L (complexity)

In computational complexity theory, L is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a logarithmic amount of memory space. Intuitively, logarithmic space is enough space to hold a constant number of pointers into the input and a logarithmic number of boolean flags. A generalization of L is NL, which is the class of languages decidable in logarithmic space on a nondeterministic Turing machine. We then trivially have <math>\mathrm{L} \subseteq \mathrm{NL}</math>. Also, a decider using <math>O(\log n)</math> space cannot use more than <math>2^{O(\log n)} = n^{O(1)}</math> time, because this is the total number of possible configurations; thus, <math>\mathrm{L} \subseteq \mathrm{P}</math>, where P is the class of problems solvable in deterministic polynomial time. Every problem in L is complete under log-space reductions; since this is useless, weaker reductions are defined which allow identification of stronger complete problems in L, but there is no generally accepted definition of L-complete. Important open problems include whether L = P, and whether L = NL. The related class of function problems is FL. FL is often used to define logspace reductions. A breakthrough October 2004 paper by Omer Reingold showed that USTCON, the problem of whether there exists a path between two vertices in a given undirected graph, is in L, establishing that L = SL, since USTCON is SL-complete. One consequence of this is a simple logical characterization of L: it contains precisely those languages expressible in first-order logic with an added commutative transitive closure operator (in graph theoretical terms, this turns every connected component into a clique). L is low for itself, because it can simulate log-space oracle queries (roughly speaking, "function calls which use log space") in log space, reusing the same space for each query.

References

• Christos Papadimitriou (1993). Computational Complexity, 1st edition, Addison Wesley. ISBN 0-201-53082-1.  Chapter 16: Logarithmic space, pp.395–408.
• Undirected ST-connectivity in Log-Space. Omer Reingold. Electronic Colloquium on Computational Complexity. No. 94.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X.  Section 8.4: The Classes L and NL, pp.294–296.
• Michael R. Garey and David S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman. ISBN 0-7167-1045-5.  Section 7.5: Logarithmic Space, pp.177–181.