On Constructivity of Galois Connections

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. 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 certified algorithms of static analysis. We show here that constructive GCs are isomorphic, in a mathematical meaning which includes 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] further provide a notion of constructive Galois connection for posets, which we prove to be mathematically isomorphic to plain GCs. Drawing on these findings, we put forward 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 in a flexible way while retaining a constructive approach to Galois connections.