Constrained Expressions and their Derivatives

First order logic without quantifiers, a.k.a. zeroth order logic, is more expressive than propositional logic. In this paper, we introduce two new operators for regular expressions allowing us to handle zeroth order boolean formula in order to increase the expressive power of expressions; We also extend the notion of interpretation and of realization using expression interpretation. We thus define the notion of constrained expression, the language denoted of which is not necessarily regular. Furthermore, we use an extension of Antimirov partial derivatives in order to solve the membership test in the general case. Finally, we show that once the interpretation is fixed, the membership test of a word in the language denoted by a constrained expression is undecidable whereas it is decidable when the interpretation is not fixed.