LECTURE 1. BASIC COMPLEXITY THEORY

Motivation

Complexity theory, at least as understood by computer scientists, is the study of the intrinsic difficulty of solving problems on a computer. In this series of lectures we attempt to give an advanced course in complexity theory with emphasis on some mathematical abstractions of computation such as nondeterminism, alternation and pushdown stores and their application to the understanding of the computational complexity of some logical theories. This course is not intended to be a survey of all that is known about the complexity of logical theories, but is instead intended to provide a student of the subject with some of the key ideas and methods so that he or she can pursue the subject further.

To clarify our definition of complexity theory we need to be more precise about what we mean by “problem”, “solving a problem”, “intrinsic difficulty” and “computer”.

Let Σ be a finite character set with at least two characters in it. The set Σ* represents the set of finite length strings in the alphabet Σ. If x µ Σ* then |x| is the length of x. Members of Σ* represent potential inputs or outputs of a computer program. A problem is a function from Σ* into Σ*. A 0–1 problem is a function from Σ* → {0, 1}. A 0–1 problem is commonly defined by a set, namely the set of strings whose image is 1 under the mapping. To solve a problem means to construct a computer program which has the same input/output behavior as the function defining the problem.