Translating IΔ0 + exp Proofs into Weaker Systems

Abstract

Abstract. The purpose of this paper is to explore the relationship between I∆0 + exp and its weaker subtheories. We give a method of translating certain classes of I∆0 + exp proofs into weaker systems of arithmetic such as Buss' systems S2. We show if IEi(exp) A with a proof P of expind-rank(P ) ≤ n + 1 where all (∀≤: right) or (∃≤: left) have bounding terms not containing function symbols, then S i 2 ⊇ IEi,2 A n . Here A is not necessarily a bounded formula. For IOpen(exp) we prove a similar result. Using our translations we show IOpen(exp) I∆0(exp). Here I∆0(exp) is a conservative extension of I∆0 + exp obtained by adding to I∆0 a symbol for 2x to the language as well as defining axioms for it.Mathematics Subject Classification: 03F30, 68Q15.

Strengths and Weaknesses of LH Arithmetic

“…This resulted in a theory that was so weak it seemed unlikely it could formalize any interesting circuit lower bounds. Despite this, some limited attempt to show one can translate proofs from stronger theories into meaningful results in Z was given in Pollett [14]. In this paper, we use a weaker language for bounded arithmetic than that given in Buss [3].…”