Algorithmic Problem Complexity

TL;DR

This paper develops a mathematical theory of algorithmic problem complexity, analyzing static measures for finite and infinite objects, and classifies problems using an inductive hierarchy of automata.

Contribution

It introduces a new framework for measuring and classifying the complexity of algorithmic problems, extending traditional models with inductive Turing machines.

Findings

Defined optimal complexity measures for finite and infinite objects.

Established an inductive hierarchy of algorithmic problems.

Classified problems like the halting problem within this hierarchy.

Abstract

People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at determination of computer abilities in solving different problems and estimation of resources that computers need to do this. Here we build the part of this theory related to static measures of algorithms. At first, we consider problems for finite words and study algorithmic complexity of such problems, building optimal complexity measures. Then we consider problems for such infinite objects as functions and study algorithmic complexity of these problems, also building optimal complexity measures. In the second part of the work, complexity of algorithmic problems, such as the halting problem for Turing machines, is measured by the classes of automata that are necessary to solve this problem. To classify different problems with respect to their complexity, inductive Turing machines, which extend possibilities of Turing machines, are used. A hierarchy of inductive Turing machines generates an inductive hierarchy of algorithmic problems. Here we specifically consider algorithmic problems related to Turing machines and inductive Turing machines, and find a place for these problems in the inductive hierarchy of algorithmic problems.