Lecture Notes on Monadic First- and Second-Order Logic on Strings (2301.10519v1)

Abstract: These notes present the essentials of first- and second-order monadic logics on strings with introductory purposes. We discuss Monadic First-Order logic and show that it is strictly less expressive than Finite-State Automata, in that it only captures a strict subset of Regular Languages -- the non-counting ones. We then introduce Monadic Second-Order logic; such a logic is, syntactically, a superset of Monadic First-Order logic and captures Regular Languages exactly. We also show how to transform an automaton into a corresponding formula and vice versa. Finally, we discuss the use of logical characterizations of classes of languages as the basis for automatic verification techniques.