L (complexity)

From Wikipedia, the free encyclopedia

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.

[edit] 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.

v • d • e Important complexity classes (more)
P • NP • co-NP • NP-C • co-NP-C • NP-hard • UP • #P • #P-C • L • NL • NC • P-C • PSPACE • PSPACE-C • EXPTIME • NEXPTIME • EXPSPACE • 2-EXPTIME • PR • RE • Co-RE • RE-C • Co-RE-C • R • BQP • BPP • RP • ZPP • PCP • IP • PH