On Process Complexity.

TL;DR: This paper defines a variant of process complexity based on Levin's definition of a process, and proves that strict process complexity does not agree within an additive constant with Schnorr's original process complexity.

Abstract: Process complexity is one of the basic variants of Kolmogorov complexity. Unlike plain Kolmogorov complexity process complexity provides a simple characterization of randomness for real numbers in terms of initial segment complexity. Process complexity was first developed in (Schnorr 1973). Schnorr's definition of a process, while simple, can be difficult to work with. In many situations, a preferable definition of a process is that given by Levin in (Levin & Zvonkin 1970). In this paper we define a variant of process complexity based on Levin's definition of a process. We call this variant strict process complexity. Strict process complexity retains the main desirable properties of process complexity. Particularly, it provides simple characterizations of Martin-Lof random real numbers, and of computable real numbers. However, we will prove that strict process complexity does not agree within an additive constant with Schnorr's original process complexity. One of the basic properties of prefix-free complexity is that it is subadditive. Subadditive means that there is some constant d such that for all strings σ, τ the complexity of στ (σ and τ concatenated) is less than or equal to the sum of the complexities of σ and τ plus d. A fundamental question about any complexity measure is whether or not it is subadditive. In this paper we resolve this question for process complexity by proving that neither of these process complexities is subadditive.