Visibly Pushdown Languages and Free Profinite Algebras

Abstract: We build a notion of algebraic recognition for visibly pushdown languages by finite algebraic objects. These come with a typical Eilenberg relationship, now between classes of visibly pushdown languages and classes of finite algebras. Building on that algebraic foundation, we further construct a topological object with one purpose being the possibility to derive a notion of equations, through which it is possible to prove that some given visibly pushdown language is not part of a certain class (or to even show decidability of the membership-problem of the class in some cases). In particular, we obtain a special instance of Reiterman's theorem for pseudo-varieties. These findings are then employed on two subclasses of the visibly pushdown languages, for which we derive concrete sets of equations. For some showcase languages, these equations are utilised to prove non-membership to the previously described classes.