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 reversalbounded 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.

On the Commutative Equivalence of Algebraic Formal Series and Languages

“…Such notion amounts to ask, for an arbitrary language L, that the language L∩L 1 is countingregular, for every regular language L 1 . A thorough analysis of the structure and of the decidability issues of this family of languages have been obtained in [35,36].…”