3 research outputs found
Resource Bounded Unprovability of Computational Lower Bounds
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.Comment: 78 page
Thoughts on the Riemann hypothesis
The simultaneous appearance in May 2003 of four books on the Riemann
hypothesis (RH) provoked these reflections. We briefly discuss whether the RH
should be added as a new axiom, or whether a proof of the RH might involve the
notion of randomness