ICALP 1979: Automata, Languages and Programming pp 431-445

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

• Burkhard Monien
• Ivan Hal Sudborough
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 71)

Abstract

A family of problems {GAP(2dS(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(2dS(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.

References

1. (1).
W. J. Savitch, "Deterministic simulation of nondeterministic Turing machines", ACM Symposium on Theory of Computing (1969), 247–248.Google Scholar
2. (2).
W. J. Savitch, "Relationships between nondeterministic and deterministic tape complexities", JCSS 4(1970), 177–192.Google Scholar
3. (3).
N. D. Jones, "Space bounded reducibility among combinatorial problems", JCSS 11(1975), 68–75.Google Scholar
4. (4).
R. E. Stearns, J. Hartmanis, and P. M. Lewis, "Hierarchies of memory limited computations", IEEE Conf. Record on Switching Circuit Theory and Logical Design (1965), 191–202.Google Scholar
5. (5).
C. Papadimitriou, "The NP-completeness of the bandwidth minimization problem", Computing 16(1976), 263–270.Google Scholar
6. (6).
M. Sipser, "Halting space bounded computations", IEEE Conf. Record on Foundations of Computer Science (1978), 73–74.Google Scholar
7. (7).
J. E. Hopcroft and J. D. Ullman, "Some results on tape bounded Turing machines", J. ACM (1969), 168–188.Google Scholar
8. (8).
J. Seiferas, "Techniques for separating space complexity classes", JCSS 14(1977), 73–99.Google Scholar
9. (9).
J. Hartmanis, M. Immerman, and S. Mahaney, "One-way log-tape reductions", IEEE Conf. Record on Foundations of Computer Science (1978), 65–72.Google Scholar