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Institute of Theoretical Computer Science Chair of Automata Theory

THE INCLUSION PROBLEM FOR WEIGHTED AUTOMATA ON INFINITE TREES

Stefan Borgwardt Rafael Pe˜ naloza

Debrecen, August 21, 2011

Introduction

• Automata on infinite trees can recognize tree-shaped models
• Emptiness test useful to decide satisfiability in logics
• Inclusion test could be used to decide entailment
• Here: generalization to lattice-weighted automata

Debrecen, August 21, 2011 Inclusion for Weighted Automata 2

Lattices

De Morgan lattice:

• Bounded distributive lattice L = (L, ⊕, ⊗, 0, 1)
• De Morgan negation − : L → L

1 x = 1 − x 1 b a c b a c

Debrecen, August 21, 2011 Inclusion for Weighted Automata 3

Trees

Infinite k-ary trees:

• Nodes are identified by their positions in K ∗, where K := {1, . . . , k}

ε 1 11 . . . . . . 12 . . . . . . 2 21 . . . . . . 22 . . . . . .

Debrecen, August 21, 2011 Inclusion for Weighted Automata 4

Trees

Infinite k-ary trees:

• Nodes are identified by their positions in K ∗, where K := {1, . . . , k}

ε 1 11 . . . . . . 12 . . . . . . 2 21 . . . . . . 22 . . . . . .

• A labeled tree t ∈ ΣK∗ is a function t : K ∗ → Σ

Debrecen, August 21, 2011 Inclusion for Weighted Automata 4

Automata

Tree automaton A = (Q, Σ, I, ∆, X):

• states Q
• input alphabet Σ
• initial state set I ⊆ Q
• transition relation ∆ ⊆ Q × Σ × Qk
• acceptance condition X ⊆ Qω

Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata

Tree automaton A = (Q, Σ, I, ∆, X):

• states Q
• input alphabet Σ
• initial state set I ⊆ Q
• transition relation ∆ ⊆ Q × Σ × Qk
• acceptance condition X ⊆ Qω

Successful run r ∈ succ(A) ⊆ QK∗: every path is in X

Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata

Tree automaton A = (Q, Σ, I, ∆, X):

• states Q
• input alphabet Σ
• initial state set I ⊆ Q
• transition relation ∆ ⊆ Q × Σ × Qk
• acceptance condition X ⊆ Qω

Successful run r ∈ succ(A) ⊆ QK∗: every path is in X Transition of r on t ∈ ΣK∗ at u ∈ K ∗: − − − → r(t, u) := (r(u), t(u), r(u1), . . . , r(uk))

Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata

Tree automaton A = (Q, Σ, I, ∆, X):

• states Q
• input alphabet Σ
• initial state set I ⊆ Q
• transition relation ∆ ⊆ Q × Σ × Qk
• acceptance condition X ⊆ Qω

Successful run r ∈ succ(A) ⊆ QK∗: every path is in X Transition of r on t ∈ ΣK∗ at u ∈ K ∗: − − − → r(t, u) := (r(u), t(u), r(u1), . . . , r(uk)) t ∈ L(A) iff

∃

r∈succ(A)

r(ε) ∈ I ∧ ∀

u∈K∗

− − − → r(t, u) ∈ ∆

Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata

Weighted tree automaton A = (Q, Σ, S, in, wt, X):

• states Q
• input alphabet Σ
• (finite) distributive lattice S
• initial distribution in : Q → S
• transition weight function wt : Q × Σ × Qk → S
• acceptance condition X ⊆ Qω

Successful run r ∈ succ(A) ⊆ QK∗: every path is in X Transition of r on t ∈ ΣK∗ at u ∈ K ∗: − − − → r(t, u) := (r(u), t(u), r(u1), . . . , r(uk)) t ∈ L(A) iff

∃

r∈succ(A)

r(ε) ∈ I ∧ ∀

u∈K∗

− − − → r(t, u) ∈ ∆

Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata

Weighted tree automaton A = (Q, Σ, S, in, wt, X):

• states Q
• input alphabet Σ
• (finite) distributive lattice S
• initial distribution in : Q → S
• transition weight function wt : Q × Σ × Qk → S
• acceptance condition X ⊆ Qω

Successful run r ∈ succ(A) ⊆ QK∗: every path is in X Transition of r on t ∈ ΣK∗ at u ∈ K ∗: − − − → r(t, u) := (r(u), t(u), r(u1), . . . , r(uk)) (A, t) =

• r∈succ(A)

in(r(ε)) ⊗

• u∈K∗

wt(− − − → r(t, u))

Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata

Weighted tree automaton A = (Q, Σ, S, in, wt, X):

• states Q
• input alphabet Σ
• (finite) distributive lattice S
• initial distribution in : Q → S
• transition weight function wt : Q × Σ × Qk → S
• acceptance condition X ⊆ Qω

Successful run r ∈ succ(A) ⊆ QK∗: every path is in X Transition of r on t ∈ ΣK∗ at u ∈ K ∗: − − − → r(t, u) := (r(u), t(u), r(u1), . . . , r(uk)) (A, t) =

• r∈succ(A)

in(r(ε)) ⊗

• u∈K∗

wt(− − − → r(t, u)) Acceptance conditions: looping, B¨ uchi, co-B¨ uchi, parity

Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata (2)

PTIME-problems for B¨ uchi automata:

• Infimum of two automata: (C, t) = (A, t) ⊗ (B, t)
• Supremum of two automata: (C, t) = (A, t) ⊕ (B, t)
• Computing the behavior

t∈ΣK∗ (A, t) [Baader, Pe˜

naloza 2010]

Debrecen, August 21, 2011 Inclusion for Weighted Automata 6

Description Logics

The description logic ALC Syntax concept name A ∈ NC role name r ∈ NR

Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics

The description logic ALC Syntax interpretation I = (·I, ∆I) concept name A ∈ NC AI ⊆ ∆I role name r ∈ NR rI ⊆ ∆I × ∆I

Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics

The description logic ALC Syntax interpretation I = (·I, ∆I) concept name A ∈ NC AI ⊆ ∆I role name r ∈ NR rI ⊆ ∆I × ∆I top concept ⊤ ∆I bottom concept ⊥ ∅ conjunction C ⊓ D CI ∩ DI disjunction C ⊔ D CI ∪ DI negation ¬C CI

Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics

The description logic ALC Syntax interpretation I = (·I, ∆I) concept name A ∈ NC AI ⊆ ∆I role name r ∈ NR rI ⊆ ∆I × ∆I top concept ⊤ ∆I bottom concept ⊥ ∅ conjunction C ⊓ D CI ∩ DI disjunction C ⊔ D CI ∪ DI negation ¬C CI existential restriction ∃r.C {x | ∃y : (x, y) ∈ rI ∧ y ∈ CI} universal restriction ∀r.C {x | ∀y : (x, y) ∈ rI → y ∈ CI}

Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics

The description logic ALC Syntax interpretation I = (·I, ∆I) concept name A ∈ NC AI ⊆ ∆I role name r ∈ NR rI ⊆ ∆I × ∆I top concept ⊤ ∆I bottom concept ⊥ ∅ conjunction C ⊓ D CI ∩ DI disjunction C ⊔ D CI ∪ DI negation ¬C CI existential restriction ∃r.C {x | ∃y : (x, y) ∈ rI ∧ y ∈ CI} universal restriction ∀r.C {x | ∀y : (x, y) ∈ rI → y ∈ CI} terminological axiom C ⊑ D CI ⊆ DI

Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics

The description logic ALC Syntax interpretation I = (·I, ∆I) concept name A ∈ NC AI ⊆ ∆I role name r ∈ NR rI ⊆ ∆I × ∆I top concept ⊤ ∆I bottom concept ⊥ ∅ conjunction C ⊓ D CI ∩ DI disjunction C ⊔ D CI ∪ DI negation ¬C CI existential restriction ∃r.C {x | ∃y : (x, y) ∈ rI ∧ y ∈ CI} universal restriction ∀r.C {x | ∀y : (x, y) ∈ rI → y ∈ CI} terminological axiom C ⊑ D CI ⊆ DI

• Consistency of a TBox T (set of axioms): Is there a model of T ?
• Satisfiability of C w.r.t. T : Is there a model I of T with CI = ∅?
• Subsumption C ⊑T D: Does CI ⊆ DI hold in all models I of T ?

Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Tree Models

• ALC has the tree model property
• Satisfiability of C w.r.t. T can be reduced to emptiness of AC,T

ë

Debrecen, August 21, 2011 Inclusion for Weighted Automata 8

Tree Models

• ALC has the tree model property
• Satisfiability of C w.r.t. T can be reduced to emptiness of AC,T
• Behavior computation can be used for axiom pinpointing

ë identifying the axioms of T that are responsible for a contradiction

Debrecen, August 21, 2011 Inclusion for Weighted Automata 8

Tree Models

• ALC has the tree model property
• Satisfiability of C w.r.t. T can be reduced to emptiness of AC,T
• Behavior computation can be used for axiom pinpointing

ë identifying the axioms of T that are responsible for a contradiction

• C ⊑T D iff C ⊓ ¬D is unsatisfiable

Debrecen, August 21, 2011 Inclusion for Weighted Automata 8

Tree Models

• ALC has the tree model property
• Satisfiability of C w.r.t. T can be reduced to emptiness of AC,T
• Behavior computation can be used for axiom pinpointing

ë identifying the axioms of T that are responsible for a contradiction

• C ⊑T D iff C ⊓ ¬D is unsatisfiable
• Inclusion test is useful for non-standard inferences

Debrecen, August 21, 2011 Inclusion for Weighted Automata 8

Inclusion and Complementation

Given two automata A, A′, does L(A′) ⊆ L(A) hold? Given an automaton A, construct an automaton A with L(A) = L(A).

Debrecen, August 21, 2011 Inclusion for Weighted Automata 9

Inclusion and Complementation

Given two automata A, A′, does L(A′) ⊆ L(A) hold? Given an automaton A, construct an automaton A with L(A) = L(A). Given two weighted automata A, A′, compute

t∈ΣK∗ (A′, t) ⊕ (A, t). Debrecen, August 21, 2011 Inclusion for Weighted Automata 9

Inclusion and Complementation

Given two automata A, A′, does L(A′) ⊆ L(A) hold? Given an automaton A, construct an automaton A with L(A) = L(A). Given two weighted automata A, A′, compute

t∈ΣK∗ (A′, t) ⊕ (A, t).

Given a weighted automaton A, construct a weighted automaton A with A = A.

Debrecen, August 21, 2011 Inclusion for Weighted Automata 9

Inclusion and Complementation

Given two automata A, A′, does L(A′) ⊆ L(A) hold? Given an automaton A, construct an automaton A with L(A) = L(A). Given two weighted automata A, A′, compute

t∈ΣK∗ (A′, t) ⊕ (A, t).

Given a weighted automaton A, construct a weighted automaton A with A = A. [Buhrke, Lescow, V¨

• ge 1996; Kupferman, Vardi 1998; Vardi, Wilke 2008]:

Inclusion is in EXPTIME for parity automata [Seidl 1989]: Inclusion is EXPTIME-hard for automata on finite trees

Debrecen, August 21, 2011 Inclusion for Weighted Automata 9

Glass-box Approach

Modify algorithms for unweighted complementation for weighted automata:

Debrecen, August 21, 2011 Inclusion for Weighted Automata 10

Glass-box Approach

Modify algorithms for unweighted complementation for weighted automata:

• [Miyano, Hayashi 1984; Muller, Schupp 1987]:

– Exponential constructions for complementing looping and co-B¨ uchi into B¨ uchi automata (powerset construction Q 2Q)

Debrecen, August 21, 2011 Inclusion for Weighted Automata 10

Glass-box Approach

Modify algorithms for unweighted complementation for weighted automata:

• [Miyano, Hayashi 1984; Muller, Schupp 1987]:

– Exponential constructions for complementing looping and co-B¨ uchi into B¨ uchi automata (powerset construction Q 2Q)

• Translation of the constructions and proofs to finite De Morgan lattices:

– from 2Q to SQ – from ∧ to ⊗ and ∨ to ⊕ – from ∀ to and ∃ to – from q ∈ I to in(q) and (. . . ) ∈ ∆ to wt(. . . ) – from x ⇒ y to x ⊕ y or x ≤ y

Debrecen, August 21, 2011 Inclusion for Weighted Automata 10

Glass-box Approach

Example: “If r is a successful run of A on t and rc is a successul run of A on t, then all paths p of length m have a node u ∈ p such that r(u) / ∈ rc(u). ”

Debrecen, August 21, 2011 Inclusion for Weighted Automata 11

Glass-box Approach

Example: “If r is a successful run of A on t and rc is a successul run of A on t, then all paths p of length m have a node u ∈ p such that r(u) / ∈ rc(u). ” r ∈ succ(A), rc ∈ succ(A) : r ∈ ∆(t) ∧ rc ∈ ∆c(t) ⇒

∀

p∈Path(K∗,m) ∃ u∈p

r(u) / ∈ rc(u)

Debrecen, August 21, 2011 Inclusion for Weighted Automata 11

Glass-box Approach

Example: “If r is a successful run of A on t and rc is a successul run of A on t, then all paths p of length m have a node u ∈ p such that r(u) / ∈ rc(u). ” r ∈ succ(A), rc ∈ succ(A) : r ∈ ∆(t) ∧ rc ∈ ∆c(t) ⇒

∀

p∈Path(K∗,m) ∃ u∈p

r(u) / ∈ rc(u) r ∈ succ(A), rc ∈ succ(A) : wt(t, r) ⊗ wtc(t, rc) ≤

• p∈Path(K∗,m)
• u∈p

rc(u)(r(u))

Debrecen, August 21, 2011 Inclusion for Weighted Automata 11

Glass-box Approach

Example: “If r is a successful run of A on t and rc is a successul run of A on t, then all paths p of length m have a node u ∈ p such that r(u) / ∈ rc(u). ” r ∈ succ(A), rc ∈ succ(A) : r ∈ ∆(t) ∧ rc ∈ ∆c(t) ⇒

∀

p∈Path(K∗,m) ∃ u∈p

r(u) / ∈ rc(u) r ∈ succ(A), rc ∈ succ(A) : wt(t, r) ⊗ wtc(t, rc) ≤

• p∈Path(K∗,m)
• u∈p

rc(u)(r(u)) Only correct for Boolean lattices

Debrecen, August 21, 2011 Inclusion for Weighted Automata 11

Black-box Approach

p ∈ S meet prime: a ⊗ b ≤ p implies a ≤ p or b ≤ p Every x ∈ S is equal to the infimum of all meet prime elements above x.

Debrecen, August 21, 2011 Inclusion for Weighted Automata 12

Black-box Approach

p ∈ S meet prime: a ⊗ b ≤ p implies a ≤ p or b ≤ p Every x ∈ S is equal to the infimum of all meet prime elements above x. Which meet prime elements of S are above

t∈ΣK∗ (A′, t) ⊕ (A, t)? Debrecen, August 21, 2011 Inclusion for Weighted Automata 12

Black-box Approach

p ∈ S meet prime: a ⊗ b ≤ p implies a ≤ p or b ≤ p Every x ∈ S is equal to the infimum of all meet prime elements above x. Which meet prime elements of S are above

t∈ΣK∗ (A′, t) ⊕ (A, t)?

• t∈ΣK∗ (A′, t) ⊕ (A, t) ≤ p

Debrecen, August 21, 2011 Inclusion for Weighted Automata 12

Black-box Approach

p ∈ S meet prime: a ⊗ b ≤ p implies a ≤ p or b ≤ p Every x ∈ S is equal to the infimum of all meet prime elements above x. Which meet prime elements of S are above

t∈ΣK∗ (A′, t) ⊕ (A, t)?

• t∈ΣK∗ (A′, t) ⊕ (A, t) ≤ p

iff ∃t ∈ ΣK∗ : (A′, t) ≥ p and (A, t) ≤ p

Debrecen, August 21, 2011 Inclusion for Weighted Automata 12

Black-box Approach

p ∈ S meet prime: a ⊗ b ≤ p implies a ≤ p or b ≤ p Every x ∈ S is equal to the infimum of all meet prime elements above x. Which meet prime elements of S are above

t∈ΣK∗ (A′, t) ⊕ (A, t)?

• t∈ΣK∗ (A′, t) ⊕ (A, t) ≤ p

iff ∃t ∈ ΣK∗ : (A′, t) ≥ p and (A, t) ≤ p iff ∃t ∈ ΣK∗ : t ∈ L(A′

≥p) and t /

∈ L(Ap)

Debrecen, August 21, 2011 Inclusion for Weighted Automata 12

Black-box Approach

p ∈ S meet prime: a ⊗ b ≤ p implies a ≤ p or b ≤ p Every x ∈ S is equal to the infimum of all meet prime elements above x. Which meet prime elements of S are above

t∈ΣK∗ (A′, t) ⊕ (A, t)?

• t∈ΣK∗ (A′, t) ⊕ (A, t) ≤ p

iff ∃t ∈ ΣK∗ : (A′, t) ≥ p and (A, t) ≤ p iff ∃t ∈ ΣK∗ : t ∈ L(A′

≥p) and t /

∈ L(Ap) iff L(A′

≥p) L(Ap) Debrecen, August 21, 2011 Inclusion for Weighted Automata 12

Black-box Approach

p ∈ S meet prime: a ⊗ b ≤ p implies a ≤ p or b ≤ p Every x ∈ S is equal to the infimum of all meet prime elements above x. Which meet prime elements of S are above

t∈ΣK∗ (A′, t) ⊕ (A, t)?

• t∈ΣK∗ (A′, t) ⊕ (A, t) ≤ p

iff ∃t ∈ ΣK∗ : (A′, t) ≥ p and (A, t) ≤ p iff ∃t ∈ ΣK∗ : t ∈ L(A′

≥p) and t /

∈ L(Ap) iff L(A′

≥p) L(Ap)

We need exponentially many inclusion tests between unweighted automata.

Debrecen, August 21, 2011 Inclusion for Weighted Automata 12

Conclusions

Summary:

• black-box (n2m) is faster and more general than this glass-box approach (2nm)
• optimizations of glass-box algorithm?

Debrecen, August 21, 2011 Inclusion for Weighted Automata 13

Conclusions

Summary:

• black-box (n2m) is faster and more general than this glass-box approach (2nm)
• optimizations of glass-box algorithm?

Applications:

• lattice-weighted automata for axiom pinpointing
• automata-based reasoning in fuzzy description logics

Debrecen, August 21, 2011 Inclusion for Weighted Automata 13

Thank You

Franz Baader and Rafael Pe˜ naloza: Automata-based axiom pinpointing.

• J. Autom. Reasoning, 45(2):91–129, 2010. Special Issue: IJCAR’08.

Stefan Borgwardt and Rafael Pe˜ naloza: Complementation and inclusion of weighted automata on infinite trees: Revised version. LTCS-Report 11-02, Technische Universit¨ at Dresden, 2011. See http://lat.inf.tu-dresden.de/research/reports.html. Stefan Borgwardt and Rafael Pe˜ naloza: Description logics over lattices with multi-valued ontologies. In Proc. IJCAI’11, pages 768–773. AAAI Press, 2011. Orna Kupferman and Yoad Lustig: Lattice automata. In Proc. VMCAI’07, volume 4349 of LNCS, pages 199–213. Springer, 2007 . Satoru Miyano and Takeshi Hayashi: Alternating finite automata on

• mega-words.
• Theor. Comput. Sci., 32:321–330, 1984.

David E. Muller and Paul E. Schupp: Alternating automata on infinite trees.

• Theor. Comput. Sci., 54(2-3):267–276, 1987

.

Debrecen, August 21, 2011 Inclusion for Weighted Automata 14

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