Monadic Second-Order Logic and Automata

Abstract

We now move to extensions of first-order logic. In this chapter we introduce second-order logic, and consider its often used fragment, monadic second-order logic,or MSO, in which one can quantify over subsets of the universe. We study the expressive power of this logic over graphs, proving that its existential fragment expresses some NP-complete problems, but at the same time cannot express graph connectivity. Then we restrict our attention to strings and trees, and show that, over them, MSO captures regular string and tree languages. We explore the connection with automata to prove further definability and complexity results.