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Temporal Logic with Cyclic Counting and the Degree of Aperiodicity of Finite Automata1
Acta Cybernetica 16 (2003) 1-28.
Abstract:
We define the degree of aperiodicity of finite automata and show that for every set of positive integers, the class of finite automata whose degree of aperiodicity belongs to the division ideal generated by 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 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 of moduli. It follows that when 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 .
Footnotes
This paper is a revised version of the author's technical report RS-01-53, BRICS, December, 2001.
Dept. of Computer Science, University of Szeged, Szeged, Hungary. Email: esik@inf.u-szeged.hu The results of this paper were obtained during the first author's visit at the Faculty of Science at the Kyoto Sangyo University and at the Department of Computer Science at the University of Aalborg. The first author was partially supported by Grant-in-Aid 10044098, Japan Society for the Promotion of Science, by BRICS (Basic Research in Computer Science), and by grant no. T30511 from the National Foundation of Hungary for Scientific Research.
Dept. of Mathematics, Kyoto Sangyo University, Kyoto, Japan. Email: ito@ksuvx0.kyoto-su.ac.jp Partially supported by Grant-in-Aid 10044098, Japan Society for the Promotion of Science.
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