Decidability of Geometricity of Regular Languages

Abstract

Abstract. Geometrical languages generalize languages introduced to model temporal validation of real-time softwares. We prove that it is decidable whether a regular language is geometrical. This result was previously known for binary languages.

Geometrically Closed Positive Varieties of Star-Free Languages

“…A geometrical closure is an operation on formal languages introduced recently by Dubernard, Guaiana, and Mignot [8]. It is defined as follows: Take any language L over some k-letter alphabet and consider the set called the figure of L in [8], which consists of all elements of N k corresponding to Parikh vectors of prefixes of words from L. The geometrical closure of L is the language γ(L) of all words w such that the Parikh vectors of all the prefixes of w lie in the figure of L. This closure operator was inspired by the previous works of Blanpain, Champarnaud, and Dubernard [4] and Béal et al [3], in which geometrical languages are studied -using the terminology from later paper [8], these can be described as languages whose prefix closure is equal to their geometrical closure. Note that this terminology was motivated by the fact that a geometrical language is completely determined by its (geometrical) figure.…”