# Divide and Congruence III: From Decomposition of Modal Formulas to Preservation of Stability and Divergence

## TL;DR

This paper extends the decomposition method for modal formulas to include stability and divergence considerations, enabling congruence formats for weak semantics that respect stability and divergence properties in transition systems.

## Contribution

It introduces a relaxation of congruence formats for stability-respecting weak semantics and links divergence-preserving semantics to stability-respecting formats through a new proof approach.

## Key findings

- Relaxed congruence formats for stability-respecting semantics
- Inclusion of priority operator within stability-respecting formats
- Establishment of a connection between divergence-preserving and stability-respecting semantics

## Abstract

In two earlier papers we derived congruence formats with regard to transition system specifications for weak semantics on the basis of a decomposition method for modal formulas. The idea is that a congruence format for a semantics must ensure that the formulas in the modal characterisation of this semantics are always decomposed into formulas that are again in this modal characterisation. The stability and divergence requirements that are imposed on many of the known weak semantics have so far been outside the realm of this method. Stability refers to the absence of a $\tau$-transition. We show, using the decomposition method, how congruence formats can be relaxed for weak semantics that are stability-respecting. This relaxation for instance brings the priority operator within the range of the stability-respecting branching bisimulation format. Divergence, which refers to the presence of an infinite sequence of $\tau$-transitions, escapes the inductive decomposition method. We circumvent this problem by proving that a congruence format for a stability-respecting weak semantics is also a congruence format for its divergence-preserving counterpart.