A note on Parikh maps, abstract languages, and decision problems

Abstract

We consider formulas similar to Presburger formulas, but extended by allowing the predicate | (for divides), but not allowing universal quantification of the resulting expressions. The solution sets for such formulas are known to be not semilinear, in general. We show that some results concerning decidability questions for classes of languages with effectively constructible semilinear Parikh maps can be extended to languages whose effectively constructible Parikh maps are solutions to these new formulas, which we call “modular”. We suggest that modularity may offer a natural extension of semilinearity, in which many previously established results may remain valid.