A Constructive Framework for Galois Connections

Abstract interpretation-based static analyses rely on abstract domains of program properties, suchas intervals or congruences for integer variables. Galois connections (GCs) between posets providethe 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 calcula-tional proof style and therefore enables the use of proof assistants like Coq and Agda for automaticallyextracting 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 partitioningGCs — an already known class of GCs which allows to cast standard set partitions as an abstract do-main. Darais and Van Horn [2016] also provide a notion of constructive GC for posets, which weprove to be isomorphic to plain GCs and therefore lose their constructive attribute. Drawing on thesefindings, we put forward and advocate the use of purely partitioning GCs, a novel class of constructiveabstract domains for a mechanized approach to abstract interpretation. We show that this class of ab-stract domains allows us to represent a set partition with more flexibility while retaining a constructiveapproach to Galois connections.