Locality and Subsumption Testing in EL and some of its extensions Viorica Sofronie-Stokkermans Max-Planck-Institut f¨ ur Informatik Saarbr¨ ucken AiML 2008, September 9–12, 2008, Nancy 1
Motivation Description logics – used for KR in databases/ontologies Provide a logical basis for modeling and reasoning about: • objects • classes of objects (concepts) • relationshops between objects (links, roles) Wide variety of description logics (various degrees of expressivity) This talk: Tractable description logic: EL , EL + and extensions [Baader’03–] used e.g. in medical ontologies (SNOMED) 2
Motivation Description logics – used for KR in databases/ontologies Focus: Tractable description logic: EL , EL + and extensions [Baader’03–] used e.g. in medical ontologies (SNOMED) Main contributions of this paper: • Alternative proof of tractability of EL , EL + based on a notion of locality �→ a hierarchical reduction to SAT in the case of EL �→ reduction to SAT of ground Horn formulae in the case of EL + • Identify tractable extensions of EL and EL + �→ with n -ary roles and/or numerical domains. 3
Overview • Introduce EL and EL + and the deduction problems (subsumption w.r.t. a TBox resp. CBox) • Algebraic semantics TBox/CBox subsumption �→ u.w.p. for SLat ∪ Mon(Σ)( ∪ Ax) • Local theories and local theory extensions • Locality results for SLat ∪ Mon(Σ)( ∪ Ax) • Tractable extensions of EL , EL + 4
Overview • Introduce EL and EL + and the deduction problems (subsumption w.r.t. a TBox resp. CBox) • Algebraic semantics TBox/CBox subsumption �→ u.w.p. for SLat ∪ Mon(Σ)( ∪ Ax) • Local theories and local theory extensions • Locality results for SLat ∪ Mon(Σ)( ∪ Ax) • Tractable extensions of EL , EL + 5
EL : Generalities Concepts: • primitive concepts N C E • complex concepts (built using concept constructors ⊓ , r ) Roles: N R • C ∈ N C �→ C I ⊆ D I Interpretations: I = ( D I , · I ) • r ∈ N R �→ r I ⊆ D I × D I Constructor name Syntax Semantics C I 1 ∩ C I conjunction C 1 ⊓ C 2 2 y (( x , y ) ∈ r I and y ∈ C I ) } E E existential restriction r . C { x | 6
EL : Generalities • primitive concepts N C Concepts: E • complex concepts (built using concept constructors ⊓ , r ) Roles: N R • C ∈ N C �→ C I ⊆ D I Interpretations: I = ( D I , · I ) • r ∈ N R �→ r I ⊆ D I × D I Problem: Given: TBox (set T of concept inclusions C i ⊑ D i ) and concepts C , D Task: test whether C ⊑ T D , then C I ⊆ D I C I ⊆ D I i.e. whether for all I if i i A C i ⊑ D i ∈ T Decidable in PTIME [Baader02] (graphs/simulation, transl. to Datalog, ...) 7
EL : Example Primitive concepts: protein, process, substance Roles: catalyzes, produces E Terminology: enzyme = protein ⊓ catalyzes.reaction E (TBox) catalyzer = catalyzes.process E reaction = process ⊓ produces.substance Query: enzyme ⊑ catalyzer? 8
Algebraic semantics for EL Translation of concept descriptions to terms: �→ C for any concept name C C C 1 ⊓ C 2 �→ C 1 ⊓ C 2 = C 1 ∧ C 2 E E r . C �→ r . C = f r ( C ) Algebraic semantics for EL . Assume that N C = { c 1 , . . . , c n } . The following are equivalent: (1) C ⊑ T D A ^ (2) BAO ∪ Jh( { f r | r ∈ N R } ) | = c 1 , . . . , c n (( C i ≤ D i ) → C ≤ D ) C i ⊑ D i ∈T A ^ (3) SLat ∪ Mon( { f r | r ∈ N R } ) | = c 1 , . . . , c n (( C i ≤ D i ) → C ≤ D ) C i ⊑ D i ∈T 9
Algebraic semantics for EL Translation of concept descriptions to terms: �→ C for any concept name C C C 1 ⊓ C 2 �→ C 1 ⊓ C 2 = C 1 ∧ C 2 E E r . C �→ r . C = f r ( C ) Algebraic semantics for EL . Assume that N C = { c 1 , . . . , c n } . The following are equivalent: (1) C ⊑ T D A ^ (2) BAO ∪ Jh( { f r | r ∈ N R } ) | = c 1 , . . . , c n (( C i ≤ D i ) → C ≤ D ) C i ⊑ D i ∈T A ^ (3) SLat ∪ Mon( { f r | r ∈ N R } ) | = c 1 , . . . , c n (( C i ≤ D i ) → C ≤ D ) C i ⊑ D i ∈T 10
EL : Algebraic semantics Primitive concepts: protein, process, substance Roles: catalyzes, produces E Terminology: enzyme = protein ⊓ catalyzes.reaction E (TBox) catalyzer = catalyzes.process E reaction = process ⊓ produces.substance Query: enzyme ⊑ catalyzer? SLat ∪ Mon | =enzyme = protein ⊓ catalyzes-some(reaction) ∧ catalyzer = catalyze-some(process) ∧ reaction = process ⊓ produces-some(substance) ⇒ enzyme ⊑ catalyzer A Mon : C , D ( C ⊑ D → catalyze-some( C ) ⊑ catalyze-some( D )) A C , D ( C ⊑ D → produces-some( C ) ⊑ produces-some( D )) 11
EL + : generalities • primitive concepts N C Concepts: E • complex concepts (built using concept constructors ⊓ , r ) Roles: N R • C ∈ N C �→ C I ⊆ D I Interpretations: I = ( D I , · I ) • r ∈ N R �→ r I ⊆ D I × D I Problem: Given: CBox C = ( T , RI ), where T set of concept inclusions C i ⊑ D i ; RI set of role inclusions r ◦ s ⊑ t or r ⊑ t concepts C , D Task: test whether C ⊑ T D , and r I ◦ s I ⊆ t I then C I ⊆ D I C I ⊆ D I i.e. whether for all I if i i A A C i ⊑ D i ∈ T r ◦ s ⊑ t ∈ RI 12
EL + : Example Primitive concepts: protein, process, substance Roles: catalyzes, produces, helps-producing E Terminology: enzyme = protein ⊓ catalyzes.reaction E (TBox) reaction = process ⊓ produces.substance Role inclusions: catalyzes ◦ produces ⊑ helps-producing E Query: enzyme ⊑ protein ⊓ helps-producing.substance ? 13
Algebraic semantics for EL + Translation of concept descriptions to terms: �→ C for any concept name C C C 1 ⊓ C 2 �→ C 1 ⊓ C 2 = C 1 ∧ C 2 E E r . C �→ r . C = f r ( C ) A �→ Ax ( RI ) = { x f r ( f s ( x )) ≤ f t ( x ) | r ◦ s ⊆ t ∈ RI } RI Algebraic semantics for EL + Assume that N C = { c 1 , . . . , c n } . T.f.a.e. for any EL + C Box C = ( T , RI ) (1) C ⊑ C D A ^ (2) SLat ∪ Mon( { f r | r ∈ N R } ) ∪ Ax ( RI ) | = c 1 , . . . , c n (( C i ≤ D i ) → C ≤ D ) C i ⊑ D i ∈T 14
EL + : Algebraic semantics Primitive concepts: protein, process, substance Roles: catalyzes, produces, helps-producing E Terminology: enzyme = protein ⊓ catalyzes.reaction E (TBox) reaction = process ⊓ produces.substance Role inclusions: catalyzes ◦ produces ⊑ helps-producing E Query: enzyme ⊑ protein ⊓ helps-producing.substance ? SLat ∪ Mon ∪ Ax ( RI ) | =enzyme = protein ⊓ catalyzes-some(reaction) ∧ reaction = process ⊓ produces-some(substance) ⇒ enzyme ⊑ protein ⊓ helps-producing-some(substance) A Mon : C , D ( C ⊑ D → catalyze-some( C ) ⊑ catalyze-some( D )) A C , D ( C ⊑ D → produces-some( C ) ⊑ produces-some( D )) A Ax ( RI ) : x (catalyzes-some(produces-some( x )) ≤ helps-producing( x )) 15
Efficient reasoning in the algebraic models • TBox subsumption in EL �→ uniform word problem w.r.t. SLat ∪ Mon(Σ) • CBox subsumption in EL + �→ uniform word problem SLat ∪ Mon(Σ) ∪ Ax , where Ax consists of axioms of the type: A A x ( f ( x ) ≤ g ( x )) and x ( f ( g ( x )) ≤ h ( x )) The uniform word problem for SLat ∪ Mon(Σ)( ∪ Ax ) : decidable in PTIME Explanation: Local theories / local theory extensions 16
Overview • Introduce EL and EL + and the deduction problems (subsumption w.r.t. a TBox resp. CBox) • Algebraic semantics TBox/CBox subsumption �→ u.w.p. for SLat ∪ Mon(Σ)( ∪ Ax) • Local theories and local theory extensions • Locality results for SLat ∪ Mon(Σ)( ∪ Ax) • Tractable extensions of EL , EL + 17
Local theories Local theories [McAllester & Givan’92; Ganzinger’01] K set of equational Horn clauses K is local, if for ground clauses G , K ∪ G | = ⊥ iff K [ G ] ∪ G | = ⊥ [McAllester, Givan ’92, ’93] Local theories capture PTIME 18
� � Local theories Local theories [McAllester & Givan’92; Ganzinger’01] K set of equational Horn clauses K is local, if for ground clauses G , K ∪ G | = ⊥ iff K [ G ] ∪ G | = ⊥ [Ganzinger’01] � Emb( K ) K local theory [Skolem’20] � � � ���������� � � [McAllester et al.’92,’93] � � [Evans’53,Burris’95] � � � � � Horn theory of K in PTIME 19
Local theories Local theories [McAllester & Givan’92; Ganzinger’01] K set of equational Horn clauses K is local, if for ground clauses G , K ∪ G | = ⊥ iff K [ G ] ∪ G | = ⊥ Example: • Axiomatization of lattices [Skolem 1920] Locality: every poset embeds into a lattice (Dedekind-MacNeille completion) • Similar results for axiomatizations of semilattices • Several other examples (algebra [Burris95] ; verification: theories of lists ...) 20
Local theories Local theories [McAllester & Givan’92; Ganzinger’01] K set of equational Horn clauses Compl: Pol. in size of G K is local, if for ground clauses G , G : f ( c )= f ( d ) Ex: K ∪ G | = ⊥ iff K [ G ] ∪ G | = ⊥ Mon f [ G ] : c ≤ d → f ( c ) ≤ f ( d ) Ψ closure operation on ground terms. Compl: Pol. in size of Ψ( G ) K is Ψ-local, if for ground clauses G , K ∪ G | = ⊥ iff K [Ψ( G )] ∪ G | = ⊥ Compl: Pol. in size of G T 0 ⊆ K is stably local, if for ground clauses G , G : f ( c )= f ( d ) Ex: = ⊥ iff T 0 ∪ K [ G ] ∪ G | T 0 ∪ K ∪ G | = ⊥ Mon [ G ] :c ≤ d → f ( c ) ≤ f ( d ) f f ( c ) ≤ d → f ( f ( c )) ≤ f ( d ) 21
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