On eliminating nondeterminism from Turing machines which use less than logarithm worktape space

Abstract

A family of problems {GAP(2^dS(n))}_d>0 is described that is log space complete for NSPACE(S(n)), for functions S(n) which grow less rapidly than the logarithm function. An algorithm is described to recognize GAP(2^dS(n)) deterministically in space S(n) × log n. Thus, we show for constructible functions S(n), with log log n ≤ S(n) ≤ log n, that:

$$\begin{gathered}(1) NSPACE(S(n)) \subseteq DSPACE(S(n) x log n), and \hfill \\(2) NSPACE(S(n)) \subseteq DSPACE( log n) iff \hfill \\\left\{ {GAP(2^{dS(n)} )} \right\}_{d > 0} \subseteq DSPACE(log n) \hfill \\\end{gathered}$$

In particular, when S(n)=log log n, we have: (1) NSPACE(log log n) ... DSPACE(log n × log log n), and (2) NSPACE(log log n) ... DSPACE(log n) iff {GAP(log n)^d)}_d>0 ... DSPACE(log n). In addition it is shown that the question of whether NSPACE(S(n)) is identical to DSPACE(S(n)), for sublogarithmic functions S(n), is closely related to the space complexity of the graph accessibility problem for graphs with bounded bandwidth.

the work of this author was supported in part by NSF Grant No. MCS 77-02494 and was performed in part while visiting the Gesamthochschule Paderborn in Paderborn, Germany.