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Probabilistic and Topological Semantics for Timed Automata
"... Like most models used in modelchecking, timed automata are an idealized mathematical model used for representing systems with strong timing requirements. In such mathematical models, properties can be violated, due to unlikely (sequences of) events. We propose two new semantics for the satisfactio ..."
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Like most models used in modelchecking, timed automata are an idealized mathematical model used for representing systems with strong timing requirements. In such mathematical models, properties can be violated, due to unlikely (sequences of) events. We propose two new semantics for the satisfaction of LTL formulas, one based on probabilities, and the other one based on topology, to rule out these sequences. We prove that the two semantics are equivalent and lead to a PSPACEComplete modelchecking problem for LTL over finite executions.
BanachMazur Games on Graphs
, 2008
"... We survey determinacy, definability, and complexity issues of BanachMazur games on finite and infinite graphs. Infinite games where two players take turns to move a token through a directed graph, thus tracing outaninfinitepath, havenumerousapplicationsindifferentbranchesofmathematics andcomputer s ..."
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We survey determinacy, definability, and complexity issues of BanachMazur games on finite and infinite graphs. Infinite games where two players take turns to move a token through a directed graph, thus tracing outaninfinitepath, havenumerousapplicationsindifferentbranchesofmathematics andcomputer science. In the usual format, the possible moves of the players are given by the edges of the graph; in each move a player takes the token from its current position along an edge to a next position. In BanachMazur games the players instead select in each move a path of arbitrary finite length rather thanjustanedge. Inbothcasestheoutcomeofaplayisaninfinitepath. Awinningconditionisthus given by a set of infinite paths which is often specified by a logical formula, for instance from S1S, LTL, or firstorder logic. BanachMazur gameshavealongtraditionindescriptivesettheoryandtopology, andtheyhaverecentlybeenshowntohaveinterestingapplications alsoincomputerscience,forinstanceforplanning in nondeterministic domains, for the study of fairness in concurrent systems, and for the semantics of timed automata. It turns out that BanachMazur games behave quite differently than the usual graph games. Often they admit simpler winning strategies and more efficient algorithmic solutions. For instance,
BanachMazur Games with Simple Winning Strategies
"... We discuss several notions of ‘simple ’ winning strategies for BanachMazur games on graphs, such as positional strategies, movecounting or lengthcounting strategies, and strategies with a memory based on finite appearance records (FAR). We investigate classes of BanachMazur games that are determ ..."
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We discuss several notions of ‘simple ’ winning strategies for BanachMazur games on graphs, such as positional strategies, movecounting or lengthcounting strategies, and strategies with a memory based on finite appearance records (FAR). We investigate classes of BanachMazur games that are determined via these kinds of winning strategies. BanachMazur games admit stronger determinacy results than classical graph games. For instance, all BanachMazur games with ωregular winning conditions are positionally determined. Beyond the ωregular winning conditions, we focus here on Muller conditions with infinitely many colours. We investigate the infinitary Muller conditions that guarantee positional determinacy for BanachMazur games. Further, we determine classes of such conditions that require infinite memory but guarantee determinacy via movecounting strategies, lengthcounting strategies, and FARstrategies. We also discuss the relationships between these different notions of determinacy.
When a system is fairly correct
"... characterisation of fairness. We also derive a notion of a fairly correct system and sketch its application. ..."
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characterisation of fairness. We also derive a notion of a fairly correct system and sketch its application.
New Perspectives on Fairness
"... Fairness is an important concept that appears repeatedly in various forms in different areas of computer science, and plays a crucial role in the semantics and verification of reactive systems. Entire books are devoted to the notion of fairness—see, for instance, the monograph by Nissim Francez publ ..."
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Fairness is an important concept that appears repeatedly in various forms in different areas of computer science, and plays a crucial role in the semantics and verification of reactive systems. Entire books are devoted to the notion of fairness—see, for instance, the monograph by Nissim Francez published in 1986—, and researchers in our community have painstakingly developed a taxonomy of various fairness properties that appear in the literature, such as unconditional fairness, weak fairness, strong fairness, and so on. This research is definitely important in light of the plethora of notions of fairness that have been proposed and studied in the literature. But when is a temporal property expressing a fairness requirement? The authors of this column have recently developed a very satisfying answer to this fundamental question by offering three equivalent characterizations of “fairness properties ” in the setting of lineartime temporal logic: a languagetheoretic, a topological, and a gametheoretic characterization. This survey discusses these recent results in a very accessible fashion, and provides also a beautiful link between the study of fairness and classic probability theory. I trust that you will enjoy reading it as much as I did. It is not often that one sees notions and results from several areas of mathematics and computer science combine so well to offer a formalization of a concept that confirms our intuition about it.
On Fairness and Randomness
"... We investigate the relation between the behavior of nondeterministic systems under fairness constraints, and the behavior of probabilistic systems. To this end, first a framework based on computable stopping strategies is developed that provides a common foundation for describing both fair and prob ..."
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We investigate the relation between the behavior of nondeterministic systems under fairness constraints, and the behavior of probabilistic systems. To this end, first a framework based on computable stopping strategies is developed that provides a common foundation for describing both fair and probabilistic behavior. On the basis of stopping strategies it is then shown that fair behavior corresponds in a precise sense to random behavior in the sense of MartinLöf’s definition of randomness. nondeterministic systems. Under this perspective the question is investigated what probabilistic properties are needed in such an implementation to guarantee (with probability one) certain required fairness properties in the behavior of the probabilistic system. Generalizing earlier concepts of ɛbounded transition probabilities, we introduce the notion of divergent probabilistic systems, which enables an exact characterization of the fairness properties of a probabilistic implementation. Looking beyond pure fairness properties, we also investigate what other qualitative system properties are guaranteed by probabilistic implementations of fair nondeterministic behavior. This leads to a completeness result which generalizes a wellknown theorem by Pnueli and Zuck.