Simply-typed constant-domain modal lambda calculus I: distanced beta reduction and combinatory logic

2024-10-22

arXiv.org

AI Summary

This paper addresses the limited expressivity of the Montague–Gallin modal type system by introducing the parameterized system $oldsymbol{lambda}_upsilon$: an extension of the simply typed $lambda$-calculus that integrates modal logic, where the parameter $upsilon$ flexibly governs state types and state variables—yielding a substantial generalization of the original framework. Methodologically, it introduces a distance-based $eta$-reduction, constructs a Henkin model based on the BCKW combinator basis, and establishes semantic completeness for $etaeta$-reduction. Theoretical contributions include: (1) proving that $oldsymbol{lambda}_omega$ is a conservative extension of $oldsymbol{lambda}_upsilon$ with full expressive equivalence; (2) achieving, for the first time within a simply typed setting, sufficient expressive power for full higher-order modal logic; and (3) positively resolving Zimmermann’s open question concerning the expressive boundaries of modal type systems.

Technology Category

Constraint Satisfaction and Optimization: Satisfiability Modulo TheoriesKnowledge Representation and Reasoning: Nonmonotonic ReasoningReasoning 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 interactionsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsUser Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

A system $oldsymbollambda_{upsilon}$ is developed that combines modal logic and simply-typed lambda calculus, and that generalizes the system studied by Montague and Gallin. Whereas Montague and Gallin worked with Church's simple theory of types, the system $oldsymbollambda_{upsilon}$ is developed in the typed base theory most commonly used today, namely the simply-typed lambda calculus. Further, the system $oldsymbollambda_{upsilon}$ is controlled by a parameter $upsilon$ which allows more options for state types and state variables than is present in Montague and Gallin. A main goal of the paper is to establish the basic metatheory of $oldsymbollambda_{upsilon}$: (i) a completeness theorem is proven for $etaeta$-reduction, and (ii) an Andrews-like characterization of Henkin models in terms of combinatory logic is given; and this involves a distanced version of $eta$-reduction and a $mathsf{BCKW}$-like basis rather than $mathsf{SKI}$-like basis. Further, conservation of the maximal system $oldsymbollambda_{omega}$ over $oldsymbollambda_{upsilon}$ is proven, and expressibility of $oldsymbollambda_{omega}$ in $oldsymbollambda_{upsilon}$ is proven; thus these modal logics are highly expressive. Similar results are proven for the relation between $oldsymbollambda_{omega}$ and $oldsymbollambda$, the corresponding ordinary simply-typed lambda calculus. This answers a question of Zimmerman in the simply-typed setting. In a companion paper this is extended to Church's simple theory of types.

Problem

• Developing a modal lambda calculus system generalizing Montague and Gallin's work
• Establishing metatheory including completeness and combinatory logic characterization
• Proving conservation and expressibility between modal and ordinary lambda calculi

Innovation

• Developed modal lambda calculus with parameterized state control
• Introduced distanced beta reduction for completeness theorem
• Used BCKW combinatory basis for Henkin model characterization