Complejidad descriptiva y computacional en maquinas de Turing pequenas

TL;DR

This paper systematically explores the complexity and behavior of small Turing machines with 2 and 3 states, analyzing their computational resources, runtimes, and patterns to understand their computational properties and structures.

Contribution

It provides a comprehensive analysis of small Turing machines, revealing how resource constraints influence their computational complexity and uncovering structural patterns.

Findings

Average runtime increases with number of states

Non-trivial machines tend to utilize all available resources

Instances of linear speed-up observed in some cases

Abstract

We start by an introduction to the basic concepts of computability theory and the introduction of the concept of Turing machine and computation universality. Then se turn to the exploration of trade-offs between different measures of complexity, particularly algorithmic (program-size) and computational (time) complexity as a mean to explain these measure in a novel manner. The investigation proceeds by an exhaustive exploration and systematic study of the functions computed by a large set of small Turing machines with 2 and 3 states with particular attention to runtimes, space-usages and patterns corresponding to the computed functions when the machines have access to larger resources (more states). We report that the average runtime of Turing machines computing a function increases as a function of the number of states, indicating that non-trivial machines tend to occupy all the resources at hand. General slow-down was witnessed and some incidental cases of (linear) speed-up were found. Throughout our study various interesting structures were encountered. We unveil a study of structures in the micro-cosmos of small Turing machines.