A Category-Theoretic Framework for Dependent Effect Systems

2026-01-21

AI Summary

Existing categorical frameworks for graded monads struggle to express effects that depend on program values—so-called dependent effects—thereby limiting their applicability in program analysis. This work proposes indexed graded monads, which, for the first time, integrate dependent effects into the categorical model of graded monads by combining fibrational indexing with semantic techniques from dependent type theory. The resulting formal framework provides a unified semantic foundation for refinement type systems featuring dependent effects and is successfully instantiated in multiple program analysis tasks, including cost analysis, probabilistic bounds, expected-value reasoning, and timing-sensitive security verification. This significantly enhances both the expressiveness and the scope of applicability of effect systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Graded monads refine traditional monads using effect annotations in order to describe quantitatively the computational effects that a program can generate. They have been successfully applied to a variety of formal systems for reasoning about effectful computations. However, existing categorical frameworks for graded monads do not support effects that may depend on program values, which we call dependent effects, thereby limiting their expressiveness. We address this limitation by introducing indexed graded monads, a categorical generalization of graded monads inspired by the fibrational"indexed"view and by classical categorical semantics of dependent type theories. We show how indexed graded monads provide semantics for a refinement type system with dependent effects. We also show how this type system can be instantiated with specific choices of parameters to obtain several formal systems for reasoning about specific program properties. These instances include, in particular, cost analysis, probability-bound reasoning, expectation-bound reasoning, and temporal safety verification.

Problem

• dependent effects
• graded monads
• effect systems
• category theory
• refinement type systems

Innovation

• indexed graded monads
• dependent effects
• refinement type system
• categorical semantics
• effect systems