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Decision problems for tag systems

Published online by Cambridge University Press:  12 March 2014

Stål Aanderaa  and
Dag Belsnes
Stål Aanderaa
Affiliation:
University of Oslo, Oslo, Norway
Dag Belsnes
Affiliation:
University of Oslo, Oslo, Norway
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Extract

The aim of this paper is to study tag systems as defined by Post [Post 1943, pp. 203–205 and Post, 1965, pp. 370–373]. The existence of a tag system with unsolvable halting problem was proved by Minsky by constructing a universal tag system [Minsky 1961, see also Cocke and Minsky 1964, Wang 1963, and Minsky 1967, pp. 267–273]. Hence the halting problem of a tag system can be of the complete degree 0′. We shall prove that the halting problem for a tag system can have an arbitrary (recursively enumerable) degree of undecidability (Corollary III).

A related problem arises when we ask if there exists a uniform procedure for determining, given a tag system, whether or not there is any word on which the tag system does not halt, an “immortal” word in the system. The alternative, of course, being that the system eventually halts on every (finite) word. It is shown here that this problem, the immortality problem for tag systems, is recursively unsolvable of degree 0″ (Corollary II).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 2 , June 1971 , pp. 229 - 239
Copyright
Copyright © Association for Symbolic Logic 1971

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References

[1]Aanderaa, S. O., Some recursively undecidable problems in automata theory and quantification theory. Notices of the American Mathematical Society, vol. 13 (1966), Abstract 66T-327.Google Scholar
[2]Axt, P. and Singletary, W. E., Decision problems for monogenie Post normal systems, Notices of the American Mathematical Society, vol. 13 (1966), Abstract pp. 640–649.Google Scholar
[3]Axt, P. and Singletary, W. E., On deterministic normal systems, Zeitschrift fär Mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 49–62.CrossRefGoogle Scholar
[4]Belsnes, D. and Aanderaa, S. O., Decision problems for tag systems, Notices of the American Mathematical Society, vol. 14 (1967), Abstract 67T-698.Google Scholar
[5]Boone, W. W., Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems, Annals of Mathematics, vol. 83 (1966), pp. 520–571.CrossRefGoogle Scholar
[6]Boone, W. W. and Rogers, H. Jr., On a problem of J. H. C. Whitehead and a problem of Alonzo Church, Mathematica Scandinavica, vol. 19 (1966), pp. 185–192.CrossRefGoogle Scholar
[7]Cocke, J. and Minsky, M. L., Universality of tag systems with P = 2, Journal of the Association for Computing Machinery, vol. 11 (1964), pp. 15–20.CrossRefGoogle Scholar
[8]Cudia, D. F. and Singletary, W. E., Post's correspondence problem and degrees of unsolvability; Degrees of unsolvability in automata and grammars, this Journal, vol. 30 (1965), pp. 267–268.Google Scholar
[9]Cudia, D. F. and Singletary, W. E., The Post correspondence problem, this Journal, vol. 33 (1968a), pp. 418–430.Google Scholar
[10]Cudia, D. F. and Singletary, W. E.Degrees of unsolvability in formal grammars, Journal of the Association for Computing Machinery, vol. 15 (1968b), pp. 680–692.CrossRefGoogle Scholar
[11]Davis, M. D., A note on universal Turing machines, Automata Studies, Princeton, 1956, pp. 167–175.Google Scholar
[12]Fischer, P. C., Quantificational variants on the halting problem for Turing machines, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15, pp. 211–218.CrossRefGoogle Scholar
[13]Hooper, Ph. K., The undecidability of the Turing machine immortality problem, this Journal, vol. 31 (1966a), pp. 219–234.Google Scholar
[14]Hooper, Ph. K., Monogenie Post normal systems of arbitrary degree, Journal of the Association for Computing Machinery, vol. 13 (1966b), pp. 359–363.CrossRefGoogle Scholar
[15]Hooper, Ph. K., The immortality problem for Post normal systems, Journal of the Association for Computing Machinery, vol. 13 (1966c), pp. 594–599.CrossRefGoogle Scholar
[16]Maslov, S. Ju., On E. L. Post's “tag problem”, Trudy Matematičeskogo Instituta im. V. A. Steklova, vol. 72 (1964), pp. 57–68.Google Scholar
[17]Minsky, M. L., Recursive unsolvability of Post's problem of tag and other topics in theory of Turing machines, Annals of Mathematics, vol. 74 (1961), pp. 437–455.CrossRefGoogle Scholar
[18]Minsky, M. L., Computation: finite and infinite machines, Prentice-Hall, 1967.Google Scholar
[19]Post, E. L., Formal reduction of the general combinatorial decision problem, American Journal of Mathematics, vol. 65 (1943), pp. 197–215.CrossRefGoogle Scholar
[20]Post, E. L., Absolutely unsolvable problems and relatively undecidable propositions—account of an anticipation, M. Davis, The Undecidable (ms. unpublished, 1941).Google Scholar
[21]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, 1967.Google Scholar
[22]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. 149–175.CrossRefGoogle Scholar
[23]Wang, H., Tag systems and lag systems, Mathematische Annalen, vol. 152 (1963), pp. 65–74.CrossRefGoogle Scholar