A Note on Induction Schemas in Bounded Arithmetic

TL;DR

This paper compares induction schemas in bounded arithmetic theories, showing that certain schemas are provable in one theory but not in another unless P equals NC, and offers alternative axiomatisations.

Contribution

It demonstrates the non-provability of specific induction schemas in Allen's D_{2}^{1} unless P=NC and provides new axiomatisations of S^{1}_{2}.

Findings

S^{1}_{2} proves certain induction schemas.

D_{2}^{1} does not prove related schemas unless P=NC.

Provides alternative axiomatisations of S^{1}_{2}.

Abstract

As is well known, Buss' theory of bounded arithmetic $S^{1}_{2}$ proves $\Sigma_{0}^{b}(\Sigma_{1}^{b})-LIND$; however, we show that Allen's $D_{2}^{1}$ does not prove $\Sigma_{0}^{b}(\Sigma_{1}^{b})-LLIND$ unless $P = NC$. We also give some interesting alternative axiomatisations of $S^{1}_{2}$.