Description Logic Knowledge Base Exchange

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Description Logic Knowledge Base ExchangeElena Botoevasupervisor: Diego CalvanesePhD Final ExaminationApril 10, 2014BolzanoElena Botoeva(FUB)Description Logic Knowledge Base Exchange1/33

Outline1 Introduction2 Summary of Work3 Results4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representationsElena Botoeva(FUB)Description Logic Knowledge Base Exchange2/33

Outline1 Introduction2 Summary of Work3 Results4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representationsElena Botoeva(FUB)Description Logic Knowledge Base Exchange3/33

Knowledge Base Exchange: a Simple ScenarioSizeCategoryColorChoose size5Choose color5BrandChoose brand5BItemOpen Shoes: (3 items .BApparelbrown sandStrappyeXXwedge sandPlateaueXXheel sandHighHeeleXX.Elena Botoeva(FUB)Description Logic Knowledge Base Exchange4/33

Knowledge Base Exchange: a Simple ScenarioSizeCategoryColorChoose size5BrandChoose color5Choose brand5BItemOpen Shoes: (3 items .brown sandStrappyeXXBApparelwedge sandPlateaueXXheel sandHighHeeleXX.Viewed as a knowledge uHighHeelPlateauHighHeelbrown sandwedge sandheel sand.Apparel.Elena Botoeva(FUB)Description Logic Knowledge Base Exchange4/33

Knowledge Base Exchange: a Simple ScenarioSizeCategoryColorChoose size5Choose color5BrandChoose brand5BItemOpen Shoes: (3 items .brown sandStrappyeXXBApparelwedge sandPlateaueXXheel sandHighHeeleXX.The website after restructuring:SizeCategoryColorChoose size5Choose color5BrandChoose brand5BProductSandals: (3 products hingbrown sandClassiceXXwedge sandPlatformeXXheel sandHeeledeXX.Elena Botoeva(FUB)Description Logic Knowledge Base Exchange4/33

Knowledge Base Exchange: a Simple ScenarioSizeCategoryColorChoose size5Choose color5BrandChoose brand5BItemOpen Shoes: (3 items Heel.brown sandStrappyeXXBApparelwedge sandPlateaueXXheel sandHighHeeleXX.The website after restructuring:SizeCategoryColorChoose size5Choose color5BrandChoose brand5BProductSandals: (3 products hingbrown sandClassiceXXwedge sandPlatformeXXheel sandHeeledeXX.Elena Botoeva(FUB)Description Logic Knowledge Base Exchange4/33

Data Exchange [Fagin et al., 2003]mapping MΣΣ11source schemaΣΣ22target schemaI1source instanceElena Botoeva(FUB)Description Logic Knowledge Base Exchange5/33

Data Exchange [Fagin et al., 2003]mapping MΣΣ11ΣΣ22target schemasource schemaI1source instanceElena Botoeva(FUB)best solutionI2target instanceDescription Logic Knowledge Base Exchange5/33

Data Exchange [Fagin et al., 2003]mapping MΣΣ11ΣΣ22target schemasource schemaI1best solutionI2target instancesource instanceMapping M is a set of inclusions of conjunctive queries (CQs) x, y q1 (x, y ) z q2 (x, z) .I1 is a complete database instance.I2 is an incomplete database instance.Elena Botoeva(FUB)Description Logic Knowledge Base Exchange5/33

Data Exchange ExampleM:I1 :Elena Botoeva(FUB) a, t.(AuthorOf (a, t) g .BookInfo(t, a, g ))AuthorOfnabokovlolitatolkienlotrDescription Logic Knowledge Base Exchange6/33

Data Exchange ExampleM:I1 : a, t.( AuthorOf (a, t)AuthorOf g .BookInfo(t, a, g )I2 kienlotrlotrtolkienfantasyI2 is a solution for I1 under M.Elena Botoeva(FUB)Description Logic Knowledge Base Exchange6/33

Data Exchange ExampleM:I1 : a, t.( AuthorOf (a, t) g .BookInfo(t, a, g )I2 omedytolkienlotrlotrtolkienfantasyI20 :BookInfololitanabokovnull1lotrtolkiennull2I2 is a solution for I1 under M.I20 is a universal solution for I1 under M.Elena Botoeva(FUB)Description Logic Knowledge Base Exchange6/33

Data Exchange ExampleM:I1 : a, t.( AuthorOf (a, t) g .BookInfo(t, a, g )I2 medytolkienlotrlotrtolkienfantasyI20 :I2 is a solution for I1 under M.I20 is a universal solution for I1 under M.Elena null2 there is a homomorphism from I20 to I2 .Description Logic Knowledge Base Exchange6/33

Incomplete Data and Knowledge ExchangeA framework for Data Exchange with incomplete data was proposed by Arenas et al.[2011].mapping MΣΣ11ΣΣ22target signaturesource signatureI1solutionI2incomplete source instanceElena Botoeva(FUB)Description Logic Knowledge Base Exchange7/33

Incomplete Data and Knowledge ExchangeA framework for Data Exchange with incomplete data was proposed by Arenas et al.[2011].mapping MΣΣ11ΣΣ22target signaturesource signaturelogical theory T1logical theory T2solutionElena Botoeva(FUB)I1I2source KBtarget KBDescription Logic Knowledge Base Exchange7/33

Incomplete Data and Knowledge ExchangeA framework for Data Exchange with incomplete data was proposed by Arenas et al.[2011].mapping MΣΣ11ΣΣ22target signaturesource signaturelogical theory T1logical theory T2solutionI1I2source KBtarget KBWe specialize this framework to Description Logics, and in particular to DL-Lite R .Elena Botoeva(FUB)Description Logic Knowledge Base Exchange7/33

Description Logic DL-LiteRDescription Logics (DLs) are decidable fragments of First-Order Logic,used as Knowledge Representation formalisms.DLs talk about the domain of interest by means of concepts (unary predicates): roles (binary predicates):Author, Book, A, BAuthorOf, WrittenBy, P, RDL-LiteR is a light-weight DL that asserts concept inclusions and disjointness of atomic concepts A,the domain P and the range P of atomic roles P,Book v WrittenBy , role inclusions and disjointness of atomic roles P andtheir inverses P , AuthorOf v WrittenBy , ground factsElena Botoeva(FUB)Author(nabokov), AuthorOf(nabokov,lolita),A(a), P(a, b).Description Logic Knowledge Base Exchange8/33

Description Logic DL-LiteRDescription Logics (DLs) are decidable fragments of First-Order Logic,used as Knowledge Representation formalisms.DLs talk about the domain of interest by means of concepts (unary predicates): roles (binary predicates):Author, Book, A, BAuthorOf, WrittenBy, P, RDL-LiteR is a light-weight DL that asserts concept inclusions and disjointness of atomic concepts A,the domain P and the range P of atomic roles P,Book v WrittenBy ,TBox T role inclusions and disjointness of atomic roles P andtheir inverses P , AuthorOf v WrittenBy , ground factsElena Botoeva(FUB)Author(nabokov), AuthorOf(nabokov,lolita),A(a), P(a, b).Description Logic Knowledge Base ExchangeABox A8/33

Description Logic DL-LiteRDescription Logics (DLs) are decidable fragments of First-Order Logic,used as Knowledge Representation formalisms.DLs talk about the domain of interest by means of concepts (unary predicates): roles (binary predicates):Author, Book, A, BAuthorOf, WrittenBy, P, RDL-LiteR is a light-weight DL that asserts concept inclusions and disjointness of atomic concepts A,the domain P and the range P of atomic roles P,Book v WrittenBy ,TBox T role inclusions and disjointness of atomic roles P andtheir inverses P , AuthorOf v WrittenBy , ground factsAuthor(nabokov), AuthorOf(nabokov,lolita),A(a), P(a, b).ABox ASatisfiability check over a DL-Lite R KB K hT , Ai can be donein polynomial time (in fact, in NLogSpace).Elena Botoeva(FUB)Description Logic Knowledge Base Exchange8/33

Outline1 Introduction2 Summary of Work3 Results4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representationsElena Botoeva(FUB)Description Logic Knowledge Base Exchange9/33

The Summary of my PhD WorkIn this thesis, we123Propose a general framework for exchanging Description LogicKnowledge Bases.Define and analyse relevant reasoning problems in this setting.Develop reasoning techniques and characterize the computationalcomplexity of the problems.Elena Botoeva(FUB)Description Logic Knowledge Base Exchange10/33

Outline1 Introduction2 Summary of Work3 Results4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representationsElena Botoeva(FUB)Description Logic Knowledge Base Exchange11/33

1. Knowledge Base Exchange Frameworkmapping MΣΣ11ΣΣ22target signaturesource signatureA1D1B1T1C1A1source KBElena Botoeva(FUB)Description Logic Knowledge Base Exchange12/33

1. Knowledge Base Exchange Frameworkmapping MΣΣ11ΣΣ22target signaturesource signatureA1D1B1Elena Botoeva(FUB)T1C1T2A2solutionB2C2A1A2source KBtarget KBDescription Logic Knowledge Base Exchange12/33

2. Reasoning ProblemsElena Botoeva(FUB)Description Logic Knowledge Base Exchange13/33

2. Reasoning ProblemsSolutionElena Botoeva(FUB)universal solutionpreserves all modelsuniversal UCQ-solutionpreserves all answers toUnions of Conjunctive QueriesUCQ-representationpreserves all answers to UCQs,independently of the ABoxDescription Logic Knowledge Base Exchange13/33

simpleA( ABa) ox, esexR(tea)nd,Rco ed(ant AB,bain o)lab xese lednulls2. Reasoning ProblemsABoxSolutionElena Botoeva(FUB)universal solutionpreserves all modelsuniversal UCQ-solutionpreserves all answers toUnions of Conjunctive QueriesUCQ-representationpreserves all answers to UCQs,independently of the ABoxDescription Logic Knowledge Base Exchange13/33

2. Reasoning Problemsnon-emptinessIs there any solutionfor K1 (resp. T1 ) under M?membershipIs K2 (resp. T2 ) a solutionfor K1 (resp. T1 ) under M?simpleA( ABa) ox, esexR(tea)nd,Rco ed(ant AB,bain o)lab xese lednullsDecision problemABoxSolutionElena Botoeva(FUB)universal solutionpreserves all modelsuniversal UCQ-solutionpreserves all answers toUnions of Conjunctive QueriesUCQ-representationpreserves all answers to UCQs,independently of the ABoxDescription Logic Knowledge Base Exchange13/33

3. ResultsUniversal solutionsMembershipNon-emptinesssimple ABoxesPTime-completePTime-completeextended ABoxesNP-completePSpace-hard, in ExpTimeUniversal UCQ-solutionsMembershipNon-emptinesssimple ABoxesPSpace-hardin ExpTimeextended ABoxesin -emptinessElena completeDescription Logic Knowledge Base Exchange14/33

3. ResultsUniversal solutionsMembershipNon-emptiness1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba43VrzZ4Wb5 Xq1Tk4 6YP6šYP6YP6šYPz6šYP6YP6šYP13—VR2UN4 WB5XQ1 TKzzZ3VRsimple ABoxesPTime-completePTime-completeextended ABoxesNP-completePSpace-hard, in ExpTime8761 games54abcdefgcircuit value problemUniversal tionsMembershipNon-emptinessElena Botoeva(FUB)ad-hoc2 automata3-colorabilityvalidity of QBFhsimple ABoxesPSpace-hardin ExpTimeextended ABoxesin Space-completeDescription Logic Knowledge Base Exchange14/33

3. ResultsUniversal solutionsMembershipNon-emptinesssimple ABoxesPTime-completePTime-completeextended ABoxesNP-completePSpace-hard, in ExpTimeUniversal UCQ-solutionsMembershipNon-emptinesssimple ABoxesPSpace-hardin ExpTimeextended ABoxesin ExpTimePSpace-hard1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba43VrzZ4Wb5 Xq1Tk4 6YP6šYP6YP6šYPz6šYP6YP6šYP13—VR2UN4 WB5XQ1 TKzzZ3VR8763 game-theoretic54abcdefghvalidity of QBFUCQ-representationsMembershipNon-emptinessElena completeDescription Logic Knowledge Base Exchange14/33

3. ResultsUniversal solutionsMembershipNon-emptinesssimple ABoxesPTime-completePTime-completeextended ABoxesNP-completePSpace-hard, in ExpTimeUniversal UCQ-solutionsMembershipNon-emptinesssimple ABoxesPSpace-hardin ExpTimeextended ABoxesin mplete4 5 graph-theoreticreachability in directed graphsElena Botoeva(FUB)Description Logic Knowledge Base Exchange14/33

3. ResultsUniversal solutionsMembershipNon-emptiness1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba43VrzZ4Wb5 Xq1Tk4 6YP6šYP6YP6šYPz6šYP6YP6šYP13—VR2UN4 WB5XQ1 TKzzZ3VRsimple ABoxesPTime-completePTime-completeextended ABoxesNP-completePSpace-hard, in ExpTime8761 games54abcdefgad-hoc2 automata3-colorabilityvalidity of QBFhcircuit value problemUniversal UCQ-solutionsMembershipNon-emptinesssimple ABoxesPSpace-hardin ExpTime1 e4 e5 2 Nf3 Nc6 3 Bb5 3. . . a6 3. . . Nf6 4 Ba4extended ABoxesin ExpTimePSpace-hard3VrzZ4Wb5 Xq1Tk4 6YP6šYP6YP6šYPz6šYP6YP6šYP13—VR2UN4 WB5XQ1 TKzzZ3VR8763 game-theoretic54abcdefghvalidity of exityNLogSpace-completeNLogSpace-complete4 5 graph-theoreticreachability in directed graphsElena Botoeva(FUB)Description Logic Knowledge Base Exchange14/33

Outline1 Introduction2 Summary of Work3 Results4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representationsElena Botoeva(FUB)Description Logic Knowledge Base Exchange15/33

The Essence of Knowledge Base ExchangeA mapping is a triple M (Σ1 , Σ2 , T12 ),where T12 is a set of DL-Lite R inclusions from Σ1 to Σ2hT1 , A1 i is a DL-Lite R knowledge base over Σ1 (source KB)hT2 , A2 i is a DL-Lite R knowledge base over Σ2 (target KB)Elena Botoeva(FUB)Description Logic Knowledge Base Exchangemapping MΣ11ΣΣ22target signaturesource signatureA1T1D1B1C1A2solutionB2T2C2A1A2source KBtarget KB16/33

The Essence of Knowledge Base ExchangeA mapping is a triple M (Σ1 , Σ2 , T12 ),where T12 is a set of DL-Lite R inclusions from Σ1 to Σ2hT1 , A1 i is a DL-Lite R knowledge base over Σ1 (source KB)hT2 , A2 i is a DL-Lite R knowledge base over Σ2 (target KB)mapping MΣ11ΣΣ22target signaturesource signatureA1T1D1B1C1T2A2solutionB2C2A1A2source KBtarget KBFor a KB K, we denote by UK the canonical model of K (chase in databases). hT2 , A2 i is a universal solution for hT1 , A1 i under M (Σ1 , Σ2 , T12 ) iff?T2 andUA2 is Σ2 -homomorphically equivalent to UhT1 T12 ,A1 i . hT2 , A2 i is a universal UCQ-solution for hT1 , A1 i under M (Σ1 , Σ2 , T12 ) iffUhT2 ,A2 i is finitely Σ2 -homomorphically equivalent to UhT1 T12 ,A1 i . T2 is a UCQ-representation for T1 under M (Σ1 , Σ2 , T12 ) iff?for each source ABox A1 ,UhT2 T12 ,A1 i is Σ2 -homomorphically equivalent to UhT1 T12 ,A1 i .? plus one more condition with little technical impact? plus one more condition for checking consistencyElena Botoeva(FUB)Description Logic Knowledge Base Exchange16/33

The Essence of Knowledge Base ExchangeA mapping is a triple M (Σ1 , Σ2 , T12 ),where T12 is a set of DL-Lite R inclusions from Σ1 to Σ2hT1 , A1 i is a DL-Lite R knowledge base over Σ1 (source KB)hT2 , A2 i is a DL-Lite R knowledge base over Σ2 (target KB)mapping MΣ11ΣΣ22target signaturesource signatureA1T1D1B1C1T2A2solutionB2C2A1A2source KBtarget KBFor a KB K, we denote by UK the canonical model of K (chase in databases). hT2 , A2 i is a universal solution for hT1 , A1 i under M (Σ1 , Σ2 , T12 ) iff?T2 andUA2 is Σ2 -homomorphically equivalent to UhT1 T12 ,A1 i . hT2 , A2 i is a universal UCQ-solution for hT1 , A1 i under M (Σ1 , Σ2 , T12 ) iffUhT2 ,A2 i is finitely Σ2 -homomorphically equivalent to UhT1 T12 ,A1 i . T2 is a UCQ-representation for T1 under M (Σ1 , Σ2 , T12 ) iff?for each source ABox A1 ,UhT2 T12 ,A1 i is Σ2 -homomorphically equivalent to UhT1 T12 ,A1 i .We present our 5 techniques that check the existence of the homomorphisms.? plus one more condition with little technical impact? plus one more condition for checking consistencyElena Botoeva(FUB)Description Logic Knowledge Base Exchange16/33

The Canonical and Generating ModelsLet K hT , Ai, whereElena Botoeva(FUB)Description Logic Knowledge Base Exchange17/33

The Canonical and Generating ModelsLet K hT , Ai, whereA {B(a)}aaBThe canonical model UKBThe generating model GKWe call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .Elena Botoeva(FUB)Description Logic Knowledge Base Exchange17/33

The Canonical and Generating ModelsLet K hT , Ai, whereA {B(a)}aT {B v RaBRawRThe canonical model UKBRwRThe generating model GKWe call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .Elena Botoeva(FUB)Description Logic Knowledge Base Exchange17/33

The Canonical and Generating ModelsLet K hT , Ai, whereA {B(a)}aT {B v R R v SaBRawRBRwRSSawR wSThe canonical model UKwSThe generating model GKWe call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .Elena Botoeva(FUB)Description Logic Knowledge Base Exchange17/33

The Canonical and Generating ModelsLet K hT , Ai, whereA {B(a)}aT {B v R R v S S aBRawRBRwRv SSSawR wSwSSawR wS wSS···The canonical model UKThe generating model GKWe call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .Elena Botoeva(FUB)Description Logic Knowledge Base Exchange17/33

The Canonical and Generating ModelsLet K hT , Ai, whereA {B(a)}aT {B v R R v SawRBR, QwR S v SR v Q}aBR, QSSawR wSwSSawR wS wSS···The canonical model UKThe generating model GKWe call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .Elena Botoeva(FUB)Description Logic Knowledge Base Exchange17/33

The Canonical and Generating ModelsLet K hT , Ai, whereA {B(a)}aaT {B v R,Q R v SawR S v SR v Q}awR wS,QwRSSwSSΣ {Q, S}awR wS wSS···The canonical model UK ΣThe generating model GK ΣWe call wR and wS witnesses of K, denoted Wit(K).Moreover, we write, e.g., a K wR , or wR K wS .Elena Botoeva(FUB)Description Logic Knowledge Base Exchange17/33

Outline1 Introduction2 Summary of Work3 Results4 Technical DevelopmentUniversal SolutionsUniversal UCQ-solutionsUCQ-representationsElena Botoeva(FUB)Description Logic Knowledge Base Exchange18/33

Membership for Simple Universal Solutions is in PTimeA2 is a universal solution for hT1 , A1 i under M (Σ1 , Σ2 , T12 ) iff? there exist a Σ2 -homomorphism from UA2 to UhT1 T12 ,A1 i , a Σ2 -homomorphism from UhT1 T12 ,A1 i to UA2 .? plus one more condition of no technical interestElena Botoeva(FUB)Description Logic Knowledge Base Exchange19/33

Membership for Simple Universal Solutions is in PTimeA2 is a universal solution for hT1 , A1 i under M (Σ1 , Σ2 , T12 ) iff? there exist a Σ2 -homomorphism from UA2 to UhT1 T12 ,A1 i ,EASY a Σ2 -homomorphism from UhT1 T12 ,A1 i to UA2 .? plus one more condition of no technical interestElena Botoeva(FUB)Description Logic Knowledge Base Exchange19/33

Membership for Simple Universal Solutions is in PTimeA2 is a universal solution for hT1 , A1 i under M (Σ1 , Σ2 , T12 ) iff? there exist a Σ2 -homomorphism from UA2 to UhT1 T12 ,A1 i ,EASY a Σ2 -homomorphism from UhT1 T12 ,A1 i to UA2 . via Reachability Games on graphs? plus one more condition of no technical interestElena Botoeva(FUB)Description Logic Knowledge Base Exchange19/33

Membership for Simple Universal Solutions is in PTimeA2 is a universal solution for hT1 , A1 i under M (Σ1 , Σ2 , T12 ) iff? there exist a Σ2 -homomorphism from UA2 to UhT1 T12 ,A1 i ,EASY a Σ2 -homomorphism from UhT1 T12 ,A1 i to UA2 . via Reachability Games on graphsFor a KB K, an ABox A, and a signature Σ,we construct a reachability game (G, F ) such thatthere exists a Σ-homomorphism from UK to UAiffDuplicator has a strategy against Spoilerto avoid F in G.? plus one more condition

Description Logic Knowledge Base Exchange Elena Botoeva supervisor: Diego Calvanese PhD Final Examination April 10, 2014 Bolzano Elena Botoeva(FUB)Description Logic Knowledge Base Exchange1/33. Outline 1 Introduction 2 Summary of Work 3 Results 4 Technical Development Universal Solutions Universal UCQ-solutions UCQ-representations Elena Botoeva(FUB)Description Logic Knowledge Base Exchange2/33 .

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