On counting functions and slenderness of languages

Abstract We study counting-regular languages — these are languages L for which there is a regular language L ′ such that the number of strings of length n in L and L ′ are the same for all n. We show that the languages accepted by unambiguous nondeterministic Turing machines with a one-way read-only input tape and a reversal-bounded worktape are counting-regular. Many one-way acceptors are a special case of this model, such as reversal-bounded deterministic pushdown automata, reversal-bounded deterministic queue automata, and many others, and therefore all languages accepted by these models are counting-regular. This result is the best possible in the sense that the claim does not hold for either 2-ambiguous NPDA 's, unambiguous NPDA 's with no reversal-bound, and other models. We also study closure properties of counting-regular languages, and we study decidability problems in regards to counting-regularity. For example, it is shown that the counting-regularity of even some restricted subclasses of NPDA 's is undecidable. Lastly, k-slender languages — where there are at most k words of any length — are also studied. Amongst other results, it is shown that it is decidable whether a language in any semilinear full trio is k-slender.

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