NL-complete

In computational complexity theory, NL-complete is a complexity class containing the languages that are complete for NL, the class of decision problems that can be solved by a nondeterministic Turing machine using a logarithmic amount of memory space. The NL-complete languages are the most "difficult" or "expressive" problems in NL. If a method exists for solving any one of the NL-complete problems in logarithmic memory space, then NL = L.

NL consists of the decision problems that can be solved by a nondeterministic Turing machine with a read-only input tape and a separate read-write tape whose size is limited to be proportional to the logarithm of the input length. Similarly, L consists of the languages that can be solved by a deterministic Turing machine with the same assumptions about tape length. Because there are only a polynomial number of distinct configurations of these machines, both L and NL are subsets of the class P of deterministic polynomial-time decision problems.

Formally, a decision problem is NL-complete when it belongs to NL, and has the additional property that every other decision problem in NL can be reduced to it. Unless otherwise specified, the reductions in this definition are assumed to be many-one reductions by a deterministic logarithmic-space algorithm.

If an NL-complete language X could belong to L, then so would every other language Y in NL. For, suppose (by NL-completeness) that there existed a deterministic logspace reduction r that maps an instance y of problem Y to an instance x of problem X, and also (by the assumption that X is in L) that there exists a deterministic logspace algorithm A for solving problem X. With these assumptions, a problem y in language Y could be solved in logarithmic space by an algorithm that simulates the behavior of algorithm A on input r(y), using the reduction algorithm to simulate each access to the read-only tape for r(y).

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