topological, automata-theoretic and logical characterizations of … · 2016-01-07 · topological,...

Topological, Automata-Theoretic andLogical Characterizations of FinitaryLanguagesPresentation for YR-CONCUR 2010

Who? Krishnendu Chatterjee & Nathanael Fijalkow

From? IST Austria (Institute of Science and Technology, Austria)

When? September 4th, 2010

Introduction: system specification

non-terminating (e.g web server);

discrete time;

non-deterministic.


non-terminating ;

discrete time;

non-deterministic.

a finite alphabetΣ represent propositions;(e.g ”available”, ”waiting”, ”critical error”)


non-terminating ;

discrete time;

non-deterministic.

a finite alphabetΣ represent propositions;

runs are infinite words:w = w0 · w1 . . . wn . . . ∈ Σω;

specification given as a languageL ⊆ Σω;


non-terminating ;

discrete time;

non-deterministic.

a finite alphabetΣ represent propositions;

runs are infinite words:w = w0 · w1 . . . wn . . . ∈ Σω;

specification given as a languageL ⊆ Σω;

ω-regular language: safety + liveness;

liveness properties: ”something good happens eventually”.

Outline

Motivations

Characterizations

Expressions

Classical liveness properties

A first example, Buchi:

a given set of propositions appears infinitely often;(e.g ”job done”)

Classical liveness properties

A first example, Buchi:

a given set of propositions appears infinitely often;

A second example, parity:

integers are assigned to propositions, representing apriority;

along an execution, some integers appear infinitely often;

parity specifies that the least priority appearing infinitelyoften is even.

A drawback of classicalω-regularspecifications

v0 v1 v2 v3 v4

v5v6v7v8v9

v10 v11 v12 v13 . . .


v0 v1 v2 v3 v4

v5v6v7v8v9

v10 v11 v12 v13 . . .

Specification: Buchi withF = {v2k | k ∈ N}.


v0 v1 v2 v3 v4

v5v6v7v8v9

v10 v11 v12 v13 . . .

Specification: Buchi withF = {v2k | k ∈ N}.

Satisfied, but the time until something good happens maygrow unbounded!

A stronger formulation of liveness: finitaryliveness [AH94]

Intuitively: there exists an unknown, fixed boundb suchthat something good happens withinb transitions.


Intuitively: there exists anunknown, fixed boundb suchthat something good happens withinb transitions.

unknown: retain independence from granularity.



It can be expressed as a finitary operator on languages:

fin(L) =⋃

{M | M closed andω-regular, M ⊆ L}




fin(L) =⋃


closed: involves Cantor topology;

ω-regular: involvesω-regularity;




fin(L) =⋃


closed: involves Cantor topology;

ω-regular: involvesω-regularity;

restriction operator: fin(L) ⊆ L.

Back to the example

Finitary Buchi:F = {v2k | k ∈ N}

v0 v1 v2 v3 v4

v5v6v7v8v9

v10 v11 v12 v13 . . .

Not satisfied!

Outline

Motivations

Characterizations

Expressions

Describing classical finitary objectives:Buchi

Let F ⊆ Σ,

Buchi(F) = {w | Inf(w) ∩ F 6= ∅}

Inf(w) is the set of propositions that appear infinitely oftenin w.


Let F ⊆ Σ,

Buchi(F) = {w | Inf(w) ∩ F 6= ∅}

We define:

nextk(w, F) = inf{k′ − k | k′ ≥ k, wk′ ∈ F}

Describing classical finitary objectives:BuchiLet F ⊆ Σ,

Buchi(F) = {w | Inf(w) ∩ F 6= ∅}

We define:


w = v0 . . . vk vk+1 . . . vk′−1︸︷︷︸

/∈F

vk′︸︷︷︸

∈F


Let F ⊆ Σ,

Buchi(F) = {w | Inf(w) ∩ F 6= ∅}

We define:


Lemma

fin(Buchi(F)) = {w | lim supk nextk(w, F) < ∞}


Let F ⊆ Σ,

Buchi(F) = {w | Inf(w) ∩ F 6= ∅}

We define:


Lemma

fin(Buchi(F)) ={w | ∃B ∈ N,∃n ∈ N,∀k ≥ n, nextk(w, F) ≤ B}

Describing classical finitary objectives:parity

Let p : Σ → N a priority function,

Parity(p) = {w | min(p(Inf(w))) is even}




We define:

distk(w, p) = inf{k′ − k |k′ ≥ k, p(wk′) even andp(wk′) ≤ p(wk)}




We define:

distk(w, p) = inf{k′ − k |k′ ≥ k, p(wk′) even andp(wk′) ≤ p(wk)}

Lemmafin(Parity(p)) = {w | lim sup

kdistk(w, p) < ∞}

Topological characterization

Theorem Buchi(F) is Π2-complete.Parity lies in the boolean closure ofΣ2 andΠ2.

Topological characterization

Theorem Buchi(F) is Π2-complete.Parity lies in the boolean closure ofΣ2 andΠ2.

Theorem1 For all p : Σ → N, we havefin(Parity(p)) ∈ Σ2.

2 For all ∅ ⊂ F ⊂ Σ, we have thatfin(Buchi(F)) isΣ2-complete.

3 There exists p: Σ → N such thatfin(Parity(p)) isΣ2-complete.

Automata-theoretic characterization

We consider automata on infinite words, whose acceptanceconditions are finitary Buchi and finitary parity.

Automata-theoretic characterization

We consider automata on infinite words, whose acceptanceconditions are finitary Buchi and finitary parity.

{DN

}

·

{ε

F(finitary)

}

·

{B(Buchi)P(parity)

}

DFB

DB

DFP

NFB = NFP

NB

Outline

Motivations

Characterizations

Expressions

ω-regular expressions

Regular expressions defines regular languages over finitewords:

L := ∅ | ε | σ | L · L | L∗ | L + L; σ ∈ Σ

ω-regular languages are finite union ofL · L′ω, whereL andL′ are regular languages of finite words.

The bound operatorB [BC06]

Lω = {u0 · u1 · . . . · uk . . . | u0, u1, . . . , uk, . . . ∈ L}


Lω = {u0 · u1 · . . . · uk . . . | u0, u1, . . . , uk, . . . ∈ L}

LB = {u0 · u1 · . . . · uk . . . |u0, u1, . . . , uk, . . . ∈ Land(|un|)n∈N is bounded

}


Lω = {u0 · u1 · . . . · uk . . . | u0, u1, . . . , uk, . . . ∈ L}

LB = {u0 · u1 · . . . · uk . . . |u0, u1, . . . , uk, . . . ∈ Land(|un|)n∈N is bounded

}

(complete definitions require the use of infinite sequencesof finite words)

Star-freeωB-regular expressions

Star-freeωB-regular languages are finite union ofL · Mω,where

L is a regular language over finite words;

M is a star-freeB-regular language over infinite words.





Star-freeB-regular languages are described by thegrammar:

M := ∅ | ε | σ | M · M | MB | M + M; σ ∈ Σ





Star-freeB-regular languages are described by thegrammar:

M := ∅ | ε | σ | M · M | MB | M + M; σ ∈ Σ

Theorem NFP (non-deterministic finitary parity) has exactly thesame expressive power as star-freeωB-regularexpressions.

Conclusion

finitary objectives is a refinment for specification purposes;

for ω-regular languages, topological, logical andautomata-theoretic characterizations were known;for finitary languages, all were missing; we established:

topological characterization;automata-theoretic characterization, comparison toω-regularlanguages, closure properties;logical characterization using byωB-regular expressions.

Conclusion

finitary objectives is a refinment for specification purposes;

for ω-regular languages, topological, logical andautomata-theoretic characterizations were known;for finitary languages, all were missing; we established:

topological characterization;automata-theoretic characterization, comparison toω-regularlanguages, closure properties;logical characterization using byωB-regular expressions.

Future work:

algorithmic issues: equivalence withωB-regularexpressions, emptiness problem of finitary automata;

a tool for finitary objectives...

Bibliography

R. Alur and T.A. Henzinger.Finitary fairness.In LICS’94, pages 52–61. IEEE, 1994.

Mikolaj Bojanczyk and Thomas Colcombet.Bounds inω-regularity.In LICS’06, pages 285–296. IEEE, 2006.

The end

Thank you for your attention!

topological, automata-theoretic and logical characterizations of … · 2016-01-07 · topological,...

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