The strength of replacement in weak arithmetic

Abstract

The replacement (or collection or choice,) axiom scheme BB(/spl Gamma/) asserts bounded quantifier exchange as follows: /spl forall/I < |a| /spl exist/x < ao(i, x) /spl rarr/ /spl exist/w /spl forall/i < |a| o (i, [w]/sub i/) where o is in the class /spl Gamma/ of formulas. The theory S/sub 2//sup 1/ proves the scheme BB(/spl Sigma//sub 1//sup b/), and thus in S/sub 2//sup 1/ every /spl Sigma//sub 1//sup b/ formula is equivalent to a strict /spl Sigma//sub 1//sup b/ formula (in which all non-sharply-bounded quantifiers are in front). Here we prove (sometimes subject to an assumption) that certain theories weaker than S/sub 2//sup 1/ do not prove either BB(/spl Sigma//sub 1//sup b/) or BB(/spl Sigma//sub 0//sup b/). We show (unconditionally) that V/sup 0/ does not prove BB(/spl Sigma//sub 1//sup B/), where V/sup 0/ (essentially I/spl Sigma//sub 0//sup 1,b/) is the two-sorted theory associated with the complexity class AC/sup 0/. We show that PV does not prove BB(/spl Sigma//sub 0//sup b/), assuming that integer factoring is not possible in probabilistic polynomial time. Johannsen and Pollet introduced the theory C/sub 2//sup 0/ associated with the complexity class TC/sup 0/, and later introduced an apparently weaker theory /spl Delta//sub 1//sup b/ - CR for the same class. We use our methods to show that /spl Delta//sub 1//sup b/ - CR is indeed weaker than C/sub 2//sup 0/, assuming that RSA is secure against probabilistic polynomial time attack. Our main tool is the KPT witnessing theorem.