Finite Vector Spaces as Model of Simply-Typed Lambda-Calculi

TL;DR

This paper explores using finite vector spaces as a computational model for linear logic, developing finite lambda-calculi and models, and analyzing their completeness and relationships.

Contribution

It introduces a finite vector space model for linear logic, extends finite lambda calculus algebraically, and compares two finite models with examples.

Findings

Finite set model is fully complete for the language.

Finite vector space model is not initially complete.

Algebraic extension recovers completeness of the vector space model.

Abstract

In this paper we use finite vector spaces (finite dimension, over finite fields) as a non-standard computational model of linear logic. We first define a simple, finite PCF-like lambda-calculus with booleans, and then we discuss two finite models, one based on finite sets and the other on finite vector spaces. The first model is shown to be fully complete with respect to the operational semantics of the language. The second model is not complete, but we develop an algebraic extension of the finite lambda calculus that recovers completeness. The relationship between the two semantics is described, and several examples based on Church numerals are presented.