Teil III: Wissensrepräsentation und Inferenz. Kap.10: Beschreibungslogiken. Beschreibungslogiken (Description Logics)

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1 Geschichte Vorlesung Künstliche Intelligenz Wintersemester 2007/08 Teil III: Wissensrepräsentation und Inferenz Kap.10: Beschreibungslogiken Ihre Enticklung urde inspiriert durch semantische Netze und Frames. Frühere Namen: KL-ONE like languages terminological logics Ziel ar eine Wissensrepräsentation mit formaler emantik. Das erste Beschreibungslogik-basierte ystem ar KL-ONE (1985). Weitere ysteme u.a. LOOM (1987), BACK (1988), KI (1991), CLAIC (1991), FaCT (1998), ACE (2001), KAON 2 (2005). Mit Material von Carsten Lutz, Uli attler: Ian Horrocks: 3 Beschreibungslogiken (Description Logics) Literatur A family of logic based Knoledge epresentation formalisms Descendants of semantic netorks and KL-ONE Describe domain in terms of concepts (classes), roles (relationships) and individuals Distinguished by: Formal semantics (typically model theoretic) Decidable fragments of FOL Closely related to Propositional Modal & Dynamic Logics Provision of inference services ound and complete decision procedures for key problems Implemented systems (highly optimised) Einfache prache zum tart: ALC (Attributive Language ith Complement) D. Nardi,. J. Brachman. An Introduction to Description Logics. In: F. Baader, D. Calvanese, D.L. McGuinness, D. Nardi, P.F. Patel-chneider (eds.): Description Logic Handbook, Cambridge University Press, 2002, F. Baader, W. Nutt: Basic Description Logics. In: Description Logic Handbook, Ian Horrocks, Peter F. Patel-chneider and Frank van Harmelen. From HIQ and DF to OWL: The making of a eb ontology language. H03a.pdf Im emantic Web ird HOIN(D n ) eingesetzt. Hierauf basiert die emantik von OWL DL. 2 4

2 ecall: Logics and Model Theory ecall: Logics and Model Theory Ontology/K languages aim to model (part of) orld Terms in language correspond to entities in orld Meaning given by, e.g.: Mapping to another formalism, such as FOL, ith on ell defined semantics or a Model Theory (MT) World Model Daisy isa Co Co kindof Animal Interpretation MT defines relationship beteen syntax and interpretations There can be many interpretations (models) of one piece of syntax Models supposed to be analogue of (part of) orld E.g., elements of model correspond to objects in orld Formal relationship beteen syntax and models tructure of models reflect relationships specified in syntax Inference (e.g., subsumption) defined in terms of MT E.g., T ² A v B iff in every model of T, ext(a) ext(b) Mary isa Person Person kindof Animal Z123ABC isa Car Mary drives Z123ABC b a {ha,bi, } 5 7 ecall: Logics and Model Theory ecall: Logics and Model Theory Many logics (including standard First Order Logic) use a model theory based on (Zermelo-Frankel) set theory The domain of discourse (i.e., the part of the orld being modelled) is represented as a set (often refered as ) Objects in the orld are interpreted as elements of Classes/concepts (unary predicates) are subsets of Properties/roles (binary predicates) are subsets of (i.e., 2 ) Ternary predicates are subsets of 3 etc. The sub-class relationship beteen classes can be interpreted as set inclusion. Formally, the vocabulary is the set of names e use in our model of (part of) the orld {Daisy, Co, Animal, Mary, Person, Z123ABC, Car, drives, } An interpretation I is a tuple h, I i is the domain (a set) I is a mapping that maps Names of objects to elements of Names of unary predicates (classes/concepts) to subsets of Names of binary predicates (properties/roles) to subsets of And so on for higher arity predicates (if any) 6 8

3 DL Architecture DL emantics Interpretation function I the obvious ay, i.e.: extends to concept expressions in Knoledge Base Tbox (schema) Man Human u Male Happy-Father Man u has-child Female u Abox (data) John : Happy-Father hjohn, Maryi : has-child Inference ystem Interface 9 11 DL Knoledge Base DL Knoledge Bases (Ontologies) A DL Knoledge Base is of the form K = ht, Ai DL Knoledge Base (KB) normally separated into 2 parts: TBox is a set of axioms describing structure of domain (i.e., a conceptual schema), e.g.: HappyFather Man haschild.female Elephant Animal Large Grey transitive(ancestor) ABox is a set of axioms describing a concrete situation (data), e.g.: John:HappyFather <John,Mary>:hasChild eparation has no logical significance But may be conceptually and implementationally convenient 10 T (Tbox) is a set of axioms of the form: C v D (concept inclusion) C D (concept equivalence) v (role inclusion) (role equivalence) + v (role transitivity) A (Abox) is a set of axioms of the form x D (concept instantiation) hx,yi (role instantiation) To sorts of Tbox axioms often distinguished Definitions C v D or C D here C is a concept name General Concept Inclusion axioms (GCIs) C v D here C is an arbitrary concept 12

4 Knoledge Base emantics yntax für DLs (ohne concrete domains) Hitzler & ure, 2005 An interpretation I satisfies (models) an axiom A (I ² A): I ² C v D iff C I D I I ² C D iff C I = D I I ² v iff I I I ² iff I = I I ² + v iff ( I ) + I I ² x D iff x I D I I ² hx,yi iff (x I,y I ) I I satisfies a Tbox T (I ² T ) iff I satisfies every axiom A in T O Q (N) ALC Concepts Atomic Not And Or Exists For all At least At most Nominal A, B C C u D C t D.C.C n.c ( n ) n.c ( n ) {i 1,,i n } Ontology (=Knoledge Base) Concept Axioms (TBox) ubclass C v D Equivalent C D ole Axioms (Box) H ubrole v Transitivity Trans() Assertional Axioms (ABox) AIFB I satisfies an Abox A (I ² A) iff I satisfies every axiom A in A oles Instance ole C(a) (a,b) I satisfies an KB K (I ² K) iff I satisfies both T and A I Atomic Inverse - ame Different a = b a b = ALC + Transitivity OWL DL = HOIN(D) (D: concrete domain) Folie Inference Tasks Knoledge is correct (captures intuitions) C subsumes D.r.t. K iff for every model I of K, C I D I Knoledge is minimally redundant (no unintended synonyms) C is equivalent to D.r.t. K iff for every model I of K, C I = D I Knoledge is meaningful (classes can have instances) C is satisfiable.r.t. K iff there exists some model I of K s.t. C I Querying knoledge x is an instance of C.r.t. K iff for every model I of K, x I C I hx,yi is an instance of.r.t. K iff for, every model I of K, (x I,y I ) I Knoledge base consistency A KB K is consistent iff there exists some model I of K

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10 Preliminaries: Negation Normal Form 2. Tableau algorithms for ALC and extensions To make our life easier, e transform each concept C 0 into an equivalent C 1 in NNF Equivalent: C 0 C 1 and C 1 C 0 We see a tableau algorithm for ALC andextenditith NNF: negation occurs only in front of concept names 1 general TBoxes and Ho? By pushing negation inards (de Morgan et. al): 2 inverse roles Goal: Design sound and complete desicion procedures for satisfiability (and subsumption) of DLs hich are ell-suited for implementation purposes (C D) C D (C D) C D C C.C. C.C. C From no on: concepts are in NNF and sub(c) denotes the set of all sub-concepts of C A tableau algorithm for the satisfiability of ALC concepts More intuition Goal: design an algorithm hich takes an ALC concept C 0 and 1. returns satisfiable iff C 0 is satisfiable and 2. terminates, on every input, i.e., hich decides satisfiability of ALC concepts. ecall: such an algorithm cannot exist for FOL since satisfiability of FOL is undecidable. Idea: our algorithm is tableau-based and tries to construct a model of C 0 by breaking C 0 don syntactically, thus inferring ne constraints on such a model. Find out hether A.B. B A.B.( B.E) Our tableau algorithm orks on a completion tree hich represents a model I: nodes represent elements of Δ I is satisfiable... each node x is labelled ith concepts L(x) sub(c 0 ) C L(x) is read as x should be an instance of C edges represent role successorship each edge x, y is labelled ith a role-name from C 0 L( x, y ) is read as (x, y) should be in I is initialised ith a single root node x 0 ith L(x 0 )={C 0 } is expanded using completion rules

11 Completion rules for ALC Properties of the completion rules for ALC The -rule is non-deterministic: -rule: if C 1 C 2 L(x) and {C 1,C 2 } L(x) -rule: if C 1 C 2 L(x) and {C 1,C 2 } L(x) then set L(x) =L(x) {C 1,C 2 } then set L(x) =L(x) {C 1,C 2 } -rule: if C 1 C 2 L(x) and {C 1,C 2 } L(x) = -rule: if C 1 C 2 L(x) and {C 1,C 2 } L(x) = then set L(x) =L(x) {C} for some C {C 1,C 2 } then set L(x) =L(x) {C} for some C {C 1,C 2 } -rule: if.c L(x) and x has no -successor y ith C L(y), -rule: if.c L(x) and x has no -successor y ith C L(y), then create a ne node y ith L( x, y ) ={} and L(y) ={C} then create a ne node y ith L( x, y ) ={} and L(y) ={C} -rule: if.c L(x) and there is an -successor y of x ith C / L(y) -rule: if.c L(x) and there is an -successor y of x ith C / L(y) then set L(y) =L(y) {C} then set L(y) =L(y) {C} Properties of the completion rules for ALC Last details on tableau algorithm for ALC We only apply rules if their application does something ne -rule: if C 1 C 2 L(x) and {C 1,C 2 } L(x) then set L(x) =L(x) {C 1,C 2 } -rule: if C 1 C 2 L(x) and {C 1,C 2 } L(x) = then set L(x) =L(x) {C} for some C {C 1,C 2 } -rule: if.c L(x) and x has no -successor y ith C L(y), then create a ne node y ith L( x, y ) ={} and L(y) ={C} -rule: if.c L(x) and there is an -successor y of x ith C / L(y) then set L(y) =L(y) {C} Clash: a c-tree contains a clash if it has a node x ith L(x) or {A, A} L(x) otherise, it is clash-free Complete: a c-tree is complete if none of the completion rules can be applied to it Anser behaviour: hen started for C 0 (in NNF!), the tableau algorithm is initialised ith a single root node x 0 ith L(x 0 )={C 0 } repeatedly applies the completion rules (in hatever order it likes) anser C 0 is satisfiable iff the completion rules can be applied in such a ay that it results in a complete and clash-free c-tree (careful: this is non-deterministic)...go back to examples

12 Properties of our tableau algorithm Extend tableau algorithm to ALC ith general TBoxes: Preliminaries Lemma: Let C 0 an ALC-concept in NNF. Then Corollary: 1. the algorithm terminates hen applied to C 0 and 2. the rules can be applied such that they generate a clash-free and complete completion tree iff C 0 is satisfiable. 1. Our tableau algorithm decides satisfiability and subsumption of ALC. 2. atisfiability (and subsumption) in ALC is decidable in Ppace. 3. ALC has the finite model property i.e., every satisfiable concept has a finite model. We extend our tableau algorithm by adding a ne completion rule: remember that nodes represent elements of Δ I and if C D T, then for each element x in a model I of T if x C I,thenx D I hence x ( C) I or x D I x ( C D) I x (NNF( C D)) I for NNF(E) the negation normal form of E 4. ALC has the tree model property i.e., every satisfiable concept has a tree model. 5. ALC has the finite tree model property i.e., every satisfiable concept has a finite tree model Extend tableau algorithm to ALC ith general TBoxes Completion rules for ALC ith TBoxes ecall: Concept inclusion: of the form C D for C, D (complex) concepts (General) TBox: a finite set of concept inclusions I satisfies C D iff C I D I I is a model of TBox T iff I satisfies each concept equation in T C 0 is satisfiable.r.t. T iff there is a model I of T ith C0 I Goal Lemma: Let C 0 an ALC-concept and T be a an ALC-TBox. Then 1. the algorithm terminates hen applied to T and C 0 and 2. the rules can be applied such that they generate a clash-free and complete completion tree iff C 0 is satisfiable.r.t. T. -rule: if C 1 C 2 L(x) and {C 1,C 2 } L(x) then set L(x) =L(x) {C 1,C 2 } -rule: if C 1 C 2 L(x) and {C 1,C 2 } L(x) = then set L(x) =L(x) {C} for some C {C 1,C 2 } -rule: if.c L(x) and x has no -successor y ith C L(y), then create a ne node y ith L( x, y ) ={} and L(y) ={C} -rule: if.c L(x) and there is an -successor y of x ith C / L(y) then set L(y) =L(y) {C} T -rule: if C 1 C2 T and NNF( C 1 C 2 ) L(x) then set L(x) =L(x) {NNF( C 1 C 2 )}

13 A tableau algorithm for ALC ith general TBoxes A tableau algorithm for ALC ith general TBoxes Example: Consider satisfiability of C.r.t. {C.C} Tableau algorithm no longer terminates! eason: size of concepts no longer decreases along paths in a completion tree -rule: if C 1 C 2 L(x), {C 1,C 2 } L(x), and x is not blocked then set L(x) =L(x) {C 1,C 2 } -rule: if C 1 C 2 L(x), {C 1,C 2 } L(x) =, and x is not blocked then set L(x) =L(x) {C} for some C {C 1,C 2 } Observation: most nodes on this path look the same and e keep repeating ourselves egain termination ith a cycle-detection technique called blocking Intuitively, henever e find a situation here y has to satisfy stronger constraints than x, e freeze x, i.e., block rules from being applied to x y L(x) L(y) x -rule: if.c L(x), x has no -successor y ith C L(y), and x is not blocked then create a ne node y ith L( x, y ) ={} and L(y) ={C} -rule: if.c L(x), thereisan-successor y of x ith C / L(y) and x is not blocked then set L(y) =L(y) {C} T -rule: if C 1 C2 T, NNF( C 1 C 2 ) L(x) and x is not blocked then set L(x) =L(x) {NNF( C 1 C 2 )} A tableau algorithm for ALC ith general TBoxes: Blocking Tableaux ules for ALC x is directly blocked if it has an ancestor y ith L(x) L(y) in this case and if y is the closest such node to x, e say that x is blocked by y anodeisblocked if it is directly blocked or one of its ancestors is blocked restrict the application of all rules to nodes hich are not blocked x {C 1 C 2,...} x {C 1 C 2,...} x {.C,...} x y x {C 1 C 2, C 1,C 2,...} x {C 1 C 2, C,...} for C {C 1,C 2 } {.C,...} {C} completion rules for ALC.r.t. TBoxes x {.C,...} x {.C,...} y {...} y {C,...} easoning ith Expressive Description Logics p. 5/27 39

14 Tableaux ule for Transitive oles x y {.C,...} {...} + x y {.C,...} {.C,...} L() ={.C.( C D).C.(.C)} Where is (i.e., ( I ) + = I ) No longer naturally terminating (e.g., if C =. ) Need blocking imple blocking suffices for ALC plus transitive roles I.e., do not expand node label if ancestor has superset label More expressive logics (e.g., ith inverse roles) need more sophisticated blocking strategies easoning ith Expressive Description Logics p. 6/ L() ={.C.( C D).C.(.C)} 41 43

15 L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} L(x) ={C} x L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} L(x) ={C} x 45 47

16 L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} L(x) ={C, C D} x L(x) ={C, ( C D), C} x L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} L(x) ={C, C D} x L(x) ={C, ( C D), C} x clash 49 51

17 L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} L(x) ={C, C D} x x L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} x x y L(y) ={C} 53 55

18 L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} x y L(y) ={C} x y L(y) ={C,.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} x y L(y) ={C,.C,.(.C)} x y L(y) ={C,.C,.(.C)} z L(z) ={C} 57 59

19 L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} x y L(y) ={C,.C,.(.C)} x y L(y) ={C,.C,.(.C)} z L(z) ={C} blocked z L(z) ={C,.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} L() ={.C,.( C D),.C,.(.C)} x y L(y) ={C,.C,.(.C)} x y L(y) ={C,.C,.(.C)} z L(z) ={C,.C,.(.C)} blocked z L(z) ={C,.C,.(.C)} Concept is satisfiable: T corresponds to model 61 63

20 A tableau algorithm for ALC ith general TBoxes: ummary The tableau algorithm presented here decides satisfiability of ALC-concepts.r.t. TBoxes, and thus also L() ={.C,.( C D),.C,.(.C)} x y L(y) ={C,.C,.(.C)} decides subsumption of ALC-concepts.r.t. TBoxes uses blocking to ensure termination, and is non-deterministic due to the -rule in the orst case, it builds a tree of depth exponential in the size of the input, and thus of double exponential size. Hence it runs in (orst case) 2NExpTime, can be implemented in various ays, order/priorities of rules data structure etc. Concept is satisfiable: T corresponds to model is amenable to optimisations more on this next eek Properties of our tableau algorithm for ALC ith TBoxes Challenges Lemma: Let T be a general ALC-Tbox and C 0 an ALC-concept. Then 1. the algorithm terminates hen applied to T and C 0 and 2. the rules can be applied such that they generate a clash-free and complete completion tree iff C 0 is satisfiable.r.t. T. Corollary: 1. atisfiability of ALC-concept.r.t. TBoxes is decidable 2. ALC ith TBoxes has the finite model property 3. ALC ith TBoxes has the tree model property Increased expressive poer Existing DL systems implement (at most) HIQ OWL extends HIQ ith datatypes and nominals calability Very large KBs easoning ith (very large numbers of) individuals Other reasoning tasks Querying Matching Least common subsumer... Tools and Infrastructure upport for large scale ontological engineering and deployment easoning ith Expressive Description Logics p. 16/27 67

21 ummary Description Logics are family of logical K formalisms Applications of DLs include DataBases and emantic Web Ontologies ill provide vocabulary for semantic markup OWL eb ontology language based on HIQ DL et to become W3C standard (OWL) & already idely adopted Use of DL provides formal foundations and reasoning support DL easoning based on tableau algorithms Highly Optimised implementations used in DL systems Challenges remain easoning ith full OWL language (Convincing) demonstration(s) of scalability Ne reasoning tasks Development of (high quality) tools and infrastructure easoning ith Expressive Description Logics p. 23/27 68 esources lides from this talk FaCT system (open source) OilEd (open source) OIL W3C Web-Ontology (WebOnt) orking group (OWL) DL Handbook, Cambridge University Press easoning ith Expressive Description Logics p. 25/27 69