@(x, y) is valid over all tree models. For the restricted case that all variables range over finite domains, the validity problem is decidable, and we present an algorithm for constructing the program whenever it exists. The algorithm is based on a new procedure for checking the emptiness of Rabin automata on infinite trees in time exponential in the number of pairs, but only polynomial in the number of states. This leads to a synthesis algorithm whose complexity is double exponential in the length of the given specification.

On the synthesis of a reactive module

Publisher Summary This chapter focuses on finite automata on infinite sequences and infinite trees. The chapter discusses the complexity of the complementation process and the equivalence test. Deterministic Muller automata and nondeterministic Buchi automata are equivalent in recognition power. Any nonempty Rabin recognizable set contains a regular tree and shows that the emptiness problem for Rabin tree automata is decidable. The chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree. A short overview of the work that studies the fine structure of the class of Rabin recognizable sets of trees is also presented in the chapter. Depending on the formalism in which tree properties are classified, the results fall in three categories: monadic second-order logic, tree automata, and fixed-point calculi.

Automata on infinite objects

Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. The problems of receptiveness, realizability, and controllability can be formulated as model-checking problems for alternating-time formulas. Depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators, we obtain the two alternating-time temporal logics ATL and ATLa.ATL and ATLa are interpreted over concurrent game structures. Every state transition of a concurrent game structure results from a choice of moves, one for each player. The players represent individual components and the environment of an open system. Concurrent game structures can capture various forms of synchronous composition for open systems, and if augmented with fairness constraints, also asynchronous composition. Over structures without fairness constraints, the model-checking complexity of ATL is linear in the size of the game structure and length of the formula, and the symbolic model-checking algorithm for CTL extends with few modifications to ATL. Over structures with weak-fairness constraints, ATL model checking requires the solution of 1-pair Rabin games, and can be done in polynomial time. Over structures with strong-fairness constraints, ATL model checking requires the solution of games with Boolean combinations of Buchi conditions, and can be done in PSPACE. In the case of ATLa, the model-checking problem is closely related to the synthesis problem for linear-time formulas, and requires doubly exponential time.

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Alternating-time temporal logic

The subject of this chapter is the study of formal languages (mostly languages recognizable by finite automata) in the framework of mathematical logic.

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Languages, automata, and logic

The verification problem for probabilistic concurrent finite-state program is to decide whether such a program satisfies its linear temporal logic specification. We describe an automata-theoretic approach, whereby probabilistic quantification over sets of computations is reduced to standard quantification over individual computations. Using new determinization construction for ω-automata, we manage to improve the time complexity of the algorithm by two exponentials. The time complexity of the final algorithm is polynomial in the size of the program and doubly exponential in the size of the specification.

Automatic verification of probabilistic concurrent finite state programs

Our main purpose is to present an algorithm which decides whether or not a condition 𝕮(X, Y) stated in sequential calculus admits a finite automata solution, and produces one if it exists. This solves a problem stated in [4] and contains, as a very special case, the answer to Case 4 left open in [6]. In an equally appealing form the result can be restated in the terminology of [7], [10], [15]: Every ω-game definable in sequential calculus is determined. Moreover the player who has a winning strategy, in fact, has a winning finite-state strategy, that is one which can effectively be played in a strong sense. The main proof, that of the central Theorem 1, will be presented at the end. We begin with a discussion of its consequences.

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Solving sequential conditions by finite-state strategies

Let D = be a relational system whereby D is a nonempty set and P i is an m i-ary relation on D. With D we associate the (weak) monadic second-order theory (W)MT[D] consisting of the first-order predicate calculus with individual variables ranging over D; monadic predicate variables ranging over (finite) subsets of D; monadic predicate quantifiers; and constants corresponding to P 1, P 2, ⋯ We will often use (W)MT[D] ambiguously to mean also the set of true sentences of (W)MT[D].

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Lawrence H. Landweber

Papers

Solving sequential conditions by finite-state strategies

Definability in the Monadic Second-Order Theory of Successor