On the Structure of Polynomial Time Reducibility
Journal of the ACM (JACM)
On isomorphisms and density of NP and other complete sets
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Some connections between mathematical logic and complexity theory
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
The consistency of "P = NP" and related problems with fragments of number theory
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Independence results in Computer Science? (Preliminary Version)
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
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In this paper we study two alternative approaches for investigating whether NP complete sets have fast algorithms. One is to ask whether there are long initial segments on which such sets are easily decidable by relatively short programs. The other approach is to ask whether there are weak fragments of arithmetic for which it is consistent to believe that P &equil; NP. We show, perhaps surprisingly, that the two questions are equivalent: It is consistent to believe that P &equil; NP in certain models of weak arithmetic theories iff it is true (in the standard model of computation) that there are infinitely many initial segments on which satisfiability is polynomially decidable by programs that are much shorter than the length of the initial segment.