UNIFICATION IN EL WITHOUT TOP CONSTRUCTOR Franz Baader, Nguyen T. - PowerPoint PPT Presentation

Institute for Theoretical Computer Science Chair for Automata Theory UNIFICATION IN EL WITHOUT TOP CONSTRUCTOR Franz Baader, Nguyen T. Binh, Stefan Borgwardt, Barbara Morawska WARU Utrecht, 26.05.2011

Outline Why unification in EL −⊤ is interesting Description Logics EL and EL −⊤ Unification in EL and EL −⊤ Unification in EL −⊤ is in PS PACE Unification in EL −⊤ is PS PACE -hard Conclusion Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 2 of 26

Why unification in EL −⊤ is interesting Practical reasons • EL is interesting, e.g. for SNOMED developers • unification should help with detecting redundancies in ontologies • SNOMED does not use ⊤ in concept definitions • unification in EL −⊤ is more appropriate for SNOMED Theoretical reasons • What is the complexity of EL -unification without ⊤ ? – unification in semigroups with monotone operators – solving linear language inclusions – finite satisfiability of anti-Horn clauses with monotone functions • Can we define local solutions for unification in EL −⊤ ? Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 3 of 26

EL −⊤ EL Description logics: concept names N c � � Description Logics EL and EL −⊤ role names N r Syntax of concept terms � � top concept ⊤ � conjunction ⊓ � � existential restriction ∃ r.C � � Example Great-grandfather Old ⊓ Happy ⊓ Man ⊓ ∃ has child. ( ∃ has child. ( ∃ has child. ⊤ )) Old ⊓ Happy ⊓ Man ⊓ ∃ has child. ( Clever ⊓ ∃ has child. ( Honest ⊓ ∃ has child.Handsome )) Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 4 of 26

Description Logics EL and EL −⊤ Semantics of concept terms and subsumption Syntax Interpretation I A I ⊆ ∆ I concept name A ∈ N c r I ⊆ ∆ I × ∆ I role name r ∈ N r whole domain ∆ I top concept ⊤ C I ∩ D I conjunction C ⊓ D existential restriction { a ∈ ∆ I | there is b ∈ ∆ I , ( a, b ) ∈ r I and b ∈ C I } ∃ r.C C I ⊆ D I subsumption C ⊑ D C I = D I equivalence C ≡ D Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 5 of 26

Description Logics EL and EL −⊤ Atoms concept names, existential restrictions Every concept C is a conjunction of atoms. Lemma C 1 ⊓ · · · ⊓ C n ⊑ D and D is an atom, iff there is C i , such that C i ⊑ D. Example Let C = A ⊓ ∃ r. ( A ⊓ B ) ⊓ ∃ r.B , At ( C ) = { A, B, ∃ r. ( A ⊓ B ) , ∃ r.B } Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 6 of 26

Unification in EL and EL −⊤ We partition concept names into: • concept variables X 1 , X 2 , . . . (can be substituted by concept terms) • concept constants A 1 , A 2 , . . . . A concept term is ground if it has no variables. Unification problem in EL −⊤ ( and in EL ) C 1 ≡ ? D 1 , . . . , C n ≡ ? D n , where C i , D i are concept terms in EL −⊤ (or in EL ). (Decision problem) Does a substitution σ , s.t. σ ( C 1 ) ≡ σ ( D 1 ) , . . . , σ ( C n ) ≡ σ ( D n ) , exist? Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 7 of 26

Unification in EL and EL −⊤ We partition concept names into: • concept variables X 1 , X 2 , . . . (can be substituted by concept terms) • concept constants A 1 , A 2 , . . . . A concept term is ground if it has no variables. Unification problem in EL −⊤ ( and in EL ) C 1 ⊑ ? D 1 , . . . , C n ⊑ ? D n , where C i , D i are concept terms in EL −⊤ ( or in EL ). (Decision problem) Does a substitution σ , s.t. σ ( C 1 ) ⊑ σ ( D 1 ) , . . . , σ ( C n ) ⊑ σ ( D n ) , exist? Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 8 of 26

Unification in EL and EL −⊤ Particles Particles • concept names, • existential restrictions: ∃ r 1 . ∃ r 2 , . . . ∃ r n .A where A is a concept name. Particles of a concept C • If C is a concept name, then Part ( C ) := { C } . • If C = ∃ r.D , then Part ( C ) := {∃ r.M | M ∈ Part ( D ) } . • If C = C 1 ⊓ C 2 , then Part ( C ) := Part ( C 1 ) ∪ Part ( C 2 ) . Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 9 of 26

Unification in EL and EL −⊤ Particles Example Let C = ∃ r. ( A ⊓ ∃ s. ( A ⊓ B ) ⊓ B ) , Part ( C ) = {∃ r.A, ∃ r.B, ∃ r. ∃ s.A, ∃ r. ∃ s.B } • r ⊓ ǫ ǫ s ⊓ A B ǫ ǫ A B Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 10 of 26

Unification in EL and EL −⊤ Particles Example Let C = ∃ r. ( A ⊓ ∃ s. ( A ⊓ B ) ⊓ B ) , Part ( C ) = {∃ r.A, ∃ r.B, ∃ r. ∃ s.A, ∃ r. ∃ s.B } Lemma Let C be an EL −⊤ -concept term and B a particle, then • If B ⊑ C, then B ≡ C. • B ∈ Part ( C ) iff C ⊑ B. Example Let A, B be constants, then the unification problem { A ⊑ ? X, B ⊑ ? X } has a solution in EL , not in EL −⊤ . Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 11 of 26

Unification in EL and EL −⊤ Example The following problem {∃ r. ( A ⊓ B ) ⊑ ? X, ∃ r. ( B ⊓ C ) ⊑ ? X } has a unifier in EL −⊤ : X �→ ∃ r.B A unifier in EL : X �→ ⊤ Notice: • Every unification problem in EL −⊤ is also a unification problem in EL . • Every solution in EL −⊤ is also a solution in EL . • If a unifier σ is a EL −⊤ unifier, then Part ( σ ( X )) � = ∅ . Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 12 of 26

Unification in EL and EL −⊤ From EL −⊤ to EL and back to EL −⊤ If γ is a unifier in EL −⊤ : • γ 1 is a unifier in EL −⊤ : S X � �� � � γ 1 ( X ) = γ 1 ( D ) ⊓ A 1 ⊓ · · · ⊓ A m D ∈ Γ where γ ( X ) ⊑ γ ( D ) • γ 2 is a local unifier in EL : S X � �� � � γ 2 ( X ) = γ 2 ( D ) D ∈ Γ • γ 3 is a unifier in EL −⊤ : S X � �� � � γ 3 ( X ) = γ 3 ( D ) ⊓ P 1 ⊓ · · · ⊓ P k D ∈ Γ P i is a particle and there is A j , such that A j ⊑ P i . Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 13 of 26

Unification in EL −⊤ is in PS PACE Algorithm 1. Guess subsumptions between atoms of the goal, such that a unifier in EL can be defined. Step 1 Guess additional subsumptions. Step 2 Construct linear language inclusions. 2. Check if the goal subsumptions and the additional subsumptions allow for any particles for each variable X . Step 3 Construct an alternating automaton. Step 4 Test the emptiness of this automaton. Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 14 of 26

Unification in EL −⊤ is in PS PACE Example 1. ∃ s.Z ⊑ ? X , 2. ∃ r. ( A ⊓ B ) ⊑ ? Y , 3. ∃ r. ( B ⊓ C ) ⊑ ? Y 4. ∃ s.Y ⊑ ? V 5. ∃ r.Z ⊓ Y ⊑ ? ∃ r.C Additional subsumptions (some of them) 6. X ⊑ ? ∃ s.Y V ⊑ ? ∃ s.Y , (guess) , . ∃ s.Z ⊑ ? ∃ s.Y , (from 1 and 6) 7 8. Z ⊑ ? Y , (from 7) 9. ∃ r.Z ⊑ ? ∃ r.C (guess from 5) 10. Z ⊑ ? C (from 9) S X = {∃ s.Y } , S V = {∃ s.Y } , S Z = { C } , S Y = ∅ Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 15 of 26

Unification in EL −⊤ is in PS PACE Subsumption mapping τ : At (Γ) 2 → { 0 , 1 } 1. Properties of subsumptions – τ ( D, D ) = 1 – τ ( A, B ) = 0, for A, B different constants – τ ( ∃ r.C 1 , ∃ s.C 2 ) = 0, for r, s different role names – τ ( A, ∃ r.C ) = τ ( ∃ r.C, A ) , for a constant A – τ ( ∃ r.C 1 , ∃ r.C 2 ) = τ ( C 1 , C 2 ) – if τ ( C 1 , C 2 ) = τ ( C 2 , C 3 ) = 1, then τ ( C 1 , C 3 ) = 1 2. Acyclicity: the assignment S τ is acyclic 3. Unification of the goal: for each goal subsumption C 1 ⊓ · · · ⊓ C n ⊑ D – if D is a non-variable atom, then there is C i with τ ( C i , D ) = 1, – if D is a variable, then τ ( D, C ) = 1, where C is a non-variable atom, implies τ ( C i , C ) = 1 for at least one C i . Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 16 of 26

Unification in EL −⊤ is in PS PACE Our goal: Check if there is a unifier in EL −⊤ where all subsumptions guessed in τ hold, i.e. if the subsumptions allow for any particles for all variables in the goal. What subsumptions? ∆ Γ := { C 1 ⊓ · · · ⊓ C n ⊑ ? X ∈ Γ | X is a variable in Γ } ∆ τ := { C ⊑ ? X | X is a variable and τ ( C, X ) = 1 } ∆ Γ , τ := ∆ Γ ∪ ∆ τ A particle P for X C 1 ⊓ · · · ⊓ C n ⊑ ? X ⊑ ? P for each C 1 ⊓ · · · ⊓ C n ⊑ ? X in ∆ Γ , τ Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 17 of 26

Unification in EL −⊤ is in PS PACE Algorithm: Step 2. Linear language inclusions A particle P for X C 1 ⊓ · · · ⊓ C n ⊑ ? X ⊑ ? P for each C 1 ⊓ · · · ⊓ C n ⊑ ? X in ∆ Γ , τ Particles are words over N r A particle P is of the form ∃ r 1 ∃ r 2 . . . ∃ r n .A ∃ r 1 r 2 . . . r n .A ∃ ω .A , for a constant A . Example constant A constant B Y ⊓ ∃ r.A ⊑ X X A ⊆ { ǫ } Y A ∪ { r } X B ⊆ { ǫ } Y B ∪ { r }∅ ∃ s.B ⊓ X ⊑ Y Y A ⊆ { s }∅ ∪ { ǫ } X A Y B ⊆ { s } ∪ { ǫ } X B ∃ s.A ⊓ B ⊑ Y Y A ⊆ { s } ∪ { ǫ }∅ Y B ⊆ { s }∅ ∪ { ǫ } Unification in EL without Top Constructor WARU Utrecht, 26.05.2011 Slide 18 of 26