Unification in the Description Logic EL EL - unification Minimal - PowerPoint PPT Presentation

Unification in EL Baader & Morawska Introduction Unification in the Description Logic EL EL - unification Minimal unifiers Franz Baader and Barbara Morawska Decision Procedure Conclusion TU Dresden, Germany UNIF 2009

Unification in EL Baader & Morawska Introduction EL - unification UNIF 2008 Unification in EL is of type zero. Minimal unifiers Decision UNIF 2009 Unification in EL is decidable and is in NP. Procedure Unification problem in EL is NP-complete. Conclusion

Outline Unification in EL 1 Introduction Baader & Morawska 2 EL -unification Introduction EL - 3 Towards a decision procedure unification Reductions and reduced form Minimal unifiers Subsumption order and its inverse Decision Procedure Minimal Unifiers Conclusion 4 Decision Procedure Computing minimal unifiers Complexity 5 Conclusion

Description Logic EL Unification in EL Baader & Morawska Concept names: City, Introduction Cathedral , EL - unification Top concept: ❏ , Minimal Conjunction: ❬ , unifiers Decision Existential restriction: Procedure ❉ has-location. ❏ Conclusion Example (concept term) City ❬ ❉ location. East-South of Germany ❬ ❉ university. ❏

Description Logic EL Unification in Semantics EL Baader & ♣ Δ , I q is an interpretation, where: Morawska Concepts are sets: if A P N C , A I ❸ Δ ; Introduction Roles are binary relations:if r P N R , r I ❸ Δ ✂ Δ ; EL - unification ❏ is the domain: ❏ I ✏ Δ ; Minimal unifiers Conjunction is intersection: ♣ C ❬ D q I ✏ C I ❳ D I ; Decision Procedure ♣❉ r . C q I ✏ t c P Δ ⑤ ❉ b P Δ . ♣ c , b q P r I and b P C I ✉ Conclusion Subsumption and equivalence Subsumption: C ❸ D iff for all interpretations C I ❸ D I . Equivalence: C ✑ D iff C ❸ D and D ❸ C

Variables in EL Unification in EL Baader & Morawska We define a set of variables N V as a subset of N C . Introduction Idea: concept names in N V may be defined differently by EL - unification different users or developers of a given ontology. Minimal unifiers Decision Procedure Conclusion Concepts from N V can be substituted with concept terms, concepts from N C cannot be substituted.

EL -Unification Unification in EL Baader & Morawska Example: Introduction EL - unification Minimal unifiers Decision Procedure Conclusion City ❬ ❉ location. East-South of Germany ❬ ❉ size. ( more-than-500000 ❬ less-than-1000000) Settlement ❬ ❉ has. Cathedral ❬ ❉ location.Saxony ❬ ❉ size. middle

EL -Unification Unification in EL Baader & Morawska EL -Unification Problem Introduction is a set of equalities, C 1 ✑ ? D 1 , . . . , C n ✑ ? D n , where C i , D i are EL - unification EL -concept terms. Minimal unifiers Decision Procedure Conclusion A substitution σ is an EL -unifier (solution) of an EL -unification problem C 1 ✑ ? D 1 , . . . , C n ✑ ? D n if σ ♣ C 1 q ✑ σ ♣ D 1 q , . . . , σ ♣ C n q ✑ σ ♣ D n q .

SLmO – semilattices with monotone operators Unification in EL Baader & SLmO ✏ t x ❫ ♣ y ❫ z q ✏ ♣ x ❫ y q ❫ z , Morawska x ❫ y ✏ y ❫ z , Introduction x ❫ x ✏ x , EL - x ❫ 1 ✏ x , unification t f i ♣ x ❫ y q ❫ f i ♣ y q ✏ f i ♣ x ❫ y q ⑤ 1 ↕ i ↕ n ✉ Minimal unifiers ✉ Decision Procedure ❬ is associative, commutative and idempotent, Conclusion ❏ is a unit for ❬ ❉ r i . ♣ C ❬ D q ❬ ❉ r i . D ✑ ❉ r i . ♣ C ❬ D q Existential restriction is not a homomorphism: ❉ r . ♣ A ❬ B q ❾ ❉ r . A ❬ ❉ r . B

EL -problem of Type Zero Unification in EL Baader & Morawska What are the unifiers of the following goal: Introduction ❉ R . Y ❸ ? X EL - unification Minimal For example: unifiers Decision r X ÞÑ ❉ R . Z 1 , Y ÞÑ Z 1 s Procedure r X ÞÑ ❉ R . Z 1 ❬ ❉ R . Z 2 , Y ÞÑ Z 1 ❬ Z 2 s Conclusion r X ÞÑ ❉ R . Z 1 ❬ ❉ R . Z 2 ❬ ❉ R . Z 3 , Y ÞÑ Z 1 ❬ Z 2 ❬ Z 3 s . . .

Reductions and reduced forms in EL Unification in EL Baader & Morawska Introduction EL - Reduction rules are applied to concept terms modulo AC unification Minimal C ❬ ❏ ù C unifiers Reductions A ❬ A ù A Subsumption inverse Minimal Unifiers if D ❸ C , then ❉ r . D ❬ ❉ r . C ù ❉ r . D Decision Procedure Conclusion

Equivalence of reduced concepts Unification in EL Baader & Morawska Introduction EL - Theorem (Küsters) unification Minimal C ✏ AC ♣ ♣ C ✑ D iff D unifiers Reductions where C ù ♣ C, D ù ♣ Subsumption inverse D Minimal Unifiers Decision Procedure Conclusion

Inverse of subsumption Unification in EL Baader & Morawska Subsumption order: C 1 → C 2 iff C 1 ⑩ C 2 . Introduction Subsumption order is not well founded. EL - unification Inverse of subsumption order: C 1 → is C 2 iff C 1 ⑨ C 2 . Minimal unifiers Reductions Lemma Subsumption inverse Minimal Unifiers There is no infinite sequence C 0 , C 1 , C 2 , . . . of EL -concept Decision terms such that C 0 ⑨ C 1 ⑨ C 2 ⑨ ☎ ☎ ☎ . Procedure Conclusion

Monotonicity of → is Unification in EL Lemma Baader & Morawska C is a reduced concept term and contains D, D → is D ✶ Introduction EL - unification Then: Minimal C → is C ✶ unifiers Reductions Subsumption inverse where C ✶ is obtained from C by relpalcing an occurrence of D by D ✶ . Minimal Unifiers Decision Procedure Proof Conclusion Induction on size of C . 1 C ✏ D , obvious. 2 C ✏ ❉ R . C 1 and D occurs in C 1 (induction). 3 C ✏ C 1 ❬ ☎ ☎ ☎ ❬ C n and D occurs in C i .

Monotonicity of → is Unification in EL Proof of the case where C ✏ C 1 ❬ ☎ ☎ ☎ ❬ C n Baader & and D occurs in C 1 . Morawska C 1 ❬ ☎ ☎ ☎ ❬ C n ù C ✶ 1 ❬ C 2 ❬ ☎ ☎ ☎ ❬ C n Introduction EL - By induction C 1 → is C ✶ 1 , i.e. C 1 ⑨ C ✶ 1 . unification Minimal and by monotonicity of ❸ : unifiers C 1 ❬ ☎ ☎ ☎ ❬ C n ❸ C ✶ 1 ❬ C 2 ❬ ☎ ☎ ☎ ❬ C n Reductions Subsumption inverse Hence Minimal Unifiers C 1 ❬ ☎ ☎ ☎ ❬ C n ⑧ → is C ✶ 1 ❬ C 2 ❬ ☎ ☎ ☎ ❬ C n Decision Procedure means C 1 ❬ ☎ ☎ ☎ ❬ C n ✑ C ✶ 1 ❬ C 2 ❬ ☎ ☎ ☎ ❬ C n Conclusion C 1 ✙ C ✶ 1 , there is i ✘ 1, such that C 1 ⑨ C ✶ 1 ✑ C i . But this means that C 1 “eats up” C i in C , and thus C is not reduced. Contradiction.

Minimal unifiers Unification in EL → is is well-founded Baader & its multiset extension → m is well-founded. Morawska Introduction S ♣ σ q as a multiset of all σ ♣ X q , X P Var ♣ Γ q . EL - unification Minimal unifiers Definition Reductions Subsumption inverse σ → γ iff S ♣ σ q → m S ♣ γ q . Minimal Unifiers σ, θ are ground, reduced unifiers of Γ . Decision Procedure Conclusion The ground, reduced unifier σ of Γ is minimal iff there is no unifer θ , such that σ → θ . Obviously, a goal is unifiable iff it has a minimal ground reduced unifier.

Atoms and flat goals Unification in EL Baader & A concept term is an atom iff it is a constant or of form ❉ r . C . Morawska Introduction A flat atom is an atom which is a constant or ❉ r . C , where C is EL - constant, variable or ❏ . unification Minimal unifiers A goal Γ is flat iff it only contains the equations of the form: Decision Procedure Algorithm Complexity X ✑ ? C with X a variable and C a non-variable flat atom, Conclusion X 1 ❬ ☎ ☎ ☎ ❬ X m ✑ ? Y 1 ❬ ☎ ☎ ☎ ❬ Y n , where X 1 , . . . , X m , Y 1 , . . . , Y n are variables.

Atoms of a unifier σ Unification in EL Baader & Morawska ↕ At ♣ σ q ✏ At ♣ σ ♣ X qq Introduction X P Var ♣ Γ q EL - unification Minimal Definition unifiers For every concept term C , define At ♣ C q : Decision Procedure if C ✏ ❏ , then At ♣ C q ✏ ❍ , Algorithm Complexity if C is a constant, then At ♣ C q ✏ t C ✉ , Conclusion if C ✏ ❉ r . D , then At ♣ C q ✏ t C ✉ ❨ At ♣ D q , if C ✏ D 1 ❬ D 2 , then At ♣ C q ✏ At ♣ D 1 q ❨ At ♣ D 2 q .

Locality of a minimal ground reduced unifier Unification in EL γ is a minimal reduced ground unifier of Γ Baader & Morawska Lemma Introduction If C is an atom of γ , EL - unification then there is a non-variable atom D in Γ , Minimal such that C ✑ γ ♣ D q unifiers Decision Procedure Proof by contradiction. Algorithm Complexity Idea: If C is maximal w. r. t. ❸ and violates the lemma, we Conclusion construct a smaller unifier γ ✶ – contradiction. C is a constant A . C is of the form ❉ r . C 1 .

Proof of the case where C is of the form ❉ r . C 1 Unification in EL Baader & D 1 , . . . , D n are all atoms in Γ , such that Morawska C ⑨ γ ♣ D 1 q , . . . , C ⑨ γ ♣ D n q . Introduction EL - unification C ⑨ γ ♣ D 1 q ❬ ☎ ☎ ☎ ❬ γ ♣ D n q . Minimal unifiers Obtain γ ✶ by replacing C with reduced form of Decision Procedure γ ♣ D 1 q ❬ ☎ ☎ ☎ ❬ γ ♣ D n q . Algorithm Complexity Conclusion Check if γ ✶ is also a unifier of Γ X ✑ ? E , X 1 ❬ ☎ ☎ ☎ ❬ X m ✑ ? Y 1 ❬ ☎ ☎ ☎ ❬ Y n ,