. There is a growing need for reliable methods of designing correct reactive systems such as computer operating systems and air traffic control systems. It is widely agreed that certain formalisms such as temporal logic, when coupled with automated reasoning support, provide the most effective and reliable means of specifying and ensuring correct behavior of such systems. This paper discusses known complexity and expressiveness results for a number of such logics in common use and describes key technical tools for obtaining essentially optimal mechanical reasoning algorithms. However, the emphasis is on underlying intuitions and broad themes rather than technical intricacies. 1 Introduction There is a growing need for reliable methods of designing correct reactive systems. These systems are characterized by ongoing, typically nonterminating and highly nondeterministic behavior. Examples include operating systems, network protocols, and air traffic control systems. There is w...

We consider two-player games played for an infinite number of rounds, with ω-regular winning conditions. The games may be concurrent, in that the players choose their moves simultaneously and independently, and probabilistic, in that the moves determine a probability distribution for the successor state. We introduce quantitative game µ-calculus, and we show that the maximal probability of winning such games can be expressed as the fixpoint formulas in this calculus. We develop the arguments both for deterministic and for probabilistic concurrent games; as a special case, we solve probabilistic turn-based games with ω-regular winning conditions, which was also open. We also characterize the optimality, and the memory requirements, of the winning strategies. In particular, we show that while memoryless strategies suffice for winning games with safety and reachability conditions, Büchi conditions require the use of strategies with infinite memory. The existence of optimal strategies, as opposed to ε-optimal, is only guaranteed in games with safety winning conditions.

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A procedure for the analysis of state spaces is called symbolic if it manipulates not individual states, but sets of states that are represented by constraints. Such a procedure can be used for the analysis of infinite state spaces, provided termination is guaranteed. We present symbolic procedures, and corresponding termination criteria, for the solution of infinite-state games, which occur in the control and modular verification of infinite-state systems. To characterize the termination of symbolic procedures for solving infinite-state games, we classify these game structures into four increasingly restrictive categories: 1. Class 1 consists of infinite-state structures for which all safety and reachability games can be solved...

We prove that every monadic second-order property of the unfolding of a transition system is a monadic second-order property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for one of them but not for the other.

Dynamic programs, or fixpoint iteration schemes, are useful for solving many problems on state spaces, including model checking on Kripke structures ("verification"), computing shortest paths on weighted graphs ("optimization"), computing the value of games played on game graphs ("control"). For Kripke structures, a rich fixpoint theory is available in the form of the -calculus. Yet few connections have been made between different interpretations of fixpoint algorithms. We study the question of when a particular fixpoint iteration scheme ' for verifying an !-regular property on a Kripke structure can be used also for solving a two-player game on a game graph with winning objective. We provide a sufficient and necessary criterion for the answer to be a rmative in the form of an extremal-model theorem for games: under a game interpretation, the dynamic program' solves the game with objective if and only if both (1) under an existential interpretation on Kripke structures,' is equivalent to 9, and (2) under a universal interpretation on Kripke structures,' is equivalent to 8. In other words,' is correct on all two-player game graphs i it is correct on all extremal game graphs, where one or the other player has no choice of moves. The theorem generalizes to quantitative interpretations, where it connects two-player games with costs to weighted graphs. While the standard translations from !-regular properties to the-calculus violate (1) or (2), we give a translation that satisfies both conditions. Our construction, therefore, yields fixpoint iteration schemes that can be uniformly applied on Kripke structures, weighted graphs, game graphs, and game graphs with costs, in order to meet or optimize a given !-regular objective.

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The Horn -calculus is a logic programming language allowing arbitrary nesting of least and greatest fixed points. The Horn -programs can naturally expresses safety and liveness properties for reactive systems. We extend the setbased analysis of classical logic programs by mapping arbitrary -programs into "uniform" -programs. Our two main results are that uniform -programs express regular sets of trees and that emptiness for uniform -programs is EXPTIME-complete. Hence we have a nontrivial decidable relaxation for the Horn -calculus. In a different reading, the results express a kind of robustness of the notion of regularity: alternating Rabin tree automata preserve the same expressiveness and algorithmic complexity if we extend them with pushdown transition rules (in the same way B uchi extended word automata to canonical systems).

Strategies in repeated games can be classified as to whether or not they use memory and/or randomization. We consider Markov decision processes and 2-player graph games, both of the deterministic and probabilistic varieties. We characterize when memory and/or randomization are required for winning with respect to various classes of-regular objectives, noting particularly when the use of memory can be traded for the use of randomization. In particular, we show that Markov decision processes allow randomized memoryless optimal strategies for all Müller objectives. Furthermore, we show that 2-player probabilistic graph games allow randomized memoryless strategies for winning with probability 1 those Müller objectives which are upward-closed. Upward-closure means that if a set of infinitely repeating vertices is winning, then all supersets of are also winning. 1

For an !-word language L, the derived tree language Path(L) is the language of trees having all their paths in L. We consider the hierarchies of deterministic automata on words and nondeterministic automata on trees with Rabin conditions in chain form. We show that L is on some level of the hierarchy of deterministic word automata iff Path(L) is on the same level of the hierarchy of nondeterministic tree automata. 1