Effect Systems as Abstract Interpretations

2026-06-17

AI Summary

This work addresses the lack of a systematic investigation into the formal relationship between effect systems and abstract interpretation, particularly whether they can be unified in general settings. It establishes, for the first time, a formal correspondence between the two within a generic framework by embedding effect quantales into abstract domains and formulating a novel perspective on abstract interpretation grounded in events rather than states or values. Drawing upon effect quantale theory, the abstract interpretation framework, and program semantics, the paper successfully reduces effect systems to an instance of abstract interpretation. This reduction not only clarifies the semantic foundations of effect systems but also provides a unified theoretical basis for effect-driven static analysis tools.

Abstract

Many forms of static reasoning about program behaviours are known in the literature, yet formal relationships are studied surprisingly infrequently. While most type systems are well-known to be captured by abstract interpretations, the situation for type-and-effect systems is, in the general case, unsettled despite strong hypotheses and occasional framing of effect systems as abstract interpretations. We develop a formal relationship between abstract interpretations and a general class of effect systems. First, we describe an embedding of effect quantales into abstract domains. Second, we recover the general form of an effect quantale as an abstract interpretation -- not on states or values, but on event occurrences.

Problem

• effect systems
• abstract interpretations
• type-and-effect systems
• effect quantales

Innovation

• abstract interpretation
• effect systems
• effect quantales
• event occurrences
• static analysis