Breast cancer remains a major concern for women’s lives worldwide and serves as evidence of the need for better classification strategies according to severity. Computer-aided diagnosis (CADx) powered by explainable artificial intelligence (XAI) offers a promising solution by minimizing diagnostic errors and fostering trust through a more transparent decision-making process. As XAI evolves, it plays a crucial role in increasing the interpretability of AI-driven diagnostics, particularly in distributed healthcare systems. XAI explains the model’s prediction, making clinicians more confident in accepting clinical outcomes. Accordingly, this study provides a comparative analysis of multiple deep learning models for breast cancer identification based on a publicly available dataset of 780 ultrasound images with their masks. Explanation of the classification result is then provided using the Grad-CAM method which improves the interpretability and tractability of the models. The proposed method lets the models explain their decisions visually, using heatmaps that show what part of an image contributes to the predictions in a valuable way when studying medical images. Obtained results demonstrate the XAI’s transformative potential in medical imaging, paving the way for more reliable, scalable, and efficient diagnostic tools. Also, providing a critical comparison of various types of deep learning models for breast cancer identification, the study underlines the advantages and limitations of the different architectures in solving the task.

Abstract Static analysis of logic programs by abstract interpretation requires designing abstract operators which mimic the concrete ones, such as unification, renaming, and projection. In the case of goal-driven analysis, where goal-dependent semantics are used, we also need a backward-unification operator, typically implemented through matching. In this paper, we study the problem of deriving optimal abstract matching operators for sharing and linearity properties. We provide an optimal operator for matching in the domain $\mathtt{ShLin}^{\omega }$ , which can be easily instantiated to derive optimal operators for the domains $\mathtt{ShLin}^2$ by Andy King and the reduced product $\mathtt{Sharing} \times \mathtt{Lin}$ .

In the last years almost all families of programming languages, from imperative to functional, logic, object-oriented and machine code, have been subject to static analysis by abstract interpretation. The use of a principled approach to static analysis based on the theory of abstract interpretation provided mathematical tools to reason about program properties and allowed for the rigorous and incremental design of precise and scalable static analyzers, ensuring soundness by construction. The large variety of abstract domains for many different programming languages, the ability to combine and refine them with standard abstract interpretation tools and the availability of mature abstract domain libraries allowed easily porting, reusing and experimenting with techniques born in a specific family to other programming languages and properties.

Abstract Sharing analysis is used to statically discover data structures which may overlap in object-oriented programs. Using the abstract interpretation framework, we show that sharing analysis greatly benefits from linearity information. A variable is linear in a program state when different field paths starting from it always reach different objects. We propose a graph-based abstract domain which can represent aliasing, linearity, and sharing information and define all the necessary abstract operators for the analysis of a Java-like language.

Reasoning on a complex system in the abstract interpretation theory starts with a formal description of the system behavior specified by a collecting semantics. We take the common point of view that a collecting semantics is a very precise semantics from which other abstractions may be derived. We elaborate on both the concepts of precision and derivability, and introduce a notion of adequacy which tell us when a collecting semantics is a good choice for a given family of abstractions. We instantiate this approach to the case of first-order functional programs by considering three common collecting semantics and some abstract properties of functions. We study their relative precision and give a constructive characterization of the classes of abstractions which are adequate for the collecting semantics.

Abstract In the analysis of logic programs, abstract domains for detecting sharing properties are widely used. Recently, the new domain ${\mathtt{ShLin}^{\omega}}$ has been introduced to generalize both sharing and linearity information. This domain is endowed with an optimal abstract operator for single-binding unification. The authors claim that the repeated application of this operator is also optimal for multibinding unification. This is the proof of such a claim.

In the theory of abstract interpretation, a domain is complete when abstract computations are as precise as concrete computations. In addition to the standard notion of completeness, we introduce the concept of observational completeness. A domain is

Abstract In the analysis of logic programs, abstract domains for detecting sharing and linearity information are widely used. Devising abstract unification algorithms for such domains has proved to be rather hard. At the moment, the available algorithms are correct but not optimal; i.e., they cannot fully exploit the information conveyed by the abstract domains. In this paper, we define a new (infinite) domainShLinωwhich can be thought of as a general framework from which other domains can be easily derived by abstraction.ShLinωmakes the interaction between sharing and linearity explicit. We provide a constructive characterization of the optimal abstract unification operator onShLinω, and we lift it to two well-known abstractions ofShLinω, namely, to the classicalSharing×Linabstract domain and to the more preciseShLin2abstract domain by Andy King. In the case of single-binding substitutions, we obtain optimal abstract unification algorithms for such domains.

Abstract We face the problems of correctness, optimality, and precision for the static analysis of logic programs, using the theory of abstract interpretation. We propose a framework with a denotational, goal-dependent semantics equipped with two unification operators for forward unification (calling a procedure) and backward unification (returning from a procedure). The latter is implemented through a matching operation. Our proposal clarifies and unifies many different frameworks and ideas on static analysis of logic programming in a single, formal setting. On the abstract side, we focus on the domain sharing by Jacobs and Langen (The Journal of Logic Programming, 1992, vol. 13, nos. 2–3, pp. 291–314) and provide the best correct approximation of all the primitive semantic operators, namely, projection, renaming, and forward and backward unifications. We show that the abstract unification operators are strictly more precise than those in the literature defined over the same abstract domain. In some cases, our operators are more precise than those developed for more complex domains involving linearity and freeness.

In this article, we show that reversible analyses of logic languages by abstract interpretation can be performed without loss of precision by systematically refining abstract domains. This is obtained by adding to the abstract domain the minimal amount of concrete semantic information so that this refined abstract domain becomes rich enough to allow goal-driven and goal-independent analyses agree. These domains are known as condensing abstract domains. Essentially, an abstract domain A is condensing when the goal-driven analysis performed on A for a program P and a given query can be retrieved with no loss of precision from the goal-independent analysis on A of P . We show that condensation is an abstract domain property and that the problem of making an abstract domain condensing boils down to the problem of making the corresponding abstract interpretation complete, in a weakened form, with respect to unification. In the case of abstract domains for logic program analysis approximating computed answer substitutions, we provide a clean logical characterization of condensing domains as fragments of propositional linear logic. We apply our methodology to the systematic design of condensing domains for freeness and independence analysis.

Completeness is an ideal, although uncommon, feature of abstract interpretations, formalizing the intuition that, relatively to the properties encoded by the underlying abstract domains, there is no loss of information accumulated in abstract computations. Thus, complete abstract interpretations can be rightly understood as optimal. We deal with both pointwise completeness, involving generic semantic operations, and (least) fixpoint completeness. Completeness and fixpoint completeness are shown to be properties that depend on the underlying abstract domains only. Our primary goal is then to solve the problem of making abstract interpretations complete by minimally extending or restricting the underlying abstract domains. Under the weak and reasonable hypothesis of dealing with continuous semantic operations, we provide constructive characterizations for the least complete extensions and the greatest complete restrictions of abstract domains. As far as fixpoint completeness is concerned, for merely monotone semantic operators, the greatest restrictions of abstract domains are constructively characterized, while it is shown that the existence of least extensions of abstract domains cannot be, in general, guaranteed, even under strong hypotheses. These methodologies, which in finite settings give rise to effective algorithms, provide advanced formal tools for manipulating and comparing abstract interpretations, useful both in static program analysis and in semantics design. A number of examples illustrating these techniques are given.

In this article we introduce the notion of Heyting completion in abstract interpretation. We prove that Heyting completion provides a model for Cousot's reduced cardinal power of abstract domains and that it supplies a logical basis to specify relational domains for program analysis and abstract interpretation. We study the algebraic properties of Heyting completion in relation with other well-known domain transformers, like reduced product and disjunctive completion. This provides a uniform algebraic setting where complex abstract domains can be specified by simple logic formulas, or as solutions of recursive abstract domain equations, involving few basic operations for domain construction, all characterized by a clean logical interpretation. We apply our framework to characterize directionality and condensing and in downward closed analysis of (constraint) logic programs.