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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.
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>L \subseteq NL<math>. Also, a decider using O(log n) space cannot use more than 2O(log n)=nO(1) time, because this is the total number of possible configurations; thus, <math>L \subseteq P<math>.
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 <math>L = P<math>, and whether <math>L = NL<math>.
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.
References
- Papadimitriou, Computational Complexity Theory.
- Undirected ST-connectivity in Log-Space (http://www.wisdom.weizmann.ac.il/~reingold/publications/sl.ps). Omer Reingold. Electronic Colloquium on Computational Complexity. No. 94.
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