Fast programs for initial segments and polynomial time computation in weak models of arithmetic

Abstract

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 = NP. We show, perhaps surprisingly, that the two questions are equivalent: It is consistent to believe that P = 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.