ExplicitMullerGamesare PTIME

Transcription

1 Foundations of Software Technology and Theoretical Computer Science (Bangalore) Editors: R. Hariharan, M. Mukund, V. Vinay; pp - ExplicitMullerGamesare PTIME Florian Horn horn@liafa.jussieu.fr LIAFA Université Paris 7 Case 7014, Paris cedex 13 France LI7 RWTH Ahornstraße Aachen Germany LABRI Université Bordeaux 1 351, cours de la Libération Talence cedex France ABSTRACT. Regular games provide a very useful model for the synthesis of controllers in reactive systems. The complexity of these games depends on the representation of the winning condition: if it is represented through a win-set, a coloured condition, a Zielonka-DAG or Emerson-Lei formulae, the winner problem is PSPACE-complete; if the winning condition is represented as a Zielonka tree, thewinnerproblembelongsto NPandco-NP.Inthispaper,weshowthatexplicitMullergamescan be solved in polynomial time, and provide an effective algorithm to compute the winning regions. 1 Introduction There has been a long history of using infinite games to model reactive processes[bl69, PR89].Thesystemisrepresentedasagamearena,i.e.agraphwhosestatesbelongeitherto Eve(controller) or to Adam(environment). The desired behaviour is represented as an ω- regular winning condition, which naturally expresses temporal specifications and fairness assumptions of transition systems[mp92]. The game is played by moving a token on the arena:whenitisinoneofeve sstates,shechoosesitsnextlocationamongthesuccessors ofthecurrentstate;whenitisinoneofadam sstates,hechoosesitsnextlocation. The resultofplayingthegamefor ωmovesisaninfinitepathofthegraph.evewinsifthepath satisfies the specification, and Adam wins otherwise. A fundamental determinacy result of Büchi and Landweber shows that from any initial state,oneoftheplayershasawinningstrategy[bl69]. Theproblemofthewinnerisin ThisworkwassupportedinpartbytheFrenchANRAVERISS. c Florian Horn; licensed under Creative Commons License-NC-ND

2 2 EXPLICIT MULLER GAMES ARE PTIME PSPACE for any reasonable representation of the winning condition[mcn93, NRY96], but its exact complexity depends on how the winning condition is represented. For example, if the winning condition is represented as a Zielonka tree[zie98], the problem of the winner isin NP co-np [DJW97].HunterandDawarlistin[HD05]fiveother generalpurpose representations: explicit Muller, win-set, Muller, Zielonka DAGs, Emerson-Lei. They show that the problem of the winner is PSPACE-hard for the last four representations, and leave thecomplexityofexplicitmullergamesasanopenquestion.inthispaper,weanswerthis question: the winner problem in explicit Muller games belongs to PTIME. We provide an effective cubic algorithm computing the winning regions of the players. Outline of the paper. Section 2 recalls the classical notions about regular games, and Section 3 gives an overview of the different representations of regular winning conditions. In Section 4, we introduce the notions of semi-alternation and sensibleness, and show that any explicit Muller game can be translated in polynomial time into a semi-alternating and sensiblegame.wealsostudythefamilyofgameswhereevewinsifallthestatesarevisited infinitely often. These games are used repeatedly in our algorithm, which is the subject of Section 5. 2 Definitions We recall here several classical notions about regular games, and refer the reader to[gtw02] for more details. Arenas. Anarena Aisadirectedgraph (Q, T )withoutdeadlockswhosestatesarepartitionedbetweeneve sstates(q E,representedas s)andadam sstates(q A,representedas s).a sub-arena A B of Aistherestrictionof AtoasubsetBof QsuchthateachstateofBhasa successor in B. Plays and Strategies. Aplayonthearena Aisa(possiblyinfinite)sequence ρ = ρ 0 ρ 1...ofstatessuchthat i < ρ 2, (ρ i, ρ i+1 ) T.ThesetofoccurringstatesisOcc(ρ) = {q i N, ρ i =q},andtheset oflimitstatesisinf(ρ) = {q i N, ρ i =q}. AstrategyofEveonthearena Aisafunction σfrom Q Q E to Qsuchthat w Q, q Q E, (q, σ(wq)) T.Strategiescanalsobedefinedasstrategieswithmemory.Inthiscase, σis atriple (M, σ u, σ n ),wheremisthe(possiblyinfinite)setofmemorystates, σ u : (M Q) M isthememoryupdatefunction,and σ n : (M Q) Qisthenext-movefunction. Adam s strategiesaredefinedinasimilarway. Astrategyisfinite-memoryifMisafiniteset,and memoryless if M is a singleton. A(finiteorinfinite)play ρisconsistentwith σif, i < ρ 2, ρ i Q E ρ i+1 = σ(ρ 0...ρ i ).

3 FLORIAN HORN FSTTCS Traps and Attractors. TheattractorofEvetothesetUinthearena A,denotedAttr E (U, A),isthesetofstatesfrom whereevecanforcethetokentogotothesetu.itisdefinedinductivelyby: U 0 = U U i+1 = U i {q Q E, r U i (q,r) T } U i {q Q A r, (q,r) E r U i } Attr E (U, A) = i>0u i ThecorrespondingattractorstrategytoUforEveisapositionalstrategy σ U suchthatfor anystateq Q E (Attr E (U, A) \U),q U i+1 σ U (q) U i. ThedualnotionofatrapforEvedenotesasetfromwhereEvecannotescape,unless Adamallowshertodoso:asetUisatrapforEveifandonlyif q U Q E, (q,r) T r Uand q U Q A, r U (q,r) T.Noticethatatrapisalwaysasub-arena. Regular Winning Conditions. Aregularwinningconditionisaspecification Φ Q ω oninfiniteplayswhichdependsonly onthesetofstatesvisitedinfinitelyoften:inf(ρ) =Inf(ν) (ρ Φ ν Φ).Evewins aplay ρif ρ Φ. Adamwinsif ρ / Φ. Regularwinningconditionscanbedescribedin different ways, which are presented in the next section. Winning Strategies. Givenawinningcondition Φandastateq Q,astrategy σiswinningforevefromqifany playstartinginqandconsistentwith σiswinningforeve.thewinningregionofeveisthe set of states from where she has a winning strategy. Adam s winning strategies and regions aredefinedinasimilarway. 3 Representations of regular conditions The most straightforward way to represent a regular condition F is to provide an explicit listofsetsofstates F 1,...,F l : F = {F i 1 i l}.aplay ρiswinningforeveifand onlyifinf(ρ) F.ThecomplexityoftheseexplicitMullergamesisthesubjectofthispaper. There are several other ways to represent regular conditions. In win-set games[mcn93], thewinnerdependsonlyonasubsetrofrelevantstates,andthewinningcondition Rlists onlysubsetsofr: ρiswinningforeveifinf(ρ) R R.Mullergamesextendthisideaby addingacolouringfunction χ,fromthestatestoasetofcolours C.Thewinningcondition Flistssubsetsof C,and ρiswinningforeveif χ(inf(ρ)) F.Emerson-Leigames[EL85] provideabooleanformula ϕ,whosevariablesarethestatesof Q.Aplay ρiswinningfor EveifthevaluationInf(ρ) trueand Q \Inf(ρ) falsesatisfies ϕ.

4 4 EXPLICIT MULLER GAMES ARE PTIME Zielonka s representation of regular conditions[zie98] proceeds from a different approach: it focuses on alternation between sets winning for Eve and sets winning for Adam. In his split tree(usually called Zielonka tree ), the nodes are labelled by sets of colours, the childrenaresubsetsoftheirparentwith Cattheroot,andachildanditsparentarenever winning for the same player. Finally, Zielonka DAGs[HD05] are the result of merging the nodesofthezielonkatreewiththesamelabels. The complexity of regular games depends directly on the representation of the winning condition: THEOREM 1.[DJW97] The problem of the winner in regular games whose winning condition is represented by a Zielonka tree is in NP co-np. THEOREM 2.[HD05] The problem of the winner in win-set games, Muller games, Zielonka DAG games, and Emerson-Lei games are PSPACE-complete. For explicit Muller games, the best complexity result so far was the membership of the winner problem in PSPACE, derived from the all-purpose algorithms of[mcn93] and [NRY96].ThemainresultofthispaperisTheorem3: THEOREM 3. The winner problem of explicit Muller games can be solved in polynomial time. 4 Useful notions for explicit Muller games We first define three properties of explicit Muller games. A game is: 1. semi-alternatingifthereisnotransitionbetweentwostatesofadam(buttherecanbe between two states of Eve); 2. sensibleifeachsetin Finducesasub-arenaof A; 3. orderedforinclusionifi <j F i F j. Our algorithm for explicit Muller games, Algorithm 1, relies on the fact that its input satisfies these three properties. However, this does not restrict the generality of our result, since any explicit Muller game can be transformed in polynomial time into an equivalent semi-alternating, sensible, and ordered game of polynomial size. The semi-alternation transformationconsistsinreplacingeachstateq Q A ofadambyapairofstatesr Q E,s Q A,asinFigure1.Eachsetcontainingqinthewinningconditionismodifiedaccordingly: F (λq.(r, s))f. This is where the classical alternation transformation fails: addingastatetoeachtransitionleadstoanexponentialblow-upinthesizeofthewinning condition. Agamecanbemadesensiblebyremovingfrom Fallthesetsthatdonotinducea sub-arenaof A:nomatterhowEveandAdamplay,thelimitoftheplayisaasub-arena, sothemodificationistransparentwithrespecttodecidingthewinningnatureofaplay,a strategy, or a state. Finally, ordering the sets for inclusion can be done in quadratic time. Thegamesoftheform (A, {Q}),whereEvewinsifandonlyifthetokenvisitsall the states infinitely often, play an important part in our solution to explicit Muller games. These games, which have also been studied in routing problems[dk00, IK02], are easy to solveandthereisalwaysonlyonewinnerinthewholegame:

5 FLORIAN HORN FSTTCS q r s (a) Original arena A (b) Semi-alternating arena A Figure 1: Semi-alternating arena construction PROPOSITION4. Let Abeanarena,and Gbethegame (A, {Q}). Either,foranystate q Q,Eve sattractortoqisequalto Q,andEvewinseverywherein G,orthereisastate q QsuchthatAttr E ({q}, A) = Q,andAdamwinseverywherein G. PROOF. Inthefirstcase,Evecanwinwithastrategywhosememorystatesarethestates of Q:inthememorystateq,sheplaystheattractorstrategytoq,untilthetokenreachesit. Sheupdatesthenhermemorytothenextstater,inacircularway.Inthesecondcase,Adam canwinsurelywithanytrappingstrategyoutofattr E ({q}, A):ifthetokenevergetsoutof Attr E ({q}, A),itnevergoesback. 5 Solving explicit Muller games in PTIME Our algorithm takes as input a semi-alternating, sensible explicit Muller game whose winning condition is ordered for inclusion; it returns the winning regions of the players. Each setin Fisconsideredatmostonce,startingwiththe(smallest)set F 1. Ateachstep,the operationofaset F i modifiesthearenaandthewinningconditioninoneofthefollowing ways: IfAdamwins (A Fi, {F i }), F i isremovedfrom F. IfEvewins (A Fi, {F i }),and F i isatrapforadamin A,Eve sattractorto F i in A, Attr E (F i, A),isremovedfrom A(andaddedtothewinningregionofEve),andallthesets intersectingattr E (F i, A)areremovedfrom F. IfEvewins (A Fi, {F i }),and F i isnotatrapforadamin A,anewstate F i,described in Figure 2, is added to A with the following attributes: F i isastateofadam; thepredecessorsof F i areallthestatesofevein F i ; thesuccessorsof F i arethesuccessorsoutside F i ofthestatesofadamin F i. Furthermore,thestate F i isaddedtoallthesupersetsof F i in F,and F i itselfisremoved from F. Theimportantcase,fromanintuitivepointofview,isthelastone:itcorrespondstoa threat ofevetowinbyvisitingexactlythestatesof F i.adamhastoanswerbygetting out,buthecanchoosehisexitfromanyofhisstates.noticethatitwouldnotdotosimply replacethewholeregion F i bythestate F i :asinfigure2,adammaybeabletoavoida

6 6 EXPLICIT MULLER GAMES ARE PTIME b a b a c {a,b} c F = {{a,b}, {a,b,c}} F = {{a,b,c, {a,b} }} (a) Before (b) After Figure2:RemovalofasetinanexplicitMullercondition stateof F i inalargerarena,evenifheisincapableofdoingsoin A Fi. Asonlyonestateisaddedeachstep,thenumberofstatesinthegameisboundedby A + F.ThewholeprocedureisdescribedasAlgorithm1. Intheproofofcorrectness,weuse typewriterfontstodenotethemodifiedarenaand condition,and calligraphfontstodenotetheoriginalgame.furthermore,wedenoteby F Fi theintersectionof Fand P(F i ),i.e.thesetsof Fthatarealsosubsetsof F i.wecannow proceed to the three main lemmas: LEMMA5.If,inthecourseofarunofAlgorithm1,thegame (A Fi, {F i })iswinningforeve atline6,thenevewinseverywhereinthegame (A Fi, F Fi ). PROOF. Let H 1,...,H k bethesetsof F Fi suchthat (A H j, {H j })waswinningforeveinthe runofalgorithm1.noticethat F i itselfisoneofthesestates,say H k.the σ j sdenoteher correspondingwinningstrategies. Webuildastrategy σforevein A Fi,whosememory statesarestacksofpairs (H j, ρ j ). Atanytime, ρ j isaplayof A H jwhichcanbeextended bythecurrentstateq.theinitialmemorystateis (H k, ε),andtheoperationof σwhenthe memorystateis (H j,w)andthecurrentstateisqisdescribedbelow: 1. Ifq/ H j,thetoppairisremoved,andtheprocedurerestartsatstep1withthenew memory.noticethatitmayinvolvefurtherpopsifqstilldoesnotbelongtothetop set. 2. IfqisastateofEve,and σ j (wq)isanewstate H h,thememoryismodifiedasfollows:w becomeswqh h,andanewpair (H h, ε)ispushedatthetopofthestack.theprocedure restartsatstep2withthenewmemory.noticethatitmayinvolvefurtherpushesif σ h (q)isalsoanewstate. 3. Thenewmemorystateis (H j,wq);ifqbelongstoeve,sheplays σ j (wq). Weclaimthat σiswinningforeveinthegame (A Fi, F Fi ).Let ρbeaplayconsistentwith σ,and H j bethehighestsetthatisneverunstacked.wedenoteby ρ j the(infinite)limitof the play part.as ρ j isconsistentwith σ j,inf(ρ j ) = H j.furthermore,inf(ρ) Inf(ρ j ) Q andinf(ρ) H j.so,inf(ρ) = H j F,andLemma5follows.

7 FLORIAN HORN FSTTCS Input:AnexplicitMullergame (A, F) Output: The winning regions of Eve and Adam A = (Q,Q E,Q A,T) A = (Q, Q E, Q A, T ); F F; W E ; whilef = do F i pop(f); ifevewins (A Fi, {F i })then iff i isatrapforadaminathen removeattr E (F i,a)fromaandaddittow E ; removeallthesetsintersectingattr E (F i,a)fromf; else addastate F i toq A ; addtransitionsfromf i Q E to F i ; addtransitionsfrom F i tot(f i Q A ) \F i ; add F i toallthesupersetsoff i inf; end end end returnw E Q,Q Q Algorithm 1: Polynomial algorithm for explicit Muller games ForAdam,theproblemisalittlemorecomplex:weneedtwolemmas,whoseproofs are mutually recursive: LEMMA6. If,inthecourseofarunofAlgorithm1,thegame (A Fi, {F i })iswinningfor Adamatline6,thenAdamwinseverywhereinthegame (A Fi, F Fi ). LEMMA7. If,inthecourseofaarunofAlgorithm1,thegame (A Fi, {F i })iswinningfor Eveatline6,thenAdamwinseverywhereinthegame (A Fi, F Fi \ {F i }). PROOF. Westartwiththe(simpler)proofofLemma7. Let H 1,...,H k bethemaximal sets,withrespecttoinclusion,of F Fi.Thereisawinningstrategy τ j foradamineach H j : ifadamwon (A H j, {H j }),itisawinningstrategyforthegame (A H j, F H j)(recursiveuseof Lemma6);ifEvewon (A H j, {H j }),itisastrategyforthegame (A H j, F H j \ H j )(recursive useoflemma7).thestrategy τforadamin (A Fi, {F Fi })usesktop-levelmemorystates toswitchbetweenthe {τ j } 1 j k.adamremainsinatop-levelmemorystatejonlyaslong asthetokenisin H j. Assoonasitgetsout,heupdatesitto (j modk) +1. Hisactions when the top-level memory state is j are described below: ifhewon (A H j, {H j }),heplays τ j ; ifevewon (A H j, {H j }),heplays τ j unlesshecangetoutof H j. Weclaimthat τiswinningforadamin (A Fi, F Fi ).Anyplay ρconsistentwith τfalls in exactly one of the three following categories:

8 8 EXPLICIT MULLER GAMES ARE PTIME The top-level memory of τ is not ultimately constant; thus Inf(ρ) is not included in anyofthe H j s,and ρiswinningforadam. Thetop-levelmemoryof τisultimatelyconstantatj,and (A H j, {H j })waswinningfor Adam; ρisultimatelyaplayof A H jconsistentwith τ j,so ρiswinningforadam. Thetop-levelmemoryof τisultimatelyconstantatj,and (A H j, {H j })waswinningfor Eve; ρisultimatelyaplayof A H jconsistentwith τ j,soevecanwinonlybyvisiting allthestatesof H j.but H j isnotatrapforadam,andthedefinitionof τimpliesthat Adamleavesassoonaspossible.So,atleastoneofthestatesof H j wasnotvisited, and ρ is winning for Adam. ThiscompletestheproofofLemma7.TheproofofLemma6ismoreinvolved,dueto thenecessitytoavoidatleastoneofthestatesof F i.byproposition4thereisastateqin F i suchthatx =Attr E ({q},a Fi )isnotequalto A Fi.Itfollowsfromthedefinitionof A Fi that neither F i Xnor F i \Xisempty.Adam sstrategyisthenexactlythesameasintheproof oflemma7,withtheprovisionthatadamnevermovesfrom F i \XtoX:thisguarantees thatthetokencannotvisitinfinitelyoftenallthestatesof F i,andcompletestheproofof Lemma 6. ThecorrectnessofAlgorithm1followsfromLemmas5,6,and7:thefirstoneguaranteesthatthestatesinW E QarewinningforEve,andtheothersthatthestatesremainingat theendofalgorithm1arewinningforadam. Aboutcomplexity,thereareatmost F loopsinarun,andthemosttime-consuming operationistocomputethewinnerofthegames (A Fi, {F i }),whicharequadraticin A ( A + F ).Thus,theworst-casetimecomplexityofAlgorithm1isO( F ( A + F ) 2 ), which completes the proof of Theorem 3. 6 Conclusion We have shown that the complexity of the winner problem in explicit Muller game belongs to PTIME, and provided a cubic algorithm computing the winning regions of both players. It follows from the usual reduction between two-player games and tree automata that the emptiness problem of explicit Muller tree automata can also be solved in polynomial time; a natural question is whether this is also the case for other automata problems. The existence of a polynomial algorithm for parity games remains an open problem: representing explicitly a parity condition incurs an exponential blow-up in size. Bibliography [BL69] J. Richard Büchi and Lawrence H. Landweber. Solving Sequential Conditions by Finite-State Strategies. Transactions of the American Mathematical Society, 138: , [DJW97] Stefan Dziembowski, Marcin Jurdziński, and Igor Walukiewicz. How Much MemoryisNeededtoWinInfiniteGames?InProceedingsofthe12thAnnualIEEE Symposium on Logic in Computer Science, LICS 97, pages IEEE Computer Society, 1997.

9 FLORIAN HORN FSTTCS [DK00] Michael J. Dinneen and Bakhadyr Khoussainov. Update Networks and Their Routing Strategies. In Proceedings of the 26th International Workshop on Graph- Theoretic Concepts in Computer Science, WG 00, volume 1928 of Lecture Notes in Computer Science, pages Springer-Verlag, [EL85] E. Allen Emerson and Chin-Laung Lei. Modalities for Model Checking: Branching Time Strikes Back. In Proceedings of the 12th Annual ACM Symposium on Principles of Programming Languages, POPL 85, pages 84 96, [GTW02] Erich Grädel, Wolfgang Thomas, and Thomas Wilke, editors. Automata, Logics, and Infinite Games: A Guide to Current Research[outcome of a Dagstuhl seminar, February 2001], volume 2500 of Lecture Notes in Computer Science. Springer-Verlag, [HD05] Paul Hunter and Anuj Dawar. Complexity Bounds for Regular Games. In Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science, MFCS 05, volume 3618 of Lecture Notes in Computer Science, pages Springer-Verlag, [IK02] Hajime Ishihara and Bakhadyr Khoussainov. Complexity of Some Infinite Games Played on Finite Graphs. In Proceedings of the 28th International Workshop on Graph- Theoretic Concepts in Computer Science, WG 02, volume 2573 of Lecture Notes in Computer Science, pages Springer-Verlag, [McN93] Robert McNaughton. Infinite Games Played on Finite Graphs. Annals of Pure and Applied Logic, 65(2): , [MP92] Zohar Manna and Amir Pnueli. The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer-Verlag, [NRY96] Anil Nerode, Jeffrey B. Remmel, and Alexander Yakhnis. McNaughton Games and Extracting Strategies for Concurrent Programs. Annals of Pure and Applied Logic, 78(1 3): , [PR89] Amir Pnueli and Roni Rosner. On the Synthesis of a Reactive Module. In Proceedings of the 16th Annual ACM Symposium on Principles of Programming Languages, POPL 89, pages , [Zie98] Wieslaw Zielonka. Infinite Games on Finitely Coloured Graphs with Applications to Automata on Infinite Trees. Theoretical Computer Science, 200(1 2): , This work is licensed under the Creative Commons Attribution- NonCommercial-No Derivative Works 3.0 License.