We discuss in this paper how connections, discovered almost forty years ago, between logics and automata can be used in practice. For such logics expressing regular sets, we have developed tools that allow efficient symbolic reasoning not attainable by theorem proving or symbolic model checking. We explain how the logic-automaton connection is already exploited in a limited way for the case of Quantified Boolean Logic, where Binary Decision Diagrams act as automata. Next, we indicate how BDD data structures and algorithms can be extended to yield a practical decision procedure for a more general logic, namely WS1S, the Weak Secondorder theory of One Successor. Finally, we mention applications of the automaton-logic connection to software engineering and program verification. 1

The main results of the present paper are the equivalence of definability by monadic second-order logic and recognizability for real trace languages, and that first-order definable, star-free, and aperiodic real trace languages form the same class of languages. This generalizes results on infinite words and on finite traces to infinite traces. It closes an important gap in the different characterizations of recognizable languages of infinite traces. 1 Introduction In the late 70's, A. Mazurkiewicz introduced the notion of trace as a suitable mathematical model for concurrent systems [16] (for surveys on this topic see also [1, 6, 10, 17]). In this framework, a concurrent system is seen as a set \Sigma of atomic actions together with a fixed irreflexive and symmetric independence relation I ` \Sigma \Theta \Sigma. The relation I specifies pairs of actions which can be carried out in parallel. It generates an equivalence relation on the set of sequential observations of the system. As ...

In concurrency theory, there are several examples where the interleaved model of concurrency can distinguish between execution sequences which are not significantly different. One such example is sequences that differ from each other by stuttering, i. e., the number of times a state can adjacently repeat. Another example is executions that differ only by the ordering of independently executed events. Considering these sequences as different is semantically rather meaningless. Nevertheless, specification languages that are based on interleaving semantics, such as linear temporal logic (LTL), can distinguish between them. This situation has led to several attempts to define languages that cannot distinguish between such equivalent sequences. In this paper, we take a different approach to this problem: we develop algorithms for deciding if a property cannot distinguish between equivalent sequences, i. e., is closed under the equivalence relation. We focus on properties represented by regular languages, !-regular languages, or propositional LTL formulae and show that for such properties there is a wide class of equivalence relations for which determining closure is decidable, in fact in PSPACE.

This paper shows the equivalence between the family of recognizable languages over infinite traces and the family of languages which are recognized by deterministic asynchronous cellular Muller automata. We thus give a proper generalization of McNaughton's Theorem from infinite words to infinite traces. Thereby we solve one of the main open problems in this field. As a special case we obtain that every closed (w.r.t. the independence relation) word language is accepted by some I-diamond deterministic Muller automaton. 1 Introduction A. Mazurkiewicz introduced the concept of traces as a suitable semantics for concurrent systems [Maz77]. A concurrent system is given by a set of atomic actions \Sigma = fa; b; c; : : :g together with an independence relation I ` \Sigma \Theta \Sigma, which specifies pairs of actions which can be performed concurrently. This leads to an equivalence relation on \Sigma generated by the independence relation I. More precisely, if a and b denote independent...

The main result of this paper is the extension of the theorem of Schützenberger, McNaughton and Papert on star-free sets of finite words to languages of words of countable length. We also give an other proof of the theorem of Büchi which establishes the equivalence between automata and monadic second-order sentences for defining sets of words of denumerable length. Büchi [Büc60] was the first to use certain formul&aelig; of logic, known as (monadic) second order formul&aelig;, in order to define sets of finite words. Monadic second-order formul&aelig; are built from (first-order) variables x; y; : : : representing positions in words, (second-order) variables X;Y; : : : representing sets of positions, an ordering relation < between positions, an unary relation X(x) whose signification is to test if the value of a first-order variable belongs to a set of positions or not, and, for every letter a of the alphabet, an unary relation R a (x) whose signification is to test if the letter at position x is an a. Th...

In Ref. [2], Ehrenfeucht et al. showed that a set L of finite words is regular if and only if L is -closed under some monotone Well-Quasi-Order (WQO) over finite words. We extend this result to regular ω-languages. That is, (1) an ω-language L is regular if and only if L is -closed under a periodic extension of some monotone WQO over finite words, and (2) an ω-language L is regular if and only if L is -closed under a WQO over ω-words which is a continuous extension of some monotone WQO over finite words.

This paper proposed the sufficient and necessary condition for the regularity of ω-languages in terms of a Well-Quasi-Order (WQO). That is, an ω-language L is regular if and only if L is -closed and convergenceclosed under some monotonic WQO .

The Myhill-Nerode Theorem (that for any regular language, there is a canonical recognizing device) is of paramount importance for the computational handling of many formalisms about finite words. For infinite