Nondeterministic space is closed under complementation
نویسنده
چکیده
In this paper we show that nondeterministic space s n is closed under com plementation for s n greater than or equal to log n It immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by Kuroda in See Hartmanis and Hunt for a discussion of the history and importance of this problem and Hopcroft and Ullman for all relevant background material and de nitions The history behind the proof is as follows In we showed that the set of rst order inductive de nitions over nite structures is closed under complementation This holds with or without an ordering relation on the structure If an ordering is present the resulting class is P Many people expected that the result was false in the absence of an ordering In we studied rst order logic with ordering with a transitive closure operator We showed that NSPACE log n is equal to FO pos TC i e rst order logic with ordering plus a transitive closure operation in which the transitive closure operator does not appear within any negation symbols Now we have returned to the issue of complementation in the light of recent results on the collapse of the log space hierarchies We have shown that the class FO pos TC is closed under complementation Our
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Two Applications of Inductive Counting for Complementation Problems
Following the recent independent proofs of Immerman [SLAM J. Comput., 17 (1988), pp. 935-938] and Szelepcs6nyi [Bull. European Assoc. Theoret. Comput. Sci., 33 (1987), pp. 96-100] that nondeterministic space-bounded complexity classes are closed under complementation, two further applications of the inductive counting technique are developed. First, an errorless probabilistic algorithm for the ...
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