On some algorithmic problems for groups and monoids

  • Sergei I. Adian
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 690)

Abstract

In 1912 Max Dehn formulated three main algorithmic problems for groups presented by defining relations: Word problem, Conjugacy problem and Isomorphism problem. Two years later A. Thue formulated the Word problem for semigroups presented by defining relations (Thue systems).

Keywords

Bedding Prefix Univer Dian Lentin 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adian S.I. (1955). Algorithmic unsolvability of the problem of recognizing certain properties of groups. Dokl. Akad Nauk SSSR 103, 533–535.MathSciNetGoogle Scholar
  2. 2.
    Adian S.I. (1957). Unsolvability of certain algorithmic problems of group theory. Trudy Moskov. Mat. Obshch. 6, 231–298.Google Scholar
  3. 3.
    Adian S.I. (1957). Finitely presented groups and algorithms. Dokl. Akad Nauk SSSR 117, 9–12.MathSciNetGoogle Scholar
  4. 4.
    Adian S.I. (1966). Defining relations and algorithmic problems for groups and semigroups. Proc. Steklov Inst. Math. 85. (English version published by the American Mathematical Society, 1967).Google Scholar
  5. 5.
    Adian S.I. (1976). Word transformations in a semigroup that is given by a system of defining relations. Algebra i Logika 15, 611–621; English transl. in Algebra and Logic 15 (1976).MathSciNetGoogle Scholar
  6. 6.
    Adian S.I. and Oganesian G.U. (1978). On the word and divisibility problems in semigroups with a single defining relation. Izv. Akad. Nauk SSSR (Ser. Mat.) 42, 219–225; English. transl. in Math. USSR Izv. 12 (1978).MathSciNetGoogle Scholar
  7. 7.
    Adian S.I. and Oganesian G.U. (1987). On the word and divisibility problems in semigroups with one defining relation. Mat. Zametki 41, 412–421; English transl. in Math. Notes 41 (1987).MathSciNetGoogle Scholar
  8. 8.
    Adian S.I. and Makanin G.S. (1984). Investigations on algorithmic questions of algebra. Proc. Steklov Inst. Math. 168, 207–226; English transl. in Proc. Steklov Inst. Math. 3 (1986).Google Scholar
  9. 9.
    Book R.V. (1987) Thue Systems as Rewriting Systems. J. Symbolic Computation 3, 39–68.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Boone W.W. (1959). The word problem. Ann. of Math. (2) 70, 207–265.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Borisov V.V. (1969). Simple examples of groups with unsolvable word problem. Mat. Zametki 6, 521–532; English transl. in Math. Notes 6 (1969).MATHMathSciNetGoogle Scholar
  12. 12.
    Collins D.J. (1969) Recursively enumerable degrees and the cojugacy problem. Acta Math. 122, 115–160.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Dehn M. (1912). Ãœber unendliche diskontinuierliche Gruppen. Math. Ann. 71.Google Scholar
  14. 14.
    Durnev V.G. (1973). Positive theory of a free semigroup. Dokl. Akad Nauk SSSR 211, 772–774; English transl. in Soviet Math. Dokl. 14 (1973).MathSciNetGoogle Scholar
  15. 15.
    Fridman A.A. (1962). Turing degrees of the word problem in finitely presented groups. Dokl. Akad Nauk SSSR 147, 805–808.MATHMathSciNetGoogle Scholar
  16. 16.
    Gol'berg A.I. (1978). On the impossibility of strenghtening certain results of Greendlinger and Lyndon. Uspekhi Mat. Nauk 33, 201–202; English transl. in Russian Math. Surveys 33 (1978).MATHMathSciNetGoogle Scholar
  17. 17.
    Howie J., Pride S. (1986). The word problem for one-relator semigroups. Math. Proc. of the Cambridge Phil. Soc. 99, 33–44.CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Lentin A. (1972). Equations dans les monoides libres. Gauthier-Villars and Mouton, Paris.MATHGoogle Scholar
  19. 19.
    Lyndon R. C. (1960). Equations in free groups. Trans. Amer. Math. Soc. 96, 445–457.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Lyndon R. C. (1966). On Dehn algorithm. Math. Ann. 166, 208–228.CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Magnus W. (1932). Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann. 106, 295–307.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Makanin G.S. (1977). The problem of solvability of equations in a free semigroups. Mat. Sb. 103 (145), 147–236; English transl. in Math. USSR Sb. 32 (1977).MathSciNetGoogle Scholar
  23. 23.
    Makanin G.S. (1982). Eguations in a free group. Izv. Akad. Nauk SSSR (Ser. Mat.) 46, 1199–1273; English. transl. in Math. USSR Izv. 21 (1983).MATHMathSciNetGoogle Scholar
  24. 24.
    Makanin G.S. (1984). Decidability of the universal and positive theories in a free groups. Izv. Akad. Nauk SSSR (Ser. Mat.) 48, 735–749; English. transl. in Math. USSR Izv. 25 (1985).MathSciNetGoogle Scholar
  25. 25.
    Marchenkov S.S. (1982). Undecidability of the positive AE-theory of a free semigroup. Sibirsk. Mat. Zh. 23, 196–198.MATHMathSciNetGoogle Scholar
  26. 26.
    Markov A.A. (1947). On the impossibility of certain algorithms in the theory of associative systems. Dokl. Akad Nauk SSSR 55, 683–586.Google Scholar
  27. 27.
    Markov A.A. (1954). The theory of Algorithms. Trudy Mat. Inst. Steklov 42; English transl., Israel Program Sci. Transl., Jerusalem (1961).Google Scholar
  28. 28.
    Matiyasevich Yu.V. (1967). Simple examples of undecidable associative calculi. Dokl. Akad Nauk SSSR 173, 1264–1266; English transl. in Soviet Math. Dokl. 8 (1967).MathSciNetGoogle Scholar
  29. 29.
    Novikov P.S. (1955). On the algotithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44; English transl. in Amer. Math. Soc. Transl. (2) 9 (1958).Google Scholar
  30. 30.
    Oganesian G.U. (1982). On semigroups with one relation and semigroups without cycles. Izv. Akad. Nauk SSSR (Ser. Mat.) 46, 88–94; English. transl. in Math. USSR Izv. 20 (1983).MathSciNetGoogle Scholar
  31. 31.
    Osipova V.A. (1968). On the word problem for finitely presented semigroups. Dokl. Akad Nauk SSSR 178, 1017–1020; English transl. in Soviet Math. Dokl. 9 (1968).MathSciNetGoogle Scholar
  32. 32.
    Post E.L. (1947). Recursive unsolvability of a problem of Thue. J. Symbolic Logic 12, 1–11.CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Quine W.V. (1946). Concatenation as a basis for arithmetic. J. Symbolic Logic 11, 105–114.CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Rabin M.O. (1958). Recursive unsolvavility of group theoretic problems. Ann. of Math. (2) 67, 172–194.CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Razborov A.A. (1984). On systems of equations in a free groups. Izv. Akad. Nauk SSSR (Ser. Mat.) 48, 779–832; English. transl. in Math. USSR Izv. 25 (1985).MathSciNetGoogle Scholar
  36. 36.
    Sarkisian O.A. (1979). Some relations between the word and divisibility problems in groups and semigroups. Izv. Akad. Nauk SSSR (Ser. Mat.) 43, 909–921; English. transl. in Math. USSR Izv. 15 (1980).MathSciNetGoogle Scholar
  37. 37.
    Thue A. (1914). Probleme über Verbandenlungen von Zeichenreihen nach gegebenen Regeln. Kristiana Videnskapsselkapets Skr. I: Mat.-Naturvid. Kl. 10. (Reprinted in his Selected mathematical papers, Universitetsforlaget, Oslo (1977), 493–524).Google Scholar
  38. 38.
    Taimanov A.D. and Khmelevskii Yu.I. (1980). Decidability of the universal theory of a free semigroup. Sibirsk. Mat. Zh. 21, 228–230.MathSciNetGoogle Scholar
  39. 39.
    Tartakovskii V.A. (1947). On the word problem for certain types of groups. Dokl. Akad Nauk SSSR 58, 1909–1910 (Russian).MathSciNetGoogle Scholar
  40. 40.
    Tartakovskii V.A. (1949). Solution of the word problem for groups with a k-reduced basis for k>6. Izv. Akad. Nauk SSSR (Ser. Mat.) 13, 483–494; English transl. in Amer. Math. Soc. Transl. (1) 1 (1962).MathSciNetGoogle Scholar
  41. 41.
    Tseitin G.S. (1956). Associative system with unsolvable word problem. Dokl. Akad Nauk SSSR 107, 370–371. (The full proof in Trudy Mat. Inst. Steklov 52 (1958), 172–189).MathSciNetGoogle Scholar
  42. 42.
    Zhang L. (1991). Conjugacy in Special Monoids. J. of Algebra 143, 487–497.CrossRefMATHGoogle Scholar
  43. 43.
    Zhang L. (1992). Applying rewriting methods to special monoids. Math. Proc. Cambridge Phil. Soc. 112, 495–505.CrossRefMATHGoogle Scholar
  44. 44.
    Zhang L. (1992). A short proof of a theorem of Adjan. Proc. of the Amer. Math. Soc. 116, 1–3.CrossRefMATHGoogle Scholar
  45. 45.
    Zhang L. (1992). On the conjugacy problem for one-relator monoids with elements of finite order. International J. of Algebra and Computations. 2, 209–220.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Sergei I. Adian
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

Personalised recommendations