Unprovably Intractable Languages: (preliminary version)

With respect to a given formal system ℱ, we investigate the class UI[ℱ] of "unprovably intractable" languages, i.e. those whose non-member ship in Ρ is independent of ℱ. We show that when ℱ is sufficiently strong, UI[ℱ] (together with P) is closed downward under polynomial-time Turing reducibility. In other senses, however, the boundary between UI[ℱ] and the complementary class PI[ℱ] is shown to be very badly behaved. All results hold when ℱ is Peano Arithmetic or any stronger system. Our investigations treat two questions of general interest: (1) How strong must a system be to constitute an "acceptable formal theory of computation"?, and (2) Which theorems in computer science can or cannot be "proved constructively"? We show that with regard to sufficiently strong ℱ and "the" natural definition of NP ∩ co-NP by a formula of TM codes M, the corresponding "provable class" equals NP ∩ co-NP. This contrasts with the theorem of [Kow84], using a natural criterion for proving membership of L(M) in NP ∩ co-NP stated to be constructive, that the resulting provable class equals NP ∩ co-NP only if NP ∩ co-NP has complete members.