Memory and Delay in Regular Infinite Games

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Memory and Delay in Regular Infinite Games Memory and Delay in Regular Infinite Games Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Informatiker Michael Holtmann aus Düsseldorf Berichter: Universitätsprofessor Dr. Dr.h.c. Wolfgang Thomas Universitätsprofessor Dr. Erich Grädel Tag der mündlichen Prüfung: 9. Mai 2011 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Zusammenfassung Unendliche Zwei-Personen-Spiele sind ein ausdrucksstarkes und anpassungsfä- higes Werkzeug bei der Modellierung und Verifikation reaktiver Systeme. Es ist allgemein bekannt, dass beispielsweise die Konstruktion eines Controllers, der beliebig lange in der Umgebung eines Systems agiert, reduziert werden kann auf die Berechnung einer Gewinnstrategie in einem unendlichen Spiel (Pnueli und Rosner, 1989). Für die Klasse der Spiele mit regulärer Gewinnbedingung haben Büchi und Landweber 1969 gezeigt, dass einer der Spieler eine Gewinnstrategie hat, die durch einen endlichen Automaten realisierbar ist. Basierend auf diesem grundlegenden Resultat hat die Forschung viele Versuche unternommen, die Lö- sungsverfahren für unendliche Spiele weiterzuentwickeln. Dies betrifft sowohl den zeitlichen Aufwand, den man benötigt, um Gewinnstrategien zu berechnen, als auch den Speicherbedarf, um diese dann zu programmieren. In der vorliegen- den Arbeit geht es hauptsächlich um den zweiten Aspekt. Es werden zwei Proble- me betrachtet, die im Hinblick auf die Konstruktion kleiner Controllerspezifika- tionen von Bedeutung sind. Im ersten Teil der Arbeit beschäftigen wir uns mit dem klassischen Problem, endlich repräsentierbare Gewinnstrategien zu berechnen, die von Automaten mit möglichst wenig Speicher (das heißt mit möglichst wenig Zuständen) realisiert werden können. Auch wenn ein Resultat von Dziembowski et al. aus dem Jah- re 1997 besagt, dass es exotische reguläre Spiele gibt, für die Gewinnstrategien nur durch Automaten implementiert werden können, die gemessen an der Größe der Spielarena mindestens exponentiell groß sind, so erfordern die meisten prak- tischen Beispiele doch weit weniger Speicherplatz. Wir stellen einen effizienten Algorithmus für die Reduktion des für die Implementierung von Gewinnstrate- gien verwendeten Speichers vor und wenden ihn auf mehrere Klassen regulärer Bedingungen an. Außerdem zeigen wir, dass mit unserem Verfahren ein exponen- tieller Gewinn bezüglich der Speichergröße erzielt werden kann. Im zweiten Teil dieser Arbeit führen wir einen verallgemeinerten Begriff von Strategien ein. Einer der Spieler darf jeden seiner Züge für eine endliche Anzahl von Schritten hinauszögern. Mit anderen Worten, er kann bei seinen Entschei- dungen einen Vorgriff auf die Züge des Gegners ausnutzen. Dieses Szenario lässt sich beispielsweise in verteilten Systemen wiederfinden, zum Beispiel, wenn Puf- ferspeicher für die Kommunikation zwischen entfernten Komponenten eingesetzt werden. Unser Konzept von Strategien erfasst die Klasse der stetigen Operatoren und ist damit eine Erweiterung der Arbeit von Hosch und Landweber (1972) und insbesondere auch der von Büchi und Landweber (1969). Wir zeigen, dass das Problem, ob eine gegebene reguläre Spezifikation durch einen stetigen Operator erfüllt werden kann, entscheidbar ist und dass jede solche Lösung auch auf ei- ne mit beschränktem Vorgriff reduziert werden kann. Aus unseren Ergebnissen leiten wir eine verallgemeinerte Determiniertheit regulärer Bedingungen ab. Abstract Infinite two-player games constitute a powerful and flexible framework for the design and verification of reactive systems. It is well-known that, for example, the construction of a controller acting indefinitely within its environment can be reduced to the computation of a winning strategy in an infinite game (Pnueli and Rosner, 1989). For the class of regular games, Büchi and Landweber (1969) showed that one of the players has a winning strategy which can be realized by a finite-state automaton. Based on this fundamental result, many efforts have been made by the research community to improve the solution methods for infinite games. This is meant with respect to both the time needed to compute winning strategies and the amount of space required to implement them. In the present work we are mainly concerned with the second aspect. We study two problems related to the construction of small controller programs. In the first part of the thesis, we address the classical problem of computing finite-state winning strategies which can be realized by automata with as little memory, i.e., number of states, as possible. Even though it follows from a result of Dziembowski et al. (1997) that there exist exotic regular games for which the size of automata implementing winning strategies is at least exponential in the size of the game arena, most practical cases require far less space. We propose an efficient algorithm which reduces the memory used for the implementation of winning strategies, for several classes of regular conditions, and we show that our technique can effect an exponential gain in the size of the memory. In the second part of this thesis, we introduce a generalized notion of a strat- egy. One of the players is allowed to delay each of his moves for a finite num- ber of steps, or, in other words, exploit in his strategy some lookahead on the moves of the opponent. This setting captures situations in distributed systems, for example, when buffers are present in communication between remote compo- nents. Our concept of strategies corresponds to the class of continuous operators, thereby extending the work of Hosch and Landweber (1972) and, in particular, that of Büchi and Landweber (1969). We show that the problem whether a given regular specification is solvable by a continuous operator is decidable and that each continuous solution can be reduced to one of bounded lookahead. From our results, we derive a generalized determinacy of regular conditions. Acknowledgments I am very grateful to all the people who have contributed to the success of this thesis, either directly or indirectly. First and foremost, I would like to thank my supervisor Wolfgang Thomas for giving me the opportunity to work as an academic researcher. His continu- ing support and helpful advice have been of great value to me throughout the past years. It was him who suggested to me to apply for the AlgoSyn Research Training Group, where I enjoyed three years of interdisciplinary exchange. I would like to thank Erich Grädel for his kind readiness to act as a co-examiner of this thesis. I would like to thank all my co-authors whom I have had the pleasure to work with: Marcus Gelderie, Łukasz Kaiser, Christof Löding and, again, Wolf- gang Thomas. I would like to thank all my colleagues at I7 and MGI, especially those who helped me with proofreading a first draft of this thesis and who made many help- ful comments on its contents: Tobias Ganzow, Marcus Gelderie, Łukasz Kaiser, Daniel Neider, Bernd Puchala, Roman Rabinovich, Frank Radmacher and Michaela Slaats. Finally, I am deeply indebted to my friends and family for their constant sup- port throughout the past years. Especially, I would like to mention my affection- ate girlfriend Franzi for continuing encouragement and plenty of patience she had with me. Contents Introduction 1 1 Preliminaries 11 1.1BasicNotation............................... 11 1.2 WordsandFormalLanguages. 11 1.3Automata.................................. 12 1.4InfiniteGames............................... 14 2 Synopsis of Winning Conditions 19 2.1 GameswithPositionalWinningStrategies . 20 2.2GameSimulation ............................. 25 2.3 Games with Non-Positional Winning Strategies . 27 I Memory Reduction for Strategies in Infinite Games 3 An Algorithm for Memory Reduction 41 3.1 Retrospection:MealyAutomata. 42 3.2 ReductionofGameGraphs . 47 3.3 Staiger-WagnerConditions . 54 3.4 Request-ResponseConditions . 61 3.5 MullerandStreettConditions. 66 3.6Discussion ................................. 74 Conclusion of Part I 87 II Infinite Games with Finite Delay 4 Games with Delay 93 4.1DelayOperators.............................. 94 4.2 DecisionProblem . 100 ix x Contents 5 Finite Delay in Regular Games 105 5.1TheBlockGame.............................. 106 5.2 TheSemigroupGame . 109 5.3DelayValues................................ 118 6 A Concurrent Setting 125 6.1 ReductiontotheTurn-basedSetting . 126 6.2 RegularSpecifications . 129 Conclusion of Part II 133 Bibliography 137 Index 145 Introduction The theory of infinite games in computer science was initiated by work of Alonzo Church, who raised a problem today referred to as the Church Synthe- sis Problem [Chu57, Chu63]: one is given a circuit and a requirement between input sequences to the circuit and the corresponding outputs, i.e., a binary rela- tion between infinite sequences over finite alphabets. Church asked whether it was possible to algorithmically synthesize a new circuit which satisfies the re- quirement in the sense that, when faced with any input, it generates an output which is in relation to the input. Alternatively, one should be able to determine that no such circuit exists. Moreover, generation of the output should be done in an on-demand fashion, i.e., the i-th output letter should depend only on the first i letters of the input sequence. The above scenario naturally arises as a simple format of an infinite game, called Gale-Stewart game [GS53, Mos80].
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