Towards separating nondeterminism from determinism

Abstract

The main result of this paper is a separation result: there is a positive integerk such that for all well-behaving functionst(n), there is a language accepted by a nondeterministic (multi-tape) Turing machine in timet(n) which cannot be accepting by any deterministic (multitape) Turing machine in timeO(t(n)) and simultaneously spaceo((t(n)) 1/k ). This implies, for example that for any positive integer,l,l ≠k, there is a language accepted by an l time bounded NDTM which cannot be accepted by a DTM in time and spaceO(n l ) andO((logn) l ′) respectively for anyl′. Such a result is not provable by direct diagonalization because we do not have time to “simulate and do the opposite". We devise a different method for accomplishing the result: We first use an alternating Turing machine to speed up the simulation of a time and space bounded DTM and then argue that if our separation result did not hold, every NDTM can itself be simulated faster by another NDTM producing a contradiction to the standard hierarchy results. Some other applications of this method are also presented.