Classical notions of computation and the Hasegawa-Thielecke theorem

2025-02-18

AI Summary

This paper addresses the challenge of establishing a unified syntactic and semantic framework for classical logic systems with computational involutive negation, supporting both call-by-value (CBV) and call-by-name (CBN) evaluation strategies while overcoming semantic difficulties arising from non-associative composition. To this end, the authors introduce *dialogue duploids*—a novel semantic structure generalizing dialogue categories to the non-associative, effectful setting, thereby providing the first rigorous semantic account of polarized classical computation. Building on the polarized effect calculus and double-categorical theory, they construct a fully complete and sound interpretation of the syntax into arbitrary dialogue duploids, proving that the syntax itself forms the initial dialogue duploid. Furthermore, they establish a bi-directional equivalence between central and thunkable morphisms, yielding a strict semantic justification of the Hasegawa–Thielecke theorem.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

In the spirit of the Curry-Howard correspondence between proofs and programs, we define and study a syntax and semantics for classical logic equipped with a computationally involutive negation, using a polarised effect calculus. A main challenge in designing a denotational semantics is to accommodate both call-by-value and call-by-name evaluation strategies, which leads to a failure of associativity of composition. Building on the work of the third author, we devise the notion of dialogue duploid, which provides a non-associative and effectful counterpart to the notion of dialogue category introduced by the second author in his 2-categorical account, based on adjunctions, of logical polarities and continuations. We show that the syntax of the polarised calculus can be interpreted in any dialogue duploid, and that it defines in fact a syntactic dialogue duploid. As an application, we establish, by semantic as well as syntactic means, the Hasegawa-Thielecke theorem, which states that the notions of central map and of thunkable map coincide in any dialogue duploid (in particular, for any double negation monad on a symmetric monoidal category).

Problem

• Define syntax for classical logic
• Handle call-by-value and call-by-name strategies
• Prove Hasegawa-Thielecke theorem

Innovation

• polarised effect calculus
• dialogue duploid concept
• Hasegawa-Thielecke theorem proof