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Hierarchies of Computable groups and the word problem1

Published online by Cambridge University Press:  12 March 2014

Frank B. Cannonito
Frank B. Cannonito*
Affiliation:
Hughes Aircraft Company, Fullerton, California
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Extract

The word problem for groups was first formulated by M. Dehn [1], who gave a solution for the fundamental groups of a closed orientable surface of genus g ≧ 2. In the following years solutions were given, for example, for groups with one defining relator [2], free groups, free products of groups with a solvable word problem and, in certain cases, free products of groups with amalgamated subgroups [3], [4], [5]. During the period 1953–1957, it was shown independently by Novikov and Boone that the word problem for groups is recursively undecidable [6], [7]; granting Church's Thesis [8], their work implies that the word problem for groups is effectively undecidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 31 , Issue 3 , 02 September 1966 , pp. 376 - 392
Copyright
Copyright © Association for Symbolic Logic 1996

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Footnotes

1

This work was supported by the Air Force Systems Command, Research and Technology Division, Rome Air Development Center, Griffiss Air Force Base, New York, 13442, under contract AF 30(602)-3339, and forms a portion of the author's doctoral dissertation at Adelphi University.

References

[I]Dehn, M., Über unendliche diskontinuerliche Gruppen, Mathematische Annalen, vol. 71 (1911), pp. 116–144.CrossRefGoogle Scholar
[2]Magnus, W., Das Identitätsproblem für Gruppen mit einer definierenden Relation, Mathematische Annalen, vol. 106 (1932), pp. 295–307.CrossRefGoogle Scholar
[3] J. Britton, L., Solution of the word problem for certain types of groups, I, Proceedings of the Glasgow Mathematical Association, vol. 3 (1956), pp. 45–54.CrossRefGoogle Scholar
[4]Greendlinger, M., Dehn's algorithm for the word problem, Communications on pure and applied mathematics, vol. XIII (1960), pp. 67–83.CrossRefGoogle Scholar
[5]Lipschutz, S., An extension of Greendlinger's results on the word problem, Proceedings of the American Mathematical Society, vol. 15 (1964), pp. 37–43.Google Scholar
[6]Novikov, P. S., On the algorithmic unsolvability of the word problem in group theory, Akademya Nauk S.S.S.R. Matematicheskii Institut Trudy, vol. 44 (1955).Google Scholar
[7]Boone, W. W., The word problem, Annals of mathematics, vol. 70 (1959), pp. 207–265.CrossRefGoogle Scholar
[8]Church, A., An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936), pp. 345–363.CrossRefGoogle Scholar
[9]Rabin, M. O., Computable algebra, general theory and theory of computable fields, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–360.Google Scholar
[10]Grzegorczyk, A., Some classes of recursive functions, Rozprawy Mathmetyczne, vol. 4 (1953), 46 pp.Google Scholar
[11]Kleene, S. C., Introduction to metamathematics, Van Nostrana (1952).Google Scholar
[12]Axt, P., Enumeration and the Grzegorczyk hierarchy, Zeitschrift für mathematischen Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 53–65.CrossRefGoogle Scholar
[13]Kleene, S. C., Recursive predicates and quantifiers, Transactions of the American Mathematical Society, vol. 53 (1943), pp. 41–73.CrossRefGoogle Scholar
[14]Boone, W. W., Partial results regarding word problems and recursively enumerable degrees of unsolvability, Bulletin of the American Mathematical Society, vol. 68 (1962), pp. 616–623.CrossRefGoogle Scholar
[15]Higman, G., Subgroups of finitely presented groups, Proceedings of the Royal Society, A, vol. 262 (1961), pp. 455–475.Google Scholar