Temporal Logic with Cyclic Counting and the Degree of Aperiodicity of Finite Automata

Zoltán Ésik
Masami Ito

December 2001

Abstract:

We define the degree of aperiodicity of finite automata and show that for every set $M$ of positive integers, the class ${\bf QA}_M$ of finite automata whose degree of aperiodicity belongs to the division ideal generated by $M$ is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize ${\bf QA}_M$ as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set $M$ of moduli. It follows that when $M$ is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in $M$

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