Finite Computational Structures and Implementations

TL;DR

This paper introduces an algebraic framework based on semigroups to analyze the limits, correctness, and power of finite computational systems across various paradigms.

Contribution

It generalizes classical computation models to semigroups, enabling unified analysis of different computing paradigms within a single algebraic framework.

Findings

Provides a formal algebraic definition of finite computation

Enables comparison of different computational paradigms

Summarizes recent advances in finite computational theory

Abstract

What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only partial answers to these questions. In order to make these problems more precise, we describe an abstract algebraic definition of classical computation, generalizing traditional models to semigroups. The mathematical abstraction also allows the investigation of different computing paradigms (e.g. cellular automata, reversible computing) in the same framework. Here we summarize the main questions and recent results of the research of finite computation.