@article{Ésik_Ito_2001, title={Temporal Logic with Cyclic Counting and the Degree of Aperiodicity of Finite Automata}, volume={8}, url={https://tidsskrift.dk/brics/article/view/21714}, DOI={10.7146/brics.v8i53.21714}, abstractNote={We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class 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 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.}, number={53}, journal={BRICS Report Series}, author={Ésik, Zoltán and Ito, Masami}, year={2001}, month={Dec.} }