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Some results on measure independent Gödel speed-ups1

Published online by Cambridge University Press:  12 March 2014

Martin K. Solomon
Martin K. Solomon*
Affiliation:
Graduate School of Business Administration, Rutgers UniversityNewark, New Jersey 07107
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Abstract

We study the measure independent character of Gödel speed-up theorems, in particular, we strengthen Arbib's necessary condition for the occurrence of a Gödel speed-up [2, p. 13] to an equivalence result and generalize Di Paola's speed-up theorem [4]. We also characterize undecidable theories as precisely those theories which possess consistent measure independent Gödel speed-ups and show that a theory τ2 is a measure independent Gödel speed-up of a theory τ1 if and only if the set of undecidable sentences of τ1 which are provable in τ2 is not recursively enumerable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 4 , December 1978 , pp. 667 - 672
Copyright
Copyright © Association for Symbolic Logic 1978

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Footnotes

1

Most of the results in this paper are taken from the author's Ph.D. Dissertation (Stevens Institute of Technology) which was written under the valuable supervision of Stephen L. Bloom.

The author is also grateful to Professor Bloom for his careful examination of this paper and to the referee for helpful suggestions.

References

REFERENCES

[1]Gödel, Kurt, Uber die Länge der Beweise, Ergeb, eines mathematischen Kolloquiums, VII (1936), pp. 23–24; English translation in M. Davis, The undecidable, Raven Press, 1965, pp. 82–83.Google Scholar
[2]Arbib, M. A., Speed-up theorems and incompleteness theorems, Automata theory (Caianiello, E. R., Editor), Academic Press, New York, 1966, pp. 6–24.Google Scholar
[3]Ehrenfeucht, A. and Mycielski, J., Abbreviating proofs by adding new axioms, Bulletin of the American Mathematical Society, vol. 77(1971), pp. 366–367.CrossRefGoogle Scholar
[4]Di Paola, R. A., A theorem on shortening the length of proof in formal systems of arithmetic, this Journal, vol. 40(1975), pp. 398–400.Google Scholar
[5]Davis, M., Computability and unsolvability, McGraw-Hill, New York, 1958.Google Scholar
[6]Davis, Martin, Speed-up theorems and Diophantine equations, Computational complexity (Rustin, R., Editor), Algorithmics Press, N. Y., 1973, pp. 87–95.Google Scholar
[7]Szmielew, W., Elementary properties of Abelian groups, Fundamenta Mathematicae, vol. 41, pp. 203–271.CrossRefGoogle Scholar
[8]Rogers, H. Jr., Theory of recursive functions and effective computability, McGrawHill, New York, 1967.Google Scholar
[9]Solomon, M. K., The Gödel speed-up phenomenon, Doctoral Dissertation, Stevens Institute of Technology, 1975.Google Scholar
[10]Blum, Manuel, A machine independent theory of the complexity of recursive functions, Journal of the Association for Computing Machinery, vol. 14, pp. 322–336.CrossRefGoogle Scholar