Algorithmically finite groups

TL;DR

This paper introduces the concept of algorithmically finite groups, constructs examples of such groups, and analyzes their properties, including the decidability of the Equality Problem on negligible input sets.

Contribution

It provides the first known examples of recursively presented infinite algorithmically finite groups and investigates their computational properties.

Findings

Equality Problem is decidable only on strongly negligible input sets

Constructed examples are infinite and recursively presented

Analyzed the properties related to algorithmic finiteness

Abstract

We call a group $G$ {\it algorithmically finite} if no algorithm can produce an infinite set of pairwise distinct elements of $G$. We construct examples of recursively presented infinite algorithmically finite groups and study their properties. For instance, we show that the Equality Problem is decidable in our groups only on strongly (exponentially) negligible sets of inputs.