On Determinisation of History-Deterministic Automata.

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1 On Deterministion of History-Deterministic Automt. Denis Kupererg Mich l Skrzypczk University of Wrsw YR-ICALP 2014 Copenhgen

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

3 History-deterministic utomt Ide : Nondeterminism cn e resolved without knowledge out the future.

4 History-deterministic utomt Ide : Nondeterminism cn e resolved without knowledge out the future. Introduced independently in symolic representtion (Henzinger, Pitermn 06) simplifiction qulittive models (Colcomet 09) replce determinism Applictions synthesis rnching time verifiction tree lnguges (Boker, K, Kupfermn, S 12)

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

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

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

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

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

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

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

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

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

14 Some properties of H-D utomt Composition with gmes: A G hs sme winner s G with condition L(A). Theorem (Boker, K, Kupfermn, S 12) Let A e n utomton for L A ω. Then the tree version of A recognizes {t : ll rnches of t re in L} if nd only if A is H-D. Fct Let A e H-D on finite words. Then A contins n equivlent deterministic utomton. Therefore, H-D is not useful concept on finite words. Wht out infinite words?

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

16 A H-D Büchi exmple Büchi condition: Run is ccepting if infinitely mny Büchi trnsitions re seen. x x x x Lnguge: [(x + x) (xx + xx)] ω

17 Determiniztion of Büchi H-D Theorem Let A H-D Büchi utomton. There exists deterministic utomton B with L(B) = L(A) nd B A 2. Proof scheme: Use rutl 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 low-up only. Is it true for ll ω-regulr conditions?

18 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.

19 Automton for L n H-D cobüchi utomton for L n with n sttes: letters σ nd π permute sttes deterministiclly. letter : stte 0 go nywhere ut py cobüchi (must e 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) H-D utomt cn provide oth succinctness nd sound ehviour with respect to gmes.

20 Automton for L n H-D cobüchi utomton for L n with n sttes: letters σ nd π permute sttes deterministiclly. letter : stte 0 go nywhere ut py cobüchi (must e 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) H-D utomt cn provide oth succinctness nd sound ehviour with respect to gmes. Question: Cn we effectively use them?

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

22 Summry nd conclusion Results H-D utomt cpture good properties of deterministic utomt. Inclusion is in P, ut Complementtion Deterministion. Conditions Büchi nd lower: H-D Deterministic. Conditions cobüchi nd higher: exponentil succinctness. Recognizing H-D cobüchi is in P. Open Prolems Cn we uild smll H-D utomt in systemtic wy? Complexity of deciding H-D-ness for prity utomt?

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