Unambiguous Finite Automata - RWTH Aachen...

Unambiguous Finite Automata

Christof LodingDepartment of Computer Science

RWTH Aachen University, Germany

17th International Conference onDevelopments in Language Theory

Paris-Est, June 20, 2013

Unambiguous Finite Automata · DLT 2013, Paris 1

Unambiguous Automata

Definition: A nondeterministic automaton is called unambiguous iffor each word there is at most one accepting run from an initialstate.

determinism

unambiguity

nondeterminism

big automata

inclusion, equivalence, universality are easy

small automatainclusion, equivalence, universality are hard


In this talk

Unambiguous automata on finite words and trees have alreadybeen investigated a lot.

This talk: Present some results and open problems concerningunambiguous automata on infinite words and trees.

finite words finite trees

infinite words infinite trees


Outline

1 Finite Words (and Trees)

2 Infinite Words

Strongly unambiguous Buchi automata

Safety and reachability automata

3 Infinite Trees

4 Conclusion


Outline


2 Infinite Words



3 Infinite Trees

4 Conclusion


Unambiguous automata on finite words

• On finite words, deterministic automata are as expressive asnondeterministic ones. Therefore each regular language canbe accepted by an unambiguous automaton.

• Unambiguous automata can be exponentially more succinctthan deterministic ones.

q0 p1 p2 · · · pn

a, b

a a, b a, b a, b


Decision Problems

Theorem (Meyer/Stockmeyer’72)

The problems of universality, equivalence, and inclusion for NFAsare PSPACE-hard.

Theorem (Stearns/Hunt’85)

The problems of universality, equivalence, and inclusion forunambiguous finite automata are in PTIME.

(Can also be derived from earlier results of Schutzenberger)


Counting Accepting Runs

Idea for the PTIME inclusion algorithm:

• Problem: test if L(A) ⊆ L(B) for given automata A and B

• Consider A∩ B (product construction), which is againunambiguous:• L(A∩ B) ⊆ L(A)• L(A) ⊆ L(B) ⇔ L(A∩ B) = L(A)

L(A) L(B) L(A) L(B)


Counting Accepting Runs

Idea for the PTIME inclusion algorithm:

• Problem: test if L(A) ⊆ L(B) for given automata A and B

• Consider A∩ B (product construction), which is againunambiguous:• L(A∩ B) ⊆ L(A)• L(A) ⊆ L(B) ⇔ L(A∩ B) = L(A)

• If for each length n, the number of accepting runs of length n isthe same in A∩ B and A, then L(A ∩ B) = L(A):• computing the number of accepting runs of increasing length is

easy (dynamic programming)• if the equality holds for runs up to length |A| · |B|, then it holds

for all lengths


Finite Trees

The results can be extended to finite trees.

Theorem (Seidl’90)

The problems of universality, equivalence, and inclusion forunambiguous finite automata on ranked trees are in PTIME.


Outline


2 Infinite Words



3 Infinite Trees

4 Conclusion


Buchi Automata

q0 q1

ab

b

a

p0 p1

aa

b

b

Buchi automaton:

• same syntax as nondeterministic finite automata (NFAs)

• accepts all infinite words that admit a run visiting infinitely oftena final state

Most decision problems for B uchi automata are at least ashard as for NFAs


Unambiguous Buchi Automata Always Exist

Deterministic Buchi automata are weaker than nondeterministicones.

A nondeterministic Buchi automaton for “finitely many b”:

q0 q1

a, b

a

a


Unambiguous Buchi Automata Always Exist

Deterministic Buchi automata are weaker than nondeterministicones.

A nondeterministic Buchi automaton for “finitely many b”:

q0 q1

a, b

a

a

Theorem (Arnold’83)

For each Buchi automaton there is an equivalent unambiguousBuchi automaton.

What about algorithmic properties of unambiguous Buchiautomata?


Methods from Finite Case do not Generalize

• Runs in a Buchi automaton are infinite, counting runs up to acertain length does not work.

• But: Regular languages of infinite words can be characterizedusing finite words: ultimately periodic words.

• Can this be used to lift the methods from the finite case?


Ultimately Periodic Words

Definition: An infinite word is called ultimately periodic if it is of theform uvω = uvvvvv · · · for finite words u, v.

Examples:

• aabaabaabaabaab · · · = a(aba)ω = aa(baa)ω

• aba bb︸︷︷︸

2

a bbb︸︷︷︸

3

a bbbb︸︷︷︸

4

a bbbbb︸︷︷︸

5

a · · · is not ultimately periodic


Ultimately Periodic Words

Definition: An infinite word is called ultimately periodic if it is of theform uvω = uvvvvv · · · for finite words u, v.

Examples:

• aabaabaabaabaab · · · = a(aba)ω = aa(baa)ω

• aba bb︸︷︷︸

2

a bbb︸︷︷︸

3

a bbbb︸︷︷︸

4

a bbbbb︸︷︷︸

5

a · · · is not ultimately periodic

Theorem (Buchi’62)

Two Buchi automata are equivalent if, and only if, they accept thesame ultimately periodic words. For inclusion the correspondingresult holds.

Can we count runs for ultimately periodic words?


Unambiguous Buchi Automata

Problems when trying to lift the counting method:

Decompositions of ultimately periodic words are not unique:

aa a

a

a

a


Unambiguous Buchi Automata

Problems when trying to lift the counting method:

Decompositions of ultimately periodic words are not unique:

aa a

a

a

a

Working with ultimately periodic words is difficult, even fordeterministic automata:

Proposition (Bousquet/L.2010)

Deciding whether a given deterministic Buchi automaton acceptsan ultimately periodic word with period of length n for given n isNP-hard.


Outline


2 Infinite Words



3 Infinite Trees

4 Conclusion


Strong Unambiguity

Definition: A Buchi automaton is called strongly unambiguous if foreach word there is at most one state from which it is accepted.

Remark 1 : Determinism does not imply strong unambiguity:

q0 q1a

a, b


Strong Unambiguity



q0 q1a

a, b

p0 p1

aa

b

b


Strong Unambiguity



q0 q1a

a, b

p0 p1

aa

b

b

Remark 2 : Strong unambiguity implies unambiguity.· · ·

· · ·not unambiguous


Strong Unambiguity



q0 q1a

a, b

p0 p1

aa

b

b

Remark 2 : Strong unambiguity implies unambiguity.· · ·

· · ·not unambiguous

not strongly unambiguous


Overview

determinism strong unambiguity

unambiguity

nondeterminism


Overview

determinism strong unambiguity

unambiguity

nondeterminism

Theorem (Carton, Michel 2003)

For every nondeterministic Buchi automaton there is an equivalentstrongly unambiguous Buchi automaton.


Reduction to finite words

Theorem (Bousquet/L. 2010)

There is a polynomial time reduction from the equivalence (resp.inclusion) problem for strongly unambiguous Buchi automata to theequivalence (resp. inclusion) problem for unambiguous automataon finite words.





Proof steps : for a strongly unambiguous Buchi automaton A

• ultimately periodic words uvω are accepted with short loops:u

vu v

v





Proof steps : for a strongly unambiguous Buchi automaton A

• ultimately periodic words uvω are accepted with short loops:u

vu v

v

• construct polynomial size unambiguous NFA A′ such thatuvω is accepted by A iff u#v is accepted by A′

uv

u # v





Corollary

The equivalence and inclusion problems for strongly unambiguousBuchi automata are in PTIME.


Outline


2 Infinite Words



3 Infinite Trees

4 Conclusion



Safety automata: Buchi automata inwhich all states are accepting, excepta rejecting sink state (usually omit-ted):

0

1

2

3

a

a

a

a

b

b

b

b

Reachability automata: Buchi auto-mata in which all states are rejecting,except one accepting sink state:

0

1 2

3

4 5

6

b

a

c

a

b

b

b

a

c

c

c

a, b

a, b, c


Result on safety an reachability automata

Theorem (Isaak/L. 2012)

For unambiguous safety and reachability automata the problems ofinclusion, equivalence, and universality can be solved in polynomialtime.


A simple case: universality of safety automata

Let A be an unambiguous safety automaton.

Observation: A accepts all infinite words iff A considered as NFAaccepts all finite words.

But: A considered as NFA needs not to be unambiguous.

qaa

qab

qba

qbb

a

a

a

a

b

b

b

b






qaa

qab

qba

qbb

a

a

a

a

b

b

b

b

Solution:

• Pick some infinite word, say aω.

• Declare those states final, fromwhich aω is accepted.

• The resulting NFA is unambiguousand accepts all finite words iff Aaccepts all infinite words.






qaa

qab

qba

qbb

a

a

a

a

b

b

b

b

Solution:

• Pick some infinite word, say aω.

• Declare those states final, fromwhich aω is accepted.

• The resulting NFA is unambiguousand accepts all finite words iff Aaccepts all infinite words.


Reachability automata

For reachability automata A the situation is dual:

• A considered as NFA is unambiguous if A is unambiguous.• However, language inclusion or equivalence is not preserved

when considering the automata as NFAs.

0 1a

a, b, c

0

1 2

3

4 5

6

b

a

c

a

b

b

b

a

c

c

c

a, b

a, b, c


Reachability automata

For reachability automata A the situation is dual:

• A considered as NFA is unambiguous if A is unambiguous.• However, language inclusion or equivalence is not preserved

when considering the automata as NFAs.

0 1a

a, b, c

0

1 2

3

4 5

6

b

a

c

a

b

b

b

a

c

c

c

a, b

a, b, c

As for safety automata, a reduction to the finite word case can beachieved (basically) by recomputing the set of final states.


The general case

Question: What is the complexity of inclusion, equivalence,universality for the full class of unambiguous Buchi automata?

The methods developed so far do not seem to generalize.


Outline


2 Infinite Words



3 Infinite Trees

4 Conclusion


Automata on infinite trees

• Automata A = (Q, Σ, q0, ∆, Acc) on infinite binary trees

• Transitions of the form (q, a, q′, q′′)

Run of A : b

a

b

a a

b

a a

b

b

b a

b

a a...





Run of A : bq0

a

b

a a

b

a a

b

b

b a

b

a a...





Run of A : bq0

(q0, b, q1, q2) ∈ ∆

aq1

b

a a

b

a a

bq2

b

b a

b

a a...





Run of A : bq0

(q0, b, q1, q2) ∈ ∆

aq1

(q1, a, q3, q4) ∈ ∆

bq3

a a

bq4

a a

bq2

b

b a

b

a a...





Run of A : bq0

(q0, b, q1, q2) ∈ ∆

aq1

(q1, a, q3, q4) ∈ ∆

bq3

a a

bq4

a a

bq2

(q2, b, q5, q6) ∈ ∆

bq5

b a

bq6

a a...





Run of A : bq0

(q0, b, q1, q2) ∈ ∆

aq1

(q1, a, q3, q4) ∈ ∆

bq3

a a

bq4

a a

bq2

(q2, b, q5, q6) ∈ ∆

bq5

b a

bq6

a a...

• Acc ⊆ Qω (conditions as for automata on infinite words can beused)

• Run accepting if state sequence on each path is in Acc.


Example

The set Ta of all trees t over {a, b} such that t contains at least onenode labeled a:

• States: qa that nondeterministically searches for an a,q for the rest of the tree

• Initial state: qa

• Transitions: (qa, b, qa, q) (qa, b, q, qa) (qa, a, q, q)

• Acceptance: on each path finally only q.

Nondeterminism is required for this language.


Ambiguous languages

Theorem (Carayol/L./Niwinski/Walukiewicz 2010)

There are languages of infinite trees that can be accepted by anondeterministic automaton but not by an unambiguous automaton.The language Ta is such a language.

Proof idea:

• An automaton for Ta can be modified, such that it marks ineach accepting run exactly one a (by a special state, say).

• An unambiguous automaton could thus be turned into anautomaton choosing for each tree in Ta exactly one a.

• Such a “choice automaton” cannot exist (Gurevich/Shelah’83,Carayol/L./Niwinski/Walukiewicz 2010)


The counterexamplesb 1

b0

· · ·b

b0

0

b1

b 0

· · ·b

b0

0

b1

...

b0

b1

b0

· · ·b

b0

0

a1

0,1 a = (0N0∗1)N

TN = unravelling of this graph


The counterexamplesb 1

b0

· · ·b

b0

0

b1

b 0

· · ·b

b0

0

b1

...

b0

b1

b0

· · ·b

b0

0

a1

0,1 a = (0N0∗1)N

TN = unravelling of this graph

Theorem (CLNW’10) For each auto-maton A there exists N such that Afails to choose a unique a on TN.


Deciding unambiguity

The result on inherently ambiguous languages of infinite treesraises a new question:

Is it decidable whether a given regular language of infinite trees canbe accepted by an unambiguous automaton?


Deciding unambiguity

The result on inherently ambiguous languages of infinite treesraises a new question:

Is it decidable whether a given regular language of infinite trees canbe accepted by an unambiguous automaton?

Recent partial results (Bilkowski/Skrzypczak’13):

• It is decidable for a deterministic tree automaton whether itscomplement is unambiguous.

• Bi-unambiguity (unambiguity of a language and itscomplement) is decidable if a certain conjecture on definabilityof choice functions on infinite trees is true.


Outline


2 Infinite Words



3 Infinite Trees

4 Conclusion


Conclusion

finite words finite trees

infinite words infinite trees

Two central open questions:

• What is the complexity of universality, equivalence, inclusiontesting for unambiguous Buchi automata?

• Is it decidable for a given language of infinite trees whetherthere is an unambiguous automaton for this language?


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