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The complexity of ranking simple languages

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Abstract

Ranking is the problem of computing for an input string its lexicographic index in a given (fixed) language. This paper concerns the complexity of ranking. We show that ranking languages accepted by 1-way unambiguous auxiliary pushdown automata operating in polynomial time is inNC (2). We also prove negative results about ranking for several classes of simple languages.C is rankable in deterministic polynomial time iffP=P #P, whereC is any of the following six classes of languages: (1) languages accepted by logtime-bounded nondeterministic Turing machines, (2) languages accepted by (uniform) families of unbounded fan-in circuits of constant depth and polynomial size, (3) languages accepted by 2-way deterministic pushdown automata, (4) languages accepted by multihead deterministic finite automata, (5) languages accepted by 1-way nondeterministic logspace-bounded Turing machines, and (6) finitely ambiguous linear context-free languages.

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This research was partially supported by the National Science Foundation under Grant DCR-8696097. A preliminary version of this paper was presented at the 3rd Annual Structure in Complexity Theory Conference, Washington, DC, June 1988.

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Huynh, D.T. The complexity of ranking simple languages. Math. Systems Theory 23, 1–19 (1990). https://doi.org/10.1007/BF02090763

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  • DOI: https://doi.org/10.1007/BF02090763

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