Resource Bounded Unprovability of Computational Lower Bounds

Tatsuaki Okamoto; Ryo Kashima

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Resource Bounded Unprovability of Computational Lower Bounds

Tatsuaki Okamoto·Ryo Kashima

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Abstract

This paper introduces new notions of asymptotic proofs, PT(polynomial-time)-extensions, PTM(polynomial-time Turing machine)-omega-consistency, etc. on formal theories of arithmetic including PA (Peano Arithmetic). This paper shows that P not= NP (more generally, any super-polynomial-time lower bound in PSPACE) is unprovable in a PTM-omega-consistent theory T, where T is a consistent PT-extension of PA. This result gives a unified view to the existing two major negative results on proving P not= NP, Natural Proofs and relativizable proofs, through the two manners of characterization of PTM-omega-consistency. We also show that the PTM-omega-consistency of T cannot be proven in any PTM-omega-consistent theory S, where S is a consistent PT-extension of T.

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Tatsuaki Okamoto, Ryo Kashima. 2005-03-31. Resource Bounded Unprovability of Computational Lower Bounds. https://arxiv.org/abs/cs/0503091

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