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Many-one degrees associated with problems of tag1

  • C. E. Hughes (a1)

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Tag systems were defined by Post [9], [10] and have been studied by a number of researchers including Minsky [7], Maslov [6] and Aanderaa and Belsnes [1]. In their recent paper Aanderaa and Belsnes demonstrated that every r.e. many-one degree (exclusive of the degree of the empty set) is represented by the general halting problem for tag systems, that is, by the family of halting problems ranging over all tag systems. Their result depends upon an informal proof of this property for Turing machines but may be seen to be correct in light of a formal proof due to Overbeek [8]. Our aim is to extend their results to the general word problem for these systems. Specifically, we shall present an effective method which, when applied to an arbitrary r.e. set S, where S is neither empty nor the set of all natural numbers, produces a tag system R′ whose word and halting problems are both of the same many-one degree as the decision problem for S. The proof is realized by first constructing, from the description of an arbitrary Turing machine M, which machine has at least one mortal and one immortal configuration, a 5-register machine R, whose word and halting problems are both of the same many-one degree as the halting problem for M. From R we then construct the desired tag system R′. This construction combined with Overbeek's [8] shows that every r.e. many-one degree (exclusive of the degrees of the empty set and the set of all natural numbers) is represented by the general word and halting problems for tag systems. Moreover our results are seen to be best possible with regard to degrees of unsolvability in that it is not the case that every nonrecursive r.e. one-one degree is represented by either of the general decision problems for tag systems which are considered here. These results were first shown in the author's thesis [3] and were announced in [4], They form part of an extensive study into the many-one equivalence of general decision problems. An overview of the initial findings of this research project may be found in [5].

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This work was partially supported by NSF grant GP-23779.

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[1]Aanderaa, S. and Belsnes, D., Decision problem for tag systems, this Journal, vol. 36 (1971), pp. 229239.
[2]Davis, M., Computability and unsolvability, McGraw-Hill, New York, 1958.
[3]Hughes, C. E., The representation of many-one degrees by Markov algorithms, semi-Thue systems and tag systems, Doctoral Dissertation, Pennsylvania State University, University Park, Pa., 1970.
[4]Hughes, C. E., Representation of many-one degrees by decision problems associated with tag systems, this Journal, vol. 36 (1971), p. 306. Abstract
[5]Hughes, C. E., Overbeek, R. and Singletary, W. E., The many-one equivalence of some general combinatorial decision problems, Bulletin of the American Mathematical Society, vol. 77 (1971), pp. 467472.
[6]Maslov, S. Ju., On E. L. Post's “tagproblem,” Trudy Matematiceskogo Instituta im V. A. Steklova, vol. 72 (1964), pp. 5768; English translation, American Mathematical Society Translations (2), vol. 97 (1970), pp. 1–14.
[7]Minsky, M. L., Recursive unsolvability of Post's problem “tag” and other topics in theory of Turing machines, Annals of Mathematics (2), vol. 74 (1961), pp. 437455.
[8]Overbeek, R., Representation of many-one degrees by decision problems associated with Turing machines, this Journal, vol. 36 (1971), p. 706. Abstract
[9]Post, E. L., Formal reduction of the general combinatorial decision problem, American Journal of Mathematics, vol. 65 (1943), pp. 197215.
[10]Post, E. L., Absolutely unsolvable problems and relatively undecidable propositions—account of an anticipation, The Undecidable (Davis, M., Editor), Raven Press, 1965. (Unpublished, 1941)
[11]Shepherdson, J. C., Machine configuration and word problems of given degree of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 11 (1965), pp. 149175.

Many-one degrees associated with problems of tag1

  • C. E. Hughes (a1)

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