Fragments of bounded arithmetic and bounded query classes

Abstract

We characterize functions and predicates Σ i + 1 b \Sigma _{i + 1}^b -definable in S 2 i S_2^i . In particular, predicates Σ i + 1 b \Sigma _{i + 1}^b -definable in S 2 i S_2^i are precisely those in bounded query class P Σ i p [ O ( log ⁡ n ) ] {P^{\Sigma _i^p}}[O(\log n)] (which equals to Log Space Σ i p \operatorname {Log}\;{\text {Space}}^{\Sigma _i^p} by [B-H, W]). This implies that S 2 i ≠ T 2 i S_2^i \ne T_2^i unless P Σ i p [ O ( log ⁡ n ) ] = Δ i + 1 p {P^{\Sigma _i^p}}[O(\log n)] = \Delta _{i + 1}^p . Further we construct oracle A A such that for all i ≥ 1 i \geq 1 : P Σ i p ( A ) [ O ( log ⁡ n ) ] ≠ Δ i + 1 p ( A ) {P^{\Sigma _i^p(A)}}[O(\log n)] \ne \Delta _{i + 1}^p(A) . It follows that S 2 i ( α ) ≠ T 2 i ( α ) S_2^i(\alpha ) \ne T_2^i(\alpha ) for all i ≥ 1 i \geq 1 . Techniques used come from proof theory and boolean complexity.