A Categorical Framework for Program Semantics and Semantic Abstraction

TL;DR

This paper introduces a categorical framework for abstract interpretation, modeling semantics as functors and abstractions as natural transformations, enabling structured reasoning about program properties.

Contribution

It extends functorial semantics to abstract interpretation by defining interpretations as oplax functors and abstractions as lax natural transformations.

Findings

Formalizes semantics as oplax functors in posets

Models abstraction relations as lax natural transformations

Provides examples from monadic semantics and discusses soundness

Abstract

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.