
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…
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Machine Learning and Algorithms · Evolutionary Algorithms and Applications
