Formalizing Abstract Computability: Turing Categories in Coq

Abstract

The concept of computable functions (as developed by Gödel, Church, Turing, and Kleene in the 1930's) has been extensively studied, leading to the modern subject of recursive function theory. However recent work by category theorists has led to a more conceptual and abstract foundation of computability theory—Turing categories. A Turing category models the notion of partial map as well as recursive computation, using methods of categorical algebra to formalize these concepts. The goal of this work is to provide a formal framework for analyzing this categorical model of computation. We use the Coq Proof Assistant, which implements the Calculus of (co)Inductive Constructions (CIC), and we build on an existing Coq library for general category theory. We focus on both formalizing Turing categories and on building a general framework in the form of a well-structured Coq library that can be further extended. We begin by formalizing definitions, propositions, and proofs pertaining to Turing categories, and then instantiate the more general Turing category formalism with a CIC description of the category which explicitly models the language of partial recursive functions.