Logspace Computation Research Articles

Logspace computations in graph products

We consider three important and well-studied algorithmic problems in group theory: the word, geodesic, and conjugacy problem. We show transfer results from individual groups to graph products. We concentrate on logspace complexity because the challenge is actually in small complexity classes, only. The most difficult transfer result is for the conjugacy problem. We have a general result for graph products, but even in the special case of a graph group the result is new. Graph groups are closely linked to the theory of Mazurkiewicz traces which form an algebraic model for concurrent processes. Our proofs are combinatorial and based on well-known concepts in trace theory. We also use rewriting techniques over traces. For the group-theoretical part we apply Bass–Serre theory. But as we need explicit formulae and as we design concrete algorithms all our group-theoretical calculations are completely explicit and accessible to non-specialists.

Undirected connectivity in log-space

We present a deterministic , log-space algorithm that solves st-connectivity in undirected graphs. The previous bound on the space complexity of undirected st-connectivity was log 4/3 (⋅) obtained by Armoni, Ta-Shma, Wigderson and Zhou (JACM 2000). As undirected st-connectivity is complete for the class of problems solvable by symmetric, nondeterministic, log-space computations (the class SL), this algorithm implies that SL = L (where L is the class of problems solvable by deterministic log-space computations). Independent of our work (and using different techniques), Trifonov (STOC 2005) has presented an O (log n log log n )-space, deterministic algorithm for undirected st-connectivity. Our algorithm also implies a way to construct in log-space a fixed sequence of directions that guides a deterministic walk through all of the vertices of any connected graph. Specifically, we give log-space constructible universal-traversal sequences for graphs with restricted labeling and log-space constructible universal-exploration sequences for general graphs.

Δ: Set-theoretic query language capturing LOGSPACE

Two versions of a set theoretic Δ-language are considered as theoretical prototypes for database query languages with “nested”, “unstructured” data where database states and queries are represented as hereditarily-finite (HF) sets and set theoretic operations, respectively. It is shown that these versions correspond exactly to (N/D)LOGSPACE computability over HF relative to a special graph representation (encoding) of HF-sets. Such languages over sets, capturing also PTIME, were introduced in previous works, however, descriptions of LOGSPACE over HF [34] were not completely satisfactory. Here we overcome the drawbacks of the previous approaches by some new partial result on definability of a linear ordering over finite extensional acyclic graphs and present a unified and simplified approach.

Monotone Separation of Logarithmic Space from Logarithmic Depth

We show that the monotone analogue of logspace computation is more powerful than monotone log-depth circuits: monotone bounded fanin circuits for a certain function in monotone logspace require depth Ω(lg2n).

Universal Traversal Sequences

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.

Positive relativizations for log space computability

We consider some open questions about log-space computability (deterministic, non-deterministic and alternating) and polynomial-time computability (deterministic and nondeterministic). In particular, the questions DL= NL?, DL= P?, and NL= NP? are considered, where DL (NL) denotes the class of sets accepted by log space-bounded deterministic (nondeterministic) Turing machines and P (NP) denotes the class of sets accepted by polynomial time-bounded deterministic (nondeterministic) Turing machines. We develop positive relativizations for the above questions in two different ways. First, DL and NL are relativized such that the oracle tape is not subject to the space bound and such that no other restrictions are placed on the oracle machines. In this case, we prove that C= D iff C T = D T for all tally sets T, where ( C, D) is a pair of classes from DL, NL, P, NP, and coNP. Next, both NL and AL (alternating log space) are relativized with the restriction introduced by Ruzzo, Simon and Tompa ( J. Comput. System Sci. 28 (1984) 216–230). We call this restriction the RST restriction, and prove that C= D iff C A RST = D A RST for all sets A, where ( C, D) is a pair of classes from DL, NL, coNL and AL, and C RST denotes the corresponding relativized class with the RST restriction.

Relativization of questions about log space computability

A notion of log space Turing reducibility is introduced. It is used to define relative notions of log space,ℒ A , and nondeterministic log space, . These classes are compared with the classes and which were originally defined by Baker, Gill, and Solovay [BGS]. It is shown that there exists a computable setA such that . Furthermore, there exists a computable setA such that and . Also a notion of log space truth table reducibility is defined and shown to be equivalent to the notion of log space Turing reducibility.