Basics in Discrete Complexity Theory

Abstract

In this chapter we give the formal definitions and basic results of discrete complexity theory, in particular, regarding the complexity classes between logarithmic space and exponential space. Some results on complexity classes of sparse sets and tally sets are included because they are closely related to the structure of the representations of real numbers under our formal definition. Other results in discrete complexity theory, not directly related to our theory, will be introduced in later chapters when they are needed. This chapter is not intended to be an introductory reading for discrete complexity theory. Many important issues in the theory are omitted and most theorems are given without proofs. The reader who wishes to learn more systematically about complexity theory is referred to Garey and Johnson [1979] and Balcázar, Diaz and Gabarró [1988, 1990].