A Categorical Model for the Lambda Calculus with Constructors

TL;DR

This paper introduces a categorical model for an extended lambda calculus with constructors, enhancing its expressiveness and confluence while maintaining simplicity, and proves its soundness and completeness for a specific fragment.

Contribution

It defines a sound categorical model for the lambda calculus with constructors and proves its completeness for the fragment without match-failure.

Findings

The calculus is expressive and confluent.

The categorical model is sound and complete for the fragment without match-failure.

The syntax remains simple despite added features.

Abstract

The lambda calculus with constructors is an extension of the lambda calculus with variadic constructors. It decomposes the pattern-matching a la ML into a case analysis on constants and a commutation rule between case and application constructs. Although this commutation rule does not match with the usual computing intuitions, it makes the calculus expressive and confluent, with a rather simple syntax. In this paper we define a sound notion of categorical model for the lambda calculus with constructors. We then prove that this definition is complete for the fragment of the calculus with no match-failure, using the model of partial equivalence relations.