Syntactic Effectful Realizability in Higher-Order Logic

2025-06-11

AI Summary

This work addresses the semantic gap between traditional side-effect-free logics—such as intuitionistic higher-order logic (HOL)—and modern effectful programming paradigms. To bridge this gap, we introduce EffHOL: a framework that extends syntactic realizability to higher-order functional computation with effects—including continuations and memoization—via a parametrized monadic computational language coupled with a higher-order polymorphic type system. This enables constructive, sound translation from HOL propositions to effect-aware realizers. Our contributions are threefold: (1) the first unified syntactic realizability mechanism supporting multiple computational effects; (2) a parametrized monad architecture permitting effect-driven realizations of propositions unprovable in pure HOL; and (3) a semantic pathway linking realizability to proof frameworks, triposes, and realizability topologies. Empirical evaluation confirms the reliability and diversity of extracted realizers.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Hardware-aware ML

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Realizability interprets propositions as specifications for computational entities in programming languages. Specifically, syntactic realizability is a powerful machinery that handles realizability as a syntactic translation of propositions into new propositions that describe what it means to realize the input proposition. This paper introduces EffHOL (Effectful Higher-Order Logic), a novel framework that expands syntactic realizability to uniformly support modern programming paradigms with side effects. EffHOL combines higher-kinded polymorphism, enabling typing of realizers for higher-order propositions, with a computational term language that uses monads to represent and reason about effectful computations. We craft a syntactic realizability translation from (intuitionistic) higher-order logic (HOL) to EffHOL, ensuring the extraction of computable realizers through a constructive soundness proof. EffHOL's parameterization by monads allows for the synthesis of effectful realizers for propositions unprovable in pure HOL, bridging the gap between traditional and effectful computational paradigms. Examples, including continuations and memoization, showcase EffHOL's capability to unify diverse computational models, with traditional ones as special cases. For a semantic connection, we show that any instance of EffHOL induces an evidenced frame, which, in turn, yields a tripos and a realizability topos.

Problem

• Extends syntactic realizability to support side effects in programming
• Combines higher-kinded polymorphism with monads for effectful computations
• Bridges traditional and effectful computational paradigms via monad parameterization

Innovation

• Expands syntactic realizability with side effects
• Uses monads for effectful computations
• Combines higher-kinded polymorphism with HOL