Strengths and Weaknesses of LH Arithmetic

Abstract

In this paper we provide a new arithmetic characterization of the levels of the log-time hierarchy (LH). We define arithmetic classes Σ log k and Π log k that correspond to Σk-LOGTIME and Πk-LOGTIME, respectively. We break Σ log k and Π log k into natural hierarchies of subclasses Σ m·log k and Π m·log k . We then define bounded arithmetic deduction systems (T Σ log k ) whose U {|id|} B(Σ log k )-definable functions are precisely B(Σk-LOGTIME). We show these theories are quite strong in that (1) LIOpen proves for any fixed m that Σ m·log k = Π m·log k , (2) TAC, a theory that is slightly stronger than k (T Σ log k ) whose Σ b 1 (LH)-definable functions are LH, proves LH is not equal to Σm-TIME(s) for any m > 0, where 2 s ∈ L, s(n) ∈ ω(log n), and (3) TAC proves LH = E |x| m+1 # l |x| Σ k log for all k and m. We then show that the theory TAC cannot prove the collapse of the polynomial hierarchy. Thus any such proof, if it exists, must be argued in a stronger systems than ours.Mathematics Subject Classification: 03F30, 68Q15.

A theory for Log-Space and NLIN versus co-NLIN

“…This is fairly close to saying (but not quite) that TLS cannot prove NLIN =co-NLIN. Using the techniques of Pollett and Pruim [16], it is possible that the latter result could be obtained with the techniques of this paper but the expense would be to make TLS a more awkward looking theory. The arguments presented for the results above can be generalized to where simply defined functions of quasi-linear growth are added to both TLS and S 1 .…”