Semantic Interpretation and Syntactic Model of Simply-Typed Lambda Calculus Using Multi-Ary Structures

Core Concepts

The multi-ary perspective provides a uniform framework to derive the semantic interpretation and syntactic model of simply-typed lambda calculus, along with soundness and completeness results, by factoring the unary categorical semantics through a more fundamental multi-ary structure.

Abstract

The paper presents an exposition of the semantics of the simply-typed λ-calculus, its linear and ordered variants, using multi-ary structures. It defines universal properties for multicategories and uses these to derive familiar rules for products, tensors, and exponentials. The author then explains how to recover both the category-theoretic syntactic model and its semantic interpretation from the multi-ary framework.
The key insights are:

Multi-ary structures, such as multicategories and clones, provide a more direct correspondence between syntax (terms-in-context) and semantics (multimaps) compared to the standard unary categorical approach.

Defining products, tensors, and exponentials in the multi-ary setting yields the usual type-theoretic rules, without the need for separate proofs of soundness.

The syntactic model of simply-typed λ-calculus can be extracted as the free cartesian closed category on a signature, by restricting the free multi-ary structure.

The semantic interpretation in a cartesian closed category arises by extending the category to a multi-ary structure, providing a mathematical explanation for the identification of contexts and product types.

Soundness and completeness follow from the universal properties and the fact that the forgetful functor from cartesian closed clones to signatures is faithful.

The multi-ary perspective thus offers a more natural and algebraic treatment of the semantics of typed lambda calculi compared to the standard unary categorical approach.

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Key Insights Distilled From

Clones, closed categories, and combinatory logic

by Philip Savil... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.01675.pdf

Deeper Inquiries

How can the multi-ary framework be extended to handle dependent types or other advanced type-theoretic features

The multi-ary framework presented in the context above can be extended to handle dependent types by incorporating the notion of context-dependent typing rules. In a dependent type system, types can depend on values, allowing for more precise and expressive type specifications. This can be achieved in the multi-ary framework by introducing rules that account for the dependencies between types and values within a given context. By extending the multi-ary structures to accommodate dependent types, the framework can capture a wider range of type-theoretic features and provide a more comprehensive semantic interpretation of typed lambda calculi.

What are the connections between the multi-ary semantics presented here and the polycategorical approach of Shulman

The connections between the multi-ary semantics discussed here and the polycategorical approach of Shulman lie in their shared focus on generalizing categorical structures to handle multiple inputs and outputs. Both frameworks aim to provide a unified and systematic way to model complex type systems and formal languages. While the multi-ary framework focuses on multi-ary structures in the context of lambda calculi and clones, Shulman's polycategorical approach extends this idea to a broader setting, encompassing a wide range of categorical structures and operations. By exploring the connections between these approaches, one can gain deeper insights into the underlying principles of categorical semantics and type theory.

Can the multi-ary perspective shed light on the relationship between lambda calculus and other computational formalisms, such as combinatory logic or abstract machines

The multi-ary perspective can offer valuable insights into the relationship between lambda calculus and other computational formalisms, such as combinatory logic or abstract machines. By viewing lambda terms as multi-ary structures with multiple inputs and outputs, one can analyze the operational semantics and computational behavior of lambda calculus in a more structured and systematic manner. This perspective can help in understanding how different computational models relate to each other and how they can be unified under a common framework. Additionally, by studying the semantic interpretation of combinatory logic and lambda calculus within the multi-ary framework, one can uncover deeper connections between these formalisms and potentially identify new avenues for research in computational theory.