Polynomial time ultrapowers and the consistency of circuit lower bounds

Abstract

A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory ∀PV of all polynomial time functions. Generalizing a theorem of Hirschfeld (1975), we show that every countable model of ∀PV is isomorphic to an existentially closed substructure of a polynomial time ultrapower. Moreover, one can take a substructure of a special form, namely a limit polynomial time ultrapower in the classical sense of Keisler (1963). Using a polynomial time ultrapower over a nonstandard Herbrand saturated model of ∀PV we show that ∀PV is consistent with a formal statement of a polynomial size circuit lower bound for a polynomial time computable function. This improves upon a recent result of Krajíček and Oliveira (2017). * Based on the first author's Master Thesis [4] written under the supervision of the second author. 1 All relevant technical notions will be defined precisely later. This is a post-peer-review, pre-copyedit version of an article published in Archive for mathematical logic.

Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

“…about 2 O(n) -time concepts Finally, it is possible to consider the provability of circuit upper bounds as well. This has been investigated systematically by Cook-Krajíček [13] and in a sequence of more recent works by Bydžovský, Krajíček, Müller and Oliveira [7,8,30].…”

Consistency of circuit lower bounds with bounded theories

“…This is of interest because the weaker theory PV already formalizes sophisticated arguments, such as a proof of the PCP Theorem [Pic15b]. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with PV.…”

“…We refer to the introduction of [MP17] for more information about this line of work, and to Appendix A for some related remarks that might be of independent interest. Theorem 1 and our techniques are more directly connected to [CK07], [KO17], and [BM18]. We review the relevant results next.…”

“…where k is any fixed integer. This consistency statement was subsequently improved by [BM18], who considered a more natural formalization of the statement that a language has circuits of size O(n k ) and adapted the argument [KO17] using polynomial-time ultrapowers. 14 All previous results refer to the consistency of lower bounds on infinitely many input lengths, and Theorem 1 part (c) strictly improves upon [KO17] and [BM18].…”

Feasibly constructive proofs of succinct weak circuit lower bounds

“…The succinct lower bound LB[SAT] for s = n k is shown in [52] to be unprovable in a theory formalizing NC 1 reasoning unless subexponential size formulas can approximate polynomial size circuits. Relatedly, LB[Q] has been shown to be consistent with PV 1 for Q = SAT in [21] (improving upon [36]) unless the polynomial hierarchy collapses to the Boolean hierarchy, and recently unconditionally for some Q ∈ P [40,13,14].…”