Immerman–Szelepcsényi theorem

Closure of nondeterministic space under complementation
In computational complexity theory, the Immerman–Szelepcsényi theorem states that nondeterministic space complexity classes are closed under complementation. It was proven independently by Neil Immerman and Róbert Szelepcsényi in 1987, for which they shared the 1995 Gödel Prize. In its general form the theorem states that NSPACE(s(n)) = co NSPACE(s(n)) for any function s(n) ≥ log n. The result is equivalently stated as NL = co NL; although this is the special case when s(n) = log n, it implies the general theorem by a standard padding argument. The result solved the second LBA problem. In other words, if a nondeterministic machine can solve a problem, another machine with the same resource bounds can solve its complement problem (with the yes and no answers reversed) in the same asymptotic amount of space. No similar result is known for the time complexity classes, and indeed it is conjectured that NP is not equal to co NP. The principle used to prove the theorem has become known as inductive counting. It has also been used to prove other theorems in computational complexity, including the closure of LOGCFL under complementation and the existence of error free randomized logspace algorithms for USTCON.

As a corollary, in the same article, Immerman proved that, using descriptive complexity's equality between NL and FO(Transitive Closure), the logarithmic hierarchy, i. e. the languages decided by an alternating Turing machine in logarithmic space with a bounded number of alternations, is the same class as NL.