Shin-ya Katsumata ; Xavier Rival ; Jérémy Dubut - A Categorical Framework for Program Semantics and Semantic Abstraction

entics:12288 - Electronic Notes in Theoretical Informatics and Computer Science, November 23, 2023, Volume 3 - Proceedings of MFPS XXXIX -
A Categorical Framework for Program Semantics and Semantic AbstractionArticle

Authors: Shin-ya Katsumata ; Xavier Rival ; Jérémy Dubut

    Categorical semantics of type theories are often characterized as structure-preserving functors. This is because in category theory both the syntax and the domain of interpretation are uniformly treated as structured categories, so that we can express interpretations as structure-preserving functors between them. This mathematical characterization of semantics makes it convenient to manipulate and to reason about relationships between interpretations. Motivated by this success of functorial semantics, we address the question of finding a functorial analogue in abstract interpretation, a general framework for comparing semantics, so that we can bring similar benefits of functorial semantics to semantic abstractions used in abstract interpretation. Major differences concern the notion of interpretation that is being considered. Indeed, conventional semantics are value-based whereas abstract interpretation typically deals with more complex properties. In this paper, we propose a functorial approach to abstract interpretation and study associated fundamental concepts therein. In our approach, interpretations are expressed as oplax functors in the category of posets, and abstraction relations between interpretations are expressed as lax natural transformations representing concretizations. We present examples of these formal concepts from monadic semantics of programming languages and discuss soundness.

    Volume: Volume 3 - Proceedings of MFPS XXXIX
    Published on: November 23, 2023
    Accepted on: October 16, 2023
    Submitted on: September 19, 2023
    Keywords: Computer Science - Programming Languages,Mathematics - Category Theory,18C50,F.3.2,D.3.1

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