The Adian-Rabin Theorem -- An English translation

TL;DR

This paper provides an English translation of six foundational articles on the algorithmic undecidability in group and monoid theory, including the original proof of the Adian-Rabin Theorem, a key result in computational algebra.

Contribution

It offers the first comprehensive English translation of Adian and Markov's seminal works, clarifying the original proofs and historical context of the Adian-Rabin Theorem.

Findings

Proves the non-existence of an algorithm to decide properties like triviality or infiniteness of finitely presented groups.

Establishes analogous undecidability results for monoids, predating the group case.

Provides historical and mathematical commentary on the original articles.

Abstract

This is an English translation of four remarkable articles, originally written in Russian, by Sergei Ivanovich Adian (1931--2020), supplemented by the translation of two closely related articles by Andrei Andreevich Markov Jr (1903--1979). All six articles concern algorithmic undecidability of various problems for groups and monoids. The articles by S. I. Adian give his proof of the famous "Adian-Rabin Theorem", which shows that there is no algorithm which takes as input a finite presentation of a group together with a "Markov" property $P$ (e.g. being the trivial group, being infinite, etc.), and which outputs whether or not the presented group has property $P$. The articles by A. A. Markov (Jr) give the analogous result for monoids (a significantly easier result), and appeared several years before the group-theoretic analogue. A preface detailing the contents of the articles and some commentary is provided, including the somewhat difficult task of deciding which of Adian's four articles should be considered as containing "the" proof of the Adian-Rabin theorem.