Computational type theory

Abstract

Computational type theory provides answers to questions such as: What is a type? What is a natural number? How do we compute with types? How are types related to sets? Can types be elements of types? How are data types for numbers, lists, trees, graphs, etc. related to the corresponding notions in mathematics? What is a real number? Are the integers a subtype of the reals? Can we form the type of all possible data types? Do paradoxes arise in formulating a theory of types as they do in formulating a theory of sets, such as the circular idea of the set of all sets or the idea of all sets that do not contain themselves as members? Is there a type of all types? What is the underlying logic of type theory? Why isn’t it the same logic in which standard set theories are axiomatized? What is the origin of the notion of a type? What distinguishes computational type theory from other type theories? In computational type theory, is there a type of all computable functions from the integers to the integers? If so, is it the same as the set of Turing computable functions from integers to integers? Is there a type of computable functions from any type to any type? Is there a type of the partial computable functions from a type to a type? Are there computable functions whose values are types? Do the notations of an implemented computational type theory include programs in the usual sense? What does it mean that type theory is a foundational theory for both mathematics and computer science? There have been controversies about the foundations of mathematics, does computational type theory resolve any of them, do these controversies impact a foundation for computing theory? This article answers some of these questions and points to literature answering all of them.