A Constructive Framework for Galois Connections

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

Transport via Partial Galois Connections and Equivalences

Multiple types can represent the same concept. For example, lists and trees can both represent sets. Unfortunately, this easily leads to incomplete libraries: some set-operations may only be available on lists, others only on trees. Similarly, subtypes and quotients are commonly used to construct new type abstractions in formal verification. In such cases, one often wishes to reuse operations on the representation type for the new type abstraction, but to no avail: the types are not the same. To address these problems, we present a new framework that transports programs via equivalences. Existing transport frameworks are either designed for dependently typed, constructive proof assistants, use univalence, or are restricted to partial quotient types. Our framework (1) is designed for simple type theory, (2) generalises previous approaches working on partial quotient types, and (3) is based on standard mathematical concepts, particularly Galois connections and equivalences. We introduce the notion of partial Galois connections and equivalences and prove their closure properties under (dependent) function relators, (co)datatypes, and compositions. We formalised the framework in Isabelle/HOL and provide a prototype. This is the extended version of "Transport via Partial Galois Connections and Equivalences", 21st Asian Symposium on Programming Languages and Systems, 2023.Comment: 18 pages; will appear at 21st Asian Symposium on Programming Languages and Systems, 202

On the existence of right adjoints for surjective mappings between fuzzy structures0

En este trabajo los autores continúan su estudio de la caracterización de la existencia de adjunciones (conexiones de Galois isótonas) cuyo codominio no está dotado de estructura en principio. En este artículo se considera el caso difuso en el que se tiene un orden difuso R definido en un conjunto A y una aplicación sobreyectiva f:A-> B compatible respecto de dos relaciones de similaridad definidas en el dominio A y en el condominio B, respectivamente. Concretamente, el problema es encontrar un orden difuso S en B y una aplicación g:B-> A compatible también con las correspondientes similaridades definidas en A y en B, de tal forma que el par (f,g) constituya un adjunción

A Galois connection between classical and intuitionistic logics. I: Syntax

In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Goedel's provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+(?!) fragment, identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic modal logic, whose modality !? is a strict lax modality in the sense of Aczel - and thus resembles the squash/bracket operation in intuitionistic type theories. The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov's problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by G\"odel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a simplified version of Isabelle's meta-logic

Abstracting Nash equilibria of supermodular games

Supermodular games are a well known class of noncooperative games which find significant applications in a variety of models, especially in operations research and economic applications. Supermodular games always have Nash equilibria which are characterized as fixed points of multivalued functions on complete lattices. Abstract interpretation is here applied to set up an approximation framework for Nash equilibria of supermodular games. This is achieved by extending the theory of abstract interpretation in order to cope with approximations of multivalued functions and by providing some methods for abstracting supermodular games, thus obtaining approximate Nash equilibria which are shown to be correct within the abstract interpretation framework