TR#: | CS0699 |

Class: | CS |

Title: | ON THE INDEPENDENCE OF $P$
VERSUS $NP$ |

Authors: | S. Ben-David and S. Halevi |

PDF - Revised | CS0699.revised.pdf |

Abstract: | We investigate the possibility that the current failure to resolve basic complexity theoretic questions stems from the independence of these questions with respect to the formal theories underlying our mathematical reasoning. We show that any question in the field of computational complexity, that can be shown to be independent of the axioms of Peano Arithmetic is `practically insignificant'. If, using any known mathematical paradigm, $P \neq NP$ can be shown to be independent of Peano Arithmetic, then $NP$ has extremely-close-to-polynomial deterministic time upper bounds. In particular, in such a case, there is a {\em DTIME}$(n^{log \ast (n)})$ algorithm that computes {\em SAT} correctly on infinitely many huge intervals of input lengths. We provide a complete characterization of the worst case behavior of languages whose location in the complexity hierarchy is independent (with respect to sufficiently strong proof systems). Such languages are, on one hand, easily computable for long stretches of inputs, and, on the other hand, they are complex infinitely often. (We also define an explicit example of such a language). Our results hold for both the Turing Machine and the Non-Uniform Circuit complexity models. |

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