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# 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 [itex]L \subseteq NL[itex]. 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, [itex]L \subseteq P[itex], 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).

## References

 Important complexity classes (more) P | NP | Co-NP | NP-C | Co-NP-C | NP-hard | UP | #P | #P-C | L | NC | P-C PSPACE | PSPACE-C | EXPTIME | EXPSPACE | BQP | BPP | RP | ZPP | PCP | IP | PH