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

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

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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

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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

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Outline

Why unification in EL−⊤ is interesting Description Logics EL and EL−⊤ Unification in EL and EL−⊤ Unification in EL−⊤ is in PSPACE Unification in EL−⊤ is PSPACE-hard Conclusion

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 2 of 26

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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−⊤?

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 3 of 26

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Description Logics EL and EL−⊤

Syntax of concept terms

Description logics: EL EL−⊤ concept names Nc

  • role names Nr
  • 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))

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Description Logics EL and EL−⊤

Semantics of concept terms and subsumption

Syntax Interpretation I concept name A ∈ Nc AI ⊆ ∆I role name r ∈ Nr rI ⊆ ∆I × ∆I top concept ⊤ whole domain ∆I conjunction C ⊓ D CI ∩ DI existential restriction ∃r.C {a ∈ ∆I | there is b ∈ ∆I, (a, b) ∈ rI and b ∈ CI} subsumption C ⊑ D CI ⊆ DI equivalence C ≡ D CI = DI

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Description Logics EL and EL−⊤

Atoms

concept names, existential restrictions Every concept C is a conjunction of atoms.

Lemma

C1 ⊓ · · · ⊓ Cn ⊑ D and D is an atom, iff there is Ci, such that Ci ⊑ D.

Example

Let C = A ⊓ ∃r.(A ⊓ B) ⊓ ∃r.B, At(C) = {A, B, ∃r.(A ⊓ B), ∃r.B}

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Unification in EL and EL−⊤

We partition concept names into:

  • concept variables X1, X2, . . . (can be substituted by concept terms)
  • concept constants A1, A2, . . . .

A concept term is ground if it has no variables.

Unification problem in EL−⊤ ( and in EL)

C1 ≡? D1, . . . , Cn ≡? Dn, where Ci, Di are concept terms in EL−⊤ (or in EL). (Decision problem) Does a substitution σ, s.t. σ(C1) ≡ σ(D1), . . . , σ(Cn) ≡ σ(Dn), exist?

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 7 of 26

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Unification in EL and EL−⊤

We partition concept names into:

  • concept variables X1, X2, . . . (can be substituted by concept terms)
  • concept constants A1, A2, . . . .

A concept term is ground if it has no variables.

Unification problem in EL−⊤ ( and in EL)

C1⊑?D1, . . . , Cn⊑?Dn, where Ci, Di are concept terms in EL−⊤ ( or in EL). (Decision problem) Does a substitution σ, s.t. σ(C1)⊑σ(D1), . . . , σ(Cn)⊑σ(Dn), exist?

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Unification in EL and EL−⊤

Particles

Particles

  • concept names,
  • existential restrictions: ∃r1.∃r2, . . . ∃rn.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 = C1 ⊓ C2, then Part(C) := Part(C1) ∪ Part(C2).

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 9 of 26

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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}

  • ⊓

A ǫ ⊓ A ǫ B ǫ s B ǫ r

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 10 of 26

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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−⊤.

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 11 of 26

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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)) = ∅ .

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 12 of 26

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Unification in EL and EL−⊤

From EL−⊤ to EL and back to EL−⊤

If γ is a unifier in EL−⊤:

  • γ1 is a unifier in EL−⊤:

γ1(X) =

SX

  • D∈Γ

γ1(D) ⊓ A1 ⊓ · · · ⊓ Am where γ(X) ⊑ γ(D)

  • γ2 is a local unifier in EL:

γ2(X) =

SX

  • D∈Γ

γ2(D)

  • γ3 is a unifier in EL−⊤:

γ3(X) =

SX

  • D∈Γ

γ3(D) ⊓P1 ⊓ · · · ⊓ Pk Pi is a particle and there is Aj, such that Aj ⊑ Pi.

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 13 of 26

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Unification in EL−⊤ is in PSPACE

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.

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Unification in EL−⊤ is in PSPACE

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) 7 . ∃s.Z ⊑? ∃s.Y, (from 1 and 6)

  • 8. Z ⊑? Y, (from 7)
  • 9. ∃r.Z ⊑? ∃r.C (guess from 5)
  • 10. Z ⊑? C (from 9)

SX = {∃s.Y}, SV = {∃s.Y}, SZ = {C}, SY = ∅

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 15 of 26

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Unification in EL−⊤ is in PSPACE

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.C1, ∃s.C2) = 0, for r, s different role names – τ(A, ∃r.C) = τ(∃r.C, A), for a constant A – τ(∃r.C1, ∃r.C2) = τ(C1, C2) – if τ(C1, C2) = τ(C2, C3) = 1, then τ(C1, C3) = 1

  • 2. Acyclicity: the assignment Sτ is acyclic
  • 3. Unification of the goal: for each goal subsumption C1 ⊓ · · · ⊓ Cn ⊑ D

– if D is a non-variable atom, then there is Ci with τ(Ci, D) = 1, – if D is a variable, then τ(D, C) = 1, where C is a non-variable atom, implies τ(Ci, C) = 1 for at least one Ci.

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 16 of 26

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Unification in EL−⊤ is in PSPACE

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 := {C1 ⊓ · · · ⊓ Cn ⊑? 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

C1 ⊓ · · · ⊓ Cn ⊑? X ⊑? P for each C1 ⊓ · · · ⊓ Cn ⊑? X in ∆Γ,τ

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 17 of 26

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Unification in EL−⊤ is in PSPACE

Algorithm: Step 2. Linear language inclusions

A particle P for X

C1 ⊓ · · · ⊓ Cn ⊑? X ⊑? P for each C1 ⊓ · · · ⊓ Cn ⊑? X in ∆Γ,τ

Particles are words over Nr

A particle P is of the form ∃r1∃r2 . . . ∃rn.A∃r1r2 . . . rn.A∃ω.A, for a constant A.

Example

constant A constant B Y ⊓ ∃r.A ⊑ X XA ⊆ {ǫ}YA ∪ {r} XB ⊆ {ǫ}YB ∪ {r}∅ ∃s.B ⊓ X ⊑ Y YA ⊆ {s}∅ ∪ {ǫ}XA YB ⊆ {s} ∪ {ǫ}XB ∃s.A ⊓ B ⊑ Y YA ⊆ {s} ∪ {ǫ}∅ YB ⊆ {s}∅ ∪ {ǫ}

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 18 of 26

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Unification in EL−⊤ is in PSPACE

Algorithm: Step 2. Linear language inclusions

Set of linear language inclusions IΓ,τ

  • For each constant A define a set of inclusions IA:

For each C1 ⊓ · · · ⊓ Cn ⊑ X in ∆Γ,τ XA ⊆ fA(C1) ∪ · · · ∪ fA(Cn) , – fA(∃r.C) = {r}fA(C), – fA(Y) = YA, – fA(A) = {ǫ}, – fA(B) = ∅

  • IΓ,τ =

A∈Γ IA

  • A solution θ is admissible if for each X, there is A such that

θ(XA) = ∅

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 19 of 26

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Unification in EL−⊤ is in PSPACE

Algorithm: Step 3. Construct an alternating automaton

Alternating Finite Automaton with ǫ-edges

AX = (Q∀, Q∃, Σ, q0, δ, F)

  • Q∀ – indeterminates in IΓ,τ plus (final) f∀
  • Q∃ – inclusions in IΓ,τ plus final f∃
  • Σ = Nr
  • q0 = XA
  • F = {f∃}

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 20 of 26

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Unification in EL−⊤ is in PSPACE

Algorithm: Step 3. Construct an alternating automaton

Alternating Finite Automaton with ǫ-edges

AX = (Q∀, Q∃, Σ, q0, δ, F)

Example

i1 XA ⊆ {ǫ}YA i2 XA ⊆ {s} i3 YA ⊆ {r} ∪ {ǫ}XA (XA, 0), s (i2, 0), s f∃, ǫ s ǫ (i1, 0), s (YA, 1), s (i3, 1), s (XA, 2), s (i2, 2), s f∃, ǫ s ǫ (i1, 2), s f∀, s ǫ ǫ ǫ ǫ ǫ ǫ

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Unification in EL−⊤ is in PSPACE

Algorithm: Step 4. Emptiness check

Theorem

The problem of deciding unifiability in EL−⊤ is in PSPACE. Step 1 Guessing τ is in NP Step 2 Constructing ∆Γ,τ and IΓ,τ is in P Step 3 Constructing polynomially many polynomial size ǫ−AFA’s is in P Step 4 Checking two-way alternating finite automaton for emptiness is in PSPACE.

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 22 of 26

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Unification in EL−⊤ is PSPACE-hard

Reduction of intersection emptiness problem for DFA

Deterministic finite automaton

A = (Q, Σ, q0, δ, F)

  • δ is a partial function: Q × Σ → Q
  • each state can be reached from q0

Unification problem ΓA

Signature:

  • one constant A
  • for each state q ∈ Q, a variable Xq,
  • Nr := Σ

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Unification in EL−⊤ is PSPACE-hard

Emptiness problem for a DFA

DFA A Unification problem ΓA q0 start q1 r1 r2 ∃r1.Xq0 ⊓ ∃r2.Xq1 ⊑? Xq0 A ⊑? Xq1

Lemma

ω ∈ L(A) iff there is a unifier γ of ΓA, such that ∃ω.A ∈ Part(γ(Xq0))

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 24 of 26

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Unification in EL−⊤ is PSPACE-hard

Reduction of intersection emptiness problem for DFA

Assume we have n DFA

We define a flat EL−⊤ unification problem Γ: Γ :=

  • i∈{1,...,n}

(ΓAi ∪ {Xq0,i ⊑? Y}) where Y is a new variable, not contained in any ΓAi .

Theorem

Γ is unifiable in EL−⊤ iff L(A1) ∩ · · · ∩ L(An) = ∅

WARU Utrecht, 26.05.2011 Unification in EL without Top Constructor Slide 25 of 26

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Conclusion

Results

Decision problem for unification in EL⊤ is PSPACE- complete.

  • In PSPACE: checking emptiness of an alternating automaton
  • PSPACE-hardness: checking emptiness of an interesection of DFA

Not presented here:

  • Definition of locality of unifiers in EL⊤
  • How to compute a local unifier of size at most exponential.

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