A Note on Induction Schemas in Bounded Arithmetic

Abstract

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