Semilinearity of Families of Languages

TL;DR

This paper develops techniques to create and analyze language families that are exclusively semilinear, showing their properties, closure, and decidability aspects, with new grammar systems and characterizations.

Contribution

It introduces general techniques for constructing and analyzing semilinear language families, including closure properties, decidability results, and new grammar systems extending known families.

Findings

Smallest full AFL containing a semilinear full trio is also semilinear.

Closure under intersection with NCM languages preserves semilinearity.

New grammar systems are shown to only generate semilinear 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 L that are semilinear full trios, the smallest full AFL containing L that is also closed under intersection with languages in NCM (where NCM is the family of languages accepted by NFAs 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.