We study the expressive power of linear propositional temporal logic interpreted on finite sequences or words. We first give a transparent proof of the fact that a formal language is expressible in this logic if and only if its syntactic semigroup is finite and aperiodic. This gives an effective algorithm to decide whether a given rational language is expressible. Our main result states a similar condition for the "restricted" temporal logic (RTL), obtained by discarding the "until" operator. A formal language is RTL-expressible if and only if its syntactic semigroup is finite and satisfies a certain simple algebraic condition. This leads

. We show a property of strings is expressible in the twovariable fragment of first-order logic if and only if it is expressible by both a \Sigma 2 and a \Pi 2 sentence. We thereby establish: UTL = FO 2 = \Sigma 2 " \Pi 2 = UL ; where UTL stands for the string properties expressible in the temporal logic with `eventually in the future' and `eventually in the past' as the only temporal operators and UL stands for the class of unambiguous languages. This enables us to show that the problem of determining whether or not a given temporal string property belongs to UTL is decidable (in exponential space), which settles a hitherto open problem. Our proof of \Sigma 2 " \Pi 2 = FO 2 involves a new combinatorial characterization of these two classes and introduces a new method of playing Ehrenfeucht-Fraiss'e games to verify identities in semigroups. While the number of variables required to express a certain graph property in first-order logic is an important measure for the descriptional...

This paper is an attempt to share with a larger audience some modern developments in the theory of finite automata. It is written for the mathematician who has a background in semigroup theory but knows next to nothing on automata and languages. No proofs are given, but the main results are illustrated by several examples and counterexamples

We reveal an intimate connection between semidirect products of finite semigroups and substitution of formulas in linear temporal logic. We use this connection to obtain an algebraic characterization of the until hierarchy of linear temporal logic. (The k-th level of that hierarchy is comprised of all temporal properties that are expressible by a formula of nesting depth k in the until operator.) Applying deep results from finite semigroup theory we are able to prove that each level of the until hierarchy is decidable. By means of Ehrenfeucht-Fraïssé games, we extend the results from linear temporal logic over finite sequences to linear temporal logic over infinite sequences.

The computation tree of a nondeterministic machine M with input x gives rise to a leaf string formed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular "leaf language" Y , the set of all x for which the leaf string of M is contained in Y . In this way, in the context of polynomial time computation, leaf languages were shown to capture many complexity classes. In this paper, we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation. We show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages, capturing also many classes within P. In contrast, logtime leaf languages basically behave like logtime reducibilities. Both cases are more subtle to handle than the polynomial time case. We also raise the issue of balanced versus non-balanced comp...

. We prove an effective characterization of languages having dot--depth 3=2. Let B 3=2 denote this class, i.e., languages that can be written as finite unions of languages of the form u0L1u1L2u2 \Delta \Delta \Delta Lnun , where u i 2 A and L i are languages of dot--depth one. Let F be a deterministic finite automaton accepting some language L. Resulting from a detailed study of the structure of B 3=2 , we identify a pattern P (cf. Fig. 2) such that L belongs to B 3=2 if and only if F does not have pattern P in its transition graph. This yields an NL--algorithm for the membership problem for B 3=2 . Due to known relations between the dot--depth hierarchy and symbolic logic, the decidability of the class of languages definable by \Sigma 2--formulas of the logic FO[!; min; max; S; P ] follows. We give an algebraic interpretation of our result. 1 Introduction We contribute to the theory of finite automata and regular languages, with consequences in logic as well as in algeb...

Straubing's wreath product principle provides a description of the languages recognized by the wreath product of two monoids. A similar principle for ordered semigroups is given in this paper. Applications to language theory extend standard results of the theory of varieties to positive varieties. They include a characterization of positive locally testable languages and syntactic descriptions of the operations L ! La and L ! LaA . Next we turn to concatenation hierarchies. It was shown by Straubing that the n-th level Bn of the dot-depth hierarchy is the variety V n LI, where LI is the variety of locally trivial semigroups and V n is the n-th level of the Straubing-Therien hierarchy. We prove that a similar result holds for the half levels. It follows in particular that a level or a half level of the dot-depth hierarchy is decidable if and only if the corresponding level of the Straubing-Therien hierarchy is decidable. 1 Introduction All semigroups and monoids considered in t...

For some fixed alphabet A with jAj 2, a language L ` A is in the class L 1=2 of the Straubing-Therien hierarchy if and only if it can be expressed as a finite union of languages A a 1 A a 2 A \Delta \Delta \Delta A anA , where a i 2 A and n 0. The class L 1 is defined as the boolean closure of L 1=2 . It is known that the classes L 1=2 and L 1 are decidable. We give a membership criterion for the single classes of the boolean hierarchy over L 1=2 . From this criterion we can conclude that this boolean hierarchy is proper and that its classes are decidable. In finite model theory the latter implies the decidability of the classes of the boolean hierarchy over the class \Sigma 1 of the FO[!]-logic. Moreover we prove a "forbidden-pattern" characterization of L 1 of the type: L 2 L 1 if and only if a certain pattern does not appear in the transition graph of a deterministic finite automaton accepting L. We discuss complexity theoretical consequences of our results. C...

Abstract not found

Let B 1/2 denote the class of languages having dot-depth 1=2, i.e., the class of languages that can we written as finite unions of languages u 0 A + u 1 A + \Delta \Delta \Delta un\Gamma1 A + un , where u i 2 A and n 0. A language has dot--depth one if and only if it is in the Boolean closure of B 1/ . We examine the structure of the class of dot--depth one languages with respect to Boolean operations and identify an infinite family of Boolean hierarchies inside this class. In particular, we show that 1. the union of these hierarchies amounts to the Boolean hierarchy over B 1/2 , 2. all emerging inclusions are strict, 3. the membership problems for all classes in each hierarchy are decidable and 4. a given language can exactly be located in this landscape.