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This is an English translation of four remarkable articles, originallywritten in Russian, by Sergei Ivanovich Adian (1931--2020), supplemented by thetranslation of two closely related articles by Andrei Andreevich Markov Jr(1903--1979). All six articles concern algorithmic undecidability of variousproblems for groups and semigroups. The articles by S. I. Adian give his proofof the famous "Adian-Rabin Theorem", which shows that there is no algorithmwhich 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 whichoutputs whether or not the presented group has property $P$. The articles by A.A. Markov (Jr) give the analogous result for semigroups (a significantly easierresult), and appeared several years before the group-theoretic analogue. Apreface detailing the contents of the articles and some commentary is provided,including the somewhat difficult task of deciding which of Adian's fourarticles should be considered as containing "the" proof of the Adian-Rabintheorem.

The Adian-Rabin Theorem -- An English translation