Temporal logic with cyclic counting and the degree of aperiodicity of finite automata

  • Authors:

  • Z. Ésik;M. Ito

  • Affiliations:

  • Dept. of Computer Science, University of Szeged, Szeged, Hungary;Dept. of Mathematics, Kyoto Sangyo University, Kyoto, Japan

  • Venue:
  • Acta Cybernetica
  • Year:
  • 2003

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Abstract

We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QAM 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 QAM 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.