Polymorphic Iterable Sequential Effect Systems

TL;DR

This paper introduces effect quantales, an algebraic structure that models sequential effect systems, providing a unified framework with well-defined properties, and demonstrates their applicability to existing systems and semantics.

Contribution

It defines effect quantales as a new algebraic foundation for sequential effect systems, deriving properties and showing their applicability to various existing systems and semantics.

Findings

Effect quantales model a range of sequential effect systems.

Induced iteration in effect quantales aligns with prior manual operators.

The framework generalizes to categorical semantics and relates to Kleene Algebras.

Abstract

Effect systems are lightweight extensions to type systems that can verify a wide range of important properties with modest developer burden. But our general understanding of effect systems is limited primarily to systems where the order of effects is irrelevant. Understanding such systems in terms of a semilattice of effects grounds understanding of the essential issues, and provides guidance when designing new effect systems. By contrast, sequential effect systems -- where the order of effects is important -- lack an established algebraic structure on effects. We present an abstract polymorphic effect system parameterized by an effect quantale -- an algebraic structure with well-defined properties that can model the effects of a range of existing sequential effect systems. We define effect quantales, derive useful properties, and show how they cleanly model a variety of known sequential effect systems. We show that for most effect quantales, there is an induced notion of iterating a sequential effect; that for systems we consider the derived iteration agrees with the manually designed iteration operators in prior work; and that this induced notion of iteration is as precise as possible when defined. We also position effect quantales with respect to work on categorical semantics for sequential effect systems, clarifying the distinctions between these systems and our own in the course of giving a thorough survey of these frameworks. Our derived iteration construct should generalize to these semantic structures, addressing limitations of that work. Finally, we consider the relationship between sequential effects and Kleene Algebras, where the latter may be used as instances of the former.