Multi-Buffer Simulations for Trace Language Inclusion Norbert - - PowerPoint PPT Presentation

Multi-Buffer Simulations for Trace Language Inclusion

Milka Hutagalung 1 Norbert Hundeshagen 1 Dietrich Kuske 2 Etienne Lozes 3 Martin Lange 1

1Universit¨

at Kassel

2TU Ilmenau 3ENS Cachan

GandALF 2016, Catania 14 September 2016

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Overview

Simulation Buffer Simulation Multi-Buffer Simulation Language Inclusion Trace Language Inclusion

PTime ??? PSpace Undecidable

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Simulation Language Inclusion

◮ simulation approximates language inclusion: L(A) ⊆ L(B) ◮ played between Spoiler (S) and Duplicator (D) ◮ construct two runs ρA, ρB stepwise ◮ D wins iff ρB accepting or ρA not accepting ◮ A ⊑ B iff D has winning strategy

A a a b Σ c Σ B a b c Σ Σ A ⊑ B A a b c Σ Σ B a a b Σ c Σ A ⊑ B

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Mazurkiewicz Trace

◮ set of words, but some letters are allowed to commute ◮ equivalence class [w]I, I ⊆ Σ × Σ independence relations ◮ Def. : ∼I is the least congruence on Σω with uabv ∼I ubav

for all (a, b) ∈ I and uv ∈ Σω

◮ Ex. : Σ = {a, b, c}, I = {(a, c), (c, a), (b, c), (c, b)}

bbcaca . . . ∼I bcbaca . . . ∼I cbbaca . . . [bb(ca)ω]I = {bbc(ac)ω, bcb(ac)ω, cbb(ac)ω, bb(ac)ω, . . .}

◮ Def. [L(A)]I = {u | u ∼I w and w ∈ L(A)} ◮ Ex.

b b c a [L(A)]I = [bb(ca)ω]I

◮ Given: A, B, and I

Question: Is [L(A)]I ⊆ [L(B)]I ? undecidable [Sakarovitch’92]

◮ Extend buffer simulation to approximate L(A) ⊆ [L(B)]I

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Multi-Buffer Simulation

◮ played with m FIFO buffers on NBA A, B over Σ ◮ letter distribution ˆ

Σ = (Σ1, . . . , Σm), Σi ⊆ Σ

◮ buffers capacities κ = (k1, . . . , km), ki ∈ N ∪ {ω} ◮ conf. : (p, w1, . . . , wm, q), wi ∈ Σ≤ki i ◮ in position (p, w1, . . . , wm, q)

• 1. S: p

a∈Σ

− − − → p′

• 2. D: q

u∈Σ∗

− − − − → q′, s.t. (awi)Σi = (w ′

i u)Σi, for all i ∈ {1, . . . , m}

◮ next position (p′, w′ 1, . . . , w′ m, q′) ◮ D wins iff

◮ ρA not accepting, or ◮ ρB accepting and |wA|a = |wB|a for all a ∈ Σ.

◮ A ⊑κ ˆ Σ B iff D has winning strategy

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Examples

◮ ˆ

Σ = ({a, b}, {a}, {b}, {c})

⇔ I = { (a, c), (c, a), (b, c), (c, b)}

b b c a c b b c a Buff 1 bb bb aa aa . . . Buff 2 aa aa . . . Buff 3 bb bb . . . Buff 4 cc cc cc . . .

◮ A ⊑2,1,2,0 ˆ Σ

B But, A ⊑2,0,2,0

ˆ Σ

B b b a c a c b b c a

◮ A′ ⊑ω,ω,2,0 ˆ Σ

B

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Properties of Multi-Buffer Simulation

◮ Thm. Given: A, B, k1, . . . , kn ∈ N, ˆ

Σ Question: Is A ⊑k1,...,kn

ˆ Σ

B? decidable in PTime

◮ reduce to A ⊑ B′, states in B′ are o.t.f (qB, w1, . . . , wn)

◮ Thm. A ⊑κ ˆ Σ B ⇒ L(A) ⊆ [L(B)]I

◮ ˆ

Σ ⇔ I

◮ Ex. : I = {(a, c), (c, a), (b, c), (c, b)}

then ˆ Σ = ({a, b}, {a}, {b}, {c})

b b c a c b b c a A ⊑2,1,2,0

ˆ Σ

B ⇒ L(A) ⊆ [L(B)]I

◮ Thm. If k1 ≤ ℓ1, . . . , kn ≤ ℓn, then and ∃i, ki < ℓi, then

⊑k1,...,kn ⊑ℓ1,...,ℓn

◮ Incremental approximation for L(A) ⊆ [L(B)]I

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Decidability of ⊑k1,...,kn

ˆ Σ

, k1, . . . , kn ∈ N ∪ {ω}

◮ Thm. Given: A, B

Question: Is A ⊑ω B? decidable. EXPTIME-complete

[H./Lange/Lozes’13]

◮ Thm. Given: A, B

Question: Is A ⊑k1,...,kn

ˆ Σ

B, k1, . . . , kn ∈ N ∪ {ω}? undecidable Question: Is A ⊑ω,0

ˆ Σ

B? highly undecidable

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⊑ω,0 is Highly Undecidable

◮ Thm. Deciding A ⊑ω,0 ˆ Σ

B is BΣ1

1-hard ◮ Reduction from Recursive B¨

uchi Game (RBG) Given: Computable graph G Question: Does D wins B¨ uchi Game on G?

1 100 00 G : 1001 010 . . . . . . . . . . . . TM ME ⊲ 1 ♯ 1 ⊳ · · · q0

| = . . . | =

⊲ 1 ♯ · · · qacc accept 9 / 14

Reduction from RBG to Multi-Buffer Simulation

◮ Given RBG on comp. graph G

1 100 00 00 1001 1001 010 . . . . . . . . . . . .

◮ construct A, B: (sketch)

ASchs Asim ADchs Asim ⊲q01♯ BSchs Bsim BDchs Bsim

Buff1 ⊲ q0 10 1 ♯0 0♯0 01 ⊳0♯ 1 ⊳ Buff2

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ADchs, BDchs

◮ D can chooses succesor node:

C C1 C0 1 C⊳ ⊳ C C C0 C1 C C⊳ C⊳ C0 C1 Σ

◮ Σ1 = {⊳, 0, 1}

(ω-buffer)

◮ Σ2 = {C⊳, C0, C1, C}

(0-buffer)

◮ Ex. suppose D want to push 0

1 1 to ω-buffer Buff1 ⊲ q0 1 ♯ 1 1 ⊳ Buff2

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Simulate computation of TM M with Asim, Bsim

◮ Derive from δ ˆ

δ : (Γ ∪ Q)4 → (Γ ∪ Q)≤5

◮ Ex. ⊲q01♯01⊳ |

= ⊲1q1♯01⊳ ˆ δ(⊲q01♯)ˆ δ(q01♯0) ˆ δ(1♯01)ˆ δ(1♯01⊳) = ⊲1q1♯01⊳

◮ Encode ˆ

δ on Bsim

Asim accept reject Bsim accept reject Σ

Buff1 “Conf1” Buff2 ⇒ “Conf1” “Conf2” . . . “ConfN” Conf1 | = Conf2 | = . . . | = ConfN

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Conclusion and Further Work

◮ Multi-buffer simulation:

◮ incrementally approximates trace inclusion ◮ with bounded buffers is decidable in PTime ◮ with unbounded buffer is highly undecidable, i.e. BΣ1

1-hard

◮ Further work: flush variant

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Literature

Mazurkiewicz, A. Introduction to trace theory. The Book of Traces 1995. Hutagalung, M. Lange, M. Lozes, E. Buffered simulation games for B¨ uchi automata. AFL 2014. Sakarovitch, L. The “last” decision problem for rational trace

• languages. LATIN 1992.
• D. L. Dill, A. J. Hu, and H. Wong-Toi. Checking for language

inclusion using simulation relations. CAV 1992.

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