Languages recognized by finite automata are exactly those definable by sentences of the sequential calculus, and also exactly those definable by rational expressions (also called regular expressions)
Howard Straubing, Pascal Weil. An introduction to finite automata and their connection to logic. Deepak D’Souza, Priti Shankar. Modern applications of automata theory, World Scientific, pp.3-43, 2012, IISc Research Monographs. hal-00541028v2
Can you please explain, what is the connection between finite automata and sequential calculus (logic)?