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G. S. Tseytin's seven-relation semigroup with undecidable word problem

Carl-Fredrik Nyberg-Brodda

Abstract

We give an introduction to the ideas behind G. S. Tseytin's 1958 construction of a seven-relation semigroup with undecidable word problem. We give a history of the ideas leading up to its construction, some intuition for the proof, and provide an overview of some subsequent results and developments which stem from this remarkable semigroup. An English translation of Tseytin's article by the author of the present article is supplemented at the end of the article (the Russian original was published in Trudy Mat. Inst. Steklov 52 (1958), pp. 172--189).

G. S. Tseytin's seven-relation semigroup with undecidable word problem

Abstract

We give an introduction to the ideas behind G. S. Tseytin's 1958 construction of a seven-relation semigroup with undecidable word problem. We give a history of the ideas leading up to its construction, some intuition for the proof, and provide an overview of some subsequent results and developments which stem from this remarkable semigroup. An English translation of Tseytin's article by the author of the present article is supplemented at the end of the article (the Russian original was published in Trudy Mat. Inst. Steklov 52 (1958), pp. 172--189).
Paper Structure (18 sections, 2 theorems, 34 equations)

This paper contains 18 sections, 2 theorems, 34 equations.

Table of Contents

  1. Background
  2. Semigroup and monoid presentations
  3. The word problem
  4. Encodings
  5. Undecidability results
  6. Tseytin's proof
  7. Codes
  8. Encoding groups
  9. Details of the encoding
  10. Specific word problem
  11. D. Scott's semigroup
  12. Subsequent developments
  13. Special monoids
  14. Five relations
  15. Three and fewer relations
  16. ...and 3 more sections

Key Result

Theorem 1

Let $M$ be a $k$-relation special monoid, and let $G$ be its group of units. Then:

Theorems & Definitions (4)

  • Example 2.1
  • Example 2.2
  • Theorem : Adian & Makanin, 1966
  • Theorem : Boone, Collins & Matiyasevich, 1971