Semilinearity of Families of Languages

Abstract

Techniques are developed for creating new and general language families of only semilinear languages, and for showing families only contain semilinear languages. It is shown that for language families [Formula: see text] that are semilinear full trios, the smallest full AFL containing [Formula: see text] that is also closed under intersection with languages in [Formula: see text] (where [Formula: see text] is the family of languages accepted by [Formula: see text]s augmented with reversal-bounded counters), is also semilinear. If these closure properties are effective, this also immediately implies decidability of membership, emptiness, and infiniteness for these general families. From the general techniques, new grammar systems are given that are extensions of well-known families of semilinear full trios, whereby it is implied that these extensions must only describe semilinear languages. This also implies positive decidability properties for the new systems. Some characterizations of the new families are also given.