Construction of models of bounded arithmetic by restricted reduced powers

Abstract

We present two constructions of models of bounded arithmetic, both in the form of a generalization of the ultrapower construction, that yield nonelementary extensions but do not introduce new lengths. As an application we show, assuming the existence of a one-way permutation g hard against polynomial-size circuits, that strictR 1 2 (g) is weaker than R 1 2 (g). In particular, if such a permutation can be defined by a term in the language of R 1 2 , then strictR 1 2 is weaker than R 1 2 .

A remark on pseudo proof systems and hard instances of the satisfiability problem

“…[34,Lemma 9.1]). The result is a restricted ultrapower of M. These have been studied for fragments of arithmetic [12,16,22,25,30,33]…”

“…The result is a restricted ultrapower of M . These have been studied for fragments of arithmetic ever since Skolem's definable ultrapower (cf. [, IV.1.…”

Polynomial time ultrapowers and the consistency of circuit lower bounds

“…In the following, by n δ for some rational δ we mean n δ . M |= |w a 2 (1 n )| = n ∧ g(w a 2 (1 n )) = eval (C a 1 (1 n ), w a 2 (1 n )) (11) Indeed, by (10) and Herbrand saturation gives w M a 2 (u) ∈ PTIME(M ) witnessing ∃x as in Theorem 3.2. Choosew(x, y) ∈ PV such that ∀PV provesw(x, y) = w( (x), y), so M |=w a 2 (1 n ) = w a 2 (1 n t ) for all n ∈ Log(M ).…”

“…One can also start with a nonstandard model M of ∀PV. Such structures are constructed in [21] and [10] who present their powers for functions restricted to some M -finite Ω ⊆ M (cf. Remark 2.21); in [10] F is additionally restricted to straight-line programs of a certain (nonstandard) length.…”

“…Of course, one can form restricted ultrapowers F/U for other function families F and such constructions have been frequently used to study arithmetic and its fragments [39,27,18,12,29,33,21,10] ever since Skolem's definable ultrapower [41]. We give a brief historical survey in Section 2.2.…”