Separating the Expressive Power of Propositional Dynamic and Modal Fixpoint Logics

TL;DR

This paper explores the expressive capabilities of program logics extending propositional dynamic logic and modal mu-calculus, demonstrating a new logic that surpasses their union in expressiveness and advancing the understanding of decidability boundaries.

Contribution

It introduces a new decidable program logic that is more expressive than both PDL and modal mu-calculus fragments, pushing the limits of decidability in program logics.

Findings

The new logic is more expressive than both fragments and their union.

Decidability boundaries have been extended to more expressive logics.

All PDL and modal fixpoint logics with various language classes are separated.

Abstract

We investigate the expressive power of the two main kinds of program logics for complex, non-regular program properties found in the literature: those extending propositional dynamic logic (PDL), and those extending the modal mu-calculus. This is inspired by the recent discovery of a decidable program logic called Visibly Pushdown Fixpoint Logic with Chop which extends both the modal mu-calculus and PDL over visibly pushdown languages, which, so far, constituted the ends of two pillars of decidable program logics. Here we show that this logic is not only more expressive than either of its two fragments, but in fact even more expressive than their union. Hence, the decidability border amongst program logics has been properly pushed up. We complete the picture by providing results separating all the PDL-based and modal fixpoint logics with regular, visibly pushdown and arbitrary context-free constructions.