every path is in X. Transition of r on t â ΣKâ at u â Kâ: ââââ r(t, u) := (r(u), t(u), r(u1), ... , r(uk)). Debrecen, August 21, 2011. Inclusion for Weighted Automata. 5 ...
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
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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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3
Trees
Infinite k-ary trees:
• Nodes are identified by their positions in K ∗ , where K := {1, . . . , k} ε 2
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Inclusion for Weighted Automata
22
...
...
21
...
...
12
...
...
...
11
...
1
4
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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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
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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
∗
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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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
Debrecen, August 21, 2011
∃ r(ε) ∈ I ∧ u∈K ∀ r∈succ(A)
Inclusion for Weighted Automata
−−−→ r(t, u) ∈ ∆ ∗
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
Debrecen, August 21, 2011
∃ r(ε) ∈ I ∧ u∈K ∀ r∈succ(A)
Inclusion for Weighted Automata
−−−→ r(t, u) ∈ ∆ ∗
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)) (kAk, t) =
M
in(r(ε)) ⊗
r∈succ(A)
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Inclusion for Weighted Automata
O
−−−→ wt(r(t, u))
u∈K ∗
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)) (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
Debrecen, August 21, 2011
L
t∈ΣK
∗
˜ (kAk, t) [Baader, Penaloza 2010]
Inclusion for Weighted Automata
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Description Logics The description logic ALC
Syntax
concept name
A ∈ NC
role name
r ∈ NR
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Description Logics The description logic ALC
Syntax
concept name
A ∈ NC
role name
Debrecen, August 21, 2011
r ∈ NR
Inclusion for Weighted Automata
interpretation I = (·I , ∆I ) AI ⊆ ∆I rI
⊆ ∆I × ∆I
7
Description Logics The description logic ALC
Syntax
concept name
A ∈ NC
role name
r ∈ NR
top concept
>
bottom concept
⊥
conjunction disjunction negation
Debrecen, August 21, 2011
interpretation I = (·I , ∆I ) AI ⊆ ∆I rI
⊆ ∆I × ∆I ∆I ∅
CuD
CI
∩ DI
CtD
CI
∪ DI
¬C
Inclusion for Weighted Automata
CI
7
Description Logics The description logic ALC
Syntax
concept name
A ∈ NC
interpretation I = (·I , ∆I ) AI ⊆ ∆I rI
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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Description Logics The description logic ALC
Syntax
concept name
A ∈ NC
interpretation I = (·I , ∆I ) AI ⊆ ∆I rI
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 }
terminological axiom
CvD
C I ⊆ DI
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Description Logics The description logic ALC
Syntax
concept name
A ∈ NC
interpretation I = (·I , ∆I ) AI ⊆ ∆I rI
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 }
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
Inclusion for Weighted Automata
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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
• 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 vT D iff C u ¬D is unsatisfiable
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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
• 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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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).
Given a weighted automaton A, construct a weighted automaton A with kAk = kAk.
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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).
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
Modify algorithms for unweighted complementation for weighted automata:
• [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
Modify algorithms for unweighted complementation for weighted automata:
• [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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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
r ∈ succ(A), rc ∈ succ(A) : wt(t, r) ⊗ wtc (t, rc ) ≤
O
M
rc (u)(r(u))
p∈Path(K ∗ ,m) u∈p
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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
r ∈ succ(A), rc ∈ succ(A) : wt(t, r) ⊗ wtc (t, rc ) ≤
O
M
rc (u)(r(u))
p∈Path(K ∗ ,m) u∈p
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
Debrecen, August 21, 2011
∗
(kA0 k, t) ⊕ (kAk, t) ≤ p
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. 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
Debrecen, August 21, 2011
∃t ∈
Inclusion for Weighted Automata
(kAk, t) ≤ p and (kAk, t) ≤ p
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. 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 iff
Debrecen, August 21, 2011
∃t ∈
∃t ∈
∗ ΣK
:t∈
L(A0≥p )
(kAk, t) ≤ p and (kAk, t) ≤ p and t ∈ / L(Ap )
Inclusion for Weighted Automata
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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 iff iff
Debrecen, August 21, 2011
∃t ∈
∃t ∈
∗ ΣK
:t∈
L(A0≥p )
(kAk, t) ≤ p and (kAk, t) ≤ p and t ∈ / L(Ap )
L(A0≥p ) * L(Ap )
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. 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 iff iff
∃t ∈
∃t ∈
∗ ΣK
:t∈
L(A0≥p )
(kAk, t) ≤ p and (kAk, t) ≤ p and t ∈ / L(Ap )
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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