This article presents a theory of classes and inheritance built on top of constructive type theory. Classes are defined using dependent and very dependent function types that are found in the Nuprl constructive type theory. Inheritance is defined in terms of a general subtyping relation over the underlying types. Among the basic types is the intersection type which plays a critical role in the applications because it provides a method of composing program components. The class theory is applied to defining algebraic structures such as monoids, groups, rings, etc. and relating them. It is also used to define communications protocols as infinite state automata. The article illustrates the role of these formal automata in defining the services of a distributed group communications system. In both applications the inheritance mechanisms allow reuse of proofs and the statement of general properties of system composition. 1

The basic concepts of type theory are fundamental to computer science, logic and mathematics. Indeed, the language of type theory connects these regions of science. It plays a role in computing and information science akin to that of set theory in pure mathematics. There are many excellent accounts of the basic ideas of type theory, especially at the interface of computer science and logic — specifically, in the literature of programming languages, semantics, formal methods and automated reasoning. Most of these are very technical, dense with formulas, inference rules, and computation rules. Here we follow the example of the mathematician Paul Halmos, who in 1960 wrote a 104-page book called Naïve Set Theory intended to make the subject accessible to practicing mathematicians. His book served many generations well. This article follows the spirit of Halmos ’ book and introduces type theory without recourse to precise axioms and inference rules, and with a minimum of formalism. I start by paraphrasing the preface to Halmos ’ book. The sections of this article follow his chapters closely. Every computer scientist agrees that every computer scientist must know some type theory; the disagreement begins in trying to decide how much is some. This article contains my partial answer to that question. The purpose of the article is to tell the beginning student of advanced computer science the basic type theoretic facts of life, and to do so with a minimum of philosophical discourse and logical formalism. The point throughout is that of a prospective computer scientist eager to study programming languages, or database systems, or computational complexity theory, or distributed systems or information discovery. In type theory, “naïve ” and “formal ” are contrasting words. The present treatment might best be described as informal type theory from a naïve point of view. The concepts are very general and very abstract; therefore they may

This article follows the spirit of Halmos' book and introduces type theory without recourse to precise axioms and inference rules, and with a minimum of formalism. I start by paraphrasing the preface to Halmos' book. The sections of this article follow his chapters closely. Every computer scientist agrees that every computer scientist must know some type theory; the disagreement begins in trying to decide how much is some. This article contains my partial answer to that question. The purpose of the article is to tell the beginning student of advanced computer science the basic type theoretic facts of life, and to do so with a minimum of philosophical discourse and logical formalism. The point throughout is that of a prospective computer scientist eager to study programming languages, or database systems, or computational complexity theory, or distributed systems or information discovery

Abstract This article presents a theory of classes and inheritance built on top of constructive typetheory. Classes are defined using dependent and very dependent function types that are found in the Nuprl constructive type theory. Inheritance is defined in terms of a general subtypingrelation over the underlying types. Among the basic types is the intersection type which plays a critical role in the applications because it provides a method of composing program components.The class theory is applied to defining algebraic structures such as monoids, groups, rings, etc. and relating them. It is also used to define communications protocols as infinite stateautomata. The article illustrates the role of these formal automata in defining the services of a distributed group communications system. In both applications the inheritance mechanismsallow reuse of proofs and the statement of general properties of system composition. 1 Introduction The results presented here were created as part of a broad effort to understand how to use computers to significantly automate the design and development of software systems. This is one of the main goals of the "PRL project " at Cornell1. One of the basic tenants of our approach to this task is that we should seek the most naturally expressive formal language in which to specify the services, characteristics and constraints that a software system must satisfy. If the formal expression of services is close to a natural one, then people can more readily use it. We also want to allow very compact notations for concepts used to describe systems, and this effect is also a consequence of expressive richness. We have discovered that it is frequently the case that the system we have built to implement one formal language will support an even richer one. So we have come to see our work as also progressively improving the reach of our tools.