Improved time and space hierarchies of one-tape off-line TMs

Abstract

This paper presents improved time and space hierarchies of one-tape off-line Turing Machines (TMs), which have a single worktape and a two-way input tape: (i) For any time-constructible functions t 1 (n) and t 2(n) such that infn→∞ t 1(n) log log t 1(n)/t 2(n) = 0 and t 1(n) = nO(1), there is a language which can be accepted by a t 2(n)-time TM, but not by any t 1(n)-time TM. (ii) For any space-constructible function s(n) and positive constant e, there is a language which can be accepted in space s(n) + log s(n) + (2 + e) log log s(n) by a TM with two worktape-symbols, but not in space s(n) by any TM with the same worktape-symbols. The (log log t 1(n))-gap in (i) substantially improves the Hartmanis and Stearns' (log t 1(n))-gap which survived more than 30 years.