Forest languages defined by counting maximal paths

TL;DR

This paper characterizes a class of forest languages defined by counting leaf-to-root paths and provides a decidable algorithm to determine membership in this class, extending to subclasses like PDL and CTL*.

Contribution

It introduces the class *D* of forest languages recognized by iterated wreath products of syntactic algebras and presents a finite algorithm to decide language membership.

Findings

Decidability of class *D* membership for regular forest languages

Algorithm constructs wreath product of syntactic algebras

Applicability to subclasses PDL and CTL*

Abstract

A leaf path language is a Boolean combination of sets of the form $\mathsf{{}^mE}^k L$, with $k \ge 1$ and $L$ a regular word language, which consist of those forests where the node labels in at least $k$ leaf-to-root paths make up a word that belongs to $L$. We look at the class $\mathsf{*D}$ of the languages recognized by iterated wreath products of syntactic algebras of leaf path languages. We prove the existence of an algorithm that, given a regular forest language, returns in finite time a sequence of such algebras; their wreath product is divided by the language's syntactic algebra if, and only if this language belongs to $\mathsf{*D}$, which makes membership in this class a decidable question. The result also applies to the subclasses $\mathsf{PDL}$ and $\mathsf{CTL^*}$.