Multi-Buffer Simulations for Trace Language Inclusion Norbert Hundeshagen 1 Dietrich Kuske 2 Milka Hutagalung 1 Etienne Lozes 3 Martin Lange 1 1 Universit¨ at Kassel 2 TU Ilmenau 3 ENS Cachan GandALF 2016, Catania 14 September 2016 1 / 14
Overview Buffer Simulation Language Inclusion Simulation PSpace PTime Trace Language Multi-Buffer Inclusion Undecidable Simulation ??? 2 / 14
Simulation � Language Inclusion ◮ simulation approximates language inclusion: L ( A ) ⊆ L ( B ) ◮ played between Spoiler ( S ) and Duplicator ( D ) ◮ construct two runs ρ A , ρ B stepwise ◮ D wins iff ρ B accepting or ρ A not accepting ◮ A ⊑ B iff D has winning strategy A B b a b Σ Σ a a Σ c Σ c A ⊑ B A B b b a Σ Σ a c a Σ Σ c A �⊑ B 3 / 14
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 Σ ω with uabv ∼ I ubav for all ( a , b ) ∈ I and uv ∈ Σ ω ◮ 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. c b b 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 4 / 14
Multi-Buffer Simulation ◮ played with m FIFO buffers on NBA A , B over Σ ◮ letter distribution ˆ Σ = (Σ 1 , . . . , Σ m ), Σ i ⊆ Σ ◮ buffers capacities κ = ( k 1 , . . . , k m ), k i ∈ N ∪ { ω } ◮ conf. : ( p , w 1 , . . . , w m , q ), w i ∈ Σ ≤ k i i ◮ in position ( p , w 1 , . . . , w m , q ) a ∈ Σ → p ′ 1. S : p − − − u ∈ Σ ∗ → q ′ , s.t. ( aw i ) Σ i = ( w ′ 2. D : q − − − − i u ) Σ i , for all i ∈ { 1 , . . . , m } ◮ next position ( p ′ , w ′ 1 , . . . , w ′ m , q ′ ) ◮ D wins iff ◮ ρ A not accepting, or ◮ ρ B accepting and | w A | a = | w B | a for all a ∈ Σ. ◮ A ⊑ κ Σ B iff D has winning strategy ˆ 5 / 14
Examples ◮ ˆ Σ = ( { a , b } , { a } , { b } , { c } ) ⇔ I = { ( a , c ) , ( c , a ), ( b , c ) , ( c , b ) } c c b b c b b a 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 B But, A �⊑ 2 , 0 , 2 , 0 B ˆ ˆ Σ Σ b a c c b c b b a a ◮ A ′ ⊑ ω,ω, 2 , 0 B ˆ Σ 6 / 14
Properties of Multi-Buffer Simulation ◮ Thm. Given: A , B , k 1 , . . . , k n ∈ N , ˆ Σ Question: Is A ⊑ k 1 ,..., k n B ? decidable in PTime ˆ Σ ◮ reduce to A ⊑ B ′ , states in B ′ are o.t.f ( q B , w 1 , . . . , w n ) ◮ Thm. A ⊑ κ Σ B ⇒ L ( A ) ⊆ [ L ( B )] I ˆ ◮ ˆ Σ ⇔ I ◮ Ex. : I = { ( a , c ) , ( c , a ) , ( b , c ) , ( c , b ) } then ˆ Σ = ( { a , b } , { a } , { b } , { c } ) c c b b c b b a a A ⊑ 2 , 1 , 2 , 0 B ⇒ L ( A ) ⊆ [ L ( B )] I ˆ Σ ◮ Thm. If k 1 ≤ ℓ 1 , . . . , k n ≤ ℓ n , then and ∃ i , k i < ℓ i , then ⊑ k 1 ,..., k n � ⊑ ℓ 1 ,...,ℓ n ◮ Incremental approximation for L ( A ) ⊆ [ L ( B )] I 7 / 14
Decidability of ⊑ k 1 ,..., k n , k 1 , . . . , k n ∈ N ∪ { ω } ˆ Σ ◮ Thm. Given: A , B Question: Is A ⊑ ω B ? decidable. EXPTIME-complete [H./Lange/Lozes’13] ◮ Thm. Given: A , B Question: Is A ⊑ k 1 ,..., k n B , k 1 , . . . , k n ∈ N ∪ { ω } ? undecidable ˆ Σ Question: Is A ⊑ ω, 0 B ? highly undecidable ˆ Σ 8 / 14
⊑ ω, 0 is Highly Undecidable ◮ Thm. Deciding A ⊑ ω, 0 B is B Σ 1 1 -hard ˆ Σ ◮ Reduction from Recursive B¨ uchi Game (RBG) Given: Computable graph G Question: Does D wins B¨ uchi Game on G ? . . . q 0 TM M E G : 00 1001 ⊲ 1 ♯ 1 0 0 ⊳ · · · . . . | = . . . 1 100 010 . . . . . . q acc accept ⊲ 1 0 0 ♯ · · · | = 9 / 14
Reduction from RBG to Multi-Buffer Simulation ◮ Given RBG on comp. graph G . . . 00 00 1001 1001 . . . . . . 1 100 010 . . . ◮ � construct A , B : (sketch) ⊲ q 0 1 ♯ A S chs A sim A D chs A sim B S chs B sim B D chs B sim Buff1 10 1 ♯ 0 0 ♯ 0 01 ⊳ 0 ♯ 0 1 ⊲ q 0 ⊳ Buff2 10 / 14
A D chs , B D chs ◮ D can chooses succesor node: C ⊳ ⊳ C 0 C 0 C 1 C 0 C C C C 1 Σ 0 C 1 1 C C ⊳ C ⊳ ◮ Σ 1 = { ⊳, 0 , 1 } ( ω -buffer) ◮ Σ 2 = { C ⊳ , C 0 , C 1 , C } (0-buffer) ◮ Ex. suppose D want to push 0 1 1 to ω -buffer Buff1 ⊲ q 0 1 0 ♯ 0 1 1 ⊳ Buff2 11 / 14
Simulate computation of TM M with A sim , B sim δ : (Γ ∪ Q ) 4 → (Γ ∪ Q ) ≤ 5 ◮ Derive from δ � ˆ ◮ Ex. ⊲ q 0 1 ♯ 01 ⊳ | = ⊲ 1q 1 ♯ 01 ⊳ ˆ δ ( ⊲ q 0 1 ♯ )ˆ δ ( q 0 1 ♯ 0) ˆ δ (1 ♯ 01)ˆ δ (1 ♯ 01 ⊳ ) = ⊲ 1q 1 ♯ 01 ⊳ ◮ Encode ˆ δ on B sim A sim B sim accept accept Σ reject reject Buff1 “Conf1” “Conf1” “Conf2” . . . “Conf N ” ⇒ Buff2 Conf1 | = Conf2 | = . . . | = Conf N 12 / 14
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. B Σ 1 1 -hard ◮ Further work: flush variant 13 / 14
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. 14 / 14
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