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

Mechanizing Abstract Interpretation

It is important when developing software to verify the absence of undesirable behavior such as crashes, bugs and security vulnerabilities. Some settings require high assurance in verification results, e.g., for embedded software in automobiles or airplanes. To achieve high assurance in these verification results, formal methods are used to automatically construct or check proofs of their correctness. However, achieving high assurance for program analysis results is challenging, and current methods are ill suited for both complex critical domains and mainstream use. To verify the correctness of software we consider program analyzers---automated tools which detect software defects---and to achieve high assurance in verification results we consider mechanized verification---a rigorous process for establishing the correctness of program analyzers via computer-checked proofs. The key challenges to designing verified program analyzers are: (1) achieving an analyzer design for a given programming language and correctness property; (2) achieving an implementation for the design; and (3) achieving a mechanized verification that the implementation is correct w.r.t. the design. The state of the art in (1) and (2) is to use abstract interpretation: a guiding mathematical framework for systematically constructing analyzers directly from programming language semantics. However, achieving (3) in the presence of abstract interpretation has remained an open problem since the late 1990's. Furthermore, even the state-of-the art which achieves (3) in the absence of abstract interpretation suffers from the inability to be reused in the presence of new analyzer designs or programming language features. First, we solve the open problem which has prevented the combination of abstract interpretation (and in particular, calculational abstract interpretation) with mechanized verification, which advances the state of the art in designing, implementing, and verifying analyzers for critical software. We do this through a new mathematical framework Constructive Galois Connections which supports synthesizing specifications for program analyzers, calculating implementations from these induced specifications, and is amenable to mechanized verification. Finally, we introduce reusable components for implementing analyzers for a wide range of designs and semantics. We do this though two new frameworks Galois Transformers and Definitional Abstract Interpreters. These frameworks tightly couple analyzer design decisions, implementation fragments, and verification properties into compositional components which are (target) programming-language independent and amenable to mechanized verification. Variations in the analysis design are then recovered by simply re-assembling the combination of components. Using this framework, sophisticated program analyzers can be assembled by non-experts, and the result are guaranteed to be verified by construction

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

On the Existence of Right Adjoints for Surjective Mappings between Fuzzy Structures

Abstract. We continue our study of the characterization of existence of adjunctions (isotone Galois connections) whose codomain is insufficiently structured. This paper focuses on the fuzzy case in which we have a fuzzy ordering ρA on A and a surjective mapping f : A, ≈A → B, ≈B compatible with respect to the fuzzy equivalences ≈A and ≈B. Specifically, the problem is to find a fuzzy ordering ρB and a compatible mapping g : B, ≈B → A, ≈A such that the pair (f, g) is a fuzzy adjunction

Implication operators generating pairs of weak negations and their algebraic structure

Negations operators have been developed and applied in many fields such as image processing, decision making, mathematical morphology, fuzzy logic, etc. One of the most effective non-monotonic operators are weak negations. This paper studies the algebraic structure and the characterization of the adjoint triples and Galois implication pairs which provides a fixed pair of weak negations. The obtained results allow the user to select the best conjunctor and implications associated with the most suitable negation to be used in the computations of the problem to be solved.Partially supported by the State Research Agency (AEI) and the European Regional Development Fund (ERDF) project TIN2016-76653-P, European Cooperation in Science & Technology (COST) Action CA17124

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

Imperative functional programs that explain their work

Program slicing provides explanations that illustrate how program outputs were produced from inputs. We build on an approach introduced in prior work by Perera et al., where dynamic slicing was defined for pure higher-order functional programs as a Galois connection between lattices of partial inputs and partial outputs. We extend this approach to imperative functional programs that combine higher-order programming with references and exceptions. We present proofs of correctness and optimality of our approach and a proof-of-concept implementation and experimental evaluation.Comment: Full version of ICFP 2017 paper, with appendice