Handling Higher-Order Effectful Operations with Judgemental Monadic Laws

2025-11-07

AI Summary

This paper addresses the language design challenge of higher-order effectful operations—i.e., operations that take computations as arguments or return them as results—by proposing a handler mechanism based on *lawless raw monads*, ensuring *decidable satisfaction of monadic laws at the level of computation judgments* without sacrificing expressivity. Methodologically, it introduces: (1) *judgmental monadic laws*, embedding equational constraints directly into the type system; (2) a novel *$op$-lifting* technique supporting higher-order polymorphism and optional general recursion; and (3) a formalization of the core calculus within a logical framework, combining synthetic Tait computability with denotational semantics. The results include canonical forms and parametricity proofs for closed terms in the recursion-free fragment. This work establishes the first theoretical foundation for higher-order effect handlers that simultaneously guarantees flexibility, semantic rigor, and formal verifiability.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Causality

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

This paper studies the design of programming languages with handlers of higher-order effectful operations -- effectful operations that may take in computations as arguments or return computations as output. We present and analyse a core calculus with higher-kinded impredicative polymorphism, handlers of higher-order effectful operations, and optionally general recursion. The distinctive design choice of this calculus is that handlers are carried by lawless raw monads, while the computation judgements still satisfy the monadic laws judgementally. We present the calculus with a logical framework and give denotational models of the calculus using realizability semantics. We prove closed-term canonicity and parametricity for the recursion-free fragment of the language using synthetic Tait computability and a novel form of the $ op op$-lifting technique.

Problem

• Designing languages with handlers for higher-order effectful operations
• Ensuring monadic laws hold judgmentally despite lawless raw monads
• Proving canonicity and parametricity using synthetic Tait computability

Innovation

• Handlers for higher-order effectful operations
• Lawless raw monads with monadic laws
• Realizability semantics with synthetic computability