RCF1: Theories of PR Maps and Partial PR Maps

TL;DR

This paper develops a simple algebraic categorical framework for primitive recursion and partial PR maps, providing new insights into Church's Thesis through a variable-free, formal approach.

Contribution

It introduces a new categorical theory of partial PR maps with a formal variable-free framework, connecting primitive recursion, partial maps, and Church's Thesis.

Findings

Categorical theory of PR maps is simplified and algebraic.

Partial PR maps are formalized within a diagonal monoidal category.

The approach offers new perspectives on Church's Thesis.

Abstract

We give to the categorical theory PR of Primitive Recursion a logically simple, algebraic presentation, via equations between maps, plus one genuine Horner type schema, namely Freyd's uniqueness of the initialised iterated. Free Variables are introduced - formally - as another names for projections. Predicates \chi: A -> 2 admit interpretation as (formal) Objects {A|\chi} of a surrounding Theory PRA = PR + (abstr) : schema (abstr) formalises this predicate abstraction into additional Objects. Categorical Theory P\hat{R}_A \sqsupset PR_A \sqsupset PR then is the Theory of formally partial PR-maps, having Theory PR_A embedded. This Theory P\hat{R}_A bears the structure of a (still) diagonal monoidal category. It is equivalent to "the" categorical theory of \mu-recursion (and of while loops), viewed as partial PR maps. So the present approach to partial maps sheds new light on Church's Thesis, "embedded" into a Free-Variables, formally variable-free (categorical) framework.