Properties
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).
It follows from the Immerman–Szelepcsényi theorem that, if a language is co-NL-complete (that is, if its complement is NL-complete) then the language is also NL-complete itself.
Read more about this topic: NL-complete
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