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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.
Introduction: system specification
non-terminating ;
discrete time;
non-deterministic.
a finite alphabetΣ represent propositions;(e.g ”available”, ”waiting”, ”critical error”)
Introduction: system specification
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;
Introduction: system specification
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 . . .
A drawback of classicalω-regularspecifications
v0 v1 v2 v3 v4
v5v6v7v8v9
v10 v11 v12 v13 . . .
Specification: Buchi withF = {v2k | k ∈ N}.
A drawback of classicalω-regularspecifications
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.
A stronger formulation of liveness: finitaryliveness [AH94]
Intuitively: there exists anunknown, fixed boundb suchthat something good happens withinb transitions.
unknown: retain independence from granularity.
A stronger formulation of liveness: finitaryliveness [AH94]
Intuitively: there exists an unknown, fixed boundb suchthat something good happens withinb transitions.
It can be expressed as a finitary operator on languages:
fin(L) =⋃
{M | M closed andω-regular, M ⊆ L}
A stronger formulation of liveness: finitaryliveness [AH94]
Intuitively: there exists an unknown, fixed boundb suchthat something good happens withinb transitions.
It can be expressed as a finitary operator on languages:
fin(L) =⋃
{M | M closed andω-regular, M ⊆ L}
closed: involves Cantor topology;
ω-regular: involvesω-regularity;
A stronger formulation of liveness: finitaryliveness [AH94]
Intuitively: there exists an unknown, fixed boundb suchthat something good happens withinb transitions.
It can be expressed as a finitary operator on languages:
fin(L) =⋃
{M | M closed andω-regular, M ⊆ 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.
Describing classical finitary objectives:Buchi
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:
nextk(w, F) = inf{k′ − k | k′ ≥ k, wk′ ∈ F}
w = v0 . . . vk vk+1 . . . vk′−1︸ ︷︷ ︸
/∈F
vk′︸︷︷︸
∈F
Describing classical finitary objectives:Buchi
Let F ⊆ Σ,
Buchi(F) = {w | Inf(w) ∩ F 6= ∅}
We define:
nextk(w, F) = inf{k′ − k | k′ ≥ k, wk′ ∈ F}
Lemma
fin(Buchi(F)) = {w | lim supk nextk(w, F) < ∞}
Describing classical finitary objectives:Buchi
Let F ⊆ Σ,
Buchi(F) = {w | Inf(w) ∩ F 6= ∅}
We define:
nextk(w, F) = inf{k′ − k | k′ ≥ k, wk′ ∈ F}
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}
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)}
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)}
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}
The bound operatorB [BC06]
Lω = {u0 · u1 · . . . · uk . . . | u0, u1, . . . , uk, . . . ∈ L}
LB = {u0 · u1 · . . . · uk . . . |u0, u1, . . . , uk, . . . ∈ Land(|un|)n∈N is bounded
}
The bound operatorB [BC06]
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-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-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
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!