Probabilistic Automata on Finite Words: Decidable ... - Semantic Scholar
Probabilistic Automata on Finite Words: Decidable and Undecidable Problems Hugo Gimbert
Youssouf Oualhadj
LaBRI, Bordeaux 1
March 25, 2010
Introduction Example a, b b
1
a, 21
b
a, 21 2 a
3
I
1 is the initial state.
I
{3} is the set of accepting states
I
PA (an b) = probability.
1 2n
the acceptance
Introduction Example a, b b
1
a, 21
b
a, 21 2 a
3
I
1 is the initial state.
I
{3} is the set of accepting states
I
PA (an b) = probability.
1 2n
the acceptance
Accepted Language[Rabin, 63] Let A a probabilistic automaton and a rational λ: LA (λ) = {w ∈ A∗ | PA (w ) ≥ λ} .
Introduction Example a, b b
1
a, 21
b
a, 21 2 a
3
I
1 is the initial state.
I
{3} is the set of accepting states
I
PA (an b) = probability.
1 2n
the acceptance
Accepted Language[Rabin, 63] Let A a probabilistic automaton and a rational λ: LA (λ) = {w ∈ A∗ | PA (w ) ≥ λ} .
Formal definition A probabilistic automaton is a tuple A = (Q, A, (Ma )a∈A , q0 , F ).
Outline
Emptiness Problem
Threshold isolation problem
]-acyclic probabilistic automata
Conclusion
The Emptiness problem
Definition Given a probabilisticautomaton A, PA (w ) ≥ 12 . decide ∃w ∈ A∗ s.t PA (w ) > 12 , in the strict version.
The Emptiness problem
Definition Given a probabilisticautomaton A, PA (w ) ≥ 12 . decide ∃w ∈ A∗ s.t PA (w ) > 12 , in the strict version.
Theorem [Paz, 71] The (strict) emptiness problem is undecidable. Proof by reduction from a problem on context free grammars.
New undecidability proof Lemma [Bertoni, 77] Given a probabilistic automaton A, it is undecidable whether ∃w ∈ A∗ such that PA (w ) = 12
New undecidability proof Lemma [Bertoni, 77] Given a probabilistic automaton A, it is undecidable whether ∃w ∈ A∗ such that PA (w ) = 12
Remark [Gimbert, O.] The emptiness problem is undecidable even for simple probabilistic automata. A probabilistic automaton is simple if (Ma )a∈A (s, t) ∈ 0, 12 , 1 , where (s, t) ∈ Q.
New undecidability proof Lemma [Bertoni, 77] Given a probabilistic automaton A, it is undecidable whether ∃w ∈ A∗ such that PA (w ) = 12
Remark [Gimbert, O.] The emptiness problem is undecidable even for simple probabilistic automata. A probabilistic automaton is simple if (Ma )a∈A (s, t) ∈ 0, 12 , 1 , where (s, t) ∈ Q.
Corollary [Gimbert, O.] The following problem is undecidable: I
Given a non deterministic automaton on finite words, is there a word such that at least half the computations are accepting?
Automata with few probabilistic states Proposition [Blondel, 03] The emptiness problem is undecidable for any probabilistic automaton of size larger than 46.
Automata with few probabilistic states Proposition [Blondel, 03] The emptiness problem is undecidable for any probabilistic automaton of size larger than 46.
Proposition [Gimbert, O.] The emptiness problem for automata with 1 probabilistic transition is decidable in PSPACE.
Automata with few probabilistic states Proposition [Blondel, 03] The emptiness problem is undecidable for any probabilistic automaton of size larger than 46.
Proposition [Gimbert, O.] The emptiness problem for automata with 1 probabilistic transition is decidable in PSPACE.
Lemma [Gimbert, O.] The following problem is undecidable: given a simple probabilistic automaton with 1 probabilistic transition and given a rational language of finite words L ⊆ A∗ , decide ∃w ∈ L s.t PA (w ) ≥ 12 .
Automata with few probabilistic states Proposition [Blondel, 03] The emptiness problem is undecidable for any probabilistic automaton of size larger than 46.
Proposition [Gimbert, O.] The emptiness problem for automata with 1 probabilistic transition is decidable in PSPACE.
Lemma [Gimbert, O.] The following problem is undecidable: given a simple probabilistic automaton with 1 probabilistic transition and given a rational language of finite words L ⊆ A∗ , decide ∃w ∈ L s.t PA (w ) ≥ 12 .
Corollary The emptiness problem for automata with 2 probabilistic states is undecidable.
Outline
Emptiness Problem
Threshold isolation problem
]-acyclic probabilistic automata
Conclusion
Threshold Isolation Problem
Definition Let A a probabilistic automaton. λ is isolated with respect to A if : ∃ε > 0, ∀w ∈ A∗ : |PA (w ) − λ| ≥ ε.
Threshold Isolation Problem
Definition Let A a probabilistic automaton. λ is isolated with respect to A if : ∃ε > 0, ∀w ∈ A∗ : |PA (w ) − λ| ≥ ε.
Theorem [Rabin, 63] Let A a probabilistic automaton and a rational 0 ≤ λ ≤ 1. If λ is isolated then the language L(λ) is rational.
Threshold Isolation Problem
Definition Let A a probabilistic automaton. λ is isolated with respect to A if : ∃ε > 0, ∀w ∈ A∗ : |PA (w ) − λ| ≥ ε.
Theorem [Rabin, 63] Let A a probabilistic automaton and a rational 0 ≤ λ ≤ 1. If λ is isolated then the language L(λ) is rational.
Theorem [Bertoni, 77] Given a probabilistic automaton A and a rational 0 < λ < 1, it is undecidable whether if λ isolated with respect to A or not.
λ ∈ {0, 1}
Bertoni’s proof does not work in this case!
λ ∈ {0, 1}
Bertoni’s proof does not work in this case! For the emptiness problem: I
When λ = 0 it is the emptiness problem for non deterministic automata on finite words.
I
When λ = 1 it is the emptiness problem for universal automata on finite words.
λ ∈ {0, 1}
Bertoni’s proof does not work in this case! For the emptiness problem: I
When λ = 0 it is the emptiness problem for non deterministic automata on finite words.
I
When λ = 1 it is the emptiness problem for universal automata on finite words.
I
The case λ = 1 is called the value 1 problem.
The Value 1 Problem Definition Let A a probabilistic automaton. A has the value 1 if: ∀ε > 0, ∃w ∈ A∗ , PA (w ) ≥ 1 − ε .
The Value 1 Problem Definition Let A a probabilistic automaton. A has the value 1 if: ∀ε > 0, ∃w ∈ A∗ , PA (w ) ≥ 1 − ε .
Theorem [Gimbert, O.] The value 1 problem is undecidable.
Proof. I
Reduction from the strict emptiness problem.
I
Inspired by Baier, Bertrand and Gr¨ oßer for the emptiness of probabilistic B¨ uchi automata.
Sketch of The Proof Let Ax the following automaton: a, b
6
b, 21 b
4
a, 1 − x a, x 5 a
0 b
a, b
b, 21
a
1
b a, x a, 1 − x
b 2
a
3
Sketch of The Proof Let Ax the following automaton: a, b
6
b, 21 b
4
a, 1 − x a, x
0 b
a, b
b, 21 b
1
a, x a, 1 − x
b
a
5
2 a
a (valA = 1) ⇐⇒
1 x> 2
.
3
Outline
Emptiness Problem
Threshold isolation problem
]-acyclic probabilistic automata
Conclusion
]-acyclic probabilistic automata Classical subset construction a, b a, 21 , b
{1} · a = {1, 2} . a,
1
1 2
{1, 2} · a = {1, 2} . 2
]-acyclic probabilistic automata Classical subset construction a, b a, 21 , b
{1} · a = {1, 2} . a,
1
1 2
{1, 2} · a = {1, 2} . 2
Operator # 1
1 2
1
1 2
2
{1, 2} · a# = {2} . The state 2 is recurrent. The state 1 is transient.
Support graph
b 1
2 b
a
a
a
a
b 3 a
4 b
a
Support graph b {1, 2} b
b 1 a
a
a
b 3 a
{1, 3} a
4 b
{2}
a
b a
a
{1}
2
b a
b
b
{1, 2, 3, 4}
a {3}
a, b
a
a {4}
a b
{3, 4} b
{2, 4}
b
a
Support graph b] {1, 2} b
b 1 a
a
a
b 3 a
{1, 3} a]
4 b
{2}
a
b a
a
{1}
2
b a
b
{1, 2, 3, 4}
a {3}
a
a] , b ] {4}
a b
b
{3, 4} b]
b
{2, 4} a
a]
Decidability of ]-acyclic probabilistic automata Theorem [Gimbert, O.] The value 1 problem for ]-acyclic probabilistic automata is decidable.
Decidability of ]-acyclic probabilistic automata Theorem [Gimbert, O.] The value 1 problem for ]-acyclic probabilistic automata is decidable.
Definition I
Reachability in the support graph is called ]-reachability.
I
A set T is said to be limit-reachable from another set S if there exists a sequence w0 , w1 , · · · ∈ A∗ such that: w
n PA (S −→ T ) −−−→ 1 .
n→∞
Decidability of ]-acyclic probabilistic automata Theorem [Gimbert, O.] The value 1 problem for ]-acyclic probabilistic automata is decidable.
Definition I
Reachability in the support graph is called ]-reachability.
I
A set T is said to be limit-reachable from another set S if there exists a sequence w0 , w1 , · · · ∈ A∗ such that: w
n PA (S −→ T ) −−−→ 1 .
n→∞
First remark I
The ]-reachability implies limit-reachability. But the converse is not true in general.
Example a 1 b
2 a
a
a b
a
b a
3
b
Example a 1
2
b
a
b
a
a b
a
3
b
a
a# , b a
{2} b#
b
a, b
{1, 2}
a
a
{3}
{2, 3} b#
b#
{1, 2, 3}
b#
b
a
b#
{1, 3}
b#
a
{1}
Example a 1
2
b
a
b
a
a b
a
3
b
a
a# , b a
{2} b#
b
a, b
{1, 2}
a
a
{3}
{2, 3} b#
b#
{1, 2, 3}
b#
(ab n )n b
bn
a
{1, 3}
b#
a
{1}
Sketch of the proof For ]-acyclic probabilistic automata, limit-reahability =⇒ ]-reachability.
Sketch of the proof For ]-acyclic probabilistic automata, limit-reahability =⇒ ]-reachability. Q
∀a ∈ A, Q.a# = Q . S If Q is limit-reachable from S then Q is ]-reachable from S.
Sketch of the proof For ]-acyclic probabilistic automata, limit-reahability =⇒ ]-reachability. Q
∀a ∈ A, Q.a# = Q . S If Q is limit-reachable from S then Q is ]-reachable from S. ∀a ∈ A, Q.a# = Q . Q is the unique set limit-reachable support from Q.
x
Q S
Sketch of the proof For ]-acyclic probabilistic automata, limit-reahability =⇒ ]-reachability. Q
∀a ∈ A, Q.a# = Q . S If Q is limit-reachable from S then Q is ]-reachable from S. ∀a ∈ A, Q.a# = Q .
x
Q S
Q is the unique set limit-reachable support from Q. ∀a ∈ A, Q.a = Q .
Q
∀a ∈ A, S.a# = S . Every limit-reachable set from Q contains S, S in unique.
S
Outline
Emptiness Problem
Threshold isolation problem
]-acyclic probabilistic automata
Conclusion
Conclusion Contribution I
Short and simple proof for the emptiness problem.
I
Strengthening Paz result.
I
Undecidability of the value one problem.
I
Defining a non trivial family of automata for which the value 1 problem is decidable.
Conclusion Contribution I
Short and simple proof for the emptiness problem.
I
Strengthening Paz result.
I
Undecidability of the value one problem.
I
Defining a non trivial family of automata for which the value 1 problem is decidable.
agenda I
Extend the result for ]-acyclic probabilistic automata to a larger class.
I
Non trivial class of probabilistic automata for which the emptiness problem is decidable.
Youssouf Oualhadj
LaBRI, Bordeaux 1
March 25, 2010
Introduction Example a, b b
1
a, 21
b
a, 21 2 a
3
I
1 is the initial state.
I
{3} is the set of accepting states
I
PA (an b) = probability.
1 2n
the acceptance
Introduction Example a, b b
1
a, 21
b
a, 21 2 a
3
I
1 is the initial state.
I
{3} is the set of accepting states
I
PA (an b) = probability.
1 2n
the acceptance
Accepted Language[Rabin, 63] Let A a probabilistic automaton and a rational λ: LA (λ) = {w ∈ A∗ | PA (w ) ≥ λ} .
Introduction Example a, b b
1
a, 21
b
a, 21 2 a
3
I
1 is the initial state.
I
{3} is the set of accepting states
I
PA (an b) = probability.
1 2n
the acceptance
Accepted Language[Rabin, 63] Let A a probabilistic automaton and a rational λ: LA (λ) = {w ∈ A∗ | PA (w ) ≥ λ} .
Formal definition A probabilistic automaton is a tuple A = (Q, A, (Ma )a∈A , q0 , F ).
Outline
Emptiness Problem
Threshold isolation problem
]-acyclic probabilistic automata
Conclusion
The Emptiness problem
Definition Given a probabilisticautomaton A, PA (w ) ≥ 12 . decide ∃w ∈ A∗ s.t PA (w ) > 12 , in the strict version.
The Emptiness problem
Definition Given a probabilisticautomaton A, PA (w ) ≥ 12 . decide ∃w ∈ A∗ s.t PA (w ) > 12 , in the strict version.
Theorem [Paz, 71] The (strict) emptiness problem is undecidable. Proof by reduction from a problem on context free grammars.
New undecidability proof Lemma [Bertoni, 77] Given a probabilistic automaton A, it is undecidable whether ∃w ∈ A∗ such that PA (w ) = 12
New undecidability proof Lemma [Bertoni, 77] Given a probabilistic automaton A, it is undecidable whether ∃w ∈ A∗ such that PA (w ) = 12
Remark [Gimbert, O.] The emptiness problem is undecidable even for simple probabilistic automata. A probabilistic automaton is simple if (Ma )a∈A (s, t) ∈ 0, 12 , 1 , where (s, t) ∈ Q.
New undecidability proof Lemma [Bertoni, 77] Given a probabilistic automaton A, it is undecidable whether ∃w ∈ A∗ such that PA (w ) = 12
Remark [Gimbert, O.] The emptiness problem is undecidable even for simple probabilistic automata. A probabilistic automaton is simple if (Ma )a∈A (s, t) ∈ 0, 12 , 1 , where (s, t) ∈ Q.
Corollary [Gimbert, O.] The following problem is undecidable: I
Given a non deterministic automaton on finite words, is there a word such that at least half the computations are accepting?
Automata with few probabilistic states Proposition [Blondel, 03] The emptiness problem is undecidable for any probabilistic automaton of size larger than 46.
Automata with few probabilistic states Proposition [Blondel, 03] The emptiness problem is undecidable for any probabilistic automaton of size larger than 46.
Proposition [Gimbert, O.] The emptiness problem for automata with 1 probabilistic transition is decidable in PSPACE.
Automata with few probabilistic states Proposition [Blondel, 03] The emptiness problem is undecidable for any probabilistic automaton of size larger than 46.
Proposition [Gimbert, O.] The emptiness problem for automata with 1 probabilistic transition is decidable in PSPACE.
Lemma [Gimbert, O.] The following problem is undecidable: given a simple probabilistic automaton with 1 probabilistic transition and given a rational language of finite words L ⊆ A∗ , decide ∃w ∈ L s.t PA (w ) ≥ 12 .
Automata with few probabilistic states Proposition [Blondel, 03] The emptiness problem is undecidable for any probabilistic automaton of size larger than 46.
Proposition [Gimbert, O.] The emptiness problem for automata with 1 probabilistic transition is decidable in PSPACE.
Lemma [Gimbert, O.] The following problem is undecidable: given a simple probabilistic automaton with 1 probabilistic transition and given a rational language of finite words L ⊆ A∗ , decide ∃w ∈ L s.t PA (w ) ≥ 12 .
Corollary The emptiness problem for automata with 2 probabilistic states is undecidable.
Outline
Emptiness Problem
Threshold isolation problem
]-acyclic probabilistic automata
Conclusion
Threshold Isolation Problem
Definition Let A a probabilistic automaton. λ is isolated with respect to A if : ∃ε > 0, ∀w ∈ A∗ : |PA (w ) − λ| ≥ ε.
Threshold Isolation Problem
Definition Let A a probabilistic automaton. λ is isolated with respect to A if : ∃ε > 0, ∀w ∈ A∗ : |PA (w ) − λ| ≥ ε.
Theorem [Rabin, 63] Let A a probabilistic automaton and a rational 0 ≤ λ ≤ 1. If λ is isolated then the language L(λ) is rational.
Threshold Isolation Problem
Definition Let A a probabilistic automaton. λ is isolated with respect to A if : ∃ε > 0, ∀w ∈ A∗ : |PA (w ) − λ| ≥ ε.
Theorem [Rabin, 63] Let A a probabilistic automaton and a rational 0 ≤ λ ≤ 1. If λ is isolated then the language L(λ) is rational.
Theorem [Bertoni, 77] Given a probabilistic automaton A and a rational 0 < λ < 1, it is undecidable whether if λ isolated with respect to A or not.
λ ∈ {0, 1}
Bertoni’s proof does not work in this case!
λ ∈ {0, 1}
Bertoni’s proof does not work in this case! For the emptiness problem: I
When λ = 0 it is the emptiness problem for non deterministic automata on finite words.
I
When λ = 1 it is the emptiness problem for universal automata on finite words.
λ ∈ {0, 1}
Bertoni’s proof does not work in this case! For the emptiness problem: I
When λ = 0 it is the emptiness problem for non deterministic automata on finite words.
I
When λ = 1 it is the emptiness problem for universal automata on finite words.
I
The case λ = 1 is called the value 1 problem.
The Value 1 Problem Definition Let A a probabilistic automaton. A has the value 1 if: ∀ε > 0, ∃w ∈ A∗ , PA (w ) ≥ 1 − ε .
The Value 1 Problem Definition Let A a probabilistic automaton. A has the value 1 if: ∀ε > 0, ∃w ∈ A∗ , PA (w ) ≥ 1 − ε .
Theorem [Gimbert, O.] The value 1 problem is undecidable.
Proof. I
Reduction from the strict emptiness problem.
I
Inspired by Baier, Bertrand and Gr¨ oßer for the emptiness of probabilistic B¨ uchi automata.
Sketch of The Proof Let Ax the following automaton: a, b
6
b, 21 b
4
a, 1 − x a, x 5 a
0 b
a, b
b, 21
a
1
b a, x a, 1 − x
b 2
a
3
Sketch of The Proof Let Ax the following automaton: a, b
6
b, 21 b
4
a, 1 − x a, x
0 b
a, b
b, 21 b
1
a, x a, 1 − x
b
a
5
2 a
a (valA = 1) ⇐⇒
1 x> 2
.
3
Outline
Emptiness Problem
Threshold isolation problem
]-acyclic probabilistic automata
Conclusion
]-acyclic probabilistic automata Classical subset construction a, b a, 21 , b
{1} · a = {1, 2} . a,
1
1 2
{1, 2} · a = {1, 2} . 2
]-acyclic probabilistic automata Classical subset construction a, b a, 21 , b
{1} · a = {1, 2} . a,
1
1 2
{1, 2} · a = {1, 2} . 2
Operator # 1
1 2
1
1 2
2
{1, 2} · a# = {2} . The state 2 is recurrent. The state 1 is transient.
Support graph
b 1
2 b
a
a
a
a
b 3 a
4 b
a
Support graph b {1, 2} b
b 1 a
a
a
b 3 a
{1, 3} a
4 b
{2}
a
b a
a
{1}
2
b a
b
b
{1, 2, 3, 4}
a {3}
a, b
a
a {4}
a b
{3, 4} b
{2, 4}
b
a
Support graph b] {1, 2} b
b 1 a
a
a
b 3 a
{1, 3} a]
4 b
{2}
a
b a
a
{1}
2
b a
b
{1, 2, 3, 4}
a {3}
a
a] , b ] {4}
a b
b
{3, 4} b]
b
{2, 4} a
a]
Decidability of ]-acyclic probabilistic automata Theorem [Gimbert, O.] The value 1 problem for ]-acyclic probabilistic automata is decidable.
Decidability of ]-acyclic probabilistic automata Theorem [Gimbert, O.] The value 1 problem for ]-acyclic probabilistic automata is decidable.
Definition I
Reachability in the support graph is called ]-reachability.
I
A set T is said to be limit-reachable from another set S if there exists a sequence w0 , w1 , · · · ∈ A∗ such that: w
n PA (S −→ T ) −−−→ 1 .
n→∞
Decidability of ]-acyclic probabilistic automata Theorem [Gimbert, O.] The value 1 problem for ]-acyclic probabilistic automata is decidable.
Definition I
Reachability in the support graph is called ]-reachability.
I
A set T is said to be limit-reachable from another set S if there exists a sequence w0 , w1 , · · · ∈ A∗ such that: w
n PA (S −→ T ) −−−→ 1 .
n→∞
First remark I
The ]-reachability implies limit-reachability. But the converse is not true in general.
Example a 1 b
2 a
a
a b
a
b a
3
b
Example a 1
2
b
a
b
a
a b
a
3
b
a
a# , b a
{2} b#
b
a, b
{1, 2}
a
a
{3}
{2, 3} b#
b#
{1, 2, 3}
b#
b
a
b#
{1, 3}
b#
a
{1}
Example a 1
2
b
a
b
a
a b
a
3
b
a
a# , b a
{2} b#
b
a, b
{1, 2}
a
a
{3}
{2, 3} b#
b#
{1, 2, 3}
b#
(ab n )n b
bn
a
{1, 3}
b#
a
{1}
Sketch of the proof For ]-acyclic probabilistic automata, limit-reahability =⇒ ]-reachability.
Sketch of the proof For ]-acyclic probabilistic automata, limit-reahability =⇒ ]-reachability. Q
∀a ∈ A, Q.a# = Q . S If Q is limit-reachable from S then Q is ]-reachable from S.
Sketch of the proof For ]-acyclic probabilistic automata, limit-reahability =⇒ ]-reachability. Q
∀a ∈ A, Q.a# = Q . S If Q is limit-reachable from S then Q is ]-reachable from S. ∀a ∈ A, Q.a# = Q . Q is the unique set limit-reachable support from Q.
x
Q S
Sketch of the proof For ]-acyclic probabilistic automata, limit-reahability =⇒ ]-reachability. Q
∀a ∈ A, Q.a# = Q . S If Q is limit-reachable from S then Q is ]-reachable from S. ∀a ∈ A, Q.a# = Q .
x
Q S
Q is the unique set limit-reachable support from Q. ∀a ∈ A, Q.a = Q .
Q
∀a ∈ A, S.a# = S . Every limit-reachable set from Q contains S, S in unique.
S
Outline
Emptiness Problem
Threshold isolation problem
]-acyclic probabilistic automata
Conclusion
Conclusion Contribution I
Short and simple proof for the emptiness problem.
I
Strengthening Paz result.
I
Undecidability of the value one problem.
I
Defining a non trivial family of automata for which the value 1 problem is decidable.
Conclusion Contribution I
Short and simple proof for the emptiness problem.
I
Strengthening Paz result.
I
Undecidability of the value one problem.
I
Defining a non trivial family of automata for which the value 1 problem is decidable.
agenda I
Extend the result for ]-acyclic probabilistic automata to a larger class.
I
Non trivial class of probabilistic automata for which the emptiness problem is decidable.
