Inductive definitions and second-order logic

Abstract

One of the questions addressed in this book is the logical characterization of complexity classes such as P or NP. We look for a logic such that a problem is definable in this logic iff it is computable within certain time or space bounds. The study of definability on finite structures has shown, for example, that connectivity on the class of graphs cannot be expressed in first-order logic. Yet this problem is decidable in polynomial time on deterministic machines. It is natural to study extensions of first-order logic in order to express inherent properties of problems of the classes P or NP.