The Inclusion Problem for Weighted Automata on Infinite ... - TU Dresden

Institute of Theoretical Computer Science Chair of Automata Theory

THE INCLUSION PROBLEM FOR WEIGHTED AUTOMATA ON INFINITE TREES Stefan Borgwardt

˜ Rafael Penaloza

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


Inclusion for Weighted Automata

2

Lattices

De Morgan lattice:

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

a

b

c

a

b

c

0 0



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Trees

Infinite k-ary trees:

• Nodes are identified by their positions in K ∗ , where K := {1, . . . , k} ε 2



22

...

...

21

...

...

12

...

...

...

11

...

1

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Trees

Infinite k-ary trees:

• Nodes are identified by their positions in K ∗ , where K := {1, . . . , k} ε 2 22

...

...

21

...

...

12

...

...

...

11

...

1

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



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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



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• • • • •


∗

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



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• • • • •


∗

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))



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• • • • •


∗

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(ε) ∈ I ∧ u∈K ∀ r∈succ(A)


−−−→ r(t, u) ∈ ∆ ∗

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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(ε) ∈ I ∧ u∈K ∀ r∈succ(A)


−−−→ r(t, u) ∈ ∆ ∗

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• • • • • •


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)) (kAk, t) =

M

in(r(ε)) ⊗

r∈succ(A)



O

−−−→ wt(r(t, u))

u∈K ∗

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• • • • • •


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)) (kAk, t) =

M

in(r(ε)) ⊗

r∈succ(A)

O

−−−→ wt(r(t, u))

u∈K ∗

Acceptance conditions: looping, Buchi, ¨ co-Buchi, ¨ parity Debrecen, August 21, 2011


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Automata (2)

PTIME-problems for Buchi ¨ automata:

• Infimum of two automata: (kCk, t) = (kAk, t) ⊗ (kBk, t) • Supremum of two automata: (kCk, t) = (kAk, t) ⊕ (kBk, t) • Computing the behavior


L

t∈ΣK

∗

˜ (kAk, t) [Baader, Penaloza 2010]


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Description Logics The description logic ALC

Syntax

concept name

A ∈ NC

role name

r ∈ NR



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Syntax

concept name

A ∈ NC

role name


r ∈ NR


interpretation I = (·I , ∆I ) AI ⊆ ∆I rI

⊆ ∆I × ∆I

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Syntax

concept name

A ∈ NC

role name

r ∈ NR

top concept

>

bottom concept

⊥

conjunction disjunction negation



⊆ ∆I × ∆I ∆I ∅

CuD

CI

∩ DI

CtD

CI

∪ DI

¬C


CI

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Syntax

concept name

A ∈ NC


r ∈ NR

role name top concept

>

bottom concept

⊥

conjunction disjunction

⊆ ∆I × ∆I ∆I ∅

CuD

CI

∩ DI

CtD

CI

∪ DI

negation

¬C

CI

existential restriction

∃r.C

{x | ∃y : (x, y) ∈ r I ∧ y ∈ C I }

universal restriction

∀r.C

{x | ∀y : (x, y) ∈ r I → y ∈ C I }



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Syntax

concept name

A ∈ NC


r ∈ NR


>

bottom concept

⊥


⊆ ∆I × ∆I ∆I ∅

CuD

CI

∩ DI

CtD

CI

∪ DI

negation

¬C

CI


∃r.C

{x | ∃y : (x, y) ∈ r I ∧ y ∈ C I }


∀r.C

{x | ∀y : (x, y) ∈ r I → y ∈ C I }

terminological axiom

CvD

C I ⊆ DI



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Syntax

concept name

A ∈ NC


r ∈ NR


>

bottom concept

⊥


⊆ ∆I × ∆I ∆I ∅

CuD

CI

∩ DI

CtD

CI

∪ DI

negation

¬C

CI


∃r.C

{x | ∃y : (x, y) ∈ r I ∧ y ∈ C I }


∀r.C

{x | ∀y : (x, y) ∈ r I → y ∈ C I }

terminological axiom

CvD

C I ⊆ 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 C I 6= ∅? • Subsumption C vT D: Does C I ⊆ DI hold in all models I of T ? Debrecen, August 21, 2011


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Tree Models

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



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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



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Tree Models




• C vT D iff C u ¬D is unsatisfiable



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Tree Models




• C vT D iff C u ¬D is unsatisfiable • Inclusion test is useful for non-standard inferences



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Inclusion and Complementation Given two automata A, A0 , does L(A0 ) ⊆ L(A) hold? Given an automaton A, construct an automaton A with L(A) = L(A).



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Inclusion and Complementation Given two automata A, A0 , does L(A0 ) ⊆ L(A) hold? Given an automaton A, construct an automaton A with L(A) = L(A). 0 Given two weighted automata N A, A , 0 ∗ compute t∈ΣK (kA k, t) ⊕ (kAk, t).



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



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



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Glass-box Approach

Modify algorithms for unweighted complementation for weighted automata:



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Glass-box Approach


• [Miyano, Hayashi 1984; Muller, Schupp 1987]: – Exponential constructions for complementing looping and co-Buchi ¨ into Buchi ¨ automata (powerset construction Q 2Q )



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Glass-box Approach


• [Miyano, Hayashi 1984; Muller, Schupp 1987]: – Exponential constructions for complementing looping and co-Buchi ¨ into Buchi ¨ automata (powerset construction Q 2Q )

• Translation of the constructions and proofs to finite De Morgan lattices: – from 2Q to S Q – from ∧ to ⊗ and ∨ to ⊕ N L – from ∀ to and ∃ to – from q ∈ I to in(q) and (. . . ) ∈ ∆ to wt(. . . ) – from x ⇒ y to x ⊕ y or x ≤ y



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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).”



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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) ⇒

∀

∃ r(u) ∈/ rc (u)

p∈Path(K ∗ ,m) u∈p



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∀

∃ r(u) ∈/ rc (u)


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

O

M

rc (u)(r(u))




11


∀

∃ r(u) ∈/ rc (u)


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

O

M

rc (u)(r(u))


Only correct for Boolean lattices



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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.



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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. N Which meet prime elements of S are above t∈ΣK ∗ (kA0 k, t) ⊕ (kAk, t)?



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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. N Which meet prime elements of S are above t∈ΣK ∗ (kA0 k, t) ⊕ (kAk, t)? N

t∈ΣK


∗

(kA0 k, t) ⊕ (kAk, t) ≤ p


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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. N Which meet prime elements of S are above t∈ΣK ∗ (kA0 k, t) ⊕ (kAk, t)? 0 ∗ t∈ΣK (kA k, t) ⊕ ∗ ΣK : (kA0 k, t) ≥ p

N iff


∃t ∈


(kAk, t) ≤ p and (kAk, t) ≤ p

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Black-box Approach


N iff iff


∃t ∈

∃t ∈

∗ ΣK

:t∈

L(A0≥p )

(kAk, t) ≤ p and (kAk, t) ≤ p and t ∈ / L(Ap )


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Black-box Approach


N iff iff iff


∃t ∈

∃t ∈

∗ ΣK

:t∈

L(A0≥p )


L(A0≥p ) * L(Ap )


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Black-box Approach


N iff iff iff

∃t ∈

∃t ∈

∗ ΣK

:t∈

L(A0≥p )


L(A0≥p ) * L(Ap )

We need exponentially many inclusion tests between unweighted automata.



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Conclusions

Summary:

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



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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



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Thank You ˜ Franz Baader and Rafael Penaloza: Automata-based axiom pinpointing. J. Autom. Reasoning, 45(2):91–129, 2010. Special Issue: IJCAR’08. ˜ Stefan Borgwardt and Rafael Penaloza: Complementation and inclusion of weighted automata on infinite trees: Revised version. ¨ Dresden, 2011. LTCS-Report 11-02, Technische Universitat See http://lat.inf.tu-dresden.de/research/reports.html. ˜ Stefan Borgwardt and Rafael Penaloza: 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 omega-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.



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