Universal Traversal Sequences

Abstract

In this article we discuss a purely combinatorial problem, the construction of short universal traversal sequences, and its relationship to questions about logspace computation. We state the problem formally, show how it arises naturally in complexity theory, and review some of the known partial results. A basic introduction to complexity theory can be found in [6]. The P vs. NP problem is recognized by the mathematical world as the central open question in the theory of computation. Less widely known outside of computer science is the fact that the analogous question for space-bounded computation was resolved long ago': Savitch [8] shows that any language accepted by a nondeterministic Turing machine that uses space O(s(n)) is also accepted by a deterministic Turing machine that uses space O((s(n))2). Hence deterministic polynomial space is equivalent (in terms of language-recognition power) to nondeterministic polynomial space: PSPACE = NPSPACE. However, one question about the relationship of nondeterministic spacebounded computation and its deterministic counterpart remains open: Is the quadratic blow-up in space complexity exhibited by Savitch's construction necessary? This question turns out to be most interesting for computations that use very little space. Let L be the class of languages accepted by deterministic Turing machines using only O(log n) space, and NL be the class of languages accepted by nondeterministic Turing machines using only O(log n) space. It could be the case that Savitch's theorem is optimal at this low end of the space-complexity spectrum. On the other hand, it could be that any language recognizable by a nondeterministic Turing machine is also recognizable by a deterministic Turing machine with the same space complexity; if that's true, then NL is exactly equal to L. The truth might also lie somewhere between these two extremes.