Semantics and Reasoning Algorithms for a Faithful Integration

Semantics and Reasoning Algorithms for a Faithful Integration of Description Logics and Rules Boris Motik, University of Oxford

Contents • Why Combine DLs with LP? • Main Challenge: OWA vs. CVA • Existing Approaches • Minimal Knowledge and Negation as Failure • MKNF Knowledge Bases • Reasoning and Complexity • Conclusion 2

Description Logics and OWL • OWL (Web Ontology Langage) § language for ontology modeling in the Semantic Web § standard of the W 3 C (http: //www. w 3. org/2004/OWL/) • OWL is based on Description Logics (DLs) § inspired by semantic networks § DLs have a precise semantics based on first-order logics § well-understood computational properties • What can we say in DLs? UK cities are in UK regions. UKCity v 9 is. In. UKRgn 8 x : UKCity(x) ! 9 y : is. In(x, y) Æ UKRgn(y) UK regions are EU regions. UKRgn v EURgn 8 x : UKRgn(x) ! EURgn(x) 9 is. In. EURgn v EUPart 8 x : [9 y : is. In(x, y) Æ EURgn(y)] ! EUPart(x) UKCity v EUPart 8 x : UKCity(x) ! EUPart(x) Things in EU are parts of EU. We can conclude: UK cities are parts of EU. 3

Missing Features (I) • Relational expressivity § OWL can express only tree-like axioms x 9 S. (9 R. C u 9 R. D) v Q , 8 x: {[9 y: S(x, y) Æ (9 x: R(y, x) Æ C(x)) Æ (9 x: R(y, x) Æ D(x))] ! Q(x)} , 8 x, x 1, x 2, x 3: { S(x, x 1) Æ R(x 1, x 2) Æ C(x 2) Æ R(x 1, x 3) Æ D(x 3) ! Q(x) } S x 1 R R x 2 x 3 • Polyadic predicates § e. g. , Flight(From, To, Airline) • Can be addressed by rules (LP or ASP) 4

Missing Features (II) – Closed Worlds flight(MAN, STR) flight(MAN, LHR) flight(MAN, FRA) flight(FRA, ZAG) Question: is there a flight from MAN to MUC? Open worlds (=OWL): Don’t know! Closed worlds (=LP): No. We did not specify that we know information about all possible flights. If we cannot prove something, it must be false. • Partial solution: close off flight 8 x, y: flight(x, y) $ (x ¼ MAN Æ y ¼ STR) Ç (x ¼ MAN Æ y ¼ LHR) Ç … § cannot express many things (e. g. , transitive closure) • Closed-world is orthogonal to closed-domain reasoning Person v 9 father. Person(Peter) > v { Peter, Paul } • CWA is available in various LP formalisms (e. g. , ASP) 5

Missing Features (III) – Constraints • “Each person must have an SSN” § naïve attempt: Person u : (9 has. SSN) v ? § in FOL, this is equivalent to: Person v 9 has. SSN § assume that only Person(Peter) is given § we expect the constraint to be violated (no SSN) § but KB is satisfiable: Peter has some unknown SSN • FOL formulae… § …speak about the general properties of worlds § …cannot reason about their own knowledge • Constraints can be expressed in LP 6

Missing Features (IV) • “The heart is usually on the left, but in some cases it is on the right” • Naïve approach: § Human v Heart. On. Left Dextrocardiac v Human Dextrocardiac v : Heart. On. Left the class Dextrocardiac is unsatisfiable § “with no contrary evidence, the heart is on the left” • Exceptions… § …cannot be expressed in FOL § …can be expressed in ASP 7

The Magic Formula DLs (= taxonomical reasoning) + LP Rules (= relational expressivity + nonmonotonic inferences) = The Winning Combination! 8

Contents • Why Combine DLs with LP? • Main Challenge: OWA vs. CVA • Existing Approaches • Minimal Knowledge and Negation as Failure • MKNF Knowledge Bases • Reasoning and Complexity • Conclusion 9

Open vs. Closed Worlds • “It is illegal to state that someone is a father without stating that he is a person” 8 x : [Father(x) Æ : Person(x) ! ? ] Father(a) • In DLs we derive Person(a) • The formula is equivalent to 8 x : [Father(x) ! Person(x)] § eliminates all models in which x is a father and not a person • In LP, : is interpreted as default negation § read as “is not provable” • The example is unsatisfiable • Negation defined using minimal knowledge 10

Idea of Minimal Knowledge Father(a) • DLs M 1 All models are of equal “quality”. Father(a) M 2 Father(a), Person(a) • LP This is the only minimal model. (There is no model M’ ½ M. ) M Father(a) 8 x : [Father(x) Æ : Person(x) ! ? ] § kills all models in which the formula does not hold • We are left with models that contain Person(a) • We are left with no model 11

Minimal Knowledge and Negation Father(a) • DLs M 1 Father(a) , Cat(a) • Rules M Father(a), Cat(a) M 2 Father(a), Person(a) 8 x : [Father(x) Æ : Person(x) ! Cat(x)] § esures Cat(x) in each model where x is a father and not a person • Does not entail Cat(a) • Does entail Cat(a) Nonmonotonic semantics typically prefer certain models. 12

Contents • Why Combine DLs with LP? • Main Challenge: OWA vs. CVA • Existing Approaches • Minimal Knowledge and Negation as Failure • MKNF Knowledge Bases • Reasoning and Complexity • Conclusion 13

First-Order Rule Formalisms • First-order combinations of DLs and rules: § SWRL, CARIN, AL-log, DL-safe rules § A 1 Æ … Æ A n à B 1 Æ … Æ Bm § concepts (classes) = unary predicates § roles (properties) = binary predicates § interpreted as first-order clauses • Semantics is standard first-order § Woman(x) ! Person(x) and : Person(Lassie) imply : Woman(Lassie) • Easily undecidable § decidability achieved by syntactic restrictions; e. g. , DL-safety • Issues addressed: ü relational expressivity and polyadic predicates û nonmonotonic features 14

Loose Integration • dl-programs § [Eiter, Ianni, Lukasiewicz, Schindlauer, Tompits, AIJ 2008] • A Ã B 1 Æ … Æ Bm Æ not Bm+1 Æ … Æ not Bn § A and Bi are first-order atoms over non-DL-predicates § Bi can additionally be a query atom of the form DL[ S 1 [+ p 1, S 2 [- p 2, S 3 Å- p 3; Q ] § § Si – DL predicates pi – non-DL-predicates Q – a DL query understand as conditional queries over a DL ontology • Rules are layered over a DL KB § they do not contribute to DL consequences 15

Strong Integration • A 1 Ç … Ç Ak à B 1 Æ … Æ Bm Æ not Bm+1 Æ … Æ not Bn • DL+log [Rosati, KR 2006] § DL-atoms cannot occur under negation as failure § semantics: § DL-predicates interpreted under OWA § non-DL-predicates interpreted under CWA no nonmonotonic reasoning over DL-predicates • dl-programs [Lukasiewicz, ESWC 2007] § no classical negation cannot capture ASP § faithful extension of LP and DLs only w. r. t. entailment of positive ground atoms § unclear how to extend the semantics to make if faithful w. r. t. arbitrary consequences 16

Autoepistemic Logics • LP can be encoded into first-order AEL § AEL by [Konolige, Fund. Inf. 1991] Use AEL as a framework for integrating FOL and LP § [de Bruijn, Eiter, Polleres, Tompits, IJCAI 2007] • Various encodings proposed with different levels of faithfulness § considers disjunctive datalog and not ASP • No proof theory yet 17

Contents • Why Combine DLs with LP? • Main Challenge: OWA vs. CVA • Existing Approaches • Minimal Knowledge and Negation as Failure • MKNF Knowledge Bases • Reasoning and Complexity • Conclusion 18

Knowledge Operator K (Researcher t Programmer)(Boris) Researcher v Employed Programmer v Employed ² Employed(Boris) ² Researcher(Boris) ² Programmer(Boris) ² K Employed(Boris) ² : K Researcher(Boris) ² : K Programmer(Boris) • K allows us to reason about FO consequences § KB ² K A iff KB ² A § KB ² : K A iff KB ² A • K is nonmonotonic § if we assert Researcher(Boris), then… § K Researcher(Boris) holds § : K Researcher(Boris) does not hold any more • Used in an algebra-like query language EQL-Lite § [Calvanese, De Giacomo, Lembo, Lenzerini, Rosati, IJCAI 2007] 19

Default Negation Operator not • Interpreted as not consequence Bird(Tweety) K Bird(Tweety) Æ not : Flies(Tweety) ! K Flies(Tweety) • Read as: § if § “Tweety is a bird” is a consequence § and § “Tweety cannot fly” is not a consequence § then § “Tweety can fly” should be a consequence 20

Minimal Knowledge and Negation as Failure (I, M, N) ² A iif A is true in I (I, M, N) ² : iif is false in I (I, M, N) ² 1 Æ 2 iif both 1 and 2 are true in I (I, M, N) ² K iif (J, M, N) ² for each J 2 M (I, M, N) ² not iif (J, M, N) ² for some J 2 N • Satisfiability defined w. r. t. an MKNF structure (I, M, N) § I – a FOL interpretation § M and N – sets of FOL interpretations • M is a model of if: Gelfond-Lifschitz reduct! § (I, M, M) ² with I 2 M and § for each M’ ¾ M, there is some I’ 2 M’ such that (I’, M) ² § [Lifschitz, IJCAI 91; Artificial Intelligence 94] • MKNF explains many nonmonotonic formalisms 21

Contents • Why Combine DLs with LP? • Main Challenge: OWA vs. CVA • Existing Approaches • Minimal Knowledge and Negation as Failure • MKNF Knowledge Bases • Reasoning and Complexity • Conclusion 22

MKNF Knowledge Bases • MKNF Rule: H 1 Ç … Ç Hn à B 1, …, Bm • Hi are first-order or K-atoms • Bi are first-order, K-, or not-atoms P(t 1, …, tn) - first-order atom K P(t 1, …, tn) - K-atom not P(t 1, …, tn) - not-atom • MKNF Knowledge Base K = (O, P) • O – a FOL KB in some language DL • P – a finite set of MKNF rules • Semantics by translation into MKNF (K) = K (O) Æ • DL-safety: Æ r 2 P 8 x 1, …, xn : H 1 Ç … Ç Hn ½ B 1 Æ … Æ Bm § the rules are applicable only to explicitly named objects 23

Example (I) • We derive seaside. City(Barcelona) default rule § assuming it does not lead to contradiction § deriving seaside. City(Hamburg) would cause a contraction • We derive Suggest(Barcelona) § this involves standard DL reasoning § we do not know the name of the beach in Barcelona 24

Example (II) constraint • We treat ¼ in a special way § we minimize equality along with other predicates § this yields intuitive consequences • The constraint is satisfied § Holy. Family is a church, § the architect of Sagrada. Familia has been specified, and § Holy. Family and Sagrada. Familia are synonyms 25

Faithfulness • MKNF KBs are fully faithful w. r. t. DLs (O, ; ) ² iff O ² for any FOL formula § to achieve this, we modified MKNF slightly § we must treat equality in a special way • MKNF KBs are fully faithful w. r. t. ASP (; , P) ² (: )A iff P ² (: )A for A a ground atom § already shown by Lifschitz • The combination seems quite intuitive 26

Contents • Why Combine DLs with LP? • Main Challenge: OWA vs. CVA • Existing Approaches • Minimal Knowledge and Negation as Failure • MKNF Knowledge Bases • Reasoning and Complexity • Conclusion 27

How to Represent Models • A MKNF model is a set of interpretations § = typically infinite! § we need a finite representation • Idea: represent models by FOL formulae § find a first-order formula such that M={I|I² } • We represent using K-atoms § (P, N) – a partition of all K-atoms into positive and negative § defines the consequences that must hold in an MKNF model § objective knowledge: ob. K, P = O [ { A | K A 2 P } § our main task is to find a partition (P, N) that defines a model 28

Characterization of MKNF Models Grounding Guess a partition that defines an MKNF model Check whether the rules are satisfied in this model. Check whether this model is consistent with the DL KB. Check whether this is the model of minimal knowledge. Check whether the query does not hold in the model. These are the extensions to the standard algorithm for disjunctive datalog. 29

Data Complexity • Reasoning is undecidable without DL-safety § different sources of undecidability than in the FO case • If rules have special form, we can… § …find (P, N) in an easier way (e. g. deterministically) and/or § …check the minimality condition easier • Data complexity of ground atom entailment: 30

Contents • Why Combine DLs with LP? • Main Challenge: OWA vs. CVA • Existing Approaches • Minimal Knowledge and Negation as Failure • MKNF Knowledge Bases • Reasoning and Complexity • Conclusion 31

Conclusion • MKNF rules… § …generalize many known combinations of DLs and rules § …are fully compatible with both DLs and LP § …are intuitive § think of K as consequence § think of not as not consequence § …have nice complexity § defined by the DL and the LP fragment (in most cases) • Future challenges § implementation § use in applications 32