Resource Bounded Unprovability of Computational Lower Bounds

This paper introduces new notions of asymptotic proofs, PT(polynomial-time)extensions, PTM(polynomial-time Turing machine)-ω-consistency, etc. on formal theories of arithmetic including PA (Peano Arithmetic). An asymptotic proof is a set of infinitely many formal proofs, which is introduced to define and characterize a property, PTM-ω-consistency, of a formal theory. Informally speaking, PTM-ω-consistency is a polynomial-time bounded version (in asymptotic proofs) of ω-consistency, and characterized in two manners: (1) (in the light of the extension of PTM to TM) the resource unbounded version of PTM-ω-consistency is equivalent to ω-consistency, and (2) (in the light of asymptotic proofs by PTM) a PTMω-inconsistent theory includes an axiom that only a super-polynomial-time Turing machine can prove asymptotically over PA, under some assumptions. This paper shows that P6=NP (more generally, any super-polynomial-time lower bound in PSPACE) is unprovable in a PTM-ω-consistent theory T , where T is a consistent PT-extension of PA (although this paper does not show that P6=NP is unprovable in PA, since PA has not been proven to be PTM-ω-consistent). This result implies that to prove P 6=NP by any technique requires a PTM-ω-inconsistent theory, which should include an axiom that only a super-polynomialtime machine can prove asymptotically over PA (or implies a super-polynomial-time computational upper bound) under some assumptions. This result is a kind of generalization of the result of “Natural Proofs” by Razborov and Rudich [21], who showed that to prove “P6=NP” by a class of techniques called “Natural Proofs” implies a super-polynomial-time (e.g., sub-exponential-time) algorithm that can break a typical cryptographic primitive, a pseudo-random generator. Our result also implies that any relativizable proof of P6=NP requires the resource unbounded version of PTM-ω-inconsistent theory, ω-inconsistent theory, which suggests another negative result by Baker, Gill and Solovay [1] that no relativizable proof can prove “P6=NP” in PA, which is a ω-consistent theory. Therefore, our result gives a unified view to the existing two major negative results on proving P6=NP, Natural Proofs and relativizable proofs, through the two manners of characterization of PTM-ω-consistency. We also show that the PTM-ω-consistency of T cannot be proven in any PTM-ω-consistent theory S, where S is a consistent PT-extension of T . That is, to prove the independence of P vs NP from T by proving the PTM-ω-consistency of T requires a PTM-ω-inconsistent theory, or implies a super-polynomial-time computational upper bound under some assumptions. This seems to be related to the results of Ben-David and Halevi [4] and Kurz, O’Donnell and Royer [17], who showed that to prove the independence of P vs NP from PA using any currently known mathematical paradigm implies an extremely-close-to-polynomial-time (but still super-polynomial-time) algorithm that can solve NP-complete problems. Based on this result, we show that the security of any computational cryptographic scheme is unprovable in the setting where adversaries and provers are modeled as polynomial-time Turing machines and only a PTM-ω-consistent theory is allowed to prove the security.