A Note on Induction Schemas in Bounded Arithmetic

As is well known, Buss’ theory of bounded arithmetic S 2 proves Σb0(Σ b 1) − LIND ; however, we show that Allen’s D 1 2 does not prove Σb0(Σ b 1) − LLIND unless P = NC . We also give some interesting alternative axiomatisations of S 2 . We assume familiarity with the theory of bounded arithmetic S 2 as introduced in Buss’ [2], as well as with the theory D 2 formulated by Allen in [1]. In particular, we use the general notation as introduced in [2] and in [1]. We denote the language of the theory S 2 by Lb , and the language of the theory D 2 by Ld . Thus, Lb = {0, S,+, ·, |x|, ⌊ 1 2x⌋,#,≤} , and Ld = Lb ∪ { . −, Bit(x, y),Msp(x, y),Lsp(x,y) }. The most basic theory for bounded arithmetic (which corresponds to Robinson’s Q in case of PA) for the language Lb is BASIC , introduced by Buss (see [2]), and for the language Ld is BASIC introduced by Allen (see [1]) which extends BASIC by a few additional axioms for the extra symbols. Following [1], we abbreviate Msp(x, ⌊ 1 2 |x|⌋) by Fh(x), and Lsp(x, ⌊ 12 |x|⌋) by Bh(x). We use the usual hierarchies of formulas to measure the (bounded) quantifier complexity of formulas in our first order theories: Σ i ,Π i and Σb0(Σ b i ). Here