# A Constructive Framework for Galois Connections

## TL;DR

This paper introduces a new class of constructive Galois connections called partitioning GCs, enabling more flexible and mechanized static analysis through proof assistants, and establishes their theoretical equivalence to existing GCs.

## Contribution

It proves the isomorphism between constructive Galois connections and partitioning GCs, and advocates for using partitioning GCs as a flexible, constructive framework for abstract interpretation.

## Key findings

- Constructive Galois connections are isomorphic to partitioning GCs.
- Partitioning GCs enable flexible set partition representations.
- The approach facilitates mechanized, verified static analysis algorithms.

## Abstract

Abstract interpretation-based static analyses rely on abstract domains of program properties, such as intervals or congruences for integer variables. Galois connections (GCs) between posets provide the most widespread and useful formal tool for mathematically specifying abstract domains. Recently, Darais and Van Horn [2016] put forward a notion of constructive Galois connection for unordered sets (rather than posets), which allows to define abstract domains in a so-called mechanized and calculational proof style and therefore enables the use of proof assistants like Coq and Agda for automatically extracting verified algorithms of static analysis. We show here that constructive GCs are isomorphic, in a precise and comprehensive meaning including sound abstract functions, to so-called partitioning GCs--an already known class of GCs which allows to cast standard set partitions as an abstract domain. Darais and Van Horn [2016] also provide a notion of constructive GC for posets, which we prove to be isomorphic to plain GCs and therefore lose their constructive attribute. Drawing on these findings, we put forward and advocate the use of purely partitioning GCs, a novel class of constructive abstract domains for a mechanized approach to abstract interpretation. We show that this class of abstract domains allows us to represent a set partition with more flexibility while retaining a constructive approach to Galois connections.