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The complexity of ranking simple languages. (English) Zbl 0692.68059
Summary: 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 in $$NC^{(2)}$$. We also prove negative results about ranking for several classes of simple languages. C is rankable in deterministic polynomial time iff $$P=P^{\#P}$$, where C 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.

##### MSC:
 68Q45 Formal languages and automata 68Q25 Analysis of algorithms and problem complexity 68Q05 Models of computation (Turing machines, etc.) (MSC2010) 03D10 Turing machines and related notions 03D15 Complexity of computation (including implicit computational complexity)
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