COMPUTING LOCAL UNIFIERS IN THE DESCRIPTION LOGIC EL WITHOUT THE TOP CONCEPT

Institute of Theoretical Computer Science Chair of Automata Theory COMPUTING LOCAL UNIFIERS IN THE DESCRIPTION LOGIC EL WITHOUT THE TOP CONCEPT Franz Baader Nguyen Thanh Binh Stefan Borgwardt Barbara Morawska Wrocław, July 31st, 2011

The Description Logic EL concept name role name Syntax A N C r N R Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 2

The Description Logic EL Syntax interpretation I = ( I, I ) concept name A N C A I I role name r N R r I I I Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 2

The Description Logic EL Syntax interpretation I = ( I, I ) concept name A N C A I I role name r N R r I I I conjunction C D C I D I Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 2

The Description Logic EL Syntax interpretation I = ( I, I ) concept name A N C A I I role name r N R r I I I conjunction C D C I D I existential restriction r.c {x y : (x, y) r I y C I } Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 2

The Description Logic EL Syntax interpretation I = ( I, I ) concept name A N C A I I role name r N R r I I I conjunction C D C I D I EL existential restriction r.c {x y : (x, y) r I y C I } top concept I EL subsumption C D C I D I equivalence C D C I = D I Description logics are used to formulate ontologies SNOMED CT is based on EL, but does not use Unification can be used to detect redundancies Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 2

Unification in EL ( ) Some concept names are variables (X N v ), all others are constants (A N c). unification problem: Γ = {C 1? D 1,..., C n? D n} Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 3

Unification in EL ( ) Some concept names are variables (X N v ), all others are constants (A N c). unification problem: Γ = {C 1? D 1,..., C n? D n} A unifier σ substitutes variables with concept terms such that σ(c 1 ) σ(d 1 ),..., σ(c n) σ(d n). Unification modulo the equational theory of bounded semilattices with monotone operators: A free constant X variable binary associative, commutative, idempotent operator r.c unary monotone operator constant; unit for Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 3

Previous Results Unification in EL is NP-complete: Matching is NP-hard [Baader, Küsters 2000]. Unification is in NP [Baader, Morawska 2009, 2010]. We can restrict the search to local unifiers of polynomial size. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 4

Previous Results Unification in EL is NP-complete: Matching is NP-hard [Baader, Küsters 2000]. Unification is in NP [Baader, Morawska 2009, 2010]. We can restrict the search to local unifiers of polynomial size. Unification in EL is PSPACE-complete [CADE 2011]. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 4

Previous Results Unification in EL is NP-complete: Matching is NP-hard [Baader, Küsters 2000]. Unification is in NP [Baader, Morawska 2009, 2010]. We can restrict the search to local unifiers of polynomial size. Unification in EL is PSPACE-complete [CADE 2011]. In this talk: Local unifiers in EL may be of exponential size. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 4

Preliminaries Atom: concept name or existential restriction Non-variable atom: concept constant or existential restriction Flat atom: atom of depth 1 Flat unification problem: All equations are of the form C 1 C n? D 1 D m for flat atoms C 1,..., C n, D 1,..., D m. Subsumption in EL (and EL ): The only atom subsumed by a concept name A is A itself. All atoms subsumed by an existential restriction r.e are of the form r.e with E E. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 5

Preliminaries Atom: concept name or existential restriction Non-variable atom: concept constant or existential restriction Flat atom: atom of depth 1 Flat unification problem: All equations are of the form C 1 C n? D 1 D m for flat atoms C 1,..., C n, D 1,..., D m. Subsumption in EL (and EL ): The only atom subsumed by a concept name A is A itself. All atoms subsumed by an existential restriction r.e are of the form r.e with E E. All concept terms subsumed by a conjunction of atoms D 1 D m are conjunctions of atoms C 1 C n such that for every D j there is a C i with C i D j. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 5

EL vs. EL Particle: atom of the form r 1.... r n.a If C is an EL -concept term and B is a particle, then B C implies B C. Part(C): Part(A r.(a r.b)) = {A, r.a, r. r.b} Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 6

EL vs. EL Particle: atom of the form r 1.... r n.a If C is an EL -concept term and B is a particle, then B C implies B C. Part(C): Part(A r.(a r.b)) = {A, r.a, r. r.b} In EL, it suffices to check for local unifiers σ: σ(x ) = σ(d 1 ) σ(d m), where D 1,..., D m are non-variable atoms of the unification problem. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 6

EL vs. EL Particle: atom of the form r 1.... r n.a If C is an EL -concept term and B is a particle, then B C implies B C. Part(C): Part(A r.(a r.b)) = {A, r.a, r. r.b} In EL, it suffices to check for local unifiers σ: σ(x ) = σ(d 1 ) σ(d m), where D 1,..., D m are non-variable atoms of the unification problem. Example: Γ : X? Y A, r.x? Y local EL-unifier σ 1 := {X A, Y } Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 6

EL vs. EL Particle: atom of the form r 1.... r n.a If C is an EL -concept term and B is a particle, then B C implies B C. Part(C): Part(A r.(a r.b)) = {A, r.a, r. r.b} In EL, it suffices to check for local unifiers σ: σ(x ) = σ(d 1 ) σ(d m), where D 1,..., D m are non-variable atoms of the unification problem. Example: Γ : X? Y A, r.x? Y local EL-unifier σ 1 := {X A, Y } allow also particles of σ(d) to occur: local EL -unifier σ 2 := {X A r.a, Y r.a} Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 6

Reduction to Linear Language Inclusions NP reduction to a system of linear language inclusions X i L 0 L 1 X 1 L nx n (L 0,..., L n are subsets of N R {ε}) A solution θ maps variables to languages over N R such that θ(x i ) L 0 L 1 θ(x 1 ) L nθ(x n). Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 7

Reduction to Linear Language Inclusions NP reduction to a system of linear language inclusions X i L 0 L 1 X 1 L nx n (L 0,..., L n are subsets of N R {ε}) A solution θ maps variables to languages over N R such that θ(x i ) L 0 L 1 θ(x 1 ) L nθ(x n). Γ : X? Y A, r.x? Y Y A X A, X A {ε} Y A, Y A {r}x A Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 7

Local Solutions A solution θ is local if every w θ(x A ) \ {ε} occurs on the right-hand side of some inclusion: Y A X A, X A {ε} Y A, Y A {r}x A θ(x A ) = {ε, r}, θ(y A ) = {r} Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 8

Local Solutions A solution θ is local if every w θ(x A ) \ {ε} occurs on the right-hand side of some inclusion: Y A X A, X A {ε} Y A, Y A {r}x A θ(x A ) = {ε, r}, θ(y A ) = {r} A solution θ is admissible if for every concept variable X there is a concept constant A such that θ(x A ) is non-empty. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 8

Local Solutions A solution θ is local if every w θ(x A ) \ {ε} occurs on the right-hand side of some inclusion: Y A X A, X A {ε} Y A, Y A {r}x A θ(x A ) = {ε, r}, θ(y A ) = {r} A solution θ is admissible if for every concept variable X there is a concept constant A such that θ(x A ) is non-empty. From any finite, local, admissible solution θ we can construct a local EL -unifier of size exponential in Γ and polynomial in θ. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 8

Automata Construction Finite, local solutions are closed under union. Check for all X whether there is A and a finite, local solution θ such that θ(x A ) is non-empty. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 9

Automata Construction Finite, local solutions are closed under union. Check for all X whether there is A and a finite, local solution θ such that θ(x A ) is non-empty. We construct an alternating automaton that accepts the maximal solution for X A : s 1 : s 2 : s 3 : Y A X A X A {ε} Y A Y A {r}x A ε ε s 1 ε X A s 2 ε ε Y A ε A s 3 r Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 9

The Size of Local EL -Unifiers Emptiness of this automaton can be checked in PSPACE [Jiang, Ravikumar 1991]. If it is not empty, we can even construct a finite, local solution θ of size at most exponential in Γ : Construct an equivalent nondeterministic automaton using a powerset construction {X A, s 2, A} {X A, s 2, Y A, s 1, s 3 } r r Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 10

The Size of Local EL -Unifiers Emptiness of this automaton can be checked in PSPACE [Jiang, Ravikumar 1991]. If it is not empty, we can even construct a finite, local solution θ of size at most exponential in Γ : Construct an equivalent nondeterministic automaton using a powerset construction Find a shortest accepting path (of possibly exponential length) Extract a local solution (of exponential size) from this path {X A, s 2, A} {X A, s 2, Y A, s 1, s 3 } r r Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 10

Summary If Γ has an EL -unifier, we can always construct a local EL -unifier of size exponential in Γ. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 11

Summary If Γ has an EL -unifier, we can always construct a local EL -unifier of size exponential in Γ. On the other hand, the size of all local EL -unifiers of Γ may grow exponentially in Γ. Future Work EL ( ) with general concept inclusion axioms? Other concept constructors? Implementation of a practical algorithm? Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 11

Thank You Franz Baader and Ralf Küsters. Matching concept descriptions with existential restrictions. In Proc. KR2000, pages 261 272. Morgan Kaufmann, 2000. Franz Baader and Barbara Morawska. SAT encoding of unification in EL. In Proc. LPAR 10, volume 6397 of LNCS, pages 97 111. Springer, 2010. Franz Baader, Nguyen Thanh Binh, Stefan Borgwardt, and Barbara Morawska. Unification in the description logic EL without the top concept. LTCS-Report 11-01, TU Dresden, 2011. See http://lat.inf.tu-dresden.de/research/reports.html. Tao Jiang and Bala Ravikumar. A note on the space complexity of some decision problems for finite automata. Inform. Process. Lett., 40:25 31, 1991. Frank Wolter and Michael Zakharyaschev. Undecidability of the unification and admissibility problems for modal and description logics. ACM Trans. Comput. Log., 9(4), 2008. Wrocław, July 31st, 2011 Local Unifiers in EL Without Top 12