Nondeterministic Space is Closed Under Complementation (1988)
| Citations: | 217 - 14 self |
BibTeX
@MISC{Immerman88nondeterministicspace,
author = {Neil Immerman},
title = {Nondeterministic Space is Closed Under Complementation},
year = {1988}
}
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Abstract
this paper we show that nondeterministic space s(n) is closed under complementation, 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 1964 [9]. See Hartmanis and Hunt [4] for a discussion of the history and importance of this problem, and Hopcroft and Ullman [5] for all relevant background material and definitions. The history behind the proof is as follows. In 1981 we showed that the set of first-order inductive definitions over finite structures is closed under complementation [6]. 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 1983 we studied first-order logic, with ordering, with a transitive closure operator. We showed that NSPACE[log n] is equal to (FO + pos TC), i.e. first-order logic with ordering, plus a transitive closure operation, in which the transitive closure operator does not appear within any negation symbols [7]. Now we have returned to the issue of complementation in the light of recent results on the collapse of the log space hierarchies [10, 2, 14]. We have shown that the class (FO + pos TC) is closed under complementation. Our
