Kolmogorov’s Algorithms or Machines

Abstract

At the beginning of the year of 1951, the 5th year student of the mechanics and mathematics department of Moscow State University V. A. Uspensky was given a page of typewritten text with the description of a certain abstract machine by his scientific advisor A. N. Kolmogorov. The states of this machine were one-dimensional topological complexes or, more precisely, complexes with certain additional information. This additional information consisted in the vertices of the complexes being supplied with functions with values in a preliminarily chosen finite set so that the complex turned out to be labeled (by values of this function). The formulation proposed by A. N. Kolmogorov had the following two goals:

• 1) to give a mathematical definition of the notion of algorithm that would be as general as possible (so general that it would directly include all algorithms);

• 2) to verify that even this more general definition does not lead to an extension of the class of computable functions which was already clearly understood by this time on the basis of previous narrower definitions (due to Turing, Post, Markov, and others) and which, if one limits oneself to functions, coincides with the class of partially recursive functions.