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Uspekhi Mat. Nauk, 2000, Volume 55, Issue 2(332), Pages 3–94 (Mi umn267)  

This article is cited in 30 scientific papers (total in 31 papers)

Decision problems for groups and semigroups

S. I. Adiana, V. G. Durnevb

a Steklov Mathematical Institute, Russian Academy of Sciences
b P. G. Demidov Yaroslavl State University

Abstract: The paper presents a detailed survey of results concerning the main decision problems of group theory and semigroup theory, including the word problem, the isomorphism problem, recognition problems, and other algorithmic questions related to them. The well-known theorems of Markov–Post, P. S. Novikov, Adian–Rabin, Higman, Magnus, and Lyndon are given with complete proofs. As a rule, the proofs presented in this survey are substantially simpler than those given in the original papers. For the sake of completeness, we first prove the insolubility of the halting problem for Turing machines, on which the insolubility of the word problem for semigroups is based. Specific attention is also paid to the simplest examples of semigroups with insoluble word problem. We give a detailed proof of the significant result of Lyndon that, in the class of groups presented by a system of defining relations for which the maximum mutual overlapping of any two relators is strictly less than one fifth of their lengths, the word problem is soluble, while insoluble word problems can occur when non-strict inequality is allowed. A proof of the corresponding result for finitely presented semigroups is also given, when the corresponding fraction is one half.

DOI: https://doi.org/10.4213/rm267

Full text: PDF file (665 kB)
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English version:
Russian Mathematical Surveys, 2000, 55:2, 207–296

Bibliographic databases:

Document Type: Article
UDC: 510.53+512.53+512.54
MSC: Primary 20F10, 20M05; Secondary 20E06, 03D40, 03D10
Received: 26.01.2000

Citation: S. I. Adian, V. G. Durnev, “Decision problems for groups and semigroups”, Uspekhi Mat. Nauk, 55:2(332) (2000), 3–94; Russian Math. Surveys, 55:2 (2000), 207–296

Citation in format AMSBIB
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    4. Leksin V.P., Chernavskii A.V., “Unrecognizability of manifolds. On Novikov's theorem on the unrecognizability of the sphere $\mathbb S^n$ for $n\ge 5$”, Dokl. Math., 68:1 (2003), 89–92  mathnet  mathscinet  mathscinet  zmath  isi  elib
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    6. Bokut L., Fong Y., Shiao L.S., “Grobner-Shirshov bases for algebras, groups, and semigroups”, Proceedings of the Third International Algebra Conference, 2003, 17–32  crossref  mathscinet  zmath  isi
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    8. Adian S.I., “Divisibility problem for one relator monoids”, Theoret. Comput. Sci., 339:1 (2005), 3–6  crossref  mathscinet  zmath  isi  elib  scopus  scopus
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    11. Chernavsky A.V., Leksine V.P., “Unrecognizability of manifolds”, Ann. Pure Appl. Logic, 141:3 (2006), 325–335  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    12. Kreuzer M., Myasnikov A., Rosenberger G., Ushakov A., “Quotient tests and Grobner bases”, Combinatorial Group Theory, Discrete Groups, and Number Theory, Contemporary Mathematics Series, 421, 2006, 187–200  crossref  mathscinet  zmath  isi
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    14. E. N. Pavlovskii, “Algoritmicheskaya raspoznavaemost svoistva konechnosti konechno-opredelennykh sistem”, Vestn. NGU. Ser. matem., mekh., inform., 6:4 (2006), 83–92  mathnet
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    16. Myasnikov A., “Generic Complexity of Undecidable Problems”, Computer Science - Theory and Applications, Lecture Notes in Computer Science, 4649, eds. Diekert V., Volkov M., Voronkov A., Springer-Verlag Berlin, 2007, 407–417  crossref  zmath  isi
    17. S. I. Adian, “Groups with Periodic Defining Relations”, Math. Notes, 83:3 (2008), 293–300  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    18. Myasnikov A.G., Rybalov A.N., “Generic Complexity of Undecidable Problems”, J. Symb. Log., 73:2 (2008), 656–673  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    19. Cain A.J., Maltcev V., “Decision problems for finitely presented and one-relation semigroups and monoids”, Internat. J. Algebra Comput., 19:6 (2009), 747–770  crossref  mathscinet  zmath  isi  scopus  scopus
    20. Karpuz E.G., Çevik A.S., “The word and generalized word problem for semigroups under wreath products”, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér., 52:2 (2009), 151–160  zmath  isi
    21. Alexei Myasnikov, Denis Osin, “Algorithmically finite groups”, Journal of Pure and Applied Algebra, 2011  crossref  mathscinet  isi  scopus
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    25. E. G. Karpuz, A. S. Cevik, “Gröbner–Shirshov Bases for Extended modular, Extended Hecke, and Picard Groups”, Math. Notes, 92:5 (2012), 636–642  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    26. V. N. Bezverkhnii, I. V. Dobrynina, “O svobodnykh podgruppakh v gruppakh Artina s drevesnoi strukturoi”, Chebyshevskii sb., 15:1 (2014), 32–42  mathnet
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