# Arithmetical complexity of some problems in computer science

Communications

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## Abstract

We show that the set of all (indices of) Turing machines running in time n^{2} is a complete π _{1} ^{0} set and that the set of all (indices of Turing machines computing characteristic functions of) recursive sets A such that P^{A} ≠ NP^{A} is a complete π _{2} ^{0} set. As corollaries we obtain results saying that some assertions concerning running time of Turing machines and some instances of the relativized P = NP problem are independent of set theory (or of another theory containing arithmetic).

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## References

- [BGS]Baker, T., Gill, J. and Solovay, R., Relativizations of the P = ? NP question, SIAM J. on Comp. 4 (1975), 431–442.CrossRefGoogle Scholar
- [D]Dekhtyar, M.I., On the relation of deterministic and non-deterministic complexity classes, Mathematical Foundations of Computer Science 1976 (A. Mazurkiewicz, ed.), Lecture Notes in Computer Science vol. 45, p. 255–259, Springer-Verlag 1976.Google Scholar
- [Fe]Feferman, S., Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae 49 (1960), 35–92.Google Scholar
- [HH]Hartmanis, J. and Hopcroft, J.E., Independence results in computer science, SIGACT News 8, Number 4 (Oct.-Dec. 1976), 13–21.CrossRefGoogle Scholar
- [Rog]Rogers, H., Jr., Theory of recursive functions and effective computability, McGrow-Hill, New York, 1967.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1977