Wreath Products of Distributive Forest Algebras

TL;DR

This paper proves that languages recognized by forest algebras satisfying a 2-fold iterated distributive law are in PDL and that this class is decidable, advancing the understanding of PDL's decidability.

Contribution

It establishes the decidability of a specific subclass of forest languages satisfying a 2-fold iterated distributive law, linking algebraic properties to PDL definability.

Findings

Languages with 2-fold iterated distributive law are in PDL

The class of such languages is decidable

Supports the approach to PDL decidability via algebraic methods

Abstract

It is an open problem whether definability in Propositional Dynamic Logic (PDL) on forests is decidable. Based on an algebraic characterization by Boja\'nczyk, et. al.,(2012) in terms of forest algebras, Straubing (2013) described an approach to PDL based on a k-fold iterated distributive law. A proof that all languages satisfying such a k-fold iterated distributive law are in PDL would settle decidability of PDL. We solve this problem in the case k=2: All languages recognized by forest algebras satisfying a 2-fold iterated distributive law are in PDL. Furthermore, we show that this class is decidable. This provides a novel nontrivial decidable subclass of PDL, and demonstrates the viability of the proposed approach to deciding PDL in general.