet SR SetElem
35. SetElementQ 2 1 amp EntailmentToEntailRelation AAPx returns the entail relation derived from the entailment AAPx EntailmentToEntailRelation Relation S_ ENT_ Module As A B0 q Bs As Union Transpose ENT 1 Bs Apply Join Map Function A Block BO Map 2 amp Select ENT Block B 1 B A amp Map A amp SetComplement PowerSet BO As J Return Relation S Bs EntailmentFromEntailRelation AAPB returns the entailment derived from the entail relation AAPB EntailmentFromEntailRelation Relation Q E_ Module As P As Union Transpose E 1 P Apply Union Map Function A Block B0 za dani A formiraj BO BO Apply Union Apply Union Select E Block A1 1 AI A amp Map Function b A b B0 As I Return Relation Q P 87 KSMP Techical Report Andrej Zaluski End EndPackage 88 KSMP Techical Report Andrej Zaluski KSMP skills module gt Title Knowledge Spaces Mathematica Package Skills Module gt Context KnowledgeSpaces Skills gt Author Andrej Zaluski gt Summary This package contains the Skills Module of the KnowledgeSpaces Mathematica package The Skills Module implements various approaches for handling latent skills and competencies within the field of knowledge spaces Copyright 20001 Andrej Zaluski See attached
36. Techical Report Andrej Zaluski KSMP querying module Title Knowledge Spaces Mathematica Package Querying Module Context KnowledgeSpaces Querying Author Andrej Zaluski gt Summary Copyright 20001 Andrej Zaluski See attached license Package Version 0 9 beta 1 Mathematica Version 4 0 History 1 0 reserved for the first public version 0 9 beta 1 version Keywords Psychology and Education Mathematical Psychology Knowledge Modeling Knowledge Space Theory Querying Experts Sources Warnings Limitations Discussion Requirements Packages KnowledgeSpaces Skills BeginPackage KnowledgeSpaces Querying KnowledgeSpaces L1 KnowledgeSpaces L2 KnowledgeSpaces Skills TODO Save the state of the package Off General spell General spell1 QEK AssertionPremise usage QEK AssertionPremise Assertion returns the premise contained in the assertion Assertion QEK AssertionConsequence usage QEK AssertionConsequence Assertion returns the consequence contained in the assertion Assertion 99 KSMP Techical Report Andrej Zaluski QEK MakeAssertions usage QEK MakeAssertions Q Premise returns all possible assertions given the premise Premise and the set Q of items assuming that only one element in the consequence
37. The aim of the components Modeling expert and Modeling student is to provide data structures for describing domain metaknowledge of an expert or domain knowledge of a student Additionally these descriptions also provide place for uncertainty parameters representing careless errors or lucky guesses of a student or an expert or a subjective certainty in a response or an answer given Expert s metaknowledge can be described either using a structure or a diagnostic but also using alternative solutions such as lists of positive negative and undecidable assertions having different uncertainty parameters Such modeled experts and students are to be used in simulation studies The aim of the components Theory of relations and order theory Knowledge structures Basis Surmise Systems and Entail relations is to implement selected concepts from the corresponding knowledge space theory topics in the package The list of implemented functions is available in the glossary of KSMP functions KSMP Skills module The aim of the Skills module is to implement various approaches for handling latent skills and competencies existing in the knowledge space theory In this package version only the competence performance approach is implemented inside the Component performance component of the module The querying module especially its component that implements querying experts on skills depends on this module The list of implemented functions is available in the g
38. True if SF is a family of subsets FamilyQ F_ SetQ F StructureSet SF returns the set forming the structure SF StructureSet Structure X_ F_ X StructureFamily SF returns the family forming the structure SF StructureFamily Structure X_ F_ F StructureReplaceSet SF X1 primitive function that replaces the set forming the structure SF 60 KSMP Techical Report Andrej Zaluski with the set X1 StructureReplaceSet Structure X_ F_ X1_ Structure X1 F StructureReplaceFamily SF F1 primitive function that replaces the family forming the structure with the family F1 StructureReplaceFamily Structure X_ F_ F1_ Structure X F1 StructureSetEmptyQ SF returns True if the set forming the structure SF is empty StructureSetEmptyQ Structure X_ F_ SetEmptyQ X StructureSetNonEmptyQ SF returns True if the set forming the structure SF is non empty StructureSetNonEmptyQ Structure X_ F_ SetNonEmptyQ X StructureFamilyEmptyQ SF returns True if the family forming the structure SF is empty StructureFamilyEmptyQ Structure X_ F_ SetEmptyQ F StructureFamilyNonEmptyQ SF returns True if the family forming the structure SF is non empty StructureFamilyNonEmptyQ Structure X_ F_ SetNonEmptyQ F StructureSetElementQ SF s tests whether s is a element of the set forming the structure SF StructureSetElementQ Structure S_ F_ s_ SetElementQ s S Struc
39. Y pf returns True iff the predicate pf gives True for all pairs from X x Y ForAllPairsQ X_List SetQ Y_ List SetQ pf Function ForAlQ SetCartesianProduct X Y pf ForAlTriplesQ X pf returns True iff the predicate pf gives True for all the ordered triples from X ForAllTriplesQ X_List SetQ pf_Function ForAllTriplesQ X X X pf ForAlTriplesQ X Y Z pf returns True iff the predicate pf returns True for all the ordered triples from X x Y x Z ForAllTriplesQ X_List SetQ Y_ List SetQ Z List SetQ pf Function Module ForAllQ SetCartesianProduct X Y Z pf End EndPackage 67 KSMP Techical Report Andrej Zaluski KSMP layer two Title Knowledge Spaces Mathematica Package Layer Two Context KnowledgeSpaces L2 Author Andrej Zaluski gt Summary See KSMP technical report Copyright 20001 Andrej Zaluski See attached license Package Version 0 9 beta 1 Mathematica Version 4 0 History 1 0 reserved for the first public version 0 9 beta 1 version Keywords Psychology and Education Mathematical Psychology Knowledge Modeling Knowledge Space Theory Sources Warnings Limitations See Limitations of KSMP layer 1 Discussion Requirements Packages DiscreteMath Combinatorica KnowledgeSpaces Skills Modeling Expe
40. a returns True iff the element a is a maximal with respect to the relation SR RelationMinimalQ RelationMinimalQ SR a returns True iff the element a is a minimal with respect to the relation SR RelationPairs RelationPairs SR returns the set R of ordered pairs forming the relation S R or S1 S2 R RelationPairsCardinality RelationPairsCardinality SR returns the cardinality of the family of ordered pairs forming the relation SR RelationPartialOrderQ RelationPartialOrderQ SR returns True if the relation SR is a partial order RelationQ RelationQ SR returns True iff SR is a relation RelationQuasiOrderQ RelationQuasiOrderQ SR returns True iff the relation SR is a quasi order 34 KSMP Techical Report Andrej Zaluski RelationRan RelationRan SR a returns all elements from the set forming the relation SR that are in the relation with the element a RelationReflexiveQ RelationReflexiveQ SR returns True iff the relation SR is reflexive RelationSet RelationSet SR returns the set forming the relation SR RelationSetCardinality RelationSetCardinality SR returns the cardinality of the set forming the relation SR RelationSets RelationSets SR returns the data structure S1 S2 consisting of the sets forming the relation SR from the set S1 to the set S2 RelationSetsCardinality RelationSetsCardinality SR returns the cardinalities of the sets forming the relation SR RelationSubsetPr
41. a data structure representing a student s domain knowledge given as the set QK of items MakeSimpleStudent usage MakeSimpleStudent QK returns an ideal student whose domain knowledge is given by the set QK of items StudentMakeAnswer usage StudentMakeAnswer QE LV CF returns the data structure for a student s answer consisting of the asked question QE logical value LC describing the correctness of the answer and the certainty factor CF describing the student s certainty in the given answer StudentAnswerltem usage StudentAnswerltem Ans returns the answered item within the student s answer Ans StudentAnswerLogicalValue usage StudentAnswerLogical Value Ans returns the logical value of the student s answer Ans StudentAnswerCertaintyFactor usage StudentAnswerCertaintyFactor Ans returns the certainty factor of the student s answer Ans RelationReflexiveQ usage RelationReflexiveQ SR returns True iff the relation SR is reflexive RelationAReflexiveQ usage RelationAReflexiveQ SR returns True iff the relation SR is areflexive RelationSymmetricQ usage RelationSymmetricQ SR returns True iff the relation SR is symmetric RelationAntiSymmetricQ usage RelationAntiSymmetricQ SR returns True iff the relation SR is antisymmetric RelationTransitiveQ usage RelationTransitiveQ SR returns True iff the relation SR is transitive RelationConnectedQ usage RelationConne
42. given diagnostic Diag CPADiagnosticInterpretationFunction usage CPADiagnosticInterpretationFunction Diag returns the interpretation function forming the given diagnostic Diag CPADiagnosticRepresentationFunction usage CPADiagnosticRepresentationFunction Diag returns the representation function forming the given diagnostic Diag CPAEqualDiagnosticsQ usage CPAEqualDiagnosticsQ Diag 1 Diag2 returns True iff the diagnostics Diag and Diag are equal CPARepresentationStructureFromInterpretationFunction usage CPARepresentationStructureFromInterpretationF unction k returns the induced performance structure from the given interpretation function k CPARepresentationStructureQ usage CPARepresentationStructureQ AP k returns True iff the given performance structure AP is the representation structure under the interpretation function k CPADiagnosticFromInterpretationFunction usage CPADiagnosticFromInterpretationFunction k returns the 6 tupel diagnostic induced by the interpretation function k It is assumed that the competence structure given by k is based on a set of elementary competencies Begin Private CSAxiom1Q EK returns True iff the structure fulfills Union K E property CSAxiom1QJ Structure E_List SetNonEmptyQ K_List FamilyQ Module EqualSetsQ Apply Union K E PSAxiom1Q AP returns True iff the structure fulfills Union P A property PSAxiom
43. its family with the family F1 StructureReplaceSet StructureReplaceSet SF X1 is a primitive function that returns the structure SF after replacing its set with the set X1 38 KSMP Techical Report Andrej Zaluski StructureSet StructureSet SF returns the set forming the structure SF StructureSetElementQ StructureSetElementQ SF s returns True iff the element s is a member of the set forming the structure SF StructureSetEmpty StructureSetEmptyQ SF returns True iff the set forming the structure SF is empty StructureSetNonEmptyQ StructureSetNonEmptyQ SF returns True iff the set forming the structure SF is non empty StructureSetSubsetQ StructureSetSubsetQ SF S1 returns True iff the set S1 is a subset of the set forming the structure SF StructureSubsetProperQ StructureSubsetProperQ SF1 SF2 returns True iff the structure SF1 is a proper subset of the structure SF2 Both structures are based on the same set StructureSubsetQ StructureSubsetQ SF1 SF2 returns True iff the structure SF1 is contained in the structure SF2 Both structures are based on the same set StructureToMatrix StructureToMatrix SF returns the matrix representation of the structure SF StructureToMatrix SF Normalized returns the matrix representation of the structure SF under assumption that the structure SF is normalized StudentAnswerCertaintyFactor StudentAnswerCertaintyFactor Ans returns the certainty factor of the studen
44. n n WriteStructure FN SF assumes KST_SpaceFile as the datafile format FX usage FX SF s returns all members of the family forming the structure SF which contain the element s of the set forming the structure n nFX SF C NC returns all members of the family forming the structure SF which contain all the elements from the set C and have empty intersectio with the set NC n nFX SF C returns all members of the family forming the structure SF which contain all the elements from the set C DataFileRead usage DataFileRead Filename creates a datafile object representing the data file Filename DataFileLength usage DataFileLength SFile returns the number of lines in the datafile object SFile LineCommentQ usage LineCommentQ L returns True iff the line L of a datafile object contains a comment CommentLines usage CommentLines SFile returns the list of line numbers from the datafile object SFile that contain comments NonCommentLines usage NonCommentLines SFile returns the list of all line numbers of the datafile object that do not contain a comment DataFileCommentExtract usage DataFileCommentExtract SFile returns the list representing comments from the datafile object SFile DataFileCommentPrint usage DataFileCommentPrint DF extracts and prints the comments from the datafile object DF DataFileCommentDrop usage DataFileCommentDrop SFile returns the list rep
45. of the assertion is present QEK MakeFullBlock usage QEK MakeFullBlock Q k returns the block list of the assertions of the subsets by items table defined by the given premise size k and the set Q of items QEKMakeSbITable usage QEK MakeSbITable Q returns the subsets by items table given the set Q of items QEK QueryTexts usage QEK QueryTexts PremiseSize TextPart returns the partial text of a query on an assertion given the text part TextPart introduction or question and the premise size PremiseSize 1 2 generic previous_to_last QEK FormulateQuery usage QEK FormulateQuery Q Items Assertion returns the query question on the given assertion Assertion formulated for a human expert given the set Q of items described in Items QEKAskExpert usage QEK AskExpert Q Items Assertion Human returns an answer from a human experts on assertion Assertion containing some items from the set Q described in Items n n QEKAskExpert Assert Exp Simulation returns an answer from the simulated expert Exp on the assertion Assert The expert s domain meta knowledge may be represented either using a structure or a diagnostic QEKExtractPositiveJudgments usage QEKExtractPositiveJudgments QH returns all positive judgments from the querying history QH QEKExtractNegativeJudgments usage QEKExtractNegativeJudgments QH returns all negative judgments from the querying history QH QEK QueryExpertStraig
46. represents a description of skills or a simulated expert SkOrExp represents a simulated expert on structure of the set E of elementary competencies and returns the established structure either as a space or assertions A valid option is QueryingAlgorithm Koppen default For additional valid options see QEK QueryExpertByKoppen QEOSQueryExpertOnDiagnostic QEOS QueryExpertOnDiagnostic E A DescOrExp opts queries a human expert DescOrExp having the form Skills Items describes skills and items or a simulated expert DescOrExp represents a simulated expert whose metaknowledge is given as a diagnostic on competence performance diagnostic The set A of items and the set E of elementary competencies are given The valid options are Expert Human default or Simulation QSRReport Result default Report or Both For other valid options see QEOSQueryExpertOnCompetenceSpace and QEOSQueryExpertOnInterpretationFunction 31 KSMP Techical Report Andrej Zaluski QEOSQueryExpertOnInterpretationFunction QEOS QueryExpertOnInterpretationFunction K A DescOrExp opts establishes an interpretation function by querying an expert given the set A of items and the competence space K After the querying only the competence based items set together with its interpretations is kept In case of a human expert DescOrExp has the form Skills Items where the used skills are described in Skills and the items in Items In case of a simulated expert De
47. structure to which the state will belong to is normalized StateBinary2Set B S1 returns the set representation of the state given its binary list representation B and the structure carrier set S1 StateBinary2Set B_List Normalized Flatten Position B 1 StateBinary2Set B_List S1_List Module Return StateBinary2Set B Normalized Map 1 gt 2 amp Transpose Range Length B S1 Length B Length S 1 StateSet2Binary f NS Normalized returns the binary representation of the state f under assumption that the normalized structure carrying set S contains NS elements StateSet2Binary f SC returns the binary representation of the state f taking into account the carrier set SC of the structure to which it belongs to StateSet2Binary f_List NS_Integer Normalized Module tmp ZeroRow ZeroRow Table 0 NS tmp ZeroRow tmp f 1 tmp StateSet2Binary f List S List Module NS SetCardinality S StateSet2Binary f Map 1 gt 2 amp Transpose S Range NS NS Normalized SubsetQ f S StructureNormalize SF rewrites normalizes the structure SF taking the set 1 2 3 as the set carrying the structure StructureNormalize SF S1 rewrites the structure SF taking into account S1 as the set carrying the structure StructureNormalize SR_Structure StructureNormalize SR Range SetCardinality StructureSet SR StructureNormalize Struct
48. 1 Keywords knowledge space theory knowledge assessment Sources Warnings BETA 1 TESTING PHASE Limitations Discussion Requirements KnowledgeSpaces L2 BeginPackage KnowledgeSpaces Assessment KnowledgeSpaces L1 KnowledgeSpaces L2 KAAskStudent usage KAAskStudent Stud QE Simulation returns an answer from the simulated student Stud on the question QE n nKAAskStudent Items QE Human returns an answer from a human student on the question QE described in Items KADSelectNextQuestion usage K A DSelectNextQuestion SF CA NCA returns the next item to be asked by the deterministic knowledge assessment procedure on the knowledge structure SF assuming already collected correct CA and incorrect NCA answers by a student n nKADSelectNextQuestion SF assumes that no previous answers have been collected KADAssessKnowledge usage K A DAssessKnowledge SF ItemsStudent options performs the deterministic assessment of a student s knowledge given the knowledge structure SF In case of a human student option Examinee value Human default the parameter ItemsStudent takes the description of 134 KSMP Techical Report Andrej Zaluski items while in case of a simulated student option Examinee value Simulation it stores a simulated student The function may return either a knowledge state option Assessm
49. 1 Q Structure A_List SetNonEmptyQ P_List FamilyQ Module 92 KSMP Techical Report Andrej Zaluski EqualSetsQ Apply Union P A CSAxiom2Q EK returns True iff the structure fulfills the property that the family K contains both and E as its subsets CSAxiom2Q Structure E_List SetNonEmptyQ K_ List FamilyQ Module SubsetQ E K CSAxiom2Q EK returns True iff the structure fulfills the property that the family K contains both and E as its subsets CSAxiom2Q Structure E_List SetNonEmptyQ K_ List FamilyQ Module SubsetQ E K PSAxiom2Q AP returns True iff the structure fulfills the property that the family P contains both and A as its subsets CSAxiom2Q Structure A_List SetNonEmptyQ P_List FamilyQ Module SubsetQ A P CSAxiom3Q EK returns True iff the structure EK fulfills the property of being stable under union CSAxiom3Q Structure E_List SetNonEmptyQ K_List FamilyQ Module ForAllPairsQ K Function pair Block f pair 1 g pair 2 SubsetQ f Union g K WM PSAxiom3Q EK returns True iff the structure EK fulfills the property of being stable under union PSAxiom3Q Structure A_List SetNonEmptyQ P_List FamilyQ Module ForAlIPairsQ P Function pair Block f pair 1 g pair 2 SubsetQ f Union g P mM CompetenceStructureQ EK returns True iff the structure EK is a competence structure CompetenceStructu
50. A SubsetProperQ A B tests whether the set A is a proper subset of the set B SubsetProperQ A0_List BO_List SubsetQ A0 B0 And A0 NotEqual B0 SetElementQ a S tests whether a is an element of the set S SetElementQ a0_ SO_List Module a a0 S S0 Which a SubsetQ S True SubsetQ List a S 53 KSMP Techical Report Andrej Zaluski SetNonEmptyQ S returs true if the set S is non empty SetNonEmptyQ S_ If Length S NotEqual 0 True False SetEmptyQ S returns true if the set S is empty SetEmptyQ S_ If Length S 0 True False SetCartesianProduct S1 S2 returns the Cartesian Product of the non empty sets S1 and S2 SetCartesianProduct S1 S2 returns the Cartesian product of the sets S1 S2 SetCartesianProduct S1_List SetNonEmptyQ S2_List SetNonEmptyQ CartesianProduct S1 S2 SetCartesianProduct L_List Module N Length L tmp If N 2 Return SetCartesianProduct L 1 L 2 tmp Map Flatten 1 amp SetCartesianProduct First L Map Flatten 1 amp SetCartesianProduct Rest L Return tmp PowerSet X returns the powerset of the set X with the ranking where all the subsets with the size k 1 appear before the subsets of the size k PowerSet X_ Module i Apply Join Table KSubsets X i 1 0 Length X SetDifference A B returns the set difference A B SetDifference A_ B_ Complement A
51. A PK KXs k Apply Union PK Return Mapping K PowerSet A CPARepresentations k CPAMakeRepresentation C Items returns the data structure containing the representation formed by the given competence state C and the Items set CPAMakeRepresentation C0_ Items_ C0 Items CPARepresentationCompetenceState PC returns the competence state of the given representation PC CPARepresentationCompetenceState PC_ PC 1 CPARepresentationItems pC returns the items of the given representation pC CPARepresentationltems r_List r 2 CPAMakeRepresentationFunction K A PCs returns the data structure containing the representation function formed from the competence structure K the set A of problems and the collection PCs of representations CPAMakeRepresentationFunction K_ A_ PCs_ Mapping K PowerSet A PCs CPARepresentationFunctionCompetenceStructure p returns the competence structure forming the given representation function p CPARepresentationFunctionCompetenceStructure Mapping K_ PA_ PCs_ K CPARepresentationFunctionRepresentations p returns the collection of representations forming the CPARepresentationFunctionRepresentations Mapping K_ PA_ PCs_ PCs CPAMakeDiagnostic K A P k p returns the data structure representing the diagnostic given by the family forming the competence structure K the set A of problems the family forming the performance structure P the interpretat
52. ARepresentationStructureFromInterpretationFunction k returns the induced performance structure from the given interpretation function k CPARepresentationStructureQ CPARepresentationStructureQ AP k returns True iff the given performance structure AP is the representation structure under the interpretation function k CSAxiom1Q CSAxiom1Q EK returns True iff the structure EK fulfills the property JK E CSAxiom2Q CSAxiom2Q EK returns True iff the structure fulfills the property that the family K contains both the empty set and the set E CSAxiom3Q CSAxiom3Q EK returns True iff the structure EK is stable under union DataFileBodyExtract DataFileBodyExtract SFL FileFormat returns the extracted body from the non commented data list SFL representing the datafile object using the file format FileFormat For the supported file formats see FileFormatsInfo DataFileBodyInterpret DataFileBodyInterpret B FF interprets the extracted body B from the datafile object in the FF format DataFileCommentDrop DataFileCommentDrop SFile returns the list representing the datafile object SFile after removing comments DataFileCommentExtract DataFileCommentExtract SFile returns the list containing comments from the datafile object SFile DataFileCommentPrint DataFileCommentPrint DF extracts and prints the comments from the datafile object DF 19 KSMP Techical Report Andrej Zaluski DataFileHeadExtract DataFileHea
53. Ass NegAss QEK InitializeFirstBlock Q Number of questions asked NofQuestions 0 SONDE Create and initialize a querying session sonde if requested If OptionQSRReport Report Or OptionQSRReport Both QSR QSRMakeSonde 0 0 QSR QSRSondeOpenQuestionsInitialReplace QSR Length OpenQ QSR QSRSondeQuestionsAskedReplace QSR NofQuestions VERBOSITY SONDE If OptionVerbosity gt 10 amp amp OptionQSRReport Report OptionQSRReport Both Print Initial querying session sonde status QSR l 115 KSMP Techical Report Andrej Zaluski While a new constructed block of questions is not empty While OpenQ VERBOSITY If OptionVerbosity gt 5 Print New block of questions constructed nOpen questions n OpenQ I VERBOSITY If OptionVerbosity gt 5 Print Known positive assertions n PosAss Print Known negative assertions n NegAss J Premises of the open questions belonging to the current block kBlockAs Map 1 amp OpenQ Fill the block in with both inferences and answers Any open questions in the current block While OpenQ NotEqual Get the next question store possible inferences using the following inference rules 14a 14b 14c 14d Koppen 1993 Next QEKSelectNextQuestion Q PosAss NegAss OpenQ NextAss Next 1 NegAnsInf Next 2 PosAnsInf Next 3 Query a human or simulated expert f
54. CPADiagnosticRepresentationFunction Diagnostic _ p_ p CPADiagnosticRepresentationFunction Diagnostic _ p_ p CPAEqualDiagnosticsQ Diag1 Diag2 returns True iff the diagnostics Diag1 and Diag2 are equal CPAEqualDiagnosticsQ Diagnostic E _ K1_ Al_ Pl_ kl pl_ Diagnostic E2_ K2_ A2_ P2_ k2 p2_ EqualSetsQ E1 E2 amp amp EqualSetsQ K1 K2 amp amp EqualSetsQ A 1 A2 amp amp EqualSetsQ P1 P2 amp amp EqualFunctionsQ k1 k2 amp amp EqualFunctionsQ p1 p2 CPARepresentationStructureFromInterpretationFunction k returnes the induced performance structure from the given interpretation function k CPARepresentationStructureFromInterpretationFunction Mapping A0_ PKO KXs0_ Module k Mapping A0 PK0 KXs0 k Apply Union PKO P P Union Map CPARepresentationOfCompetenceState k 2 amp K Return Structure A P CPADiagnosticFromInterpretationFunction k returns the 6 tupel diagnostic induced by the interpretation function k It is assumed that is the competence structure given through k is represented using a set of elementary competencies CPADiagnosticFromInterpretationFunction Mapping A_ PK KXs_ Module E K Apply Union PK P k Mapping A PK KXs p E Apply Union K p CPARepresentationFunctionFromInterpretationFunction k P CPARepresentationStructureFromInterpretationFunction k Return CPAMakeDiagnostic E K A P 2 k p End EndPackage 98 KSMP
55. EKMakeSbITable QEKMakeSbITable Q returns the subsets by items table given the set Q of items 29 KSMP Techical Report Andrej Zaluski QEKOutput QEKOutput Q PosAss NegAss options returns the output of a querying process either as a knowledge space or as assertions depending on the OutputStructure option The querying process is captured by the positive assertions PosAss and the negative assertions NegAss given the set Q of items The OutputStructure option accepts the following values SpaceStructure default and Assertions QEK QueryExpertByKoppen QEK QueryExpertByK oppen Q ItOrSkOrExp options queries an expert using Koppen s 1993 block by block algorithm A human ItOrSkOrExp represents a description of either items or skills or a simulated expert ItOrSkOrExp represents a simulated expert is queried on the set Q of items and a structure either space or assertions is returned The valid options are Expert Human default or Simulation OutputStructure SpaceStructure default or Assertions Verbosity Off default or 1 2 QueryOn KnowledgeSpace default or CompetenceSpace and QSRReport Result default Report or Both QEKQueryExpertStraightforward QEK QueryExpertStraightforward Q Items Human returns the judgments obtained while querying a human expert on structure of the items Q described in Items QEK QueryExpertStraightforward Q Expert Simulation returns the judgments obtained while querying the simul
56. FunctionInjectionQ f returns True iff the function f is an injection FunctionMapping FunctionMapping f returns the mapping rule of the function f FunctionMappingImage FunctionMappingImage f returns the function s image calculated using the mapping rule of the function f FunctionMappingOriginal FunctionMappingOriginal f returns the function s original calculated using the mapping rule of the function f FunctionQ FunctionQ F returns True iff F represents a function FunctionRestriction FunctionRestriction f D1 returns the function f restricted to its subdomain D1 FunctionSubsetImage FunctionSubsetImage f D1 returns the image of the subset D1 under the function f FunctionSubsetImage functions D 1 returns the image of the subset D1 after applying a list of functions from right to left FunctionSurjectionQ FunctionSurjectionQ f returns True iff the function f is a surjection 22 KSMP Techical Report Andrej Zaluski FunctionValue FunctionValue f x returns the function value f x of the element x FunctionValue functions x returns the value of x after applying the list of functions from right to left FunctionValue f D1 returns values of the elements from the set D1 after applying the function f FunctionValue functions D1 returns values of the elements from the set D1 after applying the list of functions from right to left FunctionWellDefinedQ FunctionWellDefinedQ f returns True iff th
57. Knowledge Spaces Mathematica Package Technical Report Andrej Zaluski University of Graz Version 1 0 May 2001 t Address Andrej Zaluski Institut fiir Psychologie Abteilung fiir Allgemeine Psychologie Karl Franzens Universitat Graz Universitatsplatz 2 3 A 8010 Graz Austria E mail andrej zaluski uni graz at KSMP Techical Report Andrej Zaluski Acknowledgements The work on Knowledge Spaces Mathematica Package was supported by the Ernst Mach fellowship BMBWK G 1621 Querying Experts on Skills to the author The author would like to thank Dietrich Albert Cord Hockemeyer and Christof K rner for support during the research KSMP Techical Report Andrej Zaluski License 1 This software package is provided as is but in hope to be useful No warranty either explicit or implicit is given no claim on fitness to any particular purpose made The author does not accept responsibility for consequences of using the package The entire risk as to the quality and performance of the program is with you The permission to run the package with no restrictions is given under these terms 2 Permission is granted to anyone to make or distribute verbatim copies of the package provided that the copyright notice and the permission notice are preserved and that the distributor grants the recipient permission for further redistribution as permitted by this notice 3 You may modify your copy or copies of the package or any por
58. MakeSimpleExpert SF returns an ideal expert possesing domains metaknowledge given by knowledge space SF MakeSimpleExpert usage MakeSimpleExpert Diag returns an ideal expert whose domain metaknowledge is given by the competence performance diagnostic Diag ExpertMakeAnswer usage ExpertMakeAnswer LV CF returns the data structure for an expert s answer consisting of the logical value LC and the certainty factor CF ExpertAnswerLogical Value usage ExpertAnswerLogicalValue Ans returns the logical value of the expert s answer Ans ExpertAnswerCertaintyFactor usage ExpertAnswerCertaintyFactor Ans returns the certainty factor of the expert s answer Ans MakeStudent usage MakeStudent QK CE LG returns the data structure representing a student whose domain knowledge is given by the set QK of items the student is capable of solving the uncertainty CE of making a careless error and the uncertainty LG of making a lucky guess during a knowledge assessment StudentKnowledge usage StudentKnowledge Stud returns the domain knowledge of the student Stud StudentCarelessErrors usage StudentCarelessErrors S returns the uncertainty that the student S makes a careless error StudentLuckyGuesses usage StudentLuckyGuesses S returns the uncertainty that the student S makes a lucky guess 70 KSMP Techical Report Andrej Zaluski MakeKnowledge usage MakeKnowledge QK Items returns
59. SSClausesOfltem SS q returns the clauses assigned to the item q by the surmise function forming the surmise system SS SSListOfClauses SSListOfClauses SS returns the list of clauses of the surmise system SS SSLofClClausesGet SSLofClClausesGet LofCl q returns the clauses assigned to the item q by the mapping LofC l of items to clauses SSSFClausesOfItem SSSFClausesOfltem SF q returns the clauses assigned to the item q by the surmise function SF SSSFListOfClauses SSSFListOfClauses SF returns the list of clauses of the surmise function SF SSSurmiseFunction SSSurmiseFunction SS returns the surmise function forming the surmise system SS SSSurmiseSet SSSurmiseSet SS returns the set forming the surmise system SS StateBinary2Set StateBinary2Set B Normalized returns the set representation of the state given its binary list representation B under assumption that the structure to which the state will belong to is normalized StateBinary2Set B SC returns the set representation of the state given its binary list representation B and the structure carrier set SC 37 KSMP Techical Report Andrej Zaluski StateSet2Binary StateSet2Binary f NS Normalized returns the binary representation of the state f under assumption that the cardinality of the normalized structure carrying set is NS StateSet2Binary f SC returns the binary representation of the state f taking into account the carrier set SC of the str
60. a careless error while querying ExpertCarelessErrors Expert_ Expert 2 MakeMetaKnowledge SF Structure returns a data structure representing experts knowledge in the form of structure for the given knowledge structure SF MakeMetaKnowledge Structure S_ F_ Structure Module SF Structure S F Return SF MakeMetaKnowledge PAss NAss UAss Assertions returns a data structure representing expert s metaknowledge in the form of assertions for separately given lists of positive assertions PAss negative assertions Nass and unknown assertions UAss MakeMetaKnowledge PosAss_ NegAss_ RestAss_ Assertions Module Return Assertions PosAss NegAss RestAss MakeMetaKnowledge Ent Assertions returns a data structure representing expert s metaknowledge in the form of assertions for the given Entailment Ent MakeMetaKnowledge Relation Q PosAss_ Assertions Module Ent Relation Q PosAss NegAss NegAss SetComplement PQxQ Q PosAss Return MakeMetaKnowledge PosAss NegAss Assertions 76 KSMP Techical Report Andrej Zaluski MetaKnowledgeAssertionsPositive Assertions MetaKnowledgeAssertionsPositive Assertions PA_ NA_ UN_ PA MetaKnowledgeAssertionsNegative Assertions MetaKnowledgeAssertionsNegative Assertions PA_ NA_ UN_ NA MetaKnowledgeAssertionsRest Assertions MetaKnowledgeAssertionsRest Assertions PA_ NA_ UN_ UN MakeMetaKnowledge DIAG Diag
61. after replacing the set S1 by the set S11 and the set S2 by the set 22 RelationChangePairs usage RelationChangePairs SR R1 is a primitive function that returns the relation based on the relation SR after replacing the set R of ordered pairs by the set R1 of ordered pairs RelationSetCardinality usage RelationSetCardinality SR returns the cardinality of the set forming the relation SR RelationSetsCardinality usage RelationSetsCardinality SR returns the cardinalities of the sets forming the relation R RelationPairsCardinality usage RelationPairsCardinality SR returns the cardinality of the family of ordered pairs forming the relation SR RelationChangeCarrier usage RelationChangeCarrier SR S 1 returns the relation SR after changing the original elements with the elements from the set S1 RelationChangeCarriers usage RelationChangeCarriers SR S 11 S22 returns the relation SR after changing the original elements from the set S1 with the elements from the set S11 and the elements from S2 with the elements from S22 respectively ElementsInRelationQ usage ElementsInRelationQ SR a b returns True iff the elements a and b are in the relation SR RelationMaximalQ usage RelationMaximalQ SR Alpha returns True iff the element Alpha is a maximal with respect to the relation SR RelationMinimalQ usage RelationMinimalQ SR Alpha returns True iff the element Alpha
62. al Report Andrej Zaluski system calculated from the basis B SurmiseFunctionFromBasis usage SurmiseFunctionFromBasis B returns the surmise function calculated from the basis B PQxQ usage PQxQ Q returns the cartesian product between the set Q and its powerset from which the empty set was substracted PQxPQ usage PQxPQ Q returns the cartesian product on the powerset of the set Q from which the empty set was substracted RelationRan usage RelationRan SR a returns all elements from the set forming the relation SR that are in the relation with the element a EntailRelationFromKnowledgeSpace usage EntailRelationFromKnowledgeSpace SF returns the entail relation given the knowledge space SF EntailRelationToKnowledgeSpace usage EntailRelationToKnowledgeSpace ESR returns the knowledge space given the entail relation ESR EntailmentFromKnowledgeSpace SF returns the entailment from the given knowledge space SF EntailmentToKnowledgeSpace ESR returns the knowledge space from the given entail relation ESR ShowsInterestingAssertions Ent returns only non redundants assertions of the given entailment Ent EntailmentToEntailRelation usage EntailmentToEntailRelation AA Px returns the entail relation derived from the entailment AAPx EntailmentFromEntailRelation usage EntailmentFromEntailRelation AAPB returns the entailment derived from the entail relation AAPB
63. an product of the non empty sets 1 S2 PowerSet usage PowerSet S returns the powerset of the set S By default ranking all subsets with the cardinality k 1 appear before the subsets with the cardinality k SetDifference usage SetDifference A B returns the set difference A B SetComplement usage SetComplement U S returns the complement of the set S in the universe U MakeRelation usage MakeRelation S R returns a data structure representing the relation given by the set S and the subset R of the cartesian product S xS n nMakeRelation S1 S2 R returns a data structure representing the relation 46 KSMP Techical Report Andrej Zaluski from the set S1 to the set S2 given by the subset R of the cartesian product S1xS2 RelationQ usage RelationQ SR returns True iff SR is a relation RelationSet usage RelationSet SR returns the set forming the relation SR RelationSets usage RelationSets SR returns the data structure S1 S2 consisting of the sets forming the relation S1 S2 R RelationPairs usage RelationPairs SR returns the set R of ordered pairs forming the relation S R or S1 S2 R RelationChangeSet usage RelationChangeSet SR S1 returns the relation based on the relation SR in which the set S was replaced by the set S1 RelationChangeSets usage RelationChangeSets SR S11 S22 is a primitive function that returns the relation based on the relation SR
64. apping LofCl of clauses under assumption that these objects fulfill the necessary formal requirements SSSurmiseSet usage SSSurmiseSet SS returns the set forming the surmise system SS SSSurmiseFunction usage SSSurmiseFunction SS returns the surmise function forming the surmise system SS SSSFListOfClauses usage SSSFListOfClauses SF returns the list of clauses of the surmise function SF SSListOfClauses usage SSListOfClauses SS returns the list of clauses of the surmise system SS SSSFClausesOfltem usage SSSFClausesOfltem SF q returns the clauses assigned to the item q by the surmise function SF SSClausesOfltem usage SSClausesOfltem SS q returns the clauses assigned to the item q by the surmise function forming the surmise system SS SurmiseSystemToBasis usage SurmiseSystemToBasis SS returns the basis calculated from the surmise system SS SurmiseSystemToKnowledgeSpace usage SurmiseSystemToKnowledgeSpace SS returns the basis calculated from the surmise system SS SurmiseFunctionFromKnowledgeSpace usage SurmiseFunctionFromKnowledgeSpace QK returns the surmise function calculated from the given knowledge space QK SurmiseSystemFromKnowledgeSpace usage SurmiseSystemFromKnowledgeSpace QK returns the surmise system representation of the given knowledge space QK SurmiseSystemFromBasis usage SurmiseSystemFromBasis B returns the surmise 73 KSMP Techic
65. at FileFormat Currently supported formats are KST_SpaceFile and KST_PatternFile The stored information on the formats is Head ForAllPairsQ ForAllPairsQ X pf returns True iff the predicate pf gives True for all ordered pairs from X ForAllPairsQ X Y pf returns True iff the predicate pf gives True for all pairs from X x Y ForAllQ ForAlQ X pf returns True iff the property pf holds for all elements of the set X ForAllTriplesQ ForAllTriplesQ X pf returns True iff the predicate pf gives True for all ordered triples from X ForAllTriplesQ X Y Z pf returns True iff the predicate pf returns True for all the ordered triples from X x Y x Z FunctionBijectionQ FunctionBijectionQ f returns True iff the function f is a bijection 21 KSMP Techical Report Andrej Zaluski FunctionChangeCodomain FunctionChangeCodomain f C 1 is a primitive function that returns the function f after replacing its codomain with the new set C1 FunctionChangeDomain FunctionChangeDomain f D 1 is a primitive function that returns the function f after replacing its domain with the new set D1 FunctionChangeMapping FunctionChangeMapping f M1 is a primitive function that returns the function f after replacing its mapping rule with the new mapping rule M1 FunctionCodomain FunctionCodomain f returns the codomain of the function f FunctionDomain FunctionDomain f returns the domain of the function f FunctionInjectionQ
66. ata structure P N consisting of positive inferences P and negative inferences N when negative answer on asswertion Ass is given The set Q of items and the list PosAss of established positive assertions are given in advance and inference rule implemented according to Koppen 14d QEKCollectInferencesForNegativeAnswer Q_ PosAss_ Ass_ Return QEKInferences14d Q PosAss Ass QEKCollectInferencesForPositiveAnswer Q PosAss NegAss Ass returns the datastructure P N consisting of positive inferences P and negative inferences N when possitive answer on assertion Ass is given The set Q of items lists PosAss and NegAss of established positive and negative assertions are given in advance and inference rule implemented according to Koppen 14a 14b 14c QEKCollectInferencesForPositiveAnswer Q_ PosAss_ NegAss_ Ass_ Module QEKInferences14a PosAss Ass Union QEKInferences14b PosAss NegAss Ass QEK Inferences 14c Q PosAss NegAss Ass QEKCountInferences Q PosAss NegAss Ass returns the data structure Neg Pos containg the number Neg of inferences that would be drawn in case 112 KSMP Techical Report Andrej Zaluski the negative answer is given on the assertion Ass and the number Pos of inferences in case of positive answer The set Q of items the established positive inferences PosAss and negative inferences NegAss are given QEKCountInferences Q_ PosAss_ NegAss_ Ass_ Modul
67. ate QEKAssertionPremise Assertion returns the premise of the Assertion QEKAssertionPremise Assertion_ Assertion 1 QEKAssertionConsequence Assertion returns the consequence of the Assertion QEKAssertionConsequence Assertion_ Assertion 2 QEKMakeAssertions Q Premise returns the list of all possible assertions for the given Premise and the set Q of items under the assumtion of only one element in the consequence of the assertion QEKMakeAssertions Q_ Premise_ Module Map Premise amp Q QEKMakeFullBlock Q k returns the block of the subsets by items table with the size of the premise k for the set Q of items QEKMakeFullBlock Q_ k_ Module Flatten Map Function A QEKMakeAssertions Q A KSubsets Q k 1 QEKMakeSbITable Q returns the subsets by items table for the given set Q of items 106 KSMP Techical Report Andrej Zaluski QEKMakeSbITable Q_ Module k n SetCardinality Q Flatten Table QEK MakeFullBlock Q k k 1 n 1 QEK QueryTexts PremiseSize TextPart returns the TextPart introduction or question for the premise of the size PremiseSize 1 2 generic previous_to_last QEKQueryTexts PremiseSize_ TextPart0_ Module QueryTexts TextParts TextPart TextParts introduction 2 question 3 QueryTexts 1 Suppose that a student under examination has just failed the problem Is it then practically certai
68. ate either on a human expert or on a simulated expert Querying reports having different extent of verbosity may easily be produced due to the global verbosity variable and the verbosity option that majority of the package functions have The list of implemented functions is available in the glossary of KSMP functions KSMP Assessment module The aim of the KSMP assessment module is to implement approaches to adaptive knowledge assessment existing within the knowledge space theory In this package version this module is included only for demonstration purposes and it consists only of a component for Deterministic knowledge assessment The list of implemented functions is available in the glossary of KSMP functions 11 KSMP Techical Report Andrej Zaluski File formats used KSMP supports three major data file formats spacefile itemsfile and skillsfile formats All of them are pure ascii formats due to the intention to preserve compatibility with the existing knowledge space tools as well as to ensure easy transfer of data between different hardware operating systems configurations and to facilitate exchange of data with other software products With respect to KSMP design the Datafile component of the package layer one is responsible for appropriate handling of data files Since other knowledge space tools primarily KST tools and SRBT tools are developing in parallel and independently of KSMP package some effort has been taken t
69. ated expert Expert on structure of the items Q QEK QueryTexts QEKQueryTexts PremiseSize TextPart returns the partial text of a query on an assertion given the text part TextPart introduction or question and the premise size PremiseSize 1 2 generic previous_to_last QEKSelectionRuleMaximin QEKSelectionRuleMaximin Q PosAss NegAss OpenQ returns the assertion which implies a maximal number of inferences according to the maximin selection rule QEKSelectionRuleWeightedSums QEKSelectionRuleWeightedSums Q PosAss NegAss OpenQ options returns the assertion which implies a maximal number of inferences according to the weighted sums selection rule The option ProbabilityYES is supported having 0 5 as the default value QEKSelectNextQuestion QEKSelectNextQuestion Q Pos Neg OpenQ options returns the data structure NextAss NegAnsInf PosAnsInf where NextAss is the next assertion for querying accompanied with its both positive and negative inferences NegAnsInf in case of a negative answer and PosAnsInf in case of a positive answer The selection is conducted on the set Q of items assuming previously collected positive inferences Pos negative inferences Neg and the items OpenQ left open The selection rule is given by the option SelectionRule that accepts the following values Maximin default and WeightedSums QEKStar QEKStar PosAss A returns the A star set of the premise A given the positive inferences PosAss QEOSAskExp
70. ault or Simulation QSRReport Result default Report or Both For other valid options see QEOS QueryExpertOnCompetenceSpace and QEOSQueryExpertOnInterpretationFunction QEOSQueryExpertOnDiagnostic E_ A_ DescOrExp_ opts__ Module qsr qsr1 qsr2 Exp Skills Items K k DIAG OptionHumOrSim Whether to query human or simulated expert OptionHumOrSim Expert opts Options QEOSQueryExpertOnDiagnostic What the function should return as its output OptionQSRReport QSRReport opts Options QEOSQueryExpertOnDiagnostic VERBOSITY If OptionVerbosity gt 1 Print Function returns Result Report or Both OptionQSRReport J Query on competence space In case of querying a human expert for competence space pass only skills description If OptionHumOrSim Human K qsr1 QEOSQueryExpertOnCompetenceSpace E DescOrExp 1 QSRReport gt Both opts l If OptionHumOrSim Simulation K qsr1 QEOSQueryExpertOnCompetenceSpace E DescOrExp Expert gt Simulation QSRReport gt Both opts Query on interpretation function k qsr2 QEOSQueryExpertOnInterpretationFunction K A DescOrExp 131 KSMP Techical Report Andrej Zaluski QSRReport gt Both opts Calculate the diagnostic DIAG CPADiagnosticFromInterpretationFunction k SONDE Combine two observations qsr qsr1 qsr2 What to return Switch OptionQSRReport Result Return DIAG R
71. cation mode Research mode will be suitable for research and teaching explorations and simulations on small sets of items while the application mode will be able to perform all the tasks on sets of items of relevant size for real life applications Discussion 45 KSMP Techical Report Andrej Zaluski Requirements Packages DiscreteMath Combinatorica KnowledgeSpaces WarpCore BeginPackage KnowledgeSpaces L1 DiscreteMath Combinatorica KnowledgeSpaces WarpCore TODO Save the state of the package Off General spell General spell1 SetQ usage SetQ L returns True iff L represents a set SetCardinality usage SetCardinality S returns the cardinality of the finite set S SetNonEmptyQ usage SetNonEmptyQ S returns True iff the set S is non empty SetEmptyQ usage SetEmptyQ S returns True iff the set S is empty SubsetQ usage SubsetQ A B returns True iff the set A is a subset of the set B EqualSetsQ usage EqualSetsQ A B returns True iff the sets A and B are equal SubsetProperQ usage SubsetProperQ A B returns True iff the set A is a proper subset of the set B SetElementQ usage SetElementQ a S returns True iff a is an element of the set S SetCartesianProduct usage SetCartesianProduct S1 S2 returns the Cartesian Product of the non empty sets S1 and S2 SetCartesianProduct S1 S2 returns the Cartesi
72. certainty factor of the expert s answer Ans ExpertAnswerCertaintyFactor Answer_ Answer 3 MakeStudent QKnowledge_ CarelessErrors_ LuckyGuesses_ QKnowledge CarelessErrors LuckyGuesses StudentKnowledge Student_ Student 1 77 KSMP Techical Report Andrej Zaluski StudentCarelessErrors Student_ Student 2 StudentLuckyGuesses Student_ Student 3 MakeKnowledge Questions_ Items Questions MakeSimpleStudent Questions_ MakeStudent MakeKnowledge Questions Items 0 0 StudentMakeAnswer question_ LogicalValue_ CertaintyFactor_ question Logical Value CertaintyFactor StudentAnswerltem Answer_ Answer 1 StudentAnswerLogicalValue Answer_ Answer 2 StudentAnswerCertaintyFactor Answer_ Answer 3 RelationReflexiveQ SR returns True iff the relation SR is reflexive RelationReflexiveQ Relation S_ R_ ForAlQ S SubsetQ R amp RelationAReflexiveQ returns True iff the relation SR is areflexive RelationAReflexiveQ SR_Relation RelationReflexiveQ SR RelationSymmetricQ SR returns True iff the relation SR is symmetric RelationSymmetricQ Relation SO_ RO_ Module S S0 R RO ForAllPairsQ S SubsetQ 1 2 R Limplies SubsetQl 2 AN RD amp RelationAntiSymmetricQ SR returns True iff the relation SR is antisymmetric RelationAntiSymmetricQ Relation SO_ RO_ Module S S0 R RO ForAlIPairsQ S SubsetQ I 1
73. concat itemsfile Table j j i 1 LinesPerltem i i LinesPerltem Changes the 1st string of each item to INTEGER or SYMBOL Return Map Join ToExpression 1 2 3 ToExpression 4 ToExpression 5 amp concat ReadsItems FN reads the information about the collection of items stored within the file FN according to the default fileformat KSMP_ItemsFile ReadItems FN_String ReadItems FN KSMP_ItemsFile WriteIltems FN Items stores the information about the Items into the file FN Writeltems FN_String Items_List DataFileWrite FN Items KSMP_ItemsFile MakeSkill Code Name Desc returns data object representing an skill consisting of the unique identifier Code thename Name and the description Desc MakeSkill code_ name_ description_ Module code name description SkillCode Skill returns the code of the given Skill SkillCode skill_ Module skill 1 SkillName Skill returns the name of the given Skill SkillName skill_ Module skill 2 SkillDescription Skill returns the description of the given Skill SkillDescription skill_ Module skill 3 SkillGet Skills Code returns the first skill its code matches Code SkillGet Skills_ code_ Module found found Select Skills Function skill SkillCode skill code If found found 1 ReadSkills FN KSMP _SkillsFile reads the informat
74. ctedQ SR returns True iff the relation SR is connected RelationEquivalenceQ usage RelationEquivalenceQ SR returns True iff the relation SR is an equivalence relation RelationQuasiOrderQ usage RelationQuasiOrderQ SR returns True if the relation SR is a quasi order RelationPartialOrderQ usage RelationPartialOrderQ SR returns True iff the relation SR is a partial order RelationLinearOrderQ usage RelationLinearOrderQ SR returns True iff the relation SR is a linear order KSAxiom1 usage KSAxiom1 SF returns True iff the structure SF contains both the empty set and the whole set S KSAxiom2S usage KSAxiom2S SF returns True iff the structure SF is stable under union 71 KSMP Techical Report Andrej Zaluski KSAxiom3S usage KSAxiom3S SF returns True iff the structure SF is stable under intersection KnowledgeStructureQ usage KnowledgeStructureQ SF returns True iff the structure SF satisfies formal requirements of a knowledge structure KnowledgeSpaceQ usage KnowledgeSpaceQ SF returns True iff the structure SF satisfies formal requirements of a knowledge space KnowledgeSpaceQuasiOrdinalQ usage KnowledgeSpaceQuasiOrdinalQ SF returns True iff the structure SF is a quasi ordinal knowledge space KSAxiom2C usage KSAxiom2C SF returns True iff the structure SF is closed under union KSAxiom3C usage KSAxiom3C SF returns True iff the struct
75. cture SF taking into account S1 as the set carrying the structure StructureFromMatrix usage StructureFromMatrix M returns the normalized structure from its matrix representation M n n StructureFromMatrix M Normalized returns the normalized structure from its matrix representation M n n StructureFromMatrix M SC returns the structure from its matrix representatition having the set SC as its carrier set StructureToMatrix usage StructureToMatrix SF returns the matrix representation of the structure SF n n StructureToMatrix SF Normalized returns the matrix representation of the structure SF assuming that the structure is already normalized 50 KSMP Techical Report Andrej Zaluski ReadStructure usage ReadStructure FN reads the structure stored in the file FN taking KST_SpaceFile as default data file format n n ReadStructure FN KST_SpaceFile reads the structure stored in the file FN using the data file format KST_SpaceFile n n ReadStructure FN SC reads the structure stored in the file FN and rewrites it using the carrier set SC assuming the default datafile format n n ReadStructure FN KST_SpaceFile SC reads the structure using KST_SpaceFile datafile format stored in the file FN and rewrites it using the carrier set SC WriteStructure usage WriteStructure FN SF KST_SpaceFile writes the structure SF in the file FN using KST_SpaceFile datafile format
76. dExtract SFL FileFormat returns the extracted head the from the non commented data list SFL representing the datafile object in accordance with the datafile format FileFormat For the supported datafile formats see FileFormatsInfo DataFileHeadInterpret DataFileHeadInterpret H FF interprets the extracted head H from a data file object in the FF format DataFileLength DataFileLength SFile returns the number of lines in the datafile object SFile DataFileRead DataFileRead Filename creates a datafile object representing the data file Filename DataFileWrite DataFileWrite FN M KST_SpaceFile writes the matrix M together with the derived head in the datafile FN using the KST_SpaceFile format DataFileWrite FN Items KSMP_ItemsFile writes the items description Items into the file FN using the KSMP_ItemsFile format DataFileWrite FN Skills KSMP_SkillsFile writes the skills description Skills into the file FN using the format KSMP_SkillsFile ElementsInRelationQ SR a b returns True iff the elements a and b are in the relation SR EntailmentFromEntailRelation EntailmentFromEntailRelation AAPB returns the entailment derived from the entail relation AAPB EntailmentToEntailRelation EntailmentToEntailRelation AAPx returns the entail relation derived from the entailment AAPx EntailRelationFromKnowledgeSpace EntailRelationFromKnowledgeSpace SF returns the entail relation given the knowledge space SF EntailRelationT
77. data files handling facilities to the package which would enable usage of KSMP in relevant empirical knowledge space theory applications respecting the memory limitations of personal computers The list of implemented functions is available in the glossary of KSMP functions KSMP Layer two The aims of the layer two of the KSMP package are to implement the selected knowledge space theory concepts to set up an appropriate environment for conducting simulations and finally to ensure access to the pools of items and skills for the package modules The layer two consists of the following components Items and skills Modeling expert Modeling student Theory of relations and order theory Knowledge structures Basis Surmise systems and Entail relations The components from the layer two are supposed to be the basis for the modules that is to serve as an interface between the core and the package modules Such design should facilitate a separation of the core from the modules ensuring that eventual future redesigns of the core will be reflected in the modules as less as possible The aim of the component tems and skills is to provide data structures and functions for describing items and skills that are to be used by modules while performing different procedures for example querying an expert on structure of a set of items or assessing knowledge of a student given the structure For additional information see the section on the used data file formats
78. e Return Apply Plus Map Length amp QEKCollectInferencesForNegativeAnswer Q PosAss Ass Apply Plus Map Length amp QEKCollectInferencesForPositiveAnswer Q PosAss NegAss Ass QEKSelectionRuleWeightedSums Q PosAss NegAss OpenQ options returns an assertion which has maximal expected number of inferences according to weighted sums selection rule The option ProbabilityYES is supported having 0 5 as the default value QEKSelectionRuleWeightedSums Q_ PosAss_ NegAss_ OpenQ _ opts___ Module PrYES PrNO candidates MaxAss i PrYES ProbabilityYES opts Options QEKSelectionRuleWeightedSums PrNO 1 PrYES candidates Map 1 2 1 PrNO 2 2 PrYES amp Map QEK CountInferences Q PosAss NegAss amp OpenQ MaxAss candidates 1 Dol If candidates i 2 gt MaxAss 2 MaxAss candidates 1 1 2 Length candidates Return MaxAss 1 QEKSelectionRuleMaximin Q PosAss NegAss OpenQ returns an assertion which has maximal expected number of inferences according to maximin selection rule QEKSelectionRuleMaximin Q_ PosAss_ NegAss_ OpenQ Module candidates MaxAss i candidates Map G 1 Min 2 1 2 2 amp Map QEK CountInferences Q PosAss NegAss amp OpenQ tmp Map QEKCountInferences Q PosAss NegAss amp OpenQ MaxAss candidates 1 Dol If candidates i 2 gt MaxAss 2 MaxAss candidates
79. e or some skills from QorE interpreted in ItemsOrSkills in case of querying on competence space Available options are QueryOn KnowledgeSpace default or CompetenceSpace QEKAskExpert Q_ List SetQ Items_ List Assertion_ Human opts _ Module answer CertaintyFactor OptionQueryOn Querying on knowledge space or competence space OptionQueryOn QueryOn opts Options QEKAskExpert Formulate a query on items If OptionQueryOn KnowledgeSpace answer Input QEK FormulateQuery Q Items Assertion J Formulate a query on skills If OptionQueryOn CompetenceSpace answer Input QEOSFormulateQuerySkills Q Items Assertion J CertaintyFactor 1 If MemberQ Y T YES TRUE ToUpperCase ToString answer answer True answer False Return ExpertMakeAnswer Assertion answer CertaintyFactor QEKAskExpert Assert Exp Simulation returns an answer from the expert Exp on assertion Assert Expert s meta 108 KSMP Techical Report Andrej Zaluski knowledge may be represented either with a structure or a diagnostic QEKAskExpert Assertion_ Expert_ Simulation Module answer ExpMK ExpMKAssPos ExpCErrors Extract the expert s meta knowledge ExpMK ExpertMetaKnowledge Expert Check the type of the stored meta knowledge and convert it to assertions if necessary If Head ExpMK Diagnostic ExpMK MakeMetaKnowledge EntailmentFromKnowledgeSpace Structur
80. e FunctionSubsetImage f D 1 returns the image of the subset D1 under the function f n nFunctionSubsetImage functions D 1 returns the image of the subset D1 after applying a list of functions from right to left FunctionInjectionQ usage FunctionInjectionQ f returns True iff the function f is an injection FunctionBijectionQ usage FunctionBijectionQ f returns True iff the function f is a bijection FunctionRestriction usage FunctionRestriction f D 1 returns the function f restricted to its subdomain D1 CompositionOfFunctions usage CompositionOfFunctions fn f1 return the composition fn o o f1 of the functions fn f1 under assumption that such composition is defined EqualFunctionsQ usage EqualFunctionsQ fl f2 returns True iff the functions fl and f2 are equal MakeStructure usage MakeStructure X F returns a data structure representing the structure consisting of the set X of items and the family F of subsets from the set X StructureQ usage StructureQ SF returns True iff SF is a structure FamilyQ usage FamilyQ F returns True iff SF is a family of subsets StructureSet usage StructureSet SF returns the set forming the structure SF StructureFamily usage StructureFamily SF returns the family forming the structure SF StructureReplaceSet usage StructureReplaceSet SF X1 is a primitive function that returns the structure SF after replaci
81. e theory constructs using different querying procedures Namely after developing the querying procedures with satisfactory properties one possibility would be to efficiently implement the procedures using one of standard languages as C C and disseminate the built tool However the alternative is to develop QEOS application having a Mathematica front end interface to applicants Such application would enable easy usage of the developed querying procedures and in turn building of the structures in a variety of knowledge domains 14 KSMP Techical Report Andrej Zaluski Bibliography Albert D and J Lukas 1999 Knowledge Spaces Theories Empirical Research and Applications Mahwah New Jersey and London Lawrence Erlbaum Associates Publishers Doignon J P and J C Falmagne 1999 Knowledge Spaces Berlin Springer Verlag Koppen M 1993 Extracting Human Expertise for Constructing Knowledge Spaces An Algorithm Journal of Mathematical Psychology 37 1 20 Korossy K 1999 Modeling Knowledge as Competence and Performace In D Albert and J Lukas Knowledge Spaces Theories Empirical Research and Applications Mahwah NJ and London Lawrence Erlbaum Associates Publishers 103 132 Maeder R E 1997 Programming in Mathematica Reading MA Addison Wesley Publishing Company Skienna S 1990 Implementing discrete Mathematics combinatorics and graph theory with Mathematica Redwood City CA Addison We
82. e CPADiagnosticElementaryCompetencies ExpMK CPADiagnosticCompetenceSpace ExpMK 1 Assertions If Head ExpMK Structure ExpMK MakeMetaKnowledge EntailmentFromKnowledgeSpace ExpMK Assertions l ExpMKAssPos MetaKnowledgeAssertionsPositive ExpMK ExpCErrors ExpertCarelessErrors Expert answer ExpertMakeAnswer Assertion MemberQ ExpMKAssPos Assertion Return answer QEKExtractPositiveJudgments QH returns extracted positive judgments from the query history QH QEKExtractPositiveJudgmentsFromQueryHistory QueryHistory_ Module Map ExpertAnswerAssertion amp select QueryHistory Function answer ExpertAnswerLogicalValue answer True I QEKExtractNegativeJudgments QH returns extracted positive judgments from the query history QH QEKExtractNegativeJudgmentsFromQueryHistory QueryHistory_ Module Map ExpertAnswerAssertion amp select QueryHistory Function answer ExpertAnswerLogical Value answer False NN QEK QueryExpertStraightforward Q Items Human returns the history of jugdments obtained while querying the human expert for structure on items Q interpreted in Items QEKQueryExpertStraightforward Q_ Items_ Human Module QueryHistory OpenQuestions PJudgments TrivialJudgments Exclude trivial assertions from the open questions OpenQuestions QEK AssertionsAxiom Drop PQxQ Q QueryHistory Function Ass QEKAskExpert Q Items Ass Human OpenQuest
83. e contained in the assertion Assertion QEKAssertionPremise QEKAssertionPremise Assertion returns the premise contained in the assertion Assertion QEKAssertionsAxiom1 Drop QEKAssertionsAxiom1Drop Ass returns the list of assertions excluding the trivial assertions according to the axiom 1 of the entailment concept QEKCollectInferencesForNegativeAnswer QEKCollectInferencesForNegativeAnswer Q PosAss Ass returns the data structure P N consisting of positive inferences P and negative inferences N when a negative answer on the assertion Ass is collected The set Q of items and the list PosAss of established positive assertions are given in advance The inference rule is implemented according to Koppen 14d QEKCollectInferencesForPositiveAnswer QEKCollectInferencesForPositiveAnswer Q PosAss NegAss Ass returns the data structure P N consisting of positive inferences P and negative inferences N when a positive answer on the assertion Ass is collected The set Q of items lists PosAss and NegAss of established positive and negative assertions are given in advance The inference rule is implemented according to Koppen 14a 14b 14c QEKCountInferences QEKCountInferences Q PosAss NegAss Ass returns the data structure Neg Pos containing the number Neg of inferences that would be drawn in case a negative answer is collected on the assertion Ass and the number Pos of drawn inferences in case of a positive answer The set Q of i
84. e forming the given representation function p CPARepresentationFunctionRepresentations usage CPARepresentationFunctionRepresentations p returns the collection of representations forming the representation function p CPAMakeDiagnostic usage CPA MakeDiagnostic K A P k p returns the data 91 KSMP Techical Report Andrej Zaluski structure representing the diagnostic given by the family forming the competence structure K the set A of items the family forming the performance structure P the interpretation function k and the representation function p n n CPAMakeDiagnostic E K A P k p returns the data structure representing the diagnostic given by the set E of elementary competencies the family forming the competence structure K the set A of items the family forming the performance structure P the interpretation function k and the representation function p CPADiagnosticElementaryCompetencies usage CPADiagnosticElementaryCompetencies Diag returns the set of elementary competencies forming the given diagnostic Diag CPADiagnosticCompetenceS pace usage CPA DiagnosticCompetenceSpace Diag returns the competence space forming the given diagnostic Diag CPADiagnosticltems usage CPADiagnosticltems Diag returns the set of items forming the given diagnostic Diag CPADiagnosticPerformanceSpace usage CPA DiagnosticPerformanceSpace Diag returns the performance space forming the
85. e original of a mapping rule of the function f is equal to the domain of the function FX FX SF s returns all members of the family forming the structure SF which contain the element s of the set forming the structure FX SF C NC returns all members of the family forming the structure SF which contain all the elements from the set C and have empty intersection with the set NC FX SF C returns all members of the family forming the structure SF which contain all the elements from the set C FXFilterOrderDown FXFilterOrderDown SF f returns all members of the family forming the structure SF which are subsets of the family member f FXFilterOrderUp FXFilterOrderUp SF f returns all members of the family forming the structure SF which are supersets of the family member f GenerateFamilyByTakingUnions GenerateFamilyByTakingUnions F returns the family generated by taking all possible unions of the elements from the family F TAssertionCompetenceState IAssertionCompetenceState IAss returns the competence state of the interpretation assertion IAss TAssertionItem IAssertionItem IAss returns the item of the interpretation assertion IAss ItemAnswers ItemAnswers Item returns the list of all admissible answers on the item Item where each answer consists of a code and a description ItemCode ItemCode Item returns the code of the given Item ItemCorrectAnswer ItemCorrectAnswer Item returns the correct answer of
86. egative interpretation assertions PosIAss NeglAss Count number of questions asked NofQuestions 0 SONDE Create and initialize a querying session sonde if requested If OptionQSRReport Report OptionQSRReport Both QSR QSRMakeSonde 0 0 QSR QSRSondeOpenQuestionsInitialReplace QSR Length OpenQ QSR QSRSondeQuestionsAskedReplace QSR NofQuestions l 128 KSMP Techical Report Andrej Zaluski VERBOSITY SONDE If OptionVerbosity gt 10 amp amp OptionQSRReport Report OptionQSRReport Both Print Initial querying session sonde status QSR Any interpretation assertion left open While OpenQ VERBOSITY If OptionVerbosity gt 5 Print Open questions n OpenQ Print Known positively judged interpretation assertions n PosIAss I Select the next open interpretation assertion for querying NextIAss QEOSSelectNextInterpretationQuestion OpenQ OpenQuestions gt Sorted RuleScope gt OptionRuleScope SelectionRule gt OptionSelectionRule Query a human or simulated expert for an answer If OptionHumOrSim Human Answer QEOSAskExpertOnInterpretation K Skills Items NextIAss Human NofQuestions J If OptionHumOrSim Simulation Answer QEOSAskExpertOnInterpretation NextIAss SimExpert Simulation NofQuestions J VERBOSITY If OptionVerbosity gt 5 Print NofQuestions question NextIAss The expert s answer
87. elation SR RelationSetCardinality Relation S R_ Module SR Relation S R SetCardinality RelationSet SR RelationSetsCardinality SR returns the cardinalities of the sets forming the relation R from the set S1 to the set S2 RelationSetsCardinality Relation SA_ SB_ R_ Module SR Relation SA SB R RelSets RelSets RelationSets SR SetCardinality RelSets 1 SetCardinality RelSets 2 RelationPairsCardinality SR returns the cardinality of the familiy of ordered pairs forming the relation SR RelationPairsCardinality SR_Relation SetCardinality RelationPairs SR Works for both S R and S1 S2 R RelationChangeCarrier SR Items rewrites the relation SR by changing relation s set with the set Items labels of the items RelationChangeCarrier Relation SO_ RO_ Items0_List Module SRO Relation S0 RO Items Items0 S R Changes SR SR SRO Items Union Items0 S RelationSet SR R RelationPairs SR Changes Table S i gt Items i i 1 Length S S S Changes R R Changes SR RelationChangeSet SR S SR RelationChangePairs SR R Return SR Length Items0 Length S0 amp amp Length Items0 SetCardinality Items0 The changing mapping is a bijective function RelationChangeCarriers SR Items 1 Items2 rewrites the relation SR by changing relation s first set with the set Items1 and the second set respectively
88. enQ B VERBOSITY If OptionVerbosity gt 15 Print kBlockAs kBlockAs i 1 Length candidates Construct temporary new block with open questions OpenQ Flatten Map Function A Map A amp Q kBlockAs 1 VERBOSITY If OptionVerbosity gt 10 Print 3 step candidates n OpenQ I VERBOSITY If OptionVerbosity gt 10 Print Filling in the new block with already known inferences Filling in the new block with already known inferences AIIBs Apply Union Table KSubsets Q i i 1 k 1 NewPosAss NewNegAss Take As one by one from the newly constructed block Scan Function A First positive inferences Calculating A plus for a given A flag True Aplus OpenQ 1 1 While flag i 0 While flag B QEKAssertionPremise AIIBs 1 Bstar QEKStar PosAss B If SubsetQ B Aplus amp amp Not SubsetQ Bstar Aplus Aplus Union Aplus Bstar Break flag False I VERBOSITY If OptionVerbosity gt 10 Print Calculating positive inferences If OptionVerbosity gt 15 Print Old NewPosAss NewPosAss Adding new positive inferences to the old ones tmp Union NewPosAss Map A amp Aplus NewPosAss tmp VERBOSITY 118 KSMP Techical Report Andrej Zaluski If OptionVerbosity gt 10 Print New positive inferences n NewPosAss Second negative inferences Calculating A minus for a given A Aminus
89. ence space based items and return the interpretation function tmp CPA InterpretationFunctionConformTo k VERBOSITY If OptionVerbosity gt 10 Print Established interpretations of competency space based items n FunctionMapping tmp SONDE Write data to the querying session sonde If OptionQSRReport Report OptionQSRReport Both QSR QSRSondeQuestionsAskedReplace QSR NofQuestions VERBOSITY SONDE If OptionVerbosity gt 10 amp amp OptionQSRReport Report OptionQSRReport Both Print Final querying session sonde status QSR 130 KSMP Techical Report Andrej Zaluski If OptionQSRReport Both Return tmp QSR If OptionQSRReport Report Return QSR If OptionQSRReport Result Return tmp End of Module QEOSInterpretationsFromlAssertions IASSes returns the list of partial interpretations formed from the list of interpretation assertions QEOSInterpretationsFromlAssertions IASSes_ Map 1 2 amp 1ASSes QEOS QueryExpertOnDiagnostic E A DescOrExp opts queryies a human expert DescOrExp equals to Skills Items describes the skills and the items or a simulated expert DescOrExp represents a simulated expert whose meta knowledge must be given as a diagnostic on competence performance diagnostic given the set A of items and the set E of elementary competencies The valid options are Expert Human def
90. enceSpace Diag returns the competence space forming the given diagnostic Diag CPADiagnosticElementaryCompetencies CPADiagnosticElementaryCompetencies Diag returns the set of elementary competencies forming the given diagnostic Diag CPADiagnosticFromInterpretationFunction CPADiagnosticFromInterpretationFunction k returns the 6 tupel diagnostic induced by the interpretation function k It is assumed that the competence structure given by k is based on a set of elementary competencies 16 KSMP Techical Report Andrej Zaluski CPADiagnosticInterpretationFunction CPADiagnosticInterpretationFunction Diag returns the interpretation function forming the given diagnostic Diag CPADiagnosticItems CPADiagnosticltems Diag returns the set of items forming the given diagnostic Diag CPADiagnosticPerformanceSpace CPADiagnosticPerformanceSpace Diag returns the performance space forming the given diagnostic Diag CPADiagnosticRepresentationFunction CPADiagnosticRepresentationFunction Diag returns the representation function forming the given diagnostic Diag CPAEqualDiagnosticsQ CPAEqualDiagnosticsQ Diag1 Diag2 returns True iff the diagnostics Diag and Diag2 are equal CPAInterpretationFunctionAxiomsQ CPAInterpretationFunctionAxiomsQ k returns True iff the function k satisfies the axiom 1 of interpretation functions CPAInterpretationFunctionConformTo CPAInterpretationFunctionConformTo k returns the interpreta
91. entQ Alpha S RelationMinimalQ SR Alpha tests whether the element Alpha is the minimal with respect to the relation SR RelationMinimalQ Relation S R_ Alpha _ Module SR Relation S R Apply And Map ElementsInRelationQ SR Alpha Implies Alpha Equal amp RelationSet SR SetElementQ Alpha S MaximalsOfRelation SR returns all the maximals for the relation SR MaximalsOfRelation Relation S_ R_ Module SR Relation S R DeleteCases Map If RelationMaximalQ SR 1 1 amp S Null MinimalsOfRelation SR returns all the minimals of the relation SR MinimalsOfRelation Relation S_ R_ Module SR Relation S R DeleteCases Map If RelationMinimalQ SR 1 1 amp S Null RelationComplement SR returns the complement of the relation SR RelationComplement Relation S_ R_ Module Return Relation S SetComplement SetCartesianProduct S S R RelationComplement Relation S1_ S2_ R_ Module Return Relation S 1 S2 SetComplement SetCartesianProduct S1 S2 R RelationIdentity S returns the identity relation on the set S RelationIdentity S_ Return Relation S Map amp S 57 KSMP Techical Report Andrej Zaluski EqualRelationsQ SR1 SR2 returns true iff the relations SR1 and SR2 are equal EqualRelationsQ Relation S1_ R1_ Relation S2_ R2_ EqualSetsQ S 1 S2 And EqualSetsQ R1 R2 EqualRelationsQ Relati
92. entResult value State default or an assessment history option AssessmentResult value History The additional valid option is Verbosity Off default 1 5 10 20 Examinee usage Examinee is an option of KADAssessKnowledge The admissible values are Human and Simulation AssessmentResult usage AssessmentResult is an option of KADAssessKnowledge The admissible values are State and History Options KADAssessKnowledge Examinee gt Human AssessmentResult gt State Verbosity gt Verbosity Begin Private KAFormulateQuestion usage KAFormulateQuestion Items QE returns the question based on the item QE described by Items KAFormulateQuestion Items_ q_ Module item text t2 n nThe possible answers are n n item ItemGet Items q text ItemText item lt gt t2 lt StringJoin Map ToString 1 lt gt lt gt 2 lt gt n amp ItemAnswers item Return text KADSelectNextQuestionAuxFirst Structure Q0O_List SetNonEmptyQ KO_ Module Q Q0 K K0 QK Structure Q0 K0 maxq balans i balans Map 1 Abs SetCardinality FX QK List 1 SetCardinality Complement K FX QK List 1 amp Q maxq First balans Dol If maxq 2 gt balans i 2 maxq balans i i Length balans Return maxq 1 KADSelectNextQuestionAuxNonFirst Structure Q0_ KO_ CAnswers_ NCAnswers_ Module QK Structure Q0 K0 Q Q0 States q
93. eport Return qsr Both Return DIAG qsr J End of Module QSRMakeSonde OQB NQ returns the querying session QSR sonde consisting of the number OQ of open questions at the beginning of the observing and the number NQ of question given to an expert during the observation QSRMakeSonde OQ_Integer NQ _Integer 0Q NQ QSRSondeOpenQuestionsInitial Sonde returns the initial number of open questions as recorded by the querying session Sonde QSRSondeOpenQuestionsInitial sonde_ sonde 1 QSRSondeQuestionsAsked Sonde returns the number of questions asked as recorded by the querying session Sonde QSRSondeQuestionsAsked sonde_ sonde 2 QSRSondeOpenQuestionsInitialReplace Sonde N returns the querying session Sonde with the initial number of open questions replaced with N QSRSondeOpenQuestionsInitialReplace Sonde_ N_ ReplacePart Sonde N 1 QSRSondeOpenQuestionsInitialAdd Sonde N returns the querying session Sonde with the initial number of open questions incremented by N QSRSondeOpenQuestionsInitialAdd Sonde_ N_ ReplacePart Sonde QSRSondeOpenQuestionsInitial Sonde N 1 QSRSondeQuestionsAskedReplace Sonde N returns the querying session Sonde with the number of questions asked replaced with N QSRSondeQuestionsAskedReplace Sonde_ N_ ReplacePart Sonde N 2 QSRSondeQuestionsAskedAdd Sonde N returns the querying session Sonde with the number of questions asked incre
94. er SR S1 returns the relation SR after changing the original elements with the elements of the set S1 RelationChangeCarriers RelationChangeCarriers SR S11 S22 returns the relation SR after changing the original elements from the set S1 with the elements from the set S11 and the elements of S2 with the elements from S22 respectively RelationChangePairs RelationChangePairs SR R1 is a primitive function that returns the relation based on the relation SR after replacing the set R of ordered pairs by the set R1 of ordered pairs 33 KSMP Techical Report Andrej Zaluski RelationChangeSet RelationChangeSet SR S1 is a primitive function that returns the relation based on the relation SR after replacing the set S by the set S1 RelationChangeSets RelationChangeSets SR S11 S22 is a primitive function that returns the relation based on the relation SR after replacing the set S1 by the set S11 and the set S2 by the set S22 RelationComplement RelationComplement SR returns the complement of the relation SR RelationConnectedQ RelationConnectedQ SR returns True iff the relation SR is connected RelationEquivalenceQ RelationEquivalenceQ SR returns True iff the relation SR is an equivalence relation RelationIdentity RelationIdentity S returns the identity relation on the set S RelationLinearOrderQ RelationLinearOrderQ SR returns True iff the relation SR is a linear order RelationMaximalQ RelationMaximalQ SR
95. ere the record describes one item Each record consists of five lines The first line contains code of the item This code is used for item identification by KSMP functions The next line consists of a description of the item For example querying module functions use this description in order to present assertions to a human expert The next line consist of a 12 KSMP Techical Report Andrej Zaluski question text of the item that is for example used by knowledge assessment module functions while presenting the item to an examinee The fourth line of each record identifies code of the correct answer for the item The fifth line of a record consist of a list of pairs storing information on possible answers to an multiple choice item Each pair consists of code and text The code identifies the answer while the text is used by KSMP functions while presenting the item This is an example of the itemsfile KnowledgeSpaces Mathematica Package KSMP v1 0 1 Description 1 Question Text 1 1 1 Answer 1 2 Answer 2 3 Answer 3 2 Description 2 Question Text 2 2 1 Answer 1 2 Answer 2 3 Answer 3 The itemsfile above describes two items For example an examinee s answer Answer 2 on the presented questions Question Text 2 will be treated as corrected Skillsfile format The skillssfile format is designed to store description of latent skills and competencies for purposes of querying experts module Th
96. erpretationsInterpretationReplace KXs_ NewKx_ Module item 94 KSMP Techical Report Andrej Zaluski item NewKx 1 If Select KXs 1 item amp NotEqual ReplacePart K Xs NewKx Position K Xs Select KXs 1 item amp 1 KXs CPAInterpretationsInterpretationAdd KXs kx returns the interpretations K Xs with added interpretation kx CPAInterpretationsInterpretationAdd KXs_ kx_ Module item item kx 1 If Select K Xs 1 item amp Union KXs kx KXs CPAInterpretationFunctionUpgrade k I returns the interpretation function k upgraded with the possibly empty list I of interpretations or an individual interpretation I CPAInterpretationFunctionUpgrade k_ I_ Module KXs kx Which I Return k Head I 1 List Return CPA InterpretationFunctionUpgradeAux 1 k 1 MemberQ Symbol Integer String Head I 1 True Return CPA InterpretationFunctionUpgradeAux2 k I HIDDEN CPAInterpretationFunctionUpgradeAux1 k_ KXs_ Module NofIASSs i NofIASSs Length KXs For i 1 i lt NofIASSs i k CPA InterpretationFunctionUpgradeAux2 k K Xs i Return k HIDDEN CPAInterpretationFunctionUpgradeAux2 Mapping A_ PsK_ KXs_ kx_ Module k Mapping A PsK KXs A1 A KXs1 KXs K a kx0 kx1 a kx 1 K Apply Union PsK Check whether the interpretation function axioms are fulfilled for kx If EqualSetsQ kx
97. ertOnInterpretation QEOSAskExpertOnInterpretation K Skills Items Ass Human returns an answer on the interpretation assertion Ass from a human expert given the competence structure K described in Skills and the interpretation assertion item described in Items QEOSAskExpertOnInterpretation IAss Expert Simulation returns an answer on the interpretation assertion Ass from the simulated expert Expert 30 KSMP Techical Report Andrej Zaluski QEOSFormulateQueryInterpretations a Items C K Skills returns the query question on the given item a described in Items given the competence state C of the competence structure K described in Skills QEOSFormulateQuerySkills QEOSFormulateQuerySkills E Skills Assertion returns the query question based on the given assertion Assertion given the set E of elementary skills described in Skills QEOSInterpretationFunctionCollectInferencesForNegativeAnswer QEOSInterpretationFunctionCollectInferencesForNegativeAnswer OpenQ Ass returns all interpretation assertion inferences given the list OpenQ of open interpretation assertions assuming the interpretation assertion Ass was negatively judged QEOSInterpretationFunctionCollectInferencesForPositiveAnswer QEOSInterpretationFunctionCollectInferencesForPositiveAnswer OpenQ Ass returns all interpretation assertion inferences given the list OpenQ of open interpretation assertions assuming the interpretation assertion IAss was positively judged
98. eryExpertByK oppen Q ItOrSkOrExp options queries an expert using Koppen s 1993 block by block algorithm A human ItOrSkOrExp represents a description of either items or skills or a simulated expert ItOrSkOrExp represents the simulated expert is queried on the set Q of items and a structure either a space or assertions is returned The valid options are Expert Human default or Simulation OutputStructure SpaceStructure default or Assertions Verbosity Off default or 1 2 QueryOn KnowledgeSpace default or CompetenceSpace and QSRReport Result default Report or Both QEOSQueryTextsInterpretations usage QEOSQueryTextsInterpretations NofSkils TextPart returns the textual part of a query on interpretation given the text part TextPart introduction middle ending and the number NofSkills 1 2 generic all_minus_one of skills QEOSFormulateQueryInterpretations usage QEOSFormulateQueryInterpretations a Items C K Skills returns the query question on the given item a described in Items given the competence state C of the competence structure K described in Skills QEOS QueryTextsSkills usage QEOSQueryTextsSkills PremiseSize TextPart returns the textual part of a query on interpretation given the text part TextPart introduction or question and the premise size PremiseSize 1 2 generic all_minus_one QEOSFormulateQuerySkills usage QEOSFormulateQuerySkills E Skills Asser
99. erying solutions are close to the solutions suitable for dissemination to applicants of knowledge space software tools Mathematica as an integrated platform suitable for computing programming and typesetting seemed as an appropriate candidate Needless to say interactive capabilities and Mathematica s potential in teaching was an additional argument facilitating the selection Unfortunately standard packages distributed with Mathematica were not found sufficient for systematic explorations research and development on the querying procedures so a need for developing a specific package that would enable has arisen KSMP was conceived as a more general environment under which major concepts and findings from the knowledge space theory would be implemented while QEOS application was conceived as a part of the package dealing especially with the querying experts issues After developing the querying procedures with satisfactory properties one possibility would be to efficiently implement the procedures using one of standard languages as C C and disseminate the built tool Alternatively the developed procedures could be joined with a Mathematica front end interface to the applicant into an user friendly application QEOS is conceived to be such an application aiming to facilitate building the knowledge space theory constructs in a variety of knowledge domains using querying experts methods Mathematica as a platform suitable for research and teaching w
100. especto to the positive inferences PosAss QEKEquivalentSubsetsQ PosAssertions_ A_ B_ Module QEKStar PosAssertions A QEK Star PosAssertions B QEK InitializeFirstBlock Q returns the data structure OQ P N representing the first block in Koppen s algorithm where OQ represents open questions P positive assertions and N negative assertions QEKInitializeFirstBlock Q_ Module OpenQ PosAss NegAss NegAss PosAss Map amp Q OpenQ QEK AssertionsAxiom Drop QEKMakeFullBlock Q 1 Return OpenQ PosAss NegAss QEKInferences 14a PosAss Ass returns the positive inferences contributed by the inference rule Apx amp BPA amp AxPy gt Bpy if APx would be added to the previous positive assertions PosAss QEKInferences14a PosAss_ Ass_ Module A QEKAssertionPremise Ass x QEKAssertionConsequence Ass Ax Ys Bs Ax Union A x Ys Union QEKStar PosAss Ax Bs Union Map QEK AssertionPremise amp Select PosAss EqualSetsQ QEKStar PosAss QEK AssertionPremise AD amp J If Bs NotEqual And Ys NotEqual CartesianProduct Bs Ys QEKInferences 14b PosAss NegAss Ass returns all the negative inferences made by the inference rule APxk amp AxPB amp ANy gt BNy if the assertion Apx would be added to the previous positive PosAss and negative NegAss assertions QEKInferences14b PosAss_ NegAss_ Ass_ Module A QEKAssertionPremise Ass x
101. estions at the start of the observation and the number NQ of questions presented to an expert during the observation QSRSondeOpenQuestionsInitial usage QSRSondeOpenQuestionsInitial Sonde returns the initial number of open questions recorded by the querying session Sonde QSRSondeQuestionsAsked usage QSRSondeQuestionsAsked Sonde returns the number of questions presented to an expert recorded by the querying session sonde Sonde QSRSondeOpenQuestionsInitialReplace usage QSRSondeOpenQuestionsInitialReplace Sonde N returns the querying session Sonde after replacing the initial numberof open questions by N QSRSondeOpenQuestionsInitialAdd usage QSRSondeOpenQuestionsInitialAdd Sonde N returns the querying session Sonde after incrementing the initial number of open questions by N QSRSondeQuestionsAskedReplace usage QSRSondeQuestionsAskedReplace Sonde N returns the querying session sonde Sonde after replacing the number of asked questions by N 104 KSMP Techical Report Andrej Zaluski QSRSondeQuestionsAskedAdd usage QSRSondeQuestionsAskedAdd Sonde N returns the querying session sonde Sonde after incrementing the number of presented questions by N QEQueryExpertOnKnowledgeSpace usage QEQueryExpertOnKnowledgeSpace Q ItOrExp options queries a human ItOrExp takes a description of items or a simulated expert ItOrExp represents a simulated expert on structure of the set Q
102. expert Expert QEOSInterpretationsOnItem usage QEOSInterpretationsOnItem IASSes q returns all interpretation assertions given the item q and the list ASSes of interpretation assertions QEOSInterpretationFunctionCollectInferencesForPositiveAnswer usage QEOS InterpretationFunctionCollectInferencesForPositiveAnswer OpenQ Ass returns all interpretation assertion inferences given the list OpenQ of open interpretation assertions assuming the interpretation assertion IAss was judged positively QEOSInterpretationFunctionCollectInferencesForNegativeAnswer usage QEOSInterpretationFunctionCollectInferencesForNegativeAnswer OpenQ IAss returns all interpretation assertion inferences given the list OpenQ of open interpretation assertions assuming the interpretation assertion IAss was judged negatively QEOSInterpretationFunctionCountInferences usage QEOSInterpretationFunctionCountInferences IASSes Ass returns the data structure Neg Pos containing the number Neg of inferences drawn if a negative answer on the interpretation assertion Ass has been collected and the number Pos of inferences drawn in case a positive answer has been collected The list ASSes of open interpretation assertions is given in advance QEOSInterpretationFunctionSelectionRuleMaximin usage QEOSInterpretationFunctionSelectionRuleMaximin OpenQ returns the interpretation assertion which implies a maximal number of infere
103. formal requirements of a knowledge space KnowledgeSpaceQuasiOrdinalQ Structure S_List SetNonEmptyQ F List KSAxiom1 Structure S F amp amp KSAxiom2C Structure S F amp amp KSAxiom3C Structure S F NA RowBox KSAxiom2C SF returns True iff the structure SF is closed under union GridBox KSAxiom2C Structure S_ F_ Module flag True N1 N2 subfamiliescollection i j N1 Length F For i 0 1 lt N1 Block subfamiliescollection KSubsets F i N2 Length subfamiliescollection For j 0 j lt N2 Block flagtmp1 flagtmp2 flagtmp1 flag flagtmp2 SubsetQ Union subfamiliescollection LeftDoubleBracket j RightDoubleBracket F flag flagtmp1 amp amp flagtmp2 If LInvisibleSpace flag Break G If LnvisibleSpace flag Break i flag b GridBaseline gt Baseline 1 1 Column Widths gt 0 999 ColumnAlignments gt Left N a RowBox KSAxiom3C 81 KSMP Techical Report Andrej Zaluski SF returns True iff the structure SF is closed under intersection n InterpretationBox GridBox GridBox KSAxiom3C Structure S_ F_ Module flag True N1 N2 subfamiliescollection i j N1 Length F For i 0 i lt N1 Block subfamiliescollection KSubsets F i N2 Length subfamiliescollection For j 0 j lt N2
104. g human Expert QEKEstablishKnowledgeSpaceStraightforward Q_ Items_ Human EntailmentToKnowledgeSpace QEKEntailmentFromQueryHistory Q QEK QueryExpertStraightforward Q Items Human QEKEstablishKnowledgeSpaceStraightforward Q Expert Simulation returns the structure on the items Q obtained by querying simulated Expert QEKEstablishKnowledgeSpaceStraightforward Q_ Expert_ Simulation EntailmentToKnowledgeSpace QEKEntailmentFromQueryHistory Q QEK QueryExpertStraightforward Q Expert Simulation QEKAssertionsAxiom 1 Drop Ass returns the list of assertions excluding the trival assertions according to the axiom of the entailment QEKAssertionsAxiom1 Drop Assertions_ Module Select Assertions Function Ass Block A Ass 1 q Ass 2 SetElementQ q A QEKAssertionsAxiom1 Q returns the list of all trivial assertions according to the axiom of the entailment QEKAssertionsAxiom1 Q_ Module Assertions Assertions PQxQ Q Select Assertions Function Ass Block A Ass 1 q Ass 2 SetElementQ q A 110 KSMP Techical Report Andrej Zaluski QEKStar PosAss A returns the A star of the premise A given the positive inferences PosAss QEKStar PosAssertions_ A_ Module Union Map 2 amp Select PosAssertions QEK AssertionPremise A amp Il QEKEquivalentSubsetsQ PosAss A B returns True iff the subsets A and B are equivalent with r
105. ge Assessment Report nDeterministic assessment procedure used on a student OptionExaminee and the result returned as OptionAssessmentResult OpenQ Q OpenK K While SetCardinality Openk gt 1 q KADSelectNextQuestion Structure Q K Qy Qn VERBOSITY If OptionVerbosity gt 10 Print Selected item q I Switch OptionExaminee Simulation ans KAAskStudent ItemsStudent q Simulation Human ans KA AskStudent ItemsStudent q Human J AppendTo Hist ans 136 KSMP Techical Report Andrej Zaluski If StudentAnswerLogicalValue ans AppendTo Qy StudentAnswerltem ans AppendTo Qn StudentAnswerltem ans J VERBOSITY If OptionVerbosity gt 5 Print The answer ans nCollected positive answers Qy nCollected negative answers Qn J OpenK FX Structure Q K Qy Qn VERBOSITY If OptionVerbosity gt 10 Print Remaining states OpenK J l Switch OptionAssessmentResult State tmp Qy History tmp Hist Return tmp End EndPackage 137
106. htforward usage QEK QueryExpertStraightforward Q Items Human returns the jugdments obtained while querying a human expert on structure of the items Q described in Items n n QEK QueryExpertStraightforward Q Expert Simulation returns the jugdments obtained while querying the simulated Expert on structure of the items Q QEKEntailmentFromQueryHistory usage QEKEntailmentFromQueryHistory Q QueryHistory returns all positive judgments from the querying history QueryHistory capturing an establishment of a structure on the set Q of items QEKEstablishKnowledgeSpaceStraightforward usage QEKEstablishKnowledgeSpaceStraightforward Q Items Human returns a structure on the items Q described in Items established by querying a human Expert n n QEKEstablishKnowledgeSpaceStraightforward Q Expert Simulation returns a structure on the items Q obtained by querying the simulated expert Expert QEKAssertionsAxiom 1 Drop usage QEK AssertionsAxiom1Drop Ass returns the list of assertions excluding the trival assertions according to the axiom of the entailment concept QEKStar usage QEKStar PosAss A returns the A star set of the premise A given the positive inferences PosAss QEKEquivalentSubsetsQ usage QEK EquivalentSubsetsQ PosAss A B returns True iff the subsets A and B are equivalent with respect to the positive inferences PosAss 100 KSMP Techical Report Andrej Zaluski QEK InitializeFir
107. ile Matrix2DataFileBody M KST_SpaceFile Lines DataFileWrite FN Items KSMP_ItemsFile writes the Items information into the file FN using the format KSMP_ItemsFile DataFileWrite FileName_String Items0_List KSMP_ItemsFile Module Items Changes the Ist integer or symbol of each item to STRING Items Map Join ToString 1 Rest amp Items0 Export FileName Join Items File KSMP v1 0 Flatten Items 1 Lines DataFileWrite FN Skills KSMP _SkillsFile writes the Skills information into the file FN using the format KSMP_SkillsFile DataFileWrite FileName_String SkillsO_List KSMP_SkillsFile 66 KSMP Techical Report Andrej Zaluski Module Skills Changes the Ist integer or symbol of each skill to STRING Skills Map Join ToString 1 Rest amp Skills0 Export FileName Join Skills File KSMP v1 0 Flatten Skills 1 Lines ForAllQ X pf returns True iff the property pf holds for all elements of the set X ForAllQ X_List p_Function Apply And Map p X ExistOneQ X pf returns True iff the property pf holds for at least one element of the set X ExistOneQ X_List p_Function Apply Or Map p X ForAllPairsQ X pf returns True iff the predicate pf gives True for all ordered pairs from X ForAllPairsQ X_List SetQ pf_ Function ForAllPairsQ X X pf ForAllPairsQ X
108. in the mapping LofCL is member of all clauses assigned to it n n SSAxiom2Q Q LofCl returns True iff each item from Q is member of all clauses assigned to it by the mapping LofCL of items to clauses SSAxiom2atItemQ usage SSAxiom2atltemQ q Clauses returns True iff the item 72 KSMP Techical Report Andrej Zaluski q is an element of all clauses that are assigned to it in the collection Clauses of clauses SSAxiom3Q usage SSAxiom3Q LofCl returns True iff for each clause C of any item q given by the mapping LofCl there exist a clause C1 assigned to each item q1 of the clause C that is a subset of the clause C SSAxiom4atClausesQ usage SSAxiom4atClausesQ clauses returns True iff any two clauses for the same item are incomparable SSAxiom4Q usage SSAxiom4Q LofCl returns True iff any two clauses of any item within the mapping LofCl are incomparable SurmiseFunctionQ usage SurmiseFunctionQ f returns True iff the function f represents a surmise function SurmiseSystemQ usage SurmiseSystemQ QS returns True iff QS represents a surmise system MakeSurmiseFunction usage MakeSurmiseFunction Q LofCl returns the data structure representing the surmise function given by the set Q of items and the mapping LofCl of clauses MakeSurmiseSystem usage MakeSurmiseSystem Q LofCl returns the data structure representing a surmise system given by the set Q of items and the m
109. interpretations on the item q from the list KXs of interpretations CPAInterpretationsInterpretationAdd CPAInterpretationsInterpretationAdd KXs kx returns the interpretations K Xs after adding the interpretation kx CPAInterpretationsInterpretationReplace CPAInterpretationsInterpretationReplace K Xs NewKx returns the interpretations K Xs after replacing the corresponding interpretation with the interpretation NexKx CPAMakeDiagnostic CPAMakeDiagnostic K A P k p returns the data structure representing the diagnostic given by the family forming the competence structure K the set A of items the family forming the performance structure P the interpretation function k and the representation function p CPAMakeDiagnostic E K A P k p returns the data structure representing the diagnostic given by the set E of elementary competencies the family forming the competence structure K the set A of items the family forming performance structure P the interpretation function k and the representation function p CPAMakelInterpretationFunction CPAMakelInterpretationFunction A K K Xs returns the interpretation function given the set A of items the competence structure K and the mapping KXs interpreting each item by assigning competence states CPAMakeRepresentation CPAMakeRepresentation C Items returns the data structure containing the representation formed by the given competence state C and the set Items CPAMakeRepresentationFunc
110. ion about the collection of skills stored within the file FN according to the fileformat KSMP_SkillsFile ReadSkills FN_String KSMP_SkillsFile Module concat LinesPerSkill 3 NofSkills skillsfile i skillsfile DataFileCommentDrop DataFileRead FN 75 KSMP Techical Report Andrej Zaluski NofSkills Length skillsfile LinesPerSkill For i 0 i lt LinesPerSkill NofSkills AppendTo concat skillsfile Tablef j i 1 LinesPerSkill i i LinesPerSkill Changes the Ist string of each skill to INTEGER or SYMBOL Return Map Join ToExpression 1 Rest amp concat ReadsSkills FN reads the information about the collection of skills stored within the file FN according to the default fileformat KSMP_SkillsFile ReadSkills FN_String ReadSkills FN KSMP_SkillsFile WriteSkills FN Skills stores the information about the Skills into the file FN WriteSkills FN_String Skills_List DataFileWrite FN Skills KSMP_SkillsFile MakeExpert MK CE returns the data structure representing an expert consisting of domain s metaknowledge MK and uncertainty measure CE of making a careless mistake while querying MakeExpert MetaKnowledge_ CarelessErrors_ MetaKnowledge CarelessErrors ExpertMetaKnowledge E returns expert s E domain metaknowledge ExpertMetaKnowledge Expert_ Expert 1 ExpertCarelessErrors E returns the uncertainty measure that the expert E makes
111. ion function k and the representation function p CPAMakeDiagnostic K_ A_ P_ k_ p_ Diagnostic K A P k p CPAMakeDiagnostic E K A P k pJreturns the data structure representing the diagnostic given by the set E of elementary competencies the family forming the competence structure K the set A of problems the family forming performance structure P the interpretation function k and the representation function p CPAMakeDiagnostic E_ K _ A_ P_ k p_ Diagnostic E K A P k p CPADiagnosticElementaryCompetencies Diag returns the set of elemntary competencies forming the given diagnostic Diag CPADiagnosticCompetenceSpace Diag returns the competence space forming the given diagnostic Diag CPADiagnosticltems Diag returns the set of items forming the given diagnostic Diag 97 KSMP Techical Report Andrej Zaluski CPADiagnosticltems Diagnostic _ A_ A CPADiagnosticltems Diagnostic _ A_ A CPADiagnosticPerformanceSpace Diag returns the performance space forming the given diagnostic Diag CPADiagnosticInterpretationFunction Diag returns the interpretation function forming the given diagnostic Diag CPADiagnosticInterpretationFunction Diagnostic _ k_ k CPADiagnosticInterpretationFunction Diagnostic _ k_ _ k CPADiagnosticRepresentationFunction Diag returns the representation function forming the given diagnostic Diag
112. ions Adds trivial assertions to the query history TrivialJudgments Map True I amp QEKAssertionsAxiom1 Q Returns the query history including trivial judgements Return Join QueryHistory TrivialJudgments 109 KSMP Techical Report Andrej Zaluski QEK QueryExpertStraightforward Q Expert Simulation returns the history of jugdmenets obtained while querying the simulated Expert for structure on items Q QEKQueryExpertStraightforward Q_ Expert_ Simulation Module QueryHistory OpenQuestions PJudgments TrivialJudgments Exclude trivial assertions from the open questions OpenQuestions QEK AssertionsAxiom Drop PQxQ Q Record all answers from the expert QueryHistory Map Function Ass QEK AskExpert Ass Expert Simulation OpenQuestions TrivialJudgments Map True I amp QEKAssertionsAxiom1 Q Returns the query history including trivial judgements Return Join QueryHistory TrivialJudgments QEKEntailmentFromQueryHistory Q QueryHistory extracts only positive judgments from the query history of the establishment of a structure of the items Q QEKEntailmentFromQueryHistory Q_ QueryHistory_ Module PJudgments PJudgments QEKExtractPositiveJudgmentsFromQueryHistory QueryHistory Return Relation Q PJudgments QEKEstablishKnowledgeSpaceStraightforward Q Items Human returns the structure on the items Q interpreted in Items established by queryin
113. is a minimal with respect to the relation SR MaximalsOfRelation usage MaximalsOfRelation SR returns all maximals of the relation SR MinimalsOfRelation usage MinimalsOfRelation SR returns all minimals of the relation SR RelationComplement usage RelationComplement SR returns the complement of the relation SR 47 KSMP Techical Report Andrej Zaluski RelationIdentity usage RelationIdentity S returns the identity relation on the set S EqualRelationsQ usage EqualRelationsQ SR1 SR2 returns True iff the relations SR1 and SR2 are equal RelationSubsetQ usage RelationSubsetQ SR1 SR2 returns True iff the relation SR1 is a subset of the relation SR2 RelationSubsetProperQ usage RelationSubsetProperQ SR1 SR2 returns True iff the relation SR1 is a proper subset of the relation SR2 MakeFunction usage MakeFunction D C M creates a function from the set D to the set C using the mapping rule M FunctionQ usage FunctionQ F returns True iff F represents a function FunctionDomain usage FunctionDomain f returns the domain of the function FunctionCodomain usage FunctionCodomain f returns the codomain of the function f FunctionMapping usage FunctionMapping f returns the mapping rule of the function f FunctionChangeDomain usage FunctionChangeDomain f D 1 is a primitive function that returns the function f after replacing its domain with the ne
114. is format is native to KSMP The header consist of the signature of a tool that produced the file The body is given as a sequence of records where each record describes one latent skill or elementary competency The record consists of three lines The first line contains code of a skill This code is used for skill identification by KSMP functions The second line consist of the skill name while the third line consists of a description of the skill For example querying module functions use the skill names while presenting assertions to a human expert This is an example of the skillsfile Skills File KSMP v1 0 1 Skill 1 Description Skill 1 2 Skill 2 Description Skill 2 3 Skill 3 Description Skill 3 13 KSMP Techical Report Andrej Zaluski Further developments Further developments of the package may be classified into three overlapping categories development of the core development of the modules and development of applications In this package version a plenty of written functions perform well given the memory limitations of personal computers only on small sets of items that is on the sets whose cardinalities do not reach numbers requested by relevant empirical investigations Therefore efforts will be taken in order to equip the package with data representations auxilary functions and swapping file mechanisms that will ensure adequate performance of the functions on the sets having the relevant sizes fo
115. ithin the field of knowledge spaces Providing integrated powerful computing typesetting and programming capabilities Mathematica offers an attractive working environment for research within the field of knowledge space theory Extending this with a possibility of presenting both the standard findings as well as the cutting edge results in the form of interactive notebooks to students makes Mathematica recommendable In addition to these major benefits several other characteristics are found quite useful Existence of mechanisms MathLink and the interprocess communication mechanism enables integrating Mathematica with existing Unix C knowledge space tools allowing possibilities of combining them with functions developed in Mathematica or invoking them from the front end in a transparent way These tools are optimized for computations on structures having sizes that are found in relevant empirical applications within the field therefore it would be desirable to have a possibility to use these tools Additionally possibilities of redirecting computation output to separate notebooks and automatic export to different file formats both realizable by front end programming further ease various routine tasks Probably for the research on knowledge spaces the most valuable part of the standard Mathematica distribution would be Skienna s DiscreteMath Combinatorica package Skienna 1990 3 This package served as starting point for developing the KSMP
116. ize A K returnes initialized interpretation function consisting of non interpreted items from the set A of problems given the competence space K CPAInterpretationFunctionInitialize A_ K_ Module MakeFunction A PowerSet K Map amp A CPAInterpretationFunctionAxiomsQ k returns True iff the function k satisfies the axiom of interpretation functions i e all the items are interpreted and informative CPAInterpretationFunctionAxiomsQ Mapping D_ C_ M_ Module f Mapping D C M KXs K K Apply Union C KXs FunctionMappingImage f Apply And Map Function kx EqualSetsQ kx And EqualSetsQ K kx KXs CPAInterpretationFunctionQ f returns True iff the function f is an interpretation function CPAInterpretationFunctionQ Mapping D_ C_ M_ Module f Mapping D C M FunctionQ f And FunctionWellDefinedQ f And CPAInterpretationFunctionAxiomsQ f CPAInterpretationFunctionItems k returns the set of items contained in the interpretation function k CPAInterpretationFunctionlItems Mapping A_ PsK_ KXs_ A CPAInterpretationFunctionInterpretations k returns the list of interpretations contained in the interpretation function k CPAInterpretationFunctionInterpretations Mapping A_ PsK_ KXs_ KXs CPAInterpretationsInterpretationReplace K Xs NewKx returns the interpretations KXs with the replaced interpretation NewKx for the corresponding item CPAInt
117. king only appropriate candidates Increase the premise size k First step take only appropriate candidates from the previous block Koppen 1993 S4 2 1 candidates Dol A kBlockAs i Xs SetComplement Q QEKStar PosAss A tmp Join candidates Map Union A amp Xs candidates Union tmp 1 1 Length kBlockAs VERBOSITY If OptionVerbosity gt 10 Print Constructing the next block of questions n Print 1 step premise candidates n candidates Second step take only one representative of each equivalent class Koppen 1993 S4 2 2 classes Dol A candidates i tmp Select candidates QEKEquivalentSubsetsQ PosAss A amp classes Join classes tmp 1 1 Length candidates VERBOSITY If OptionVerbosity gt 15 Print Equivalence classes of premise candidates n classes J Collect temporary candidates candidates Map 1 amp Union classes VERBOSITY If OptionVerbosity gt 10 Print 2 step premise candidates n candidates 117 KSMP Techical Report Andrej Zaluski Third step only candidates Bs for which A is subset of B and B is subset of A star does not hold Koppen 1993 4 2 3 kBlockAs Do B candidates i flag True Do A PosAss j AStar QEK Star PosAss A If SubsetQ A B And SubsetQ B Astar flag False Break 4j 1 Length PosAss If flag kBlockAs Join Op
118. le object that contain comments CommentLines DataFile SFile_ Flatten Position Map LineCommentQ amp SFile True NonCommentLines SFile returns a list of line numbers of the datafile object that do not contain a comment NonCommentLines SFile_DataFile Complement Range DataFileLength SFile CommentLines SFile DataFileCommentExtract SFile returns a list representing the comments within the datafile object Sfile DataFileCommentExtract DataFile SFileList_ SFileList CommentLines DataFile SFileList DataFileCommentPrint DF extracts and prints the comment from a datafile object DF DataFileCommentPrint DF_DataFile DataFileCommentExtract DF TableForm DataFileCommentDrop SFile returns the list representing the datafile object SFile with the 64 KSMP Techical Report Andrej Zaluski comments removed DataFileCommentDrop DataFile SFileList_ Return SFileList NonCommentLines DataFile SFileList FileFormatsInfo FileFormat Info returns the information Info about the requested file format FileFormat Currently supported formats are KST_SpaceFile KST_PatternFile KSMP_ItemsFile and KSMP_SkillsFile The stored information Info about the formats are CommentSign and Head FileFormatsInfo FileFormat_ InfoType0_ Module FileFormatData KST_SpaceFile 1 2 KST_PatternFile 123 KSMP_ItemsFile KSMP_SkillsFile
119. license Package Version 0 9 beta Mathematica Version 4 0 History 1 0 reserved for the first public version 0 9 beta 1 version Keywords Psychology and Education Mathematical Psychology Knowledge Modeling Knowledge Space Theory Skills Modeling Sources Warnings BETA 1 TESTING PHASE Limitations Discussion Requirements Packages KnowledgeSpaces L2 TO DO Add also KSMP_ Skills m BeginPackage KnowledgeSpaces Skills KnowledgeSpaces L2 TODO Save the state of the package Off General spell General spell1 CSAxiom1Q usage CSAxiom1Q EK returns True iff the structure EK fulfills the property Union K E PSAxiom1Q usage PSAxiom1Q AP returns True iff the structure AP fulfills the property Union P A 89 KSMP Techical Report Andrej Zaluski CSAxiom2Q usage CSAxiom2Q EK returns True iff the structure fulfills the property that the family K contains both the empty set and the set E PSAxiom2Q usage PSAxiom2Q AP returns True iff the structure AP fulfills the property that the family P contains both the empty set and the set A CSAxiom3Q usage CSAxiom3Q EK returns True iff the structure EK is stable under union PSAxiom3Q usage PSAxiom3Q AP returns True iff the structure AP is stable under union CompetenceStructureQ usage CompetenceStr
120. lossary of KSMP functions 10 KSMP Techical Report Andrej Zaluski KSMP Querying module The aim of the Querying module is to implement existing approaches for establishing various knowledge space theory constructs by querying experts The module in this package version consists of the following components Koppen s algorithm Querying experts on competence performance diagnostic Querying Session Sonde and General querying experts component The Koppen s algorithm component implements establishing of knowledge structures based on the concepts of entail relation and entailment as well as a full version of Koppen s block by block algorithm as published in Koppen 1993 The Querying Experts on competence performance diagnostic currently implements only a straightforward procedure for querying experts on diagnostic The most important parts of this component are the procedure for querying on competence space based on Koppen s block by block algorithm and the straightforward procedure for querying on interpretation function The Querying Session Sonde introduces the concept of observation sonde whose goal is to be a method for exploring the behavior and performance of querying procedures The General querying experts component is conceived as a component that unites all implemented querying approaches and represents an interface between the module and applications belonging to the fourth layer of the package The querying procedures may oper
121. mented by N QSRSondeQuestionsAskedAdd Sonde_ N_ ReplacePart Sonde QSRSondeQuestionsAsked Sonde N 2 QEQueryExpertOnKnowledgeSpace Q ItOrExp options queryies a human 132 KSMP Techical Report Andrej Zaluski ItOrExp represents items description or a simulated expert ItOrExp represents a simulated expert on the set Q of items and a structure either space or assertions is returned The valid option is QueryingAlgorithm Koppen default For additional valid options see QEK QueryExpertByKoppen QEQueryExpertOnKnowledgeSpace Q_ ItOrExp_ opts__ Module OptionQueryingAlgorithm ReturnedValue OptionQueryingAlgorithm QueryingAlgorithm opts Options QEQueryExpertOnKnowledgeSpace If OptionQueryingAlgorithm K oppen QueryOn gt KnowledgeSpace unchangable ReturnedValue QEK QueryExpertByKoppen Q ItOrExp QueryOn gt KnowledgeSpace opts J Return ReturnedValue End EndPackage 133 KSMP Techical Report Andrej Zaluski KSMP assessment module Title KnowledgeSpaces Mathematica Package Assessment Module Context KnowledgeSpaces Assessment Author Andrej Zaluski gt Summary Deterministic Knowledge Assessment Copyright 20001 Andrej Zaluski See attached license Package Version 0 91 beta 1 Mathematica Version 4 0 History 1 0 the first published version 0 91 beta
122. n 2 Block Textl QEKQueryTexts 2 introduction Text2 StringJoin ItemDescription ItemGet Items A 1 and ItemDescription ItemGet Items A 2 tmp QEK QueryTexts 2 question Text3 Apply StringJoin tmp ItemDescription ItemGet Items b QueryText StringJoin Text1 Text2 Text3 107 KSMP Techical Report Andrej Zaluski nQ nAssertion 1 Block Textl QEKQueryTexts previous_to_last introduction Text2 tmp QEKQueryTexts previous_to_last question Text3 Apply StringJoin tmp ItemDescription ItemGet Items b QueryText StringJoin Text1 Text2 Text3 True GENERIC Block Textl QEKQueryTexts generic introduction tmp Map ItemDescription ItemGet Items amp A Ntmp Length tmp Text2 Apply StringJoin Table tmp i lt gt i 1 Ntmp 1 Text2 Apply StringJoin Text2 tmp Ntmp tmp QEK QueryTexts generic question Text3 Apply StringJoin tmp ItemDescription ItemGet Items b QueryText StringJoin Text1 Text2 Text3 1 Text4 You may assume that chance factors such as careless errors and lucky guesses play no rule in the student s performance Return StringJoin QueryText Text4 QEKAskExpert QorE ItemsOrSkills Assertion Human options returns an answer from a human experts on assertion containing some items from QorE interpreted in ItemsOrSkills in case of querying on knowledge spac
123. n usage QEK SelectionRuleMaximin Q PosAss NegAss OpenQ returns the assertion which implies a maximal number of inferences according to the maximin selection rule QEK SelectNextQuestion usage QEK SelectNextQuestion Q Pos Neg OpenQ options returns the data structure NextAss NegAnsInf PosAnsInf where NextAss is the next assertion for querying accompanied with its both positive and negative inferences NegAnsInf in case of a negative answer and PosAnsInf in case of 101 KSMP Techical Report Andrej Zaluski a positive answer The selection is conducted on the set Q of items assuming previously collected positive inferences Pos negative inferences Neg and the items OpenQ left open The selection rule is given by the option SelectionRule that accepts the following values Maximin default WeightedSums QEK Output usage QEK Output Q PosAss NegAss options returns the output of a querying proces either as a knowledge space or as assertions depending on the OutputStructure option The querying process is captured by the positive assertions PosAss and the negative assertions NegAss given the set Q of items The OutputStructure option accepts the following values SpaceStructure default and Assertions QEKAIlAssertionsToSpace usage QEKAlIlAssertionsToSpace Q PosAss returns the knowledge space derived from the positive assertions PosAss QEK QueryExpertByK oppen usage QEK Qu
124. n that this student will also fail the problem 2 Suppose that a student under examination has just failed both the problems Ts it then practically certain that this student will also fail the problem generic Suppose that a student under examination has just failed all the problems Is it then practically certain that this student will also fail the problem previous_to_last Suppose you are tutoring a student who does not master any of the problems in the given problem set Could you then start by teaching him how to solve the problem hh TextPart Select TextParts 1 TextPart0 amp 1 1 2 Return Select QueryTexts 1 PremiseSize amp 1 1 TextPart QEKFormulateQuery Q Items Assertion returns the query question on the given assertion Assertion formulated for a human expert given the set Q of items described in Items QEKFormulateQuery Q_ Items_ Assertion_ Module nQ SetCardinality Q nAssertion SetCardinality QEK AssertionPremise Assertion A QEKAssertionPremise Assertion b QEKAssertionConsequence Assertion QueryTexts QueryText Text1 Text2 Text3 Text4 tmp Ntmp Which nAssertion 1 Block Textl QEK QueryTexts 1 introduction Text2 ItemDescription ItemGet Items A 1 tmp QEK QueryTexts 1 question Text3 Apply StringJoin tmp ItemDescription ItemGet Items b QueryText StringJoin Text1 Text2 Text3 nAssertio
125. n the given item a interpreted by Items and the competence state C of the competence structure K described through Skills QEOSFormulateQueryInterpretations a_ Items_ C_ K_ Skills_ Module NofSkills nK QueryTexts QueryText Textl Text2 Text3 Text4 TextEnding tmp Ntmp NofSkills SetCardinality C Number of elementary competencies nK SetCardinality Apply Union K Which NofSkills 1 Text1 QEOSQueryTextsInterpretations NofSkills introduction Text2 SkillName SkillGet Skills C 1 Text3 QEOSQueryTextsInterpretations NofSkills middle Text4 ItemDescription ItemGet Items a QueryText StringJoin Text1 Text2 Text3 Text4 NofSkills 2 121 KSMP Techical Report Andrej Zaluski Text1 QEOSQueryTextsInterpretations NofSkills introduction Text2 StringJoin SkillName SkillGet Skills C 1 and SkillName SkillGet Skills C 2 Text3 QEOSQueryTextsInterpretations NofSkills middle Text4 ItemDescription ItemGet Items a QueryText StringJoin Text1 Text2 Text3 Text4 all skills except one nK NofSkills 1 Text1 QEOSQueryTextsInterpretations all_minus_one introduction Text2 SkillName SkillGet Skills SetDifference A pply Union K C 1 Text3 QEOSQueryTextsInterpretations all_minus_one middle Text4 ItemDescription ItemGet Items a QueryText StringJoin Text1 Text2 Text3 Text4 generic True Text1 QEOSQueryTextsInte
126. n the interpretation function k CPARepresentationOfCompetenceState usage CPARepresentationOfCompetenceState k C returns the data structure C p C where p C is a set of items assigned by the interpretation function k to the given competence state C CPARepresentations usage CPARepresentations k returns the list of all representations of the competence structure derived from the interpretation function k CPARepresentationFunctionFromInterpretationFunction usage CPARepresentationFunctionFromInterpretationFunction k returns the representation function given the interpretation function k CPAMakeRepresentation usage CPA MakeRepresentation C Items returns the data structure containing the representation formed by the given competence state C and the set Items CPARepresentationCompetenceState usage CPA RepresentationCompetenceState PC returns the competence state of the given representation PC CPARepresentationltems usage CPARepresentationItems pC returns the set of items for the given representation pC CPAMakeRepresentationFunction usage CPAMakeRepresentationFunction K A PCs returns the data structure containing the representation function formed from the competence structure K the set A of items and the collection PCs of representations CPARepresentationFunctionCompetenceStructure usage CPARepresentationFunctionCompetenceStructure p returns the competence structur
127. nces according to the maximin selection rule The list OpenQ of open interpretation assertions is given QEOSInterpretationFunctionSelectionRuleWeightedSums usage QEOSInterpretationFunctionSelectionRuleWeightedSums OpenQ returns the interpretation assertion which implies a maximal number of inferences according to the weighted sums selection rule The list OpenQ of open interpretation assertions is given QEOSSelectNextInterpretationQuestion usage QEOSSelectNextInterpretationQuestion OpenQ opts returns the next interpretation assertion for querying given the sorted list OpenQ of open interpretation assertions respecting the given options The valid options are OpenQuestions Sorted default Unsorted in case the list OpenQ is not sorted it will be sorted with respect to the cardinality of competence states RuleScope Block default All whether to search for the next 103 KSMP Techical Report Andrej Zaluski interpretation assertion only in the next block defined by the cardinality of competence states or to search all interpretation assertions in OpenQ SelectionRule Maximin default WeightedSums which selection rule is to be used QEOSQueryExpertOnInterpretationFunction usage QEOSQueryExpertOnInterpretationFunction K A DescOrExp opts establishes an interpretation function by querying an expert given the set A of items and the competence space K After the quer
128. nd negative inferences N when a negative answer on the assertion Ass is collected The set Q of items lists PosAss and NegAss of established positive and negative assertions are given in advance The inference rule is implemented according to Koppen 1993 14d QEKCollectInferencesForPositiveAnswer usage QEKCollectInferencesForPositiveAnswer Q PosAss NegAss Ass returns the data structure P N consisting of positive inferences P and negative inferences N when a positive answer on the assertion Ass is collected The set Q of items lists PosAss and NegAss of established positive and negative assertions are given in advance The inference rule is implemented according to Koppen 1993 14a 14b 14c QEKCountInferences usage QEKCountInferences Q PosAss NegAss Ass returns the data structure Neg Pos containg the number Neg of inferences that would be drawn in case a negative answer is collected on the assertion Ass and the number Pos of inferences in case of a positive answer The set Q of items the established positive inferences PosAss and negative inferences NegAss are given in advance QEKSelectionRuleWeightedSums usage QEKSelectionRuleWeightedSums Q PosAss NegAss OpenQ options returns the assertion which implies a maximal number of inferences according to the weighted sums selection rule The option ProbabilityYES is supported having 0 5 as the default value QEKSelectionRuleMaximi
129. ng the set forming the structure by the set X1 StructureReplaceFamily usage StructureReplaceFamily SF F 1 is a primitive function that returns the structure SF after replacing its family by the family F1 StructureReplaceCarrier usage StructureReplaceCarrier SF SC returns the rewritten structure SF using the set SC as the new carrier StructureSetEmpty usage StructureSetEmptyQ SF returns True iff the set forming the structure SF is empty StructureSetNonEmptyQ usage StructureSetNonEmptyQ SF returns True iff the set forming the structure SF is non empty StructureFamilyEmptyQ usage StructureFamilyEmptyQ SF returns True iff the 49 KSMP Techical Report Andrej Zaluski family forming the structure SF is empty StructureFamilyNonEmptyQ usage StructureFamilyNonEmptyQ SF returns True iff the family forming the structure SF is non empty StructureSetElementQ usage StructureSetElementQ SF s returns True iff the element s is an element of the set forming the structure SF StructureSetSubsetQ usage StructureSetSubsetQ SF S1 returns True iff the set S1 is a subset of the set forming the structure SF StructureFamilyElementQ usage StructureFamilyElementQ SF f returns True iff the set f is an element of the family forming the structure SF StructureFamilySubsetQ usage StructureFamilySubsetQ SF F1 returns True iff the family F1 is a subset of the family forming the
130. nostic returns a data structure representing experts knowledge in the form of competence performance diagnostic for the given diagnostic DIAG MakeMetaKnowledge Diagnostic E_ K_ A_ P_ k p_ Diagnostic Module DIAG Diagnostic E K A P k p Return DIAG MakeMetaKnowledge Diagnostic K_ A_ P_ k p_ Diagnostic Module DIAG Diagnostic K A P k p Return DIAG MakeSimpleExpert SF returns an ideal expert possesing domain s metaknowledge given by the knowledge space SF MakeSimpleExpert Structure S_ F_ Module SF Structure S F MakeExpert MakeMetaKnowledge EntailmentFromKnowledgeSpace SF Assertions 0 MakeSimpleExpert DIAG returns an ideal expert possesing domain s metaknowledge given by the CPA diagnostic structure DIAG MakeSimpleExpert DIAG_Diagnostic Module MakeExpert MakeMetaKnowledge DIAG Diagnostic 0 ExpertMakeAnswer Ass LV CF returns a data structure for expert s answer on assertion Ass consisting of the logical value LC and the certainty factor CF ExpertMakeAnswer Assertion_ LogicalValue_ CertaintyFactor_ Assertion LogicalValue CertaintyFactor ExpertAnswerLogicalValue Ans returns the logical value of the expert s answer Ans ExpertAnswerAssertion Answer_ Answer 1 ExpertAnswerLogicalValue Ans returns the logical value of the expert s answer Ans ExpertAnswerLogicalValue Answer_ Answer 2 ExpertAnswerCertaintyFactor Ans returns the
131. nswer CertaintyFactor QEOSAskExpertOnInterpretation IAss Expert Simulation returns an answer data structure on the interpretation assertion IAss from the simulated Expert QEOSAskExpertOnInterpretation IAss_ Expert_ Simulation Module answer ExpCErrors CertaintyFactor answer SetElementQ IAssertionCompetenceState IAss CPAInterpretationOfltem CPADiagnosticInterpretationFunction ExpertMetaKnowledge Expert AssertionItem IAss ExpCErrors ExpertCarelessErrors Expert CertaintyFactor 1 Return ExpertMakeAnswer IAss answer CertaintyFactor QEOSInterpretationsOnItem IASSes 124 KSMP Techical Report Andrej Zaluski q returns all the interpretation assertions for the item q from the list IASSes QEOSInterpretationsOnItem IASSes_ q_ Module Select IASSes IAssertionItem q amp QEOSInterpretationFunctionCollectInferencesForPositiveAnswer OpenQ IAss returns all the interpretation assertion inferences given the list OpenQ of open interpretation assertions and interpretation assertion Ass that was judged positively QEOSInterpretationFunctionCollectInferencesForPositiveAnswer OpenQ0_ IAss_ Module OpenQ OpenQ0 q IAssertionItem IAss OpenQ QEOSInterpretationsOnItem OpenQ q Select OpenQ IAssertionItem IA ssertionItem IAss amp amp SubsetQ IAssertionCompetenceState IAss 1AssertionCompetenceState amp QEOSInterpretationFunctionCollec
132. o equip KSMP with facilities for supporting possibly different versions of the mentioned file formats Spacefile format The spacefile format is designed to store information on families of subsets particulary knowledge structures competence structures and examinees observable response patterns This format was inherited from already developed KST and SRBT tools The header consists of the signature of the tool that produced the file and metainformation while the body consists of the actual family of subsets The metainformation is given by the two lines of the header after the comment lines It consists of two integers representing the overall number of elements from which the subsets are created and the actual number of subsets present in the file Further lines consist of binary string where each line represents a subset while binary digits stands for absence or presence of a particular element e g item or elementary competence in the subset Here is an example of the spacefile KSMP v1 0 3 4 000 100 110 111 The spacefile above represents the following family of subsets 1 1 2 1 2 3 taken from the set 1 2 3 Itemsfile format The itemsfile format is designed to store description of items for purposes of querying experts or knowledge assessment This format is native to KSMP The header consist of the signature of a tool that produced the file The body is given as a sequence of records wh
133. oKnowledgeSpace EntailRelationToKnowledgeSpace ESR returns the knowledge space given the entail relation ESR EqualFunctionsQ EqualFunctionsQ fl f2 returns True iff the functions fl and f2 are equal EqualRelationsQ EqualRelationsQ SR1 SR2 returns True iff the relations SR1 and SR2 are equal EqualSetsQ EqualSetsQ A B returns True iff the sets A and B are equal 20 KSMP Techical Report Andrej Zaluski EqualStructuresQ EqualStructuresQ SF 1 SF2 returns True iff the structure SF1 is equal to the structure SF2 ExistOneQ ExistOneQ X pf returns True iff the property pf holds for at least one element of the set X ExpertAnswerCertaintyF actor ExpertAnswerCertaintyFactor Ans returns the certainty factor of an expert s answer Ans ExpertAnswerLogicalValue ExpertAnswerLogicalValue Ans returns the logical value of the expert s answer Ans ExpertCarelessErrors ExpertCarelessErrors E returns the uncertainty that the expert E makes a careless error during a querying session ExpertMakeAnswer ExpertMakeAnswer LV CF returns the data structure describing an expert s answer consisting of the logical value LC and the certainty factor CF ExpertMetaKnowledge ExpertMetaKnowledge E returns a domain metaknowledge of the expert E FamilyQ FamilyQ F returns True iff SF is a family of subsets FileFormatsInfo FileFormatsInfo FileFormat Info returns the information Info about the requested file form
134. of competence states or searching all the interpretation assertions in OpenQ SelectionRule Maximin defult WeightedSums QEOSSelectNextInterpretationQuestion OpenQ0_ opts_ _ Module OptionSelectionRule OptionRuleScope OptionOpenQuestions tmp min OpenQ OpenQ0 NextlAss TflOpenQ Return if the list of open questions is unsorted sort it according to the size of interpretation assertion competence states OptionOpenQuestions OpenQuestions opts Options QEOSSelectNextInterpretationQuestion If OptionOpenQuestions Sorted tmp Sort OpenQ Length 1 2 lt Length 2 2 amp OpenQ tmp J Ifthe scope of the selection rule is Block extract the block of 126 KSMP Techical Report Andrej Zaluski interpretation assertions with the minimal competence states OptionRuleScope RuleScope opts Options QEOSSelectNextInterpretationQuestion If OptionRuleScope Block min Length OpenQ 1 2 tmp Select OpenQ Length 2 min amp OpenQ tmp J Select the next interpretation assertion question OptionSelectionRule SelectionRule opts Options QEOSSelectNextInterpretationQuestion Switch OptionSelectionRule WeightedSums NextIAss QEOSInterpretationFunctionSelectionRuleWeightedSums OpenQ Maximin NextIAss QEOSInterpretationFunctionSelectionRuleMaximin OpenQ J Return NextIAss QEOSQueryExpertOnInterpretationFunction EK A DescOrEx
135. of items The established structure is returned either as a space or assertions A valid option is QueryingAlgorithm Koppen default For additional valid options see QEK QueryExpertByKoppen Expert usage Expert is an option of QEK QueryExpertByKoppen QEOSQueryExpertOnDiagnostic and QEOSQueryExpertOnInterpretationFunction The possible values are Human default and Simulation OpenQuestions usage OpenQuestions is an option of QEOSSelectNextInterpretationQuestion The possible values are Sorted default and Unsorted OutputStructure usage OutputStructure is an option of QEKOutput QEK QueryExpertByKoppen The possible values are SpaceStructure default and Assertions Probability Y ES usage ProbabilityYES is an option of QEKSelectionRuleWeightedSums and QEOSInterpretationFunctionSelectionRuleWeightedSums The default value is 0 5 5 QueryingA gorithm usage Querying algorithm is an option of QEOSQueryExpertOnCompetenceState and QEQueryExpertOnKnowledgeSpace The possible value is Koppen default QueryOn usage QueryOn is an option of QEKAskExpert QEK QueryExpertByKoppen The possible values are KnowledgeSpace default and CompetenceSpace QSRReport usage QSRReport is an option of QEK QueryExpertByKoppen QEOSQueryExpertOnDiagnostic and QEOSQueryExpertOnInterpretationFunction The possible values are Result default Report and Both SelectionRule usage SelectionR
136. ofCl returns true iff each item from the mapping LofCL is member of all clauses assigned to it SSAxiom2Q LofCl_ Module Q Transpose LofCl 1 SSAxiom2Q Q LofCl SSAxiom2Q Q LofCl returns true iff each item from Q is member of all clauses assigned to it by the mapping LofCL of items to clauses SSAxiom2Q Q_ LofCl_ ForAlQ Q Function item SSAxiom2atItemQ item Select LofCl 1 item amp 1 2 83 KSMP Techical Report Andrej Zaluski SSAxiom2atItemQ q clauses returns true iff the item q is an element of all its assigned clauses SSAxiom2atItemQ q_ clauses_ ForAlIQ clauses SetElementQ q amp SSAxiom3Q LofCl returns true iff for each clause C of any item q given by the mapping LofCl there exist some clause C1 assigned to each item q1 of the clause C that is subset of the clause C SSAxiom3Q LofCl_ Module q1 Apply And Map Function clause0 Block clause clause0 2 C Apply And Map Function C Apply And Map Function q1 Block qlclauses qiclauses SSLofClClausesGet LofCl q1 Apply Or Map SubsetQ C amp q1clauses C clause LofCl SSAxiom4atClausesQ clauses returns true iff any two clauses for the same item are incomparable SSAxiom4atClausesQ clauses0_ Module clauses clausesO 2 ForAllPairsQ clauses Function pair Block C1 pair 1 C2 pair 2 SubsetQ C1 C2 Implies EqualSetsQ C1 C2 MM
137. on S11_ S12_ R1_ Relation S21_ S22_ R2_ EqualSetsQ S11 S21 And EqualSetsQ S12 S22 And EqualSetsQ R1 R2 RelationSubsetQ SR1 SR2 returns true iff the relation SR1 is a subset of relation SR2 RelationSubsetQ Relation S1_ R1_ Relation S2_ R2_ EqualSetsQ S1 S2 And SubsetQ R1 R2 RelationSubsetQ Relation S11_ S12_ R1_ Relation S21_ S22 R2_ EqualSetsQ S11 S21 And EqualSetsQ S 12 S22 And SubsetQ R1 R2 RelationSubsetProperQ SR1 SR2 returns true iff the relation SR1 is a proper subset of relation SR2 RelationSubsetProperQ Relation SA_ RA Relation SB_ RB_ Module SRA Relation SA RA SRB Relation SB RB RelationSubsetQ SRA SRB And EqualRelationsQ SRA SRB RelationSubsetProperQ Relation SAl_ SA2_ RA Relation SB1_ SB2_ RB_ Module SRA Relation SA1 SA2 RA SRB Relation SB1 SB2 RB RelationSubsetQ SRA SRB And EqualRelationsQ SRA SRB MakeFunction D C M creates a function from the set D to the set C using the mapping M MakeFunction D_List C_List M_List Mapping D C M Length D Length M FunctionQ F returns True if F is a data structure containing a function FunctionQ F_ If Head F Mapping True False FunctionDomain f returns the domain of the function f FunctionDomain Mapping d_ c_ m_ d FunctionCodomain f returns the codomain of the function f FunctionCodomain Mapping D_ C_ M_ C FunctionMapping f returns the mapping r
138. operQ RelationSubsetProperQ SR1 SR2 returns True iff the relation SR1 is a proper subset of the relation SR2 RelationSubsetQ RelationSubsetQ SR1 SR2 returns True iff the relation SR1 is a subset of the relation SR2 RelationSymmetricQ RelationSymmetricQ SR returns True iff the relation SR is symmetric RelationTransitiveQ RelationTransitiveQ SR returns True if the relation SR is transitive RowList2String RowList2String R rewrites the list R of numbers as the string RowString2List RowString2List R rewrites the string R of digits as the list of numbers SetCardinality SetCardinality S returns the cardinality of the finite set S SetCartesianProduct SetCartesianProduct 1 S2 returns the cartesian product of the non empty sets S1 and S2 SetCartesianProduct S1 S2 Sk returns the cartesian product of the non empty sets S1 S2 Sk 35 KSMP Techical Report Andrej Zaluski SetComplement SetComplement U S returns the complement of the set S in the universe U SetDifference SetDifference A B returns the set difference A B SetElementQ SetElementQ a S returns True iff a is an element of the set S SetEmptyQ SetEmptyQ S returns True iff the set S is empty SetNonEmptyQ SetNonEmptyQ S returns True iff the set S is non empty SetQ SetQ L returns True iff L represents a set SkillCode SkillCode Skill returns the code of the given skill Skill SkillDescription
139. or the answer on either items or skills If OptionHumOrSim Simulation Answer QEKAskExpert NextAss ItemsExpert Simulation I If OptionHumOrSim Human amp amp OptionQueryOn KnowledgeSpace Answer QEKAskExpert Q ItemsExpert NextAss Human QueryOn gt KnowledgeSpace If OptionHumOrSim Human amp amp OptionQueryOn CompetenceSpace Answer QEKAskExpert Q ItemsExpert NextAss Human QueryOn gt CompetenceSpace NofQuestions VERBOSITY If OptionVerbosity gt 5 Print NofQuestions question NextAss The answer Answer Add the answer and the appropriate inferences to the established assertions If ExpertAnswerLogicalValue Answer AppendTo PosAss ExpertAnswerAssertion Answer If PosAnsInf 1 NotEqual AppendTo PosAss PosAnsInf 1 If PosAnsInf 2 NotEqual AppendTo NegAss PosAnsInf 2 116 KSMP Techical Report Andrej Zaluski AppendTo NegAss ExpertAnswerAssertion Answer If NegAnsInf 1 NotEqual AppendTo PosAss NegAnsInf 1 If NegAnsInf 2 NotEqual AppendTo NegAss NegAnsInf 2 J Update the list of open questions tmp SetDifference OpenQ NextAss OpenQ tmp VERBOSITY If OptionVerbosity gt 10 Print Block filled with anwers and inferences Print Known positive assertions n PosAss Print Known negative assertions n NegAss End of While Construct the next block of open questions by ta
140. otivation for developing the package is presented and Mathematica is described as a platform suitable for research and teaching within the field of knowledge spaces The software and hardware requirements are presented together with portability issues and instructions for installing the package The forms of package documentation are briefly described and some hints on using the on line documentation given The package design is presented by describing the package architecture and used file formats The package is conceived as consisting of four abstract layers In this package version the first two layers together with the warp core module forms the core of the package The layer three is comprised of different modules i e skills module querying module and assessment module Used data file formats are described in the sequel A plan to equip the fourth layer with Querying Experts on Skills application together with further development of the core enhancements and extensions of the modules are briefly presented The alphabetical glossary containing all public package functions makes the central part of the paper Finally a bibliography and the source code of the package are provided KSMP Techical Report Andrej Zaluski Contents Acknowledgements 2 License 3 Overview 4 Contents 5 Introduction 6 Motivation and development of the package 6 Mathematica as a platform suitable for research and teaching within the field of kno
141. p opts establish an interpretation function by querying an expert given the set A of items and the competence space EK After the querying only the competence based items set together with its interpretations is kept In case of human expert DescOrExp represents structure Skills Items where the used skills are described through Skills and the items through Items In case of simulated expert DescOrExp describes the expert The possible options are Verbosity Off Default 1 5 10 15 QSRReport Result default Report or Both Expert Human default Simulation SelectionRule Maximin default WeightedSums and RuleScope Block default All QEOSQueryExpertOnInterpretationFunction EK_ A_ DescOrExp opts__ Module K OpenQ PosIAss NegIAss Answer Skills Items SimExpert tmp PosAnsInf NegAnsInf k QSR OptionRuleScope OptionSelectionRule OptionVerbosity OptionHumOrSim OptionQSRReport VERBOSITY LEVEL OptionVerbosity Verbosity opts Options QEOSQueryExpertOnInterpretationFunction What the function should return as its output OptionQSRReport QSRReport opts Options QEOS QueryExpertOnInterpretationFunction Whether to query human or simulated expert OptionHumOrSim Expert opts Options QEOSQueryExpertOnInterpretationFunction The rule used for selecting the next question OptionSelectionRule SelectionRule opts Options QEOS QueryExpertOnInterpretationF
142. r empirical investigations Probably the future versions of the package will include two operating modes of the package research mode and application mode Research mode will be suitable for research teaching explorations and simulations on small sets of items while the application mode will be suitable for usage in empirical investigations The KSMP Warp Core addition to the layer one represents an effort in this direction Besides the further development of the warp core itself the other functions belonging to the package core will be also upgraded to support the warp core solutions In addition a mechanism for automatic selection between research and application mode based primarily on the available memory will also be implemented Another category of developments consists of extending and enhancing the existing modules as well as of developing additional modules Extending the existing modules will include implementing other approaches to skills as additional components inside the KSMP Skills module and implementing other querying procedures as additional components inside the KSMP Querying module The last but not the least category of further developments consists of developing applications that can be seen as members of the abstract layer four of the package Within the QEOS project there is a plan to equip the layer four with the QUEOS application whose aim is to provide an user friendly environment for establishing the knowledge spac
143. re S F KQ RQ1 RQ2 KQ FX SF x0 RQL Map Function pair Block x pair 1 y pair 2 If SubsetQ x y AppendTo RQ1 pair CartesianProduct KQ KQ RQ2 MakeRelation KQ RQ1 MinimalsOfRelation RQ2 SubsetQl x0 S Atoms SF returns all the subfamilies of the family F which are atoms by at least one element of the set S Atoms Structure S_ F_ Module SF Structure S F Union Flatten Map AtomsAt SF amp S 1 BasisFromSpace SF calculates the basis of the space SF BasisFromSpace SF_Structure KnowledgeSpaceQ Atoms SF BasisToSpace returns the space for a given basis BasisToSpace F_List FamilyQ Module F1 F1 Union Union GenerateFamilyByTakingUnions F Return Structure Take F 1 1 1 F1 SSLofClClausesGet LofCl q returns the clauses of the item q that are assigned to it by the mapping LofCl of items to clauses SSLofClClausesGet LofCL_ q_ Module clauses clauses Select LofCl 1 q amp If clauses NotEqual clauses 1 2 SSAttributionQ LofCl returns true iff a non empty collection of clauses is assigned to each item within the mapping LofCL of items to clauses SSAttributionQ LofCl_ Apply And Map Function pair SetNonEmptyQ pair 2 LofC1 SSAxiom1Q LofCl returns true iff the mapping LofCl of items to clauses is an attribution SSAxiom1 Q LofCl_ SSAttributionQ LofCl SSAxiom2Q L
144. reQ Structure E_List SetNonEmptyQ K_ List FamilyQ Module EK Structure E K CSAxiom1Q EK PerformanceStructureQ AP returns True iff the structure AP is a performance structure PerformanceStructureQ Structure E_List SetNonEmptyQ K_ List FamilyQ Module EK Structure E K PSAxiom1Q EK CompetenceSpaceQ EK returns True iff the structure EK is a competence space CompetenceSpaceQ Structure E_List SetNonEmptyQ K List FamilyQ Module EK Structure E K CSAxiom1Q EK amp amp CSAxiom2Q EK amp amp CSAxiom3Q EK PerformanceSpaceQ AP returns True iff the structure AP is a competence space PerformanceSpaceQ Structure E_List SetNonEmptyQ K_ List FamilyQ Module EK Structure E K CSAxiom1Q EK amp amp CSAxiom2Q EK amp amp CSAxiom3Q EK CPAMakelInterpretationFunction A K KXs returns the interpretation function give the set A of items the competence structure K 93 KSMP Techical Report Andrej Zaluski and the mapping KXs interpreting each item by assigning competence states CPAMakelnterpretationFunction A_ K_ KXs_ Module f f MakeFunction A PowerSet K K Xs If FunctionWellDefinedQ f Return f CPAInterpretationFunctionInitialize K returns empty interpretation function for given competence space K and empty set of problems CPAInterpretationFunctionInitialize K_ Module MakeFunction PowerSet K CPAInterpretationFunctionInitial
145. resenting the datafile object SFile after removing comments FileFormatsInfo usage FileF ormatsInfo FileF ormat Info returns the information Info about the requested file format FileFormat Currently supported formats are KST_SpaceFile KST_PatternFile The stored information about the formats is Head DataFileHeadExtract usage DataF ileHeadExtract SFL FileFormat returns the extracted head from the non commented data list SFL representing the datafile object in accordance with the datafile format FileFormat n nFor the supported datafile formats see FileFormatsInfo DataFileBodyExtract usage DataFileBodyExtract SFL FileFormat returns the 51 KSMP Techical Report Andrej Zaluski extracted body from the non commented data list SFL representing the datafile object using the file format FilFormat n nFor the supported datafile formats see FileFormatsInfo DataFileHeadInterpret usage DataFileHeadInterpret H FF interprets the extracted head H from a datafile object in the FF format DataFileBodyInterpret usage DataFileBodyInterpret B FF interprets the extracted body B from the data file object of the FF format RowList2String usage RowList2String R rewrites the list R of numbers as the string RowString2List usage RowString2List R rewrites the string R of digits as the list of numbers Matrix2DataFileBody usage Matrix2DataFileBody M_List KST_SpaceFile
146. rpretations generic introduction construct the list of skill names given a competence state C tmp Map SkillName SkillGet Skills amp C Ntmp Length tmp Text2 Apply StringJoin Table tmp i lt gt 1 1 Ntmp 1 Text2 Apply StringJoin Text2 tmp Ntmp Text3 QEOSQueryTextsInterpretations generic middle Text4 ItemDescription ItemGet Items a QueryText StringJoin Text1 Text2 Text3 Text4 TextEnding You may assume that chance factors such as careless errors and lucky guesses play no role in the student s performance Return StringJoin QueryText TextEnding QEOSQueryTextsSkills PremiseSize TextPart returns the TextPart introduction or question for the premise of the size PremiseSize 1 2 generic all_minus_one QEOSQueryTextsSkills PremiseSize_ TextPart0_ Module QueryTexts TextParts TextPart TextParts introduction 2 question 3 QueryTexts 1 Suppose that a student under examination does not posses the competence Is it then practically certain that this student also does not posses the competence 2 Suppose that a student under examination does not posses both the competencies Is it then practically certain that this student also does not posses the competence generic Suppose that a student under examination does not posses the following competencies Is it then practically certain that this student al
147. rts Component TO DO Add also KSMP_ Skills m BeginPackage KnowledgeSpaces L2 KnowledgeSpaces L1 KnowledgeSpaces Skills TODO Save the state of the package Off General spell General spell1 Makeltem usage Makeltem Code Desc QText CorrAns LofDist returns a data object representing the item consisting of the unique identifier Code description Desc text QText of the question correct Answer CorrAns and the list LofDist of admissible answers where each answer consists of a code and a textual description ItemCode usage ItemCode Item returns the code of the given Item 68 KSMP Techical Report Andrej Zaluski ItemDescription usage ItemDescription Item returns the description of the given Item ItemText usage ItemText Item returns the textual question of the given Item ItemCorrectAnswer usage ItemCorrectAnswer Item returns the correct answer of the given Item ItemAnswers usage ItemAnswers Item returns the list of all admissible answers on the item Item where each answer consists of a code and a description ItemGet usage ItemGet Items code returns the first item in the description whose code matches the code Code ReadItems usage ReadItems FN KSMP_ItemsFile reads the description of items stored in the file FN using the file format KSMP_ItemsFile n n ReadItems FN assumes the file format KSMP_ItemsFile
148. s contained in the interpretation function k CPAInterpretationFunctionInterpretations usage CPAInterpretationFunctionInterpretations k returns the list of interpretations contained in the interpretation function k CPAInterpretationsInterpretationReplace usage CPAInterpretationsInterpretationReplace K Xs NewKx returns the interpretations K Xs after replacing the corresponding interpretation with 90 KSMP Techical Report Andrej Zaluski the interpretation NewKx CPAInterpretationsInterpretationAdd usage CPA InterpretationsInterpretationAdd K Xs kx returns the interpretations KXs after adding the interpretation kx CPAInterpretationFunctionUpgrade usage CPA InterpretationFunctionUpgrade k returns the interpretation function k upgraded with the possibly empty list I of interpretations or an individual interpretation I CPAInterpretationsDeleteltemInterpretations usage CPAInterpretationsDeleteltemInterpretations K Xs q returns the list of interpratations after deleting all interpretations on the item q from the list KXs of interpretations CPAInterpretationFunctionConformTo usage CPA InterpretationFunctionConformTo k returns the interpretation function obtained from the given interpretation function k after reducing its set of items to a competence based item set CPAInterpretationOfltem usage CPA InterpretationOfltem k x returns the interpretation of the item x give
149. s the surmise system calculated from the basis B SurmiseSystemFromKnowledgeSpace SurmiseSystemFromKnowledgeSpace QK returns the surmise system representation of the given knowledge space QK SurmiseSystemQ SurmiseSystemQ QS returns True iff the data structure QS represents a surmise system SurmiseSystemToBasis SurmiseSystemToBasis SS returns the basis calculated from the surmise system SS SurmiseSystemToKnowledgeSpace SurmiseSystemToKnowledgeSpace SS returns the basis calculated from the surmise system SS WarpQ WarpQ X returns True iff X is the warp representation of a list of positive integers Writeltems Writeltems FN Items stores the description on the items Items into the file FN 40 KSMP Techical Report Andrej Zaluski WriteSkills WriteSkills FN Skills stores the description on the skills Skills into the file FN WriteStructure WriteStructure FN SF KST_SpaceFile writes the structure SF in the file FN using KST_SpaceFile datafile format WriteStructure FN SF assumes KST_SpaceFile as the datafile format 41 KSMP Techical Report Andrej Zaluski KSMP source code version 0 91 beta 1 42 KSMP Techical Report Andrej Zaluski KSMP warp core Title Knowledge Spaces Mathematica Package Warp Core gt Context KnowledgeSpaces WarpCore Author Andrej Zaluski Summary KSMP Warp Core Provides data struc
150. sage Verbosity is an option of many functions from the upper layers and modules of the KSMP package The admissible values are Off default numbers e g 1 5 10 15 52 KSMP Techical Report Andrej Zaluski FORMAT Relation S R OutputForm Format SR Relation S_ R_ OutputForm Module lt gt ToString S lt lt gt ToString Map If EmptySet gt ToString 1 lt gt lt gt ToString 2 lt gt amp RJS FORMAT Structure S F OutputForm Format SF Structure S_ F_ OutputForm Module S1 F1 S1 S gt EmptySet F1 F gt EmptySet lt ToString S1 lt gt lt gt ToString F 1 lt gt Begin Private SetQ L returns True if L is a set SetQ L_ If Head L List True False SetCardinality S returns the cardinal number of the finite set S SetCardinality S_List Length Union S SetNonEmptyQ S returs true if the set S is non empty SetNonEmptyQ S_ If Length S NotEqual 0 True False SetEmptyQ S returns true if the set S is empty SetEmptyQ S_ If Length S 0 True False SubsetQ A B tests whether the set A is a subset of the set B SubsetQ A0_List B0O_List Module A A0 B B0 A Union A0 B Union B0 Which A True B False True Intersection A B Equal A EqualSetsQ A B tests whethther the sets A and B are equal EqualSetsQ A_List B_List SubsetQ A B And SubsetQ B
151. scOrExp describes the expert The valid options are Verbosity Off Default 1 5 10 15 QSRReport Result default Report or Both Expert Human default Simulation SelectionRule Maximin default WeightedSums and RuleScope Block default All QEOSQueryTextsInterpretations QEOS QueryTextsInterpretations NofSkils TextPart returns the textual part of a query on interpretation given the text part TextPart introduction middle ending and the number NofSkills 1 2 generic all_minus_one of skills QEOSQueryTextsSkills QEOS QueryTextsSkills PremiseSize TextPart returns the textual part of a query on dependencies among skills given the text part TextPart introduction or question and the premise size PremiseSize 1 2 generic all_minus_one QEOSSelectNextInterpretationQuestion QEOSSelectNextInterpretationQuestion OpenQ options returns the next interpretation assertion for querying given the sorted list OpenQ of open interpretation assertions respecting the given options The valid options are OpenQuestions Sorted default Unsorted in case the list OpenQ is not sorted it will be sorted with respect to the cardinality of competence states RuleScope Block default All whether to search for the next interpretation assertion only in the next block defined by the cardinality of competence states or to search all the interpretation assertions in OpenQ SelectionRule Maximin default WeightedSums which selection r
152. se system SS SSSurmiseSet SS_ SS 1 SSSurmiseFunction SS returns the surmise function forming the surmise system SS SSSurmiseFunction SS_ SS 2 SSSFListOfClauses SF returns the list of clauses of the surmise function SF SSSFListOfClauses SF_Mapping SF 3 SSListOfClauses SS returns the list of clauses of the surmise system SS SSListOfClauses SS_ SSSFListOfClauses SSSurmiseFunction SS SSSFClausesOfltem SF q returns the clauses assigned to the item q by the surmise function SF SSSFClauseOfltem SF_ q_ Select SSSFListOfClauses SF 1 q amp 1 2 SSClausesOflItem SS q returns the clauses assigned to the item q by the surmise function forming the surmise system SS SSClausesOfltem SS_ q_ SSSFClauseTake SSSurmiseFunction SS q SurmiseSystemToBasis SS returns the basis representation of the surmise system SS SurmiseSystemToBasis SS_ Module S SSSurmiseSet SS Apply Union Map SSClausesOfltem SS amp S SurmiseSystemToKnowledgeSpace SS returns the basis representation of the surmise system SS SurmiseSystemToKnowledgeSpace SS_ BasisToSpace SurmiseSystemToBasis SS SurmiseFunctionFromKnowledgeSpace QK returns the surmise function representing the given knowledge space QK SurmiseFunctionFromKnowledgeSpace Structure Q_ K Module QK Structure Q K LofCl LofCl Map AtomsAt QK amp Q Return Mapping Q Null
153. sertions QEKOutput Q_ PosAss_ NegAss_ opts__ Module choice choice OutputStructure opts Options QEK Output Switch choice Assertions PosAss NegAss _ QEKAIAssertionsToSpace Q PosAss QEKAIAssertionsToSpace Q PosAss returns the knowledge space established by positive assertions PosAss obtained through completed querying QEKAIIAssertionsToSpace Q_ PosAss_ Module F F Union Map SetComplement Q QEKStar PosAss amp PowerSet Q Return Structure Q F QEK QueryExpertByKoppen Q ItOrSkOrExp options queries an expert using Koppen s 1993 block algorithm A human ItOrSkOrExp represents interpretation of items or skills or a simulated expert ItOrSkOrExp represents the simulated expert is queried on the set Q of items and a structure either space or assertions is returned The acceptable options are Expert Human default or Simulation OutputStructure SpaceStructure default or Assertions Verbosity Off default or 1 2 QueryOn KnowledgeSpace default or CompetenceSpace and QSRReport Result default Report or Both QEKQueryExpertByKoppen Q_ ItemsExpert_ opts__ Module OpenQ PosAss NegAss Next NextAss Answer NegAnsInf PosAnsInf tmp kBlockAs k A B Xs classes NewNegAss NewPosAss NofQuestions QSR OptionVerbosity OptionHumOrSim OptionOutputStructure OptionQueryOn OptionQSRReport Whether to query expert on a knowledge space or a competence space
154. ski DataFileWrite Filename StructureToMatrix SF KST_SpaceFile FX SF s returns the subfamily of the family F such that its elements contains s Element S FX SFO_Structure s0_ Module SF SFO0 s s0 FX SF s Not Head s0 List FX SF C NC returns the subfamily of the family SF such that its elements contain the C SubsetEqual S and do not contain neither one of the elements NC SubsetEqual S FX Structure S_ F_ C_List NC_List Module Select Select F SubsetQ C amp Intersection NC amp V Apply And Map SubsetQ S amp Union C NC And Intersection C NC FX SF C returns the subfamily of the family SF such that its elements contain the C SubsetEqual S FX SFO_Structure CO_List Module SF SF0 C C0 FX SF C DataFileRead Filename creates the DataFile object representing the file Filename containing the data DataFileRead FileName_String DataFile Import FileName Lines DataFileLength SFile returns the number of lines within the datafile object SFile DataFileLength DataFile SFile_ Length SFile LineCommentQ L tests whether the line L of a datafile object contains a comment LineCommentQ Line_ LineCommentQ Line KST_SpaceFile LineCommentQ Line_ FileFormat_ StringTake Line 1 FileFormatsInfo FileFormat CommentSign CommentLines SFile returns a list of line numbers of a datafi
155. sley Publishing Co Skienna S 1998 The algorithm design manual New York Springer Verlag Wolfram S 1999 The Mathematica Book Cambridge University Press 15 KSMP Techical Report Andrej Zaluski Glossary of KSMP functions version 1 0 This glossary contains all public functions of the KSMP package listed alphabetically The majority of functions are of interest both to applicants and developers while some functions are probably of interest only to a developer Atoms Atoms SF returns all subfamilies from the family forming the structure SF that are atoms by at least one element from the set forming the structure AtomsAt AtomsAt SF x returns all atoms at the element xi S given the structure SF BasisFromSpace BasisFromSpace SF returns the basis of the space SF BasisToSpace BasisToSpace F returns the space constructed from the given basis F CommentLines CommentLines SFile returns the list of all line numbers from the datafile object SFile that contain comments CompetenceSpaceQ CompetenceSpaceQ EK returns True iff the structure EK is a competence space CompetenceStructureQ CompetenceStructureQ EK returns True iff the structure EK is a competence structure CompositionOfFunctions CompositionOfFunctions fn f1 returns the composition fn o o fl of the functions fn f1 under assumption that such composition is defined CPADiagnosticCompetenceSpace CPADiagnosticCompet
156. so does not posses the competence all_minus_one Suppose you are tutoring a student who does not posses any of the competencies in the given set of competencies Could you then start by teaching him the competence TextPart Select TextParts 1 TextPart0 amp 1 1 2 Return Select QueryTexts 1 PremiseSize amp 1 1 TextPart 122 KSMP Techical Report Andrej Zaluski QEOSFormulateQuerySkills E Skills Assertion returns the query question on the given Assertion formulated for the human expert given the set E of elementary skills and their interpretation Skills QEOSFormulateQuerySkills E_ Skills_ Assertion_ Module nE SetCardinality E nAssertion SetCardinality QEK AssertionPremise Assertion A QEKAssertionPremise Assertion b QEKAssertionConsequence Assertion QueryTexts QueryText Textl Text2 Text3 Text4 tmp Ntmp Which nAssertion 1 Block Textl QEOSQueryTextsSkills 1 introduction Text2 SkillName SkillGet Skills A 1 tmp QEOSQueryTextsSkills 1 question Text3 Apply StringJoin tmp SkillName SkillGet Skills b QueryText StringJoin Text1 Text2 Text3 nAssertion 2 Block Textl QEOSQueryTextsSkills 2 introduction Text2 StringJoin SkilIName SkillGet Skills A 1 and SkillIName SkillGet Skills A 2 tmp QEOSQueryTextsSkills 2 question Text3 Apply StringJoin tmp SkillName SkillGet Skills b
157. stBlock usage QEK InitializeFirstBlock Q returns the data structure OQ P N representing the first block in the Koppens algorithm consisting of open questions OQ positive assertions P and negative assertions N QEK Inferences 4a usage QEK Inferences 1 4a PosAss Ass returns the positive inferences contributed by the inference rule Apx amp BPA amp AxPy gt Bpy when the assertion Ass is positively judged APx and added to the previous positive assertions PosAss QEK Inferences 1 4b usage QEK Inferences14b PosAss NegAss Ass returns the negative inferences contributed by the inference rule APk amp AxPB amp ANy gt BNy when the assertion APx is added to the previous positive PosAss and negative NegAss assertions QEK Inferences 1 4c usage QEK Inferences 14c Q PosAss NegAss Ass returns the negative inferences contributed by the inference rule APx amp ByPA amp BNx gt BNy when the assertion APx is added to the previous positive PosAss and negative NegAss assertions given the set Q of items QEK Inferences 1 4d usage QEK Inferences 14d Q PosAss Ass returns the negative inferences contributed by the inference rule ANx amp APB amp ByPx gt BNy if ANx is added to the previous negative assertions NegAss given the set Q of items QEKCollectInferencesForNegativeAnswer usage QEKCollectInferencesForNegativeAnswer Q PosAss Ass returns the data structure P N consisting of positive inferences P a
158. structure SF StructureSubsetQ usage StructureSubsetQ SF1 SF2 returns True iff the structure SF1 is contained in the structure SF2 Both structures are based on the same set EqualStructuresQ usage EqualStructuresQ SF1 SF2 returns True iff the structure SF1 is equal to the structure SF2 StructureSubsetProperQ usage StructureSubsetProperQ SF 1 SF2 returns True iff the structure SF1 is a proper subset of the structure SF2 Both structures are based on the same set StateBinary2Set usage StateBinary2Set B Normalized returns the set representation of the state given its binary list representation B under assumption that the structure to which the state will belong to is normalized n n StateBinary2Set B SC returns the set representation of the state given its binary list representation B and the structure carrier set SC StateSet2Binary usage StateSet2Binary f NS Normalized returns the binary representation of the state f under assumption that the cardinality of the normalized structure carrying set S is NS StateSet2Binary f SC returns the binary representation of the state f taking into account the carrier set SC of the structure to which it belongs to StructureNormalize usage StructureNormalize SF returns the normalized structure SF i e rewrites the structure SF taking the set 1 2 as the set carrying the structure n n StructureNormalize SF S 1 returns the rewritten stru
159. t 120 KSMP Techical Report Andrej Zaluski Result Return tmp Report Return QSR Both Return tmp QSR l End of Module QEOSQueryTextsInterpretations NofSkils TextPart returns the TextPart introduction middle ending for the given number NofSkills 1 2 generic all_minus_one of skills QEOSQueryTextsInterpretations NofSkills_ TextPart0_ Module QueryTexts TextParts TextPart TextParts introduction 2 middle 3 ending 4 QueryTexts 1 Suppose that a student under examination posses the competence Is possesing this skill and none of the queried skills more sufficient for solving the problem 2 Suppose that a student under examination posses both the competences Is possesing these skills and none of the queried skills more sufficient for solving the problem generic Suppose that a student under examination posses all the skills Is possessing all of these skills and none of the queried skills more sufficient for solving the problem all_minus_one Suppose that a student under examination posses all of the queried skills except the skill Is possesing these skills sufficient for solving the problem ys TextPart Select TextParts 1 TextPart0 amp 1 1 2 Return Select QueryTexts 1 NofSkills amp 1 1 TextPart QEOSFormulateQueryInterpretations a Items C K Skills returns the query question o
160. t Andrej Zaluski Map QEOSInterpretationFunctionCountInferences OpenQ amp OpenQ MaxIAss candidates 1 Dol If candidates i 2 gt MaxIAss 2 MaxIAss candidates i i 2 Length candidates Return MaxIAss 1 QEOSInterpretationFunctionSelectionRuleWeightedSums OpenQ returns an interpretation assertion which has maximal exected number of inferences according to the weighted sums selection rule QEOSInterpretationFunctionSelectionRuleWeightedSums OpenQ opts_ _ Module PrYES PrNO candidates MaxIAss i PrYES ProbabilityYES opts Options QEOSInterpretationFunctionSelectionRuleWeightedSums PrNO 1 PrYES candidates Map 1 PrNO 2 PryYES 2 2 amp Map QEOSInterpretationFunctionCountInferences OpenQ amp OpenQ MaxIAss candidates 1 Dol If candidates i 2 gt MaxIAss 2 MaxIAss candidates i 1 2 Length candidates Return MaxIAss 1 QEOSSelectNextInterpretationQuestion OpenQ opts returns the next interpretation assertion question given the sorted list OpenQ of open interpretation assertions respecting the given options Possible options are OpenQuestions Sorted default Unsorted in case the list OpenQ is not sorted sorts it according to the cardinality of competence states RuleScope Block default All whether to search for the next question only in the next block defined by the cardinality
161. t L of positive integers KSMPWarpToBinary KSMPWarpToBinary W returns the binary list representation of the given warp W KSMPWarpToList KSMPWarpToList W returns the list of positive integers from its warp representation W LineCommentQ LineCommentQ L returns True iff the line L of a datafile object contains a comment MakeExpert MakeExpert MK CE returns the data structure representing an expert consisting of the domain s metaknowledge MK and the uncertainty CE of making a careless error during a querying session MakeFunction MakeFunction D C M creates a function from the set D to the set C using the mapping rule M Makel Assertion MakelAssertion CS Item returns the interpretation assertion consisting of the competence state CS and the item Item Makeltem Makeltem Code Desc QText CorrAns LofDist returns a data structure representing the item consisting of the unique identifier Code description Desc text QText of a question correct Answer CorrAns and the list LofDist of admissible answers where each answer consists of a code and a textual description MakeKnowledge MakeKnowledge QK Items returns a data structure representing a student s domain knowledge given as the set QK of items 25 KSMP Techical Report Andrej Zaluski MakeMetaKnowledge MakeMetaKnowledge Diag Diagnostic returns a data structure representing an expert s metaknowledge in terms of the competence performance approach
162. t s answer Ans StudentAnswerltem StudentAnswerltem Ans returns the logical value of the student s answer Ans StudentAnswerLogicalValue StudentAnswerLogicalValue Ans returns the logical value of the student s answer Ans StudentCarelessErrors StudentCarelessErrors S returns the uncertainty that the student S makes a careless error StudentKnowledge StudentKnowledge S returns domain knowledge of the student S 39 KSMP Techical Report Andrej Zaluski StudentLuckyGuesses StudentLuckyGuesses S returns the uncertainty that the student S makes a lucky guess StudentMakeAnswer StudentMakeAnswer QE LV CF returns the data structure for a student s answer consisting of the asked question QE logical value LC describing the correctness of the answer and the certainty factor CF describing the student s certainty in the given answer SubsetProperQ SubsetProperQ A B returns True iff the set A is a proper subset of the set B SubsetQ SubsetQ A B returns True iff the set A is a subset of the set B SurmiseFunctionFromBasis SurmiseFunctionFromBasis B returns the surmise function calculated from the basis B SurmiseFunctionFromKnowledgeSpace SurmiseFunctionFromKnowledgeSpace QK returns the surmise function calculated from the knowledge space QK SurmiseFunctionQ SurmiseFunctionQ f returns True iff the function f represents a surmise function SurmiseSystemFromBasis SurmiseSystemFromBasis B return
163. tDifference K Xs0 K Xs form the competence based item set A Union Map 1 amp KXs return the interpretation function based on the competence based item set Return Mapping A PK KXs CPAInterpretationOfltem k x returns the interpretation of the item x given the interpretation function k CPAInterpretationOfltem Mapping A_ PK_ KXs_ x_ Module int int Select KXs 1 x amp 1 If int NotEqual int 1 2 CPARepresentationOfCompetenceState k C returns the data structure C p C where p C is the subset of all the items induced the interpretation function k that represent the given competence state C CPARepresentationOfCompetenceState Mapping A_ PK KXs_ C_ Module k Mapping A PK KXs Return C Select A SetElementQ C CPA InterpretationOfltem k amp CPARepresentations k returns the list of all representations on the set of items interpreted in thecompetence structure given through the interpretation function k CPARepresentations Mapping A_ PK_ KXs_ Module k Mapping A PK KXs k Apply Union PK Map CPARepresentationOfCompetenceState k amp K CPARepresentationFunctionFromInterpretationFunction k returns the representation function from the given interpretation function k 96 KSMP Techical Report Andrej Zaluski CPARepresentationFunctionFromInterpretationFunction Mapping A_ PK_ KXs_ Module k Mapping
164. tInferencesForNegativeAnswer OpenQ IAss returns all the interpretation assertion inferences given the list OpenQ of open interpretation assertions and interpretation assertion Ass that was judged negatively QEOSInterpretationFunctionCollectInferencesForNegativeAnswer OpenQ0_ IAss_ Module OpenQ OpenQ0 q IAssertionItem IAss OpenQ QEOSInterpretationsOnItem OpenQ q Select OpenQ AssertionItem IAssertionItem IAss amp amp SubsetQ IAssertionCompetenceState 1AssertionCompetenceState IAss amp QEOSInterpretationFunctionCountInferences IASSes IAss returns the data structure Neg Pos containing the number Neg of inferences when negative answer on the interpretation assertion IAss was obtained given the list ASSes of open interpretation assertions and the number Pos of inferences in case of positive answer QEOSInterpretationFunctionCountInferences IASSes_ IAss_ Return Length QEOSInterpretationFunctionCollectInferencesForNegativeAnswer IASSes IAss Length QEOSInterpretationFunctionCollectInferencesForPositiveAnswer IASSes IAss QEOSInterpretationFunctionSelectionRuleMaximin OpenQ returns an interpretation assertion which has maximal exected number of inferences according to the maximin selection rule QEOSInterpretationFunctionSelectionRuleMaximin OpenQ_ Module candidates MaxIAss i candidates Map 1 Min 2 1 2 2 amp 125 KSMP Techical Repor
165. ta structures and functions at one side and Mathematica implemented procedures representing concepts and procedures from the knowledge space theory on the other Data structures and functions from this abstract data types layer of the package are to be used by other layers of the KnowledgeSpaces package The layer consists of the components representing sets relations functions structures here defined as ordered tuples consisting of sets and families of subsets a component with functions representing miscellaneous operations and property tests and finally a component dealing with data input and output of results and reports The reading and writing of data files is done in a way compatible with already developed Unix C C Bash tools for dealing with knowledge spaces primarily KST tools written by Cord Hockemeyer Copyright 20001 Andrej Zaluski See attached license Package Version 0 9 beta Mathematica Version 4 0 History 1 0 reserved for the first public version 0 9 beta 1 version Keywords Keyword Psychology and Education Mathematical Psychology Knowledge Modeling Knowledge Space Theory Sources Warnings Limitations Currently on personal computers large part of writen functions works well only for small sets of items Future plans include devloping two operating modes of the package research mode and appli
166. taKnowledge usage MakeMetaKnowledge SF Structure returns a data structure representing experts knowledge in the form of structure for the 69 KSMP Techical Report Andrej Zaluski given knowledge structure SF MakeMetaKnowledge usage MakeMetaKnowledge PAss NAss UAss Assertions returns a data structure representing expert s metaknowledge in the form of assertions for separately given lists of positive assertions PAss negative assertions Nass and unknown assertions UAss MakeMetaKnowledge usage MakeMetaKnowledge Ent Assertions returns a data structure representing expert s metaknowledge in the form of assertions for the given Entailment Ent MetaKnowledgeAssertionsPositive usage MetaKnowledgeAssertionsPositive Assertions returns all positive assertions contained in the expert s metaknowledge Assertions MetaKnowledgeAssertionsNegative usage MetaKnowledgeAssertionsNegative Assertions returns all negative assertions contained in the expert s metaknowledge Assertions MetaKnowledgeAssertionsRest usage MetaKnowledgeAssertionsRest Assertions returns all neither positive nor negative assertions contained in the expert s metaknowledge Assertions MakeMetaKnowledge usage MakeMetaKnowledge DIAG Diagnostic returns a data structure representing an expert s metaknowledge in terms of the competence performance approach using the diagnostic Diag MakeSimpleExpert usage
167. tems the positive inferences PosAss and negative inferences NegAss are given in advance QEK EntailmentFromQueryHistory QEKEntailmentFromQueryHistory Q QueryHistory returns all positive judgments from the querying history QueryHistory capturing an establishment of a structure on the set Q of items QEKEquivalentSubsetsQ QEKEquivalentSubsetsQ PosAss A B returns True iff the subsets A and B are equivalent with respect to the positive inferences PosAss QEKEstablishKnowledgeSpaceStraightforward QEKEstablishKnowledgeSpaceStraightforward Q Items Human returns a structure on the items Q described in Items established by querying a human expert QEKEstablishKnowledgeSpaceStraightforward Q Expert Simulation returns a structure on the items Q obtained by querying the simulated expert Expert 28 KSMP Techical Report Andrej Zaluski QEKExtractNegativeJudgments QEKExtractNegativeJudgments QH returns all negative judgments from the querying history QH QEKExtractPositiveJudgments QEKExtractPositiveJudgments QH returns all positive judgments from the querying history QH QEKFormulateQuery QEK FormulateQuery Q Items Assertion returns the query question on the given assertion Assertion formulated for a human expert given the set Q of items described in Items QEKInferences14a QEK Inferences 14a PosAss Ass returns the positive inferences contributed by the inference rule Apx amp BPA amp AxPy gt Bpy when the assertion
168. the given Item 23 KSMP Techical Report Andrej Zaluski ItemDescription ItemDescription Item returns the description of the given Item ItemGet ItemGet Items Code returns the first item in the description Items whose code matches the code Code ItemText ItemText Item returns the textual question of the given Item KAAskStudent KAAskStudent Stud QE Simulation returns an answer from the simulated student Stud on the question QE KAAskStudent Items QE Human returns an answer from a human student on the question QE described in Items KADAssessKnowledge KADAssessKnowledge SF ItemsStudent options performs the deterministic assessment of a student s knowledge given the knowledge structure SF In case of a human student option Examinee value Human default the parameter ItemsStudent takes the description of items while in case of a simulated student option Examinee value Simulation it stores a simulated student The function may return either a knowledge state option AssessmentResult value State default or an assessment history option AssessmentResult value History The additional valid option is Verbosity Off default 1 5 10 20 KADSelectNextQuestion KADSelectNextQuestion SF CA NCA returns the next item to be asked by the deterministic knowledge assessment procedure on the knowledge structure SF assuming already collected correct CA and incorrect NCA answers by a student KADSelectNe
169. the querying session sonde Sonde after incrementing the number of presented questions by N QSRSondeQuestionsAskedReplace QSRSondeQuestionsAskedReplace Sonde N returns the querying session sonde Sonde after replacing the number of asked questions by N ReadItems ReadItems FN KSMP_ItemsFile reads the description of items stored in the file FN using the file format KSMP_ItemsFile ReadItems FN assumes the file format KSMP_ItemsFile is used ReadSkills ReadSkills FN KSMP_SkillsFile reads the description of skills stored in the file FN using the file format KSMP_SkillsFile ReadSkills FN assumes the file format KSMP_SkillsFile is used ReadStructure ReadStructure FN reads the structure stored in the file FN using KST_SpaceFile as default datafile format ReadStructure FN KST_SpaceFile reads the structure stored in the file FN using KST_SpaceFile datafile format ReadStructure FN SC reads the structure stored in the file FN and returns it rewritten using the carrier set SC assuming the default datafile format ReadStructure FN KST_SpaceFile SC reads the structure using KST_SpaceFile datafile format stored in the file FN and rewrites it using the carrier set SC RelationAntiSymmetricQ RelationAntiSymmetricQ SR returns True iff the relation SR is antisymmetric RelationAReflexiveQ RelationAReflexiveQ SR returns True iff the relation SR is areflexive RelationChangeCarrier RelationChangeCarri
170. tion CPAMakeRepresentationFunction K A PCs returns the data structure containing the representation function formed from the competence structure K the set A of items and the collection PCs of representations CPARepresentationCompetenceState CPARepresentationCompetenceState PC returns the competence state of the given representation PC CPARepresentationFunctionCompetenceStructure CPARepresentationFunctionCompetenceStructure p returns the competence structure forming the given representation function p CPARepresentationFunctionFromInterpretationFunction CPARepresentationFunctionFromInterpretationFunction k returns the representation function given the interpretation function k CPARepresentationFunctionRepresentations CPARepresentationFunctionRepresentations p returns the collection of representations forming the representation function p CPARepresentationItems CPARepresentationItems pC returns the set of items for the given representation pC 18 KSMP Techical Report Andrej Zaluski CPARepresentationOfCompetenceState CPARepresentationOfCompetenceState k C returns the data structure C p C where p C is a set of items assigned by the interpretation function k to the given competence state C CPARepresentations CPARepresentations k returns the list of all representations of the competence structure derived from the interpretation function k CPARepresentationStructureFromInterpretationFunction CP
171. tion returns the query question based on the given assertion Assertion given the set E of elementary skills described in Skills QEOSQueryExpertOnCompetenceSpace usage QEOS QueryExpertOnCompetenceSpace E SkOrExp options queries a human SkOrExp represents a description of skills or a simulated expert SkOrExp represents a simulated expert on structure of the set E of elementary competencies and returns the established structure either as a space or assertions An valid option is QueryingAlgorithm Koppen default For additional valid options see QEK QueryExpertByKoppen MakelAssertion usage MakelAssertion CS Item returns the interpretation 102 KSMP Techical Report Andrej Zaluski assertion consisting of the competence state CS and the item Item IAssertionItem usage AssertionItem IAss returns the item of the interpretation assertion Ass IAssertionCompetenceState usage AssertionCompetenceState IAss returns the competence state of the interpretation assertion IAss QEOSAskExpertOnInterpretation usage QEOSAskExpertOnInterpretation K Skills Items Ass Human returns an answer on the interpretation assertion Ass from a human expert given the competence structure K described in Skills and the interpretation assertion item desribed in Items n n QEOSAskExpertOnInterpretation IAss Expert Simulation returns an answer on the interpretation assertion IAss from the simulated
172. tion for example QEOSQueryExpertOnCompetenceSpace QEOSQueryExpertOnCompetenceSpace E SkOrExp options queries a human SkOrExp represents a description of skills or a simulated expert SkOrExp represents a simulated expert on structure of the set E of elementary competencies and returns the established structure either as a space or assertions A valid option is QueryingAlgorithm Koppen default For additional valid options see QEK QueryExpertByKoppen The Ctrl Shift K combination may ease and speed up writing of calls using the functions KSMP Techical Report Andrej Zaluski Package design Architecture of the package In general the package has been developing under assumption that it will be used by two essentially different class of users applicants and developers An applicant would probably be only interested in possibilities of using the package in empirical investigations theoretical explorations simulation studies and teaching A developer in contrast to an applicant probably will also want to suit the package to his or her own needs therefore to build own solutions extensions and applications based on the package Having this in mind together with a whole variety of knowledge space theory concepts that potentially might be of interest and therefore implemented in the package an idea of a package that would allow further developments in serveral directions independently emerged It is believed that such design would
173. tion function obtained from the given interpretation function k after reducing its set of items to a competence based item set CPAInterpretationFunctionInitialize CPAInterpretationFunctionInitialize K returns the empty interpretation function given the competence space K and the empty set of problems CPAInterpretationFunctionInitialize A K returns the initialized interpretation function consisting of the non interpreted items from the set A of items given the competence space K CPAInterpretationFunctionInterpretations CPAInterpretationFunctionInterpretations k returns the list of interpretations contained in the interpretation function k CPAInterpretationFunctionItems CPAInterpretationFunctionltems k returns the set of items contained in the interpretation function k CPAInterpretationFunctionQ CPAInterpretationFunctionQ f returns True iff the function fis an interpretation function CPAInterpretationFunctionUpgrade CPAInterpretationFunctionUpgrade k returns the interpretation function k upgraded with the possibly empty list I of interpretations or an individual interpretation I CPAInterpretationOfltem CPAInterpretationOfltem k x returns the interpretation of the item x given the interpretation function k 17 KSMP Techical Report Andrej Zaluski CPAInterpretationsDeleteItemInterpretations CPAInterpretationsDeleteItemInterpretations K Xs q returns the list of interpretations after deleting all
174. tion of it thus forming a work based on the package and copy and distribute such modifications or work under the terms of section 2 provided that you also meet the following conditions a You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change b You must cause any work that you distribute or publish that in whole or in parts contains or is derived from the package or any part thereof to be licensed as a whole at no charge to all third parties under the terms of this License You may not run copy modify sublicense or distribute the package except as expressly provided under this license Any attempt otherwise to run copy modify sublicense or distribute the package is void and will automatically terminate your rights under this License You are not required to accept this License since you have not signed it However nothing else grants you permission to run modify or distribute the package or its derivative works These actions are prohibited by law if you do not accept this License Therefore by running distributing or modifying the package or any work based on the package you indicate your acceptance of this License to do so and all its terms and conditions for copying distributing or modifying the package or works based on it KSMP Techical Report Andrej Zaluski Overview This paper describes Knowledge Spaces Mathematica Package version 1 0 Firsty m
175. tructure representing a surmise system given by the set Q of items and the mapping LofCl of clauses under assumption that these objects fulfill the necessary formal requirements Matrix2DataFileBody Matrix2DataFileBody M KST_SpaceFile returns the datafile object body from the matrix M using KST_SpaceFile format Matrix2DataFileHead Matrix2DataFileHead M KST_SpaceFile returns the head of a datafile object from the matrix M using KST_SpaceFile format MaximalsOfRelation MaximalsOfRelation SR returns all maximals of the relation SR 26 KSMP Techical Report Andrej Zaluski MetaKnowledgeAssertionsNegative MetaKnowledgeAssertionsNegative Assertions returns all negative assertions contained in the expert s metaknowledge Assertions MetaKnowledgeAssertionsPositive MetaKnowledgeAssertionsPositive Assertions returns all positive assertions contained in the expert s metaknowledge Assertions MetaKnowledgeAssertionsRest MetaKnowledgeAssertionsRest Assertions returns all neither positive nor negative assertions contained in the expert s metaknowledge Assertions MinimalsOfRelation MinimalsOfRelation SR returns all minimals of the relation SR NonCommentLines NonCommentLines SFile returns the list of all line numbers of the datafile object that do not contain a comment PerformanceSpaceQ PerformanceSpaceQ AP returns True iff the structure AP is a performance space PerformanceStructureQ PerformanceStr
176. tureSetSubsetQ SF S1 tests whether the set S1 is a subset of the set forming the structure SF StructureSetSubsetQ Structure S_ F _ S1_ SubsetQ S1 S StructureFamilyElementQ SF f tests whether the set f is a element of the family forming the structure SF StructureFamilyElementQ Structure S_ F_ f_ SetElementQ f F StructureFamilySubsetQ SF F1 tests whether the family F1 is a subset of the family forming the structure SF StructureFamilySubsetQ Structure S_ F_ F1_ SubsetQ F1 F StructureSubsetQ SF 1 SF2 tests whether the structure SF1 is contained in the sense S1 8S2 F1 SubsetEqual F2 in the structure SF2 StructureSubsetQ Structure S1_ F1_ Structure S2_ F2_ Module If SubsetQ F 1 F2 True False EqualSetsQ S1 S2 EqualStructuresQ SF1 SF2 tests whether the structure SF1 is equal to the structure SF2 EqualStructuresQ Structure S1_ F1_ Structure S2_ F2_ Module EqualSetsQ F1 F2 EqualSetsQ S1 S2 61 KSMP Techical Report Andrej Zaluski StructureSubsetProperQ SF 1 SF2 tests whether the structure SF1 is a proper subset in the sense S1 S2 F1 SubsetEqual F2 of the structure SF2 StructureSubsetProperQ SF1_ Structure SF2_Structure StructureSubsetQ SF 1 SF2 And EqualStructuresQ SF 1 SF2 StateBinary2Set B Normalized returns the set representation of the state given its binary list representation B under assumption that the
177. tures to the layer one that are suitable for environments with limited memory Copyright 20001 Andrej Zaluski See attached license Package Version 0 91 beta 1 Mathematica Version 4 0 History 1 0 the first public release 0 1 first draft Keywords Sources Warnings BETA TESTING PHASE Limitations Discussion Requirements NONE BeginPackage KnowledgeSpaces WarpCore WarpQ usage WarpQ X returns True iff X is the warp representation of a list of positive integers KSMPListToBinary usage KSMPListToBinary L returns the binary representation list for the list L consisting of positive integers KSMPBinaryToList usage KSMPBinaryToList BL returns the list of positive integers calculated from the binary list BL KSMPListToWarp usage KSMPListToWarp L returns the warp representation of the list L of positive integers KSMPBinaryToWarp usage KSMPBinaryToWarp BL returns the warp representation 43 KSMP Techical Report Andrej Zaluski of the binary representation list BL representing a list of positive integers KSMPWarpToList usage KSMPWarpToList W returns the list of positive integers from its warp representation W KSMPWarpToBinary usage KSMPWarpToBinary W returns the binary list representation of the given warp W Begin Private WarpQ W_ Return IntegerQ W
178. ucture to which it belongs to StructureFamily StructureFamily SF returns the family forming the structure SF StructureFamilyElementQ StructureFamilyElementQ SF f returns True iff the set f is a element of the family forming the structure SF StructureFamilyEmptyQ StructureFamilyEmptyQ SF returns True iff the family forming the structure SF is empty StructureFamilyNonEmptyQ StructureFamilyNonEmptyQ SF returns True iff the family forming the structure SF is non empty StructureFamilySubsetQ StructureFamilySubsetQ SF F1 returns True iff the family F1 is a subset of the family forming the structure SF StructureFromMatrix StructureFromMatrix M Normalized returns the normalized structure from its matrix representation M StructureFromMatrix M SC returns the structure from its matrix representation having the set SC as the carrier set StructureNormalize StructureNormalize SF returns the normalized structure SF that is rewrites the structure SF using the set 1 2 for carrying the structure StructureNormalize SF S1 returns the rewritten structure SF using the set S1 for carrying the structure StructureQ StructureQ SF returns True iff SF is a structure StructureReplaceCarrier StructureReplaceCarrier SF SC returns the rewritten structure SF using the set SC as the new carrier StructureReplaceFamily StructureReplaceFamily SF F 1 is a primitive function that returns the structure SF after replacing
179. uctureQ AP returns True iff the structure AP is a performance structure PowerSet PowerSet S returns the powerset of the set S By default ranking all subsets with the cardinality k 1 appear before the subsets with the cardinality k PQxPQ PQxPQ Q returns the cartesian product on the powerset of the set Q from which the empty set was substracted PQxQ PQxQ Q returns the cartesian product between the set Q and its powerset from which the empty set was substracted PSAxiom1Q PSAxiom1Q AP returns True iff the structure AP fulfills the property JP A PSAxiom2Q PSAxiom2Q AP returns True iff the structure fulfills the property that the family P contains both the empty set and the set A as its subsets PSAxiom3Q PSAxiom3Q AP returns True iff the structure AP is stable under union QEKAllAssertionsToSpace QEKAIAssertionsToSpace Q PosAss returns the knowledge space derived from the positive assertions PosAss 27 KSMP Techical Report Andrej Zaluski QEKAskExpert QEKAskExpert Q Items Assertion Human returns an answer from a human expert on the assertion Assertion containing some items from the set Q described in Items QEKAskExpert Assert Exp Simulation returns an answer from the simulated expert Exp on the assertion Assert The expert s domain metaknowledge may be represented either using a structure or a diagnostic QEK AssertionConsequence QEKAssertionConsequence Assertion returns the consequenc
180. uctureQ EK returns True iff the structure EK is a competence structure PerformanceStructureQ usage PerformanceStructureQ AP returns True iff the structure AP is a performance structure CompetenceSpaceQ usage CompetenceSpaceQ EK returns True iff the structure EK is a competence space PerformanceSpaceQ usage PerformanceSpaceQ AP returns True iff the structure AP is a performance space CPAMakelnterpretationFunction usage CPAMakelnterpretationFunction A K K Xs returns the interpretation function given the set A of items the competence structure K and the mapping KXs interpreting each item by assigning competence states CPAInterpretationFunctionInitialize usage CPAInterpretationFunctionInitialize K returns the empty interpretation function given the competence space K and the empty set of problems n n CPAInterpretationFunctionInitialize A K returnes the initialized interpretation function consisting of the non interpreted items from the set A of items given the competence space K CPAInterpretationFunctionAxiomsQ usage CPA InterpretationFunctionAxiomsQ k returns True iff the function k satisfies the axiom 1 of interpretation functions CPAInterpretationFunctionQ usage CPA InterpretationFunctionQ f returns True iff the function fis an interpretation function CPAInterpretationFunctionItems usage CPA InterpretationFunctionltems k returns the set of item
181. ule is an option of QEKSelectNextQuestion QEOSSelectNextInterpretationQuestion and QEOSQueryExpertOnInterpretationFunction The possible values are Maximin default and WeightedSums RuleScope usage RuleScope is an option of QEOSSelectNextInterpretationQuestion and QEOSQueryExpertOnInterpretationFunction The possible values are Block default and All 105 KSMP Techical Report Andrej Zaluski Options QEQueryExpertOnKnowledgeSpace QueryingAlgorithm gt Koppen Options QEKAskExpert QueryOn gt KnowledgeSpace Options QEK QueryExpertByK oppen Verbosity gt Verbosity Expert gt Human OutputStructure gt SpaceStructure QueryOn gt KnowledgeSpace QSRReport gt Result Options QEKSelectionRuleWeightedSums ProbabilityYES gt 0 5 Options QEKSelectNextQuestion SelectionRule gt Maximin Options QEK Output OutputStructure gt SpaceStructure Options QEOSInterpretationFunctionSelectionRuleWeightedSums ProbabilityYES gt 5 Options QEOSSelectNextInterpretationQuestion OpenQuestions gt Sorted RuleScope gt Block SelectionRule gt Maximin Options QEOSQueryExpertOnCompetenceSpace QueryingAlgorithm gt Koppen Options QEOSQueryExpertOnDiagnostic Expert gt Human QSRReport gt Result Options QEOSQueryExpertOnInterpretationFunction Verbosity gt Verbosity Expert gt Human RuleScope gt Block SelectionRule gt Maximin QSRReport gt Result Begin Priv
182. ule is to be used QEQueryExpertOnKnowledgeSpace QEQueryExpertOnKnowledgeSpace Q ItOrExp options queries a human ItOrExp takes a description of items or a simulated expert ItOrExp represents a simulated expert on structure of the set Q of items The established structure is returned either as a space or assertions A valid option is QueryingAlgorithm Koppen default For additional valid options see QEK QueryExpertByKoppen QSRMakeSonde QSRMakeSonde OQ NQ returns a querying session observation sonde consisting of the number OQ of open questions at the start of the observation and the number NQ of questions presented to an expert during the observation QSRSondeOpenQuestionsInitial QSRSondeOpenQuestionsInitial Sonde returns the initial number of open questions recorded by the querying session observation sonde Sonde QSRSondeOpenQuestionsInitialAdd QSRSondeOpenQuestionsInitialAdd Sonde N returns the querying session sonde Sonde after incrementing the initial number of open questions by N QSRSondeOpenQuestionsInitialReplace QSRSondeOpenQuestionsInitialReplace Sonde N returns the querying session sonde Sonde after replacing the initial number of open questions with N QSRSondeQuestionsAsked QSRSondeQuestionsAsked Sonde returns the number of questions presented to an expert recorded by the querying session sonde Sonde 32 KSMP Techical Report Andrej Zaluski QSRSondeQuestionsAskedAdd QSRSondeQuestionsAskedAdd Sonde N returns
183. ule of the function f FunctionMapping Mapping D_ C_ M_ M FunctionChangeDomain f D1 replaces the domain of the function f with the new set D1 FunctionChangeDomain F_Mapping D1_ Mapping D1 FunctionCodomain F FunctionMapping F FunctionChangeCodomain f C1 replaces the domain of the function f with the new set C1 FunctionChangeCodomain F_Mapping C1_ Mapping FunctionDomain F C1 FunctionMapping F FunctionChangeMapping f 58 KSMP Techical Report Andrej Zaluski M1 replaces the mapping rule of the function f with the new mapping tule M1 FunctionChangeMapping F_Mapping M1_ Mapping FunctionDomain F FunctionCodomain F M1 FunctionMappingOriginal f returns the original of the function s mapping rule FunctionMappingOriginal F_Mapping Transpose FunctionMapping F 1 FunctionMappingImage f returns the function s image FunctionMappingImage F_Mapping Union Transpose FunctionMapping F 2 FunctionWellDefinedQ f returns true if the original of the function s mapping rule is equal to the domain of the function FunctionWellDefinedQ f_Mapping EqualSetsQ FunctionDomain f FunctionMappingOriginal f FunctionSurjectionQ f returns true if the function fis a surjection FunctionSurjectionQ f_Mapping EqualSetsQ FunctionMappingImage f FunctionCodomain f FunctionValue f x returns the f x FunctionValue Mapping D0_ CO_ MO_ x_ Module M M0
184. unction Scope of the selection rulu while considering candidates 127 KSMP Techical Report Andrej Zaluski OptionRuleScope RuleScope opts Options QEOS QueryExpertOnInterpretationFunction VERBOSITY If OptionVerbosity gt 1 Print QEOS v1 0 Querying Session Report n A Straightforward procedure for querying an expert OptionHumOrSim on interpretation function n OptionSelectionRule selection rule used with the OptionRuleScope scope Print Function returns Result QSRReport or Both OptionQSRReport J Parameter DescOrExp means represents either an expert or a pair S I where S stands for skills description and I for items description If OptionHumOrSim Human Skills DescOrExp 1 Items DescOrExp 2 If OptionHumOrSim Simulation SimExpert DescOrExp VERBOSITY If OptionHumOrSim Human amp amp OptionVerbosity gt 5 Print Description of used skills n Skills Print Description of used items n Items l If OptionHumOrSim Simulation amp amp OptionVerbosity gt 5 Print Simulated expert n SimExpert l Creating the list of all open questions and sorting them according to their premise sizes K StructureFamily EK OpenQ SetCartesianProduct A SetDifference K tmp Sort OpenQ Length 1 2 lt Length 2 2 amp OpenQ tmp TO EXCLUDE THE WHOLE SET KAPA ALSO Initializing positive and n
185. unctionInjectionQ Mapping D0O_ CO_ M0_ Module D D0 C C0 M M0 DxD CartesianProduct D0 D0 f Mapping D0 C0 MO Apply And Map 1 1 NotEqual 1 2 Implies 2 1 NotEqual 2 2 D amp Table DxD i Function Value f DxD i 1 FunctionValue f DxD i 2 i 1 Length DxD FunctionWellDefinedQ Mapping D0 C0 M0 CompositionOfFunctions functions returns the composition fn SmallCircle SmallCircle f2 SmallCircle f1 of the functions taking from right to left in case that such composition is defined CompositionOfFunctions functions_List Module i k Length functions A1 f A1 FunctionDomain functions k f Mapping Al FunctionCodomain functions 1 Transpose A1 FunctionValue functions A 1 Return f Length functions GreaterEqual 2 And Apply And Map Head Mapping amp functions And Apply And Table SubsetQ FunctionCodomain functions i 1 FunctionDomain functions i i Length functions 1 1 1 EqualFunctionsQ f1 f2 returns true iff the functions fl and f2 are equal EqualFunctionsQ Mapping D1_ C1_ R1_ Mapping D2_ C2_ R2_ EqualSetsQ D1 D2 amp amp EqualSetsQ C1 C2 amp amp EqualSetsQ R1 R2 MakeStructure X F creates a structure X F MakeStructure X_List F_List Structure X F StructureQ SF returns True if SF is a structure StructureQ SF_ If Head SF Structure True False FamilyQ F returns
186. ure SF is closed under intersection FXFilterOrderDown usage FXFilterOrderDown SF f returns all members of the family forming the structure SF which are subsets of the family member f FXFilterOrderUp usage FXFilterOrderUp SF f returns all members of the family forming the structure SF which are supersets of the family member f GenerateFamilyByTaking Unions usage GenerateFamilyByTakingUnions F returns the family generated by taking all possible unions on the elements from the family F AtomsAt usage AtomsAt SF x returns all atoms at the element x Element S given the structure SF Atoms usage Atoms SF returns subfamilies from the family forming the structure SF that are atoms by at least one element from the set forming the structure BasisFromSpace usage BasisFromSpace SF returns the basis of the space SF BasisToSpace usage BasisToSpace F returns the space constructed from the given basis F SSLofClClausesGet usage SSLofClClausesGet LofCl q returns the clauses assigned to the item q by the mapping LofCl of items to clauses SSAttributionQ usage SSAttributionQ LofCl returns True iff a non empty collection of clauses is assigned to each item within the mapping LofCL of items to clauses SSAxiom1Q usage SSAxiom1Q LofCl returns True iff the mapping LofCl of items to clauses is an attribution SSAxiom2Q usage SSAxiom2Q LofCl returns True iff each item
187. ure S_ R_ S1_List Module SR Structure S R Return SR Map 1 gt 2 amp Transpose S S1 SetCardinality S SetCardinality S 1 StructureFromMatrix M returns the normalized structure from its matrix representation M StructureFromMatrix M Normalized returns the normalized structure from its matrix representation M StructureFromMatrix M SC returns the structure from its matrix representatition having the set SC as the carrier set StructureFromMatrix M_List SC_List Module SF 62 KSMP Techical Report Andrej Zaluski SF Structure SC Map StateBinary2Set Normalized amp M Map 1 gt 2 amp Transpose Range Length M 1 SC l Return SF Length M 1 Length SC StructureFromMatrix M_List Normalized StructureFromMatrix M Range Length M 1 StructureFromMatrix M_List StructureFromMatrix M Normalized StructureToMatrix SF returns the matrix representation of a structure SF StructureToMatrix SF Normalized returns the matrix representation of the structure SF under assumption that the structure is already normalized StructureToMatrix SF_ Structure StructureToMatrix StructureNormalize SF Normalized StructureToMatrix Structure S_ F_ Normalized Module tmp ZeroRow CS SetCardinality S i ZeroRow Table 0 CS Table tmp ZeroRow tmp F i J 1 tmp i 1 Length F ReadStructure FN
188. using the diagnostic Diag MakeRelation MakeRelation S R returns a data structure representing the relation given by the set S and the subset R of the cartesian product SxS MakeRelation S1 S2 R returns a data structure representing the relation from the set S1 to the set S2 given by the subset R of the cartesian product S1xS2 MakeSimpleExpert MakeSimpleExpert Diag returns an ideal expert whose domain metaknowledge is given by the competence performance diagnostic Diag MakeSimpleStudent MakeSimpleStudent QK returns an ideal student whose domain knowledge is given by the set QK of items MakeSkill MakeSkill Code Name Desc returns the data structure representing a skill consisting of the unique identifier Code the name Name and the description Desc MakeStructure MakeStructure X F returns a data structure representing the structure consisting from the set X and the family F of subsets MakeStudent MakeStudent QK CE LG returns the data structure representing a student whose domain knowledge is given by the set QK of items the student is capable of solving the uncertainty CE of making a careless error and the uncertainty LG of making a lucky guess during a knowledge assessment MakeSurmiseFunction MakeSurmiseFunction Q LofCl returns the data structure representing the surmise function given by the set Q of items and the mapping LofCl of clauses MakeSurmiseSystem MakeSurmiseSystem Q LofCl returns the data s
189. w set D1 FunctionChangeCodomain usage FunctionChangeCodomain f C 1 is a primitive function that returns the function f after replacing its codomain with the new set Cl FunctionChangeMapping usage FunctionChangeMapping f M1 is a primitive function that returns the function f after replacing its mapping rule with the new mapping rule M1 FunctionMappingOriginal usage FunctionMappingOriginal f returns the function s original calculated using the mapping rule of the function f FunctionMappingImage usage FunctionMappingImage f returns the function s image calculated using the mapping rule of the function f FunctionWellDefinedQ usage FunctionWellDefinedQ f returns True iff the original of a mapping rule of the function f is equal to the domain of the function FunctionSurjectionQ usage FunctionSurjectionQ f returns True iff the function f is a surjection Function Value usage FunctionValue f x returns the function value f x of the element x n nFunctionValue functions x returns the value of x after applying the list of functions from right to left n nFunctionValue f D1 returns values of the elements from the set D1 after applying the function f 48 KSMP Techical Report Andrej Zaluski n nFunctionValue functions D1 returns values of the elements from the set D1 after applying the list of functions from right to left FunctionSubsetImage usag
190. wledge spaces 6 Installation 7 Software and hardware requirements portability and versioning 7 Installing the package 7 Package documentation 8 Package design 9 Architecture of the package 9 KSMP Layer one 9 KSMP Warp core 10 KSMP Layer two 10 KSMP Skills module 10 KSMP Querying module 11 KSMP Assessment module 11 File formats used 12 Spacefile format 12 Itemsfile format 12 Skillsfile format 13 Further developments 14 Bibliography 15 Glossary of Functions 16 Source code 42 KSMP Techical Report Andrej Zaluski Introduction Motivation and development of the package Historically Knowledge Spaces Mathematica Package KSMP started as a part of Querying Experts on Skills QEOS project The selection of knowledge space theory concepts currently implemented in the package may be seen as a simple consequence of this fact The initial task within the project was to find a suitable software platform that would facilitate research and development of querying procedures that besides observable problem solving behavior also take into account latent skills and competencies Such a platform should primarily be suitable for theoretical explorations and simulation studies The platform should also be suitable for applications of the querying procedures but only to the extent that ensures adequate empirical validation of the developed procedures Additionally it would be desirable if the developed qu
191. xpertOnCompetenceSpace E_ SkOrExp opts__ Module OptionQueryingAlgorithm ReturnedValue OptionQueryingAlgorithm QueryingAlgorithm opts Options QEOSQueryExpertOnCompetenceSpace If OptionQueryingAlgorithm K oppen QueryOn gt CompetenceSpace unchangable Returned Value QEK QueryExpertByK oppen E SkOrExp QueryOn gt CompetenceSpace opts l Return ReturnedValue MakelAssertion MakelAssertion CompetenceState_ Item_ Item CompetenceState AssertionItem IAss returns the item of the interpretation assertion Ass IAssertionItem IAss_ IAss 1 AssertionCompetenceState IAss returns the competence state of the interpretation assertion Ass IAssertionCompetenceState IAss_ I Ass 2 QEOSAskExpertOnInterpretation K Skills Items IAss Human returns an answer data structure on the interpretation assertion IAss from a human expert given a competence structure K with its description stored in the data file Skills and the interpretation assertion item desribed through Items QEOSAskExpertOnInterpretation K_ Skills_ Items_ I Ass_ Human Module answer CertaintyFactor a C E E Apply Union K a lAssertionItem IAss C IAssertionCompetenceState IAss answer Input QEOSFormulateQueryInterpretations a Items C K Skills CertaintyFactor 1 If MemberQ Y T YES TRUE ToUpperCase ToString answer answer True answer False Return ExpertMakeAnswer IAss a
192. xtQuestion SF assumes that no previous answers have been collected KnowledgeSpaceQ KnowledgeSpaceQ SF returns True iff the structure SF satisfies formal requirements of a knowledge space KnowledgeSpaceQuasiOrdinalQ KnowledgeSpaceQuasiOrdinalQ SF returns True iff the structure SF satisfies formal requirements of a knowledge space KnowledgeStructureQ KnowledgeStructureQ SF returns True iff the structure SF satisfies formal requirements of a knowledge structure KSAxiom1 KSAxiom1 SF returns True iff the structure SF contains both the empty set and the whole set S KSAxiom2C KSAxiom2C SF returns True iff the structure SF is closed under union KSAxiom2S KSAxiom2S SF returns True iff the structure SF is stable under union 24 KSMP Techical Report Andrej Zaluski KSAxiom3C KSAxiom3C SF returns True iff the structure SF is closed under intersection KSAxiom3S KSAxiom3S SF returns True iff the structure SF is stable under intersection KSMPBinaryToList KSMPBinaryToList BL returns the list of positive integers calculated from the binary list BL KSMPBinaryToWarp KSMPBinaryToWarp BL returns the warp representation of the binary representation list BL representing a list of positive integers KSMPListToBinary KSMPListToBinary L returns the binary representation list of the list L consisting of positive integers KSMPListToWarp KSMPListToWarp L returns the warp representation of the lis
193. ying only the competence based items set together with its interpretations is kept In case of a human expert DescOrExp has the form Skills Items where the used skills are described in Skills and the items in Items In case of a simulated expert DescOrExp describes the expert The valid options are Verbosity Off Default 1 5 10 15 QSRReport Result default Report or Both Expert Human default Simulation SelectionRule Maximin default WeightedSums and RuleScope Block default All QEOSInterpretationsFromIAssertions usage QEOS InterpretationsFromIAssertions IASSes returns the list of interpretations derived from the list ASSes of interpretation assertions QEOSQueryExpertOnDiagnostic usage QEOS QueryExpertOnDiagnostic E A DescOrExp opts queries a human expert DescOrExp having the form Skills Items describes skills and items or a simulated expert DescOrExp represents a simulated expert whose metaknowledge is given as a diagnostic on competence performance diagnostic The set A of items and the set E of elementary competencies are given The valid options are Expert Human default or Simulation QSRReport Result default Report or Both For other valid options see QEOSQueryExpertOnCompetenceSpace and QEOSQueryExpertOnInterpretationFunction QSRMakeSonde usage QSRMakeSonde OQB NQ returns a querying session observation sonde consisting of the number OQ of open qu

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