Theorems on the time hierarchy for random access machines

Abstract

In this paper we consider questions of the time of recognition of sets of words on a machine with arbitrary access to memory. The basic result obtained in the paper asserts that for any function T(n), which can be calculated on a random access machine in time T(n), there exists a set of words A in a two-letter alphabet, recognizable by some machine in time 21T(n) + 6T(n) + 25n, but which is not recognizable in time T(n). The proof of this result consists of constructing such a set, using a diagonal procedure. It is a refinement of the theorem of Cook-Reckhow on the time hierarchy. Then we define a class of functions

, containing many functions (e.g., polynomials of degree higher than one, n log n,$2^{\sqrt n }$, etc.), which are of interest as complexity estimates. For the class

we prove a theorem refining the basic result for functions of this class. It consists of the fact that for any function T(n)

and any c>1 there exists a set of words, recognizable in time cT(n) by some random access machine, but not recognizable in time T(n). We also consider random access machines with a bound on the length of register; results are given connecting the time of work of such a machine with the time of work of a machine without restriction on the length of register.