Computing Local Unifiers in the Description Logic EL ... - TU Dresden

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 Syntax concept name

A ∈ NC

role name

r ∈ NR

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The Description Logic EL Syntax concept name role name

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interpretation I = (·I , ∆I ) AI ⊆ ∆I

A ∈ NC r ∈ NR

rI

⊆ ∆I × ∆I

Local Unifiers in EL Without Top

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The Description Logic EL Syntax concept name

AI ⊆ ∆I

A ∈ NC

role name

r ∈ NR

conjunction

CuD

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interpretation I = (·I , ∆I ) rI

⊆ ∆I × ∆I C I ∩ DI

Local Unifiers in EL Without Top

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The Description Logic EL Syntax concept name

interpretation I = (·I , ∆I ) AI ⊆ ∆I

A ∈ NC rI

⊆ ∆I × ∆I

role name

r ∈ NR

conjunction

CuD

C I ∩ DI

∃r.C

{x | ∃y : (x, y) ∈ r I ∧ y ∈ C I }

existential restriction

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The Description Logic EL Syntax

interpretation I = (·I , ∆I )

concept name

A ∈ NC

AI ⊆ ∆I

role name

r ∈ NR

conjunction

CuD

existential restriction top concept

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∃r.C >

       r I ⊆ ∆I × ∆I   I I C ∩D EL   {x | ∃y : (x, y) ∈ r I ∧ y ∈ C I }       ∆I

Local Unifiers in EL Without Top

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The Description Logic EL Syntax

interpretation I = (·I , ∆I )

concept name

A ∈ NC

AI ⊆ ∆I

role name

r ∈ NR

conjunction

CuD

existential restriction top concept

Wrocław, July 31st, 2011

∃r.C >

       r I ⊆ ∆I × ∆I   I I C ∩D EL   {x | ∃y : (x, y) ∈ r I ∧ y ∈ C I }       ∆I

Local Unifiers in EL Without Top

     

EL−>

    

2

The Description Logic EL Syntax

interpretation I = (·I , ∆I )

concept name

A ∈ NC

AI ⊆ ∆I

role name

r ∈ NR

conjunction

CuD

existential restriction

∃r.C

       r I ⊆ ∆I × ∆I   I I C ∩D EL   {x | ∃y : (x, y) ∈ r I ∧ y ∈ C I }       ∆I

top concept

>

subsumption

CvD

C I ⊆ DI

equivalence

C≡D

C I = DI

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Local Unifiers in EL Without Top

     

EL−>

    

2

The Description Logic EL Syntax

interpretation I = (·I , ∆I )

concept name

A ∈ NC

AI ⊆ ∆I

role name

r ∈ NR

conjunction

CuD

existential restriction

∃r.C

       r I ⊆ ∆I × ∆I   I I C ∩D EL   {x | ∃y : (x, y) ∈ r I ∧ y ∈ C I }       ∆I

top concept

>

subsumption

CvD

C I ⊆ DI

equivalence

C≡D

C I = DI

     

EL−>

    

• 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

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Unification in EL(−>) Some concept names are variables (X ∈ Nv ), all others are constants (A ∈ Nc ). → unification problem: Γ = {C1 ≡? D1 , . . . , Cn ≡? Dn }

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Unification in EL(−>) Some concept names are variables (X ∈ Nv ), all others are constants (A ∈ Nc ). → unification problem: Γ = {C1 ≡? D1 , . . . , Cn ≡? Dn } A unifier σ substitutes variables with concept terms such that σ(C1 ) ≡ σ(D1 ), . . . , σ(Cn ) ≡ σ(Dn ).

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Unification in EL(−>) Some concept names are variables (X ∈ Nv ), all others are constants (A ∈ Nc ). → unification problem: Γ = {C1 ≡? D1 , . . . , Cn ≡? Dn } A unifier σ substitutes variables with concept terms such that σ(C1 ) ≡ σ(D1 ), . . . , σ(Cn ) ≡ σ(Dn ). Unification modulo the equational theory of bounded semilattices with monotone operators: A

→

free constant

X

→

variable

u

→

binary associative, commutative, idempotent operator

∃r.C

→

unary monotone operator

>

→

constant; unit for u

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Previous Results

Unification in EL is NP-complete:

• Matching is NP-hard [Baader, Kusters ¨ 2000]. • Unification is in NP [Baader, Morawska 2009, 2010]. We can restrict the search to local unifiers of polynomial size.

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Previous Results

Unification in EL is NP-complete:

• Matching is NP-hard [Baader, Kusters ¨ 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].

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Previous Results

Unification in EL is NP-complete:

• Matching is NP-hard [Baader, Kusters ¨ 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.

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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 C1 u · · · u Cn ≡? D1 u · · · u Dm for flat atoms C1 , . . . , Cn , D1 , . . . , Dm .

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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 C1 u · · · u Cn ≡? D1 u · · · u Dm for flat atoms C1 , . . . , Cn , D1 , . . . , Dm . Subsumption in EL (and EL−> ):

• The only atom subsumed by a concept name A is A itself.

Wrocław, July 31st, 2011

Local Unifiers in EL Without Top

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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 C1 u · · · u Cn ≡? D1 u · · · u Dm for flat atoms C1 , . . . , Cn , D1 , . . . , Dm . 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 0 with E 0 v E.

Wrocław, July 31st, 2011

Local Unifiers in EL Without Top

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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 C1 u · · · u Cn ≡? D1 u · · · u Dm for flat atoms C1 , . . . , Cn , D1 , . . . , Dm . 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 0 with E 0 v E.

• All concept terms subsumed by a conjunction of atoms D1 u · · · u Dm are conjunctions of atoms C1 u · · · u Cn such that for every Dj there is a Ci with Ci v Dj . Wrocław, July 31st, 2011

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EL vs. EL−> Particle: atom of the form ∃r1 . . . . ∃rn .A If C is an EL−> -concept term and B is a particle, then B v C implies B ≡ C. Part(C): Part(A u ∃r.(A u ∃r.B)) = {A, ∃r.A, ∃r.∃r.B}

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EL vs. EL−> Particle: atom of the form ∃r1 . . . . ∃rn .A If C is an EL−> -concept term and B is a particle, then B v C implies B ≡ C. Part(C): Part(A u ∃r.(A u ∃r.B)) = {A, ∃r.A, ∃r.∃r.B} In EL, it suffices to check for local unifiers σ: σ(X ) = σ(D1 ) u · · · u σ(Dm ), where D1 , . . . , Dm are non-variable atoms of the unification problem.

Wrocław, July 31st, 2011

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EL vs. EL−> Particle: atom of the form ∃r1 . . . . ∃rn .A If C is an EL−> -concept term and B is a particle, then B v C implies B ≡ C. Part(C): Part(A u ∃r.(A u ∃r.B)) = {A, ∃r.A, ∃r.∃r.B} In EL, it suffices to check for local unifiers σ: σ(X ) = σ(D1 ) u · · · u σ(Dm ), where D1 , . . . , Dm are non-variable atoms of the unification problem. Example:

Γ:

X ≡? Y u A,

∃r.X v? Y

local EL-unifier σ1 := {X 7→ A, Y 7→ >}

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EL vs. EL−> Particle: atom of the form ∃r1 . . . . ∃rn .A If C is an EL−> -concept term and B is a particle, then B v C implies B ≡ C. Part(C): Part(A u ∃r.(A u ∃r.B)) = {A, ∃r.A, ∃r.∃r.B} In EL, it suffices to check for local unifiers σ: σ(X ) = σ(D1 ) u · · · u σ(Dm ), where D1 , . . . , Dm are non-variable atoms of the unification problem. Example:

Γ:

X ≡? Y u A,

∃r.X v? Y

local EL-unifier σ1 := {X 7→ A, Y 7→ >} → allow also particles of σ(D) to occur: local EL−> -unifier σ2 := {X 7→ A u ∃r.A, Y 7→ ∃r.A} Wrocław, July 31st, 2011

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Reduction to Linear Language Inclusions NP reduction to a system of linear language inclusions Xi ⊆ L0 ∪ L1 X1 ∪ · · · ∪ Ln Xn (L0 , . . . , Ln are subsets of NR ∪ {ε}) A solution θ maps variables to languages over NR such that θ(Xi ) ⊆ L0 ∪ L1 θ(X1 ) ∪ · · · ∪ Ln θ(Xn ).

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Reduction to Linear Language Inclusions NP reduction to a system of linear language inclusions Xi ⊆ L0 ∪ L1 X1 ∪ · · · ∪ Ln Xn (L0 , . . . , Ln are subsets of NR ∪ {ε}) A solution θ maps variables to languages over NR such that θ(Xi ) ⊆ L0 ∪ L1 θ(X1 ) ∪ · · · ∪ Ln θ(Xn ).

Γ:

X ≡? Y u A,

∃r.X v? Y

↓ YA ⊆ XA ,

Wrocław, July 31st, 2011

XA ⊆ {ε} ∪ YA ,

Local Unifiers in EL Without Top

YA ⊆ {r}XA

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Local Solutions

A solution θ is local if every w ∈ θ(XA ) \ {ε} occurs on the right-hand side of some inclusion: YA ⊆ XA ,

XA ⊆ {ε} ∪ YA ,

YA ⊆ {r}XA ↑

θ(XA ) = {ε, r},

Wrocław, July 31st, 2011

Local Unifiers in EL Without Top

θ(YA ) = {r}

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Local Solutions

A solution θ is local if every w ∈ θ(XA ) \ {ε} occurs on the right-hand side of some inclusion: YA ⊆ XA ,

XA ⊆ {ε} ∪ YA ,

YA ⊆ {r}XA ↑

θ(XA ) = {ε, r},

θ(YA ) = {r}

A solution θ is admissible if for every concept variable X there is a concept constant A such that θ(XA ) is non-empty.

Wrocław, July 31st, 2011

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Local Solutions

A solution θ is local if every w ∈ θ(XA ) \ {ε} occurs on the right-hand side of some inclusion: YA ⊆ XA ,

XA ⊆ {ε} ∪ YA ,

YA ⊆ {r}XA ↑

θ(XA ) = {ε, r},

θ(YA ) = {r}

A solution θ is admissible if for every concept variable X there is a concept constant A such that θ(XA ) 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|.

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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(XA ) is non-empty.

Wrocław, July 31st, 2011

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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(XA ) is non-empty.

We construct an alternating automaton that accepts the maximal solution for XA : s1 : s2 : s3 :

YA ⊆ XA XA ⊆ {ε} ∪ YA YA ⊆ {r}XA

ε s1 ε XA

ε

ε

s2 ε

YA ε

A

s3

r

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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|:

ε s1 ε XA

ε

ε

s2 ε

YA ε

A

s3

r

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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

{XA , s2 , A}

r

{XA , s2 , YA , s1 , s3 } r

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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 {XA , s2 , A}

r

{XA , s2 , YA , s1 , s3 } r

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Summary

If Γ has an EL−> -unifier, we can always construct a local EL−> -unifier of size exponential in |Γ|.

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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|.

Wrocław, July 31st, 2011

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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?

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Thank You Franz Baader and Ralf Kusters. ¨ 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

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