We investigate extensions of temporal logic by connectives de ned by nite automata on in nite words. We consider three di erent logics, corresponding to three di erent types of acceptance conditions (nite, looping and repeating) for the automata. It turns out, however, that these logics all have the same expressive power and that their decision problems are all PSPACE-complete. We also investigate connectives de ned by alternating automata and show that they do not increase the expressive power of the logic or the complexity of the decision problem.

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Translating linear temporal logic formulas to automata has proven to be an effective approach for implementing linear-time model-checking, and for obtaining many extensions and improvements to this verification method. On the other hand, for branching temporal logic, automata-theoretic techniques have long been thought to introduce an exponential penalty, making them essentially useless for model-checking. Recently, Bernholtz and Grumberg [1993] have shown that this exponential penalty can be avoided, though they did not match the linear complexity of non-automata-theoretic algorithms. In this paper, we show that alternating tree automata are the key to a comprehensive automata-theoretic framework for branching temporal logics. Not only can they be used to obtain optimal decision procedures, as was shown by Muller et al., but, as we show here, they also make it possible to derive optimal model-checking algorithms. Moreover, the simple combinatorial structure that emerges from the automata-theoretic approach opens up new possibilities for the implementation of branching-time model checking and has enabled us to derive improved space complexity bounds for this long-standing problem.

This article addresses the problem of designing memory-efficient algorithms for the verification of temporal properties of finite-state programs. Both the programs and their desired temporal properties are modeled as automata on infinite words (Biichi automata). Verification is then reduced to checking the emptiness of the automaton resulting from the product of the program and the property. This problem is usually solved by computing the strongly connected components of the graph representing the product automaton. Here, we present algorithms that solve the emptiness problem without explicitly constructing the strongly connected components of the product graph. By allowing the algorithms to err with some probability, we can implement them with a randomly accessed memory of size O(n) bits, where n is the number of states of the graph, instead of O(n log n) bits that the presently known algorithms require.

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