Good-for-Games Automata versus Deterministic Automata.

Transcription

1 Good-for-Gmes Automt versus Deterministic Automt. Denis Kuperberg 1,2 Mich l Skrzypczk 1 1 University of Wrsw 2 IRIT/ONERA (Toulouse) Séminire MoVe 12/02/2015 LIF, Luminy

2 Introduction Deterministic utomt re centrl tool in utomt theory: Polynomil lgorithms for inclusion, complementtion. Sfe composition with gmes, trees. Solutions of the synthesis problem (verifiction). Esily implemented. Problems : exponentil stte blow-up technicl constructions (Sfr) Cn we weken the notion of determinism while preserving some good properties?

3 Good-for-Gmes utomt Ide : Nondeterminism cn be resolved without knowledge bout the future.

4 Good-for-Gmes utomt Ide : Nondeterminism cn be resolved without knowledge bout the future. Introduced independently in symbolic representtion (Henzinger, Pitermn 06) simplifiction qulittive models (Colcombet 09) replce determinism Applictions synthesis brnching time verifiction tree lnguges (Boker, K, Kupfermn, S 12)

5 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 System:

6 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 System: O 1

7 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 I 2 System: O 1

8 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 I 2 System: O 1 O 2

9 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 I 2 I 3 System: O 1 O 2

10 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 I 2 I 3 System: O 1 O 2 O 3

11 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 I 2 I 3 System: O 1 O 2 O 3 System wins iff (I 1, O 1 ), (I 2, O 2 ), (I 3, O 3 ),... = ϕ.

12 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 I 2 I 3 System: O 1 O 2 O 3 System wins iff (I 1, O 1 ), (I 2, O 2 ), (I 3, O 3 ),... = ϕ. Clssicl pproch: ϕ A det then solve gme on A det.

13 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 I 2 I 3 System: O 1 O 2 O 3 System wins iff (I 1, O 1 ), (I 2, O 2 ), (I 3, O 3 ),... = ϕ. Clssicl pproch: ϕ A det then solve gme on A det. Wrong pproch: ϕ A non det : no plyer cn guess the future.

14 Evluting gme Synthesis : design system responding to environment, while stisfying constrint ϕ. Environment: I 1 I 2 I 3 System: O 1 O 2 O 3 System wins iff (I 1, O 1 ), (I 2, O 2 ), (I 3, O 3 ),... = ϕ. Clssicl pproch: ϕ A det then solve gme on A det. Wrong pproch: ϕ A non det : no plyer cn guess the future. New pproch: ϕ A GFG.

15 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: Prover: controls trnsitions, b, c b, b, c q 0 q 1 q 2 b, c c

16 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: Prover: controls trnsitions, b, c b, b, c q 0 q 1 q 2 b, c c

17 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: Prover: controls trnsitions, b, c b, b, c q 0 q 1 q 2 b, c c

18 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: b Prover: controls trnsitions, b, c b, b, c q 0 q 1 q 2 b, c c

19 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: b c Prover: controls trnsitions, b, c b, b,c q 0 q 1 q 2 b, c c

20 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: b c c Prover: controls trnsitions, b, c b, b, c q 0 q 1 q 2 b, c c

21 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: b c c... = w Prover: controls trnsitions, b, c b, b, c q 0 q 1 q 2 b, c c Plyer GFG wins if: w L Run ccepting.

22 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: b c c... = w Prover: controls trnsitions, b, c b, b, c q 0 q 1 q 2 b, c c Plyer GFG wins if: w L Run ccepting. A GFG mens tht there is strtegy σ : A Q, for ccepting words of L(A).

23 Definition vi gme A utomton on finite or infinite words. Refuter plys letters: b c c... = w Prover: controls trnsitions, b, c b, b, c q 0 q 1 q 2 b, c c Plyer GFG wins if: w L Run ccepting. A GFG mens tht there is strtegy σ : A Q, for ccepting words of L(A). How close is this to determinism?

24 Some properties of GFG utomt Composition with gmes: A G hs sme winner s G with condition L(A).

25 Some properties of GFG utomt Composition with gmes: A G hs sme winner s G with condition L(A). Theorem If A is nondeterministic nd B is GFG, it is in P to decide whether L(A) L(B).

26 Some properties of GFG utomt Composition with gmes: A G hs sme winner s G with condition L(A). Theorem If A is nondeterministic nd B is GFG, it is in P to decide whether L(A) L(B). Theorem (Boker, K, Kupfermn, Skrzypczk, ICALP 12) If A nd B re GFG for L nd L, there is deterministic utomton for L of size A B.

27 Some properties of GFG utomt Composition with gmes: A G hs sme winner s G with condition L(A). Theorem If A is nondeterministic nd B is GFG, it is in P to decide whether L(A) L(B). Theorem (Boker, K, Kupfermn, Skrzypczk, ICALP 12) If A nd B re GFG for L nd L, there is deterministic utomton for L of size A B. Theorem (Boker, K, Kupfermn, Skrzypczk, ICALP 12) Let A be n utomton for L A ω. Then the tree version of A recognizes {t : ll brnches of t re in L} if nd only if A is GFG.

28 An utomton tht is not GFG This utomton for L = ( + b) ω is not GFG:, b p, b q Opponent strtegy: ply until Eve goes in q, then ply b ω.

29 An utomton tht is not GFG This utomton for L = ( + b) ω is not GFG:, b p, b q Opponent strtegy: ply until Eve goes in q, then ply b ω. Fct GFG utomt with condition C hve sme expressivity s deterministic utomt with condition C. Therefore, GFG could improve succinctness but not expressivity.

30 An utomton tht is not GFG This utomton for L = ( + b) ω is not GFG:, b p, b q Opponent strtegy: ply until Eve goes in q, then ply b ω. Fct GFG utomt with condition C hve sme expressivity s deterministic utomt with condition C. Therefore, GFG could improve succinctness but not expressivity. But GFG on finite words deterministic (+useless trnsitions). Wht bout infinite words?

31 A GFG Büchi exmple Büchi condition: Run is ccepting if infinitely mny Büchi trnsitions re seen. x x x b b b x b Lnguge: [(x + xb) (xx + xbxb)] ω

32 Determiniztion of Büchi GFG Theorem Let A GFG Büchi utomton. There exists deterministic Büchi utomton B with L(B) = L(A) nd B A 2. Proof scheme: Use brutl powerset deterministion, rnk signtures of Wlukiewicz itertive normliztion of A dependency grph over the utomton Conclusion: the utomton cn use itself s memory structure qudrtic blow-up only. Is it true for ll ω-regulr conditions?

33 CoBüchi counter-exmple : the lnguge L n n pths: σ, π permute pths, cuts the current 0-pth. Here for n = 5: α: σ π σ π σ π DAG: time: The word α is in L n if it contins n infinite pth.

34 Automton for L n GFG cobüchi utomton for L n with n sttes: letters σ nd π permute sttes deterministiclly. letter : stte 0 go nywhere but py cobüchi (must be finitely mny times) sttes 1,..., n: do nothing Strtegy σ: try pths one fter the other. Uses memory 2 n, to ensure tht ll pths re visited.

35 Automton for L n GFG cobüchi utomton for L n with n sttes: letters σ nd π permute sttes deterministiclly. letter : stte 0 go nywhere but py cobüchi (must be finitely mny times) sttes 1,..., n: do nothing Strtegy σ: try pths one fter the other. Uses memory 2 n, to ensure tht ll pths re visited. Theorem Any deterministic utomton for L n hs Ω(2 n ) sttes. CoBüchi (nd prity) GFG utomt cn provide both succinctness nd sound behviour with respect to gmes. Question: Cn we effectively use them?

36 Recognizing GFG utomt Question: Given n utomton A, is it GFG? Theorem The complexity of deciding GFG-ness is in NP for Büchi utomt P for cobüchi utomt (involved proof) t lest s hrd s solving prity gmes (NP conp) for prity utomt. Open Problems Is it in P for ny fixed cceptnce condition? Is it equivlent to prity gmes in the generl cse?

37 Summry nd conclusion Results GFG utomt cpture good properties of deterministic utomt. Inclusion is in P, but Complementtion Deterministion. Conditions Büchi nd lower: GFG Deterministic. Conditions cobüchi nd higher: exponentil succinctness. Recognizing GFG cobüchi is in P. Open Problems Cn we build smll GFG utomt in systemtic wy? Complexity of deciding GFG-ness for prity utomt? Exct cost of Büchi GFG deterministion?