Second-Order Logic over Strings: Regular and Non-regular Fragments

Abstract

By a well-known result due to Büchi and Trakhtenbrot, all monadic second-order sentences over words describe regular languages. In this paper, we investigate prefix classes of general second-order logic. Such a prefix class is called regular, if each of its sentences describes a regular language, and nonregular otherwise. Recently, the regular and nonregular prefix classes of existential second order logic (Σ¹ ₁) were exhaustively determined. We briefly recall these results and continue this line of research by systematically investigating the syntactically more complex prefix classes Σ¹ _k(Q) of second-order logic for each integer k > 1 and for each first-order quantifier prefix Q. We give an exhaustive classification of the regular and nonregular prefix classes of this form, and derive of complexity results for the corresponding model checking problems. We also give a brief survey of recent results on the complexity of evaluating existential second-order logic over graphs, and a list of interesting open problems.