A Modern Perspective on Type Theory

Transcription

1 A Modern Perspective on Type Theory

2 APPLIED LOGIC SERIES VOLUME 29 Managing Editor Dov M. Gabbay, Department of Computer Science, King s College, London, U.K. Co-Editor Jon Barwise Editorial Assistant Jane Spurr, Department of Computer Science, King s College, London, U.K. SCOPE OF THE SERIES Logic is applied in an increasingly wide variety of disciplines, from the traditional subjects of philosophy and mathematics to the more recent disciplines of cognitive science, computer science, artificial intelligence, and linguistics, leading to new vigor in this ancient subject. Kluwer, through its Applied Logic Series, seeks to provide a home for outstanding books and research monographs in applied logic, and in doing so demonstrates the underlying unity and applicability of logic. The titles published in this series are listed at the end of this volume.

3 A Modern Perspective on Type Theory From its Origins until Today by FAIROUZ KAMAREDDINE Heriot Watt University, Edinburgh, Scotland TWAN LAAN Eindhoven, The Netherlands and ROB NEDERPELT Eindhoven University of Technology, The Netherlands KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

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5 Contents Preface Preliminaries Overview of this book Part I Part II Part III Acknowledgements 0 Introduction Avoiding the paradox in set theory Avoiding the paradox in type theory The approach ix ix ix ix xi xii xiii I The Evolution of Type Theory until the 1940s 1 Prehistory 1a Paradox threats 1b Paradox threats in formal systems 1b1 Functions and their course of values 1b2 The Russell Paradox in Grundgesetze 1b3 How wrong was Frege? 1b4 The importance of Russell s Paradox 2 Type theory in Principia Mathematica 2a Principia s propositional functions 2a1 Definition 2a2 Principia s propositional functions as 2a3 Principia s pfs and their translation in 2a4 Principia s related notions iii

6 iv CONTENTS 2a5 Principia s substitution 2b The Ramified Theory of Types RTT 2b1 Types 2b2 Formalisation of the Ramified Theory of Types 2b3 Discussion and examples 2c Properties of RTT 2c1 Types and free variables 2c2 Strong normalisation 2c3 Subterm property 2d Legal propositional functions Conclusions 3 Deramification 3a History of the deramification 3a1 The problematic character of RTT 3a2 The Axiom of Reducibility 3a3 Deramification 3b The Simple Theory of Types STT 3b1 3b2 3b3 3b4 Constructing the Simple Theory of Types from RTT Church s simply typed Comparison of RTT with Comparison of STT with 3c Are orders to be blamed? 3c1 Kripke s Theory of Truth KTT 3c2 RTT in KTT 3c3 Orders and types Conclusions II Propositions as Types, Pure Type Systems, AUTOMATH Propositions as Types and Pure Type Systems 4a Propositions as Types and Proofs as Terms (PAT) 4a1 Intuitionistic logic 4a2 The discovery of PAT: Curry 4a3 The discovery of PAT: Howard 4a4 The discovery of PAT: de Bruijn 4b Lambda calculus 4c Pure Type Systems 4c1 4c2 The Barendregt cube Metaproperties of PTSs

7 CONTENTS 5 The pre-pat RTT and STT in PAT-style 5a RTT in PAT style 5a1 An introduction to 5a2 The system 5a3 Meta-properties of 5a4 Interpreting RTT in 5a5 Logic in RTT and 5a6 Various implementations of PAT 5b STT in PAT style Conclusions 6 A Correspondence between RTT and the system Nuprl 6a On the role of orders 6b The Nuprl type system 6b1 A fragment of Nuprl in PTS-style 6b2 Orders in Nuprl 6b3 Evaluating the order of a Nuprl term 6c RTT in Nuprl 6c1 Ramified Types in Nuprl 6c2 Propositional Functions of RTT in Nuprl Conclusions 7 Automath 7a Description of AUTOMATH 7a1 Books, lines and expressions 7a2 Correct books 7a3 Definitional equality 7a4 Some elementary properties 7b From AUT-68 towards a PTS 7c 7d 7b1 7b2 7b3 The choice of the correct formation (II) rules and the parameter types The different treatment of constants and variables The definition system and the translation using 7c1 7c2 7c3 7c4 7c5 Definition and elementary properties Reduction and conversion Subject reduction Strong normalisation The formal relation between AUT-68 and More Suitable Pure Type Systems for AUTOMATH 7d1 with parameters, and v

8 vi CONTENTS 7d2 7d3 Conclusions AUT-QE III Extensions of Pure Type Systems Pure Type Systems with definitions 233 8a Definitions in contexts 234 8a1 Comparison with the definitions of AUTOMATH 237 8b Definitions in the terms and the contexts 238 8b1 Comparison with the definitions of AUTOMATH 238 Conclusions The Barendregt cube with parameters 9a On parameters in the Barendregt cube 9b The Barendregt cube refined with parameters Conclusions Pure Type Systems with parameters and definitions a 10b Parametric constants and definitions Properties of terms 10b1 Basic properties 10b2 Church-Rosser for 10b3 Strong normalisation for 10c 10d Properties of legal terms Restrictive use of parameters 10d1 with restricted parameters 10d2 Imitating parameters by 10d3 Refined Barendregt cubes 10e Systems in the refined Barendregt cube 10e1 ML 10e2 LF 10e3 and AUT-68 10e4 and AUT-QE 10e5 PAL 10f First-order predicate logic Conclusions: Yet another extension of PTSs? Practical motivation The heart of type theory Future work

9 CONTENTS vii A Type systems in this book Aa Pure Type Systems Ab The Barendregt cube Ac The Ramified Theory of Types Ac1 RTT Ac2 Ad The Simple Theory of Types Ad1 STT Ad2 Ae Af Ag Church s simply typed A fragment of Nuprl in PTS-style AUTOMATH Ag1 Ag2 AUT-68 Ah Pure Type Systems with definitions Ah1 Ah2 PTSs with definitions in contexts PTSs with definitions in the terms and the contexts Ai Pure Type Systems with parametric constants Aj A and its subsystems Aj1 Aj2 PTSs with parameters and definitions PTSs with restricted parameters and definitions Bibliography Subject Index Name Index List of Figures

10 Preface Preliminaries We assume that the reader is more or less familiar with the basics of typed and type theory. We give a survey of the most important topics concerning typed in Chapter 4. So, it may be a good idea to read Sections 4b and 4c of that chapter first. We used the historical timeline and presented the (Section 4b) and Pure Type Systems (Section 4c) after our discussion of the prehistory and of Frege, Russell and Ramsey s accounts. Throughout the book, we use to represent the set of the natural numbers (including 0), and to represent the set minus the 0. Overview of this book The book is divided into three parts: Part I The first part consists of three chapters and deals with the evolution of type theory until the 1940s. We study the prehistory of type theory up to 1910 and its development between Russell and Whitehead s Principia Mathematica ([145], ) and Church s simply typed of We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first formulation of a theory of types, the Ramified Type Theory (RTT) (based on 1 Russell discovered his paradox when he read Cantor s work. ix

11 x Preface Russell s work [131] of 1908 which succeeded [130]; this work is highly influenced by Frege s concept of types). We present RTT formally using the modern notation for type theory. Ramified types have a double hierarchy: one of orders and the other of types. Ramified types had limitations and for this reason, Russell and Whitehead added the so-called axiom of reducibility. Ramified types and the reducibility axiom were not fully accepted. This led to various calls for deramification by many including Hilbert and Ackermann, Ramsey and Leon Chwistek. In this book, we concentrate on the simple theory of types (STT) as envisaged by Ramsey, 1926 (and also independently by Hilbert and Ackermann) which is a simplification of the ramified theory of types by removing the orders. In 1940, Church developed, using his and the simple theory of types, the first most influential type theory: Church s simply typed We present STT and 2 Church s own simply typed 3 ) and we finish by comparing RTT, STT and Chapter 1. In the first chapter we discuss the prehistory of type theory. That is, we study the way in which types implicitly occurred in logic and mathematics before there was an explicit theory of types. We pay special attention to the formalisation of logic that is made in Frege s Begriffsschrift [49] and Grundgesetze der Arithmetik [52, 56], as in this formalisation many basic ideas are presented that are later used in type theory. Moreover, the system of Grundgesetze der Arithmetik is the one for which Russell derives his famous paradox, and this paradox has been the reason for Russell to introduce the first theory of types. Chapter 2. This first type theory is the subject of the second chapter. Whitehead and Russell present their theory, the Ramified Type Theory (RTT), in an informal way. Several rough descriptions of this theory have been given in the literature (see for instance [124, 70, 29, 31]) but we present a formalisation of RTT that is directly based on the presentation of RTT in Whitehead and Russell s Principia Mathematica ([145], ). The construction of this formalisation is not a simple task. Whitehead and Russell do not present a clear syntax for their socalled propositional functions in [145], neither do they make a clear difference between syntax and semantics. We present a formal definition of propositional functions that is faithful to the original ideas exposed in Principia Mathematica. A second technical problem is the notion of substitution, which is totally undefined in Principia Mathematica. The formalisation of the notion of propositional function 2 Note that the simple theory of types (1926) existed before the was invented by Church in 1932 and hence before Church s simply typed of Nevertheless, nowadays, when one refers to simple type theory, one usually means Church s simply typed of It should be noted furthermore, that Russell s type structure was different from that of Church. The former was set-based with linear sequences of types. The latter was function-based. 3 We write for the original calculus of Church as presented in [29]. Note that this is different from the calculus used in frameworks like the Barendregt cube and the Pure Type Systems found in [3].

12 Overview of this book xi makes it possible to express the notion of substitution of Principia Mathematica in terms of We use techniques from typed and untyped to give a precise description of substitution, and to show that substitution is well-defined as long as we restrict ourselves to well-typed propositional functions of RTT. Chapter 3. In 1926, Ramsey [124] proposed an important simplification of RTT, the simple theory of types (STT). This simple type theory has become the basis for many modern type systems, and for the simply typed of Church [29]. The simplification consisted of the removal of one of the two hierarchies from RTT. The hierarchy of types is maintained, while the hierarchy of orders is removed. In Chapter 3 we discuss this process, known as deramification. An important observation of this chapter is that though the orders do not occur in the mainstream of type theories, they still provide an important intuition for logicians. We show that there is a close link between the hierarchy of orders in RTT and the hierarchy of truths that was introduced by Kripke [96]. We also show that Kripke s use of orders is more flexible than Russell s, and that this is due to the fact that orders occur at the semantical level in Kripke s theory, while they occur at the level of syntax in RTT. Part II The second part of the book consists of four chapters and deals with major developments of type theory since the 1940s. Although type theory was not active in the 1950s, in the 1960s there was a revival of interest in types and their applications to computer science and mathematics. But then, both RTT and turned out to be too restrictive for mathematics and computer science where fixed points (to mention one example) play an important role. Since the 1960s, we have seen many new type systems and the influence of types in logic, mathematics and computation continues to grow. Both logicians and computer scientists have developed several branches of typed Also mathematics has benefitted from typed especially since 1967 when de Bruijn used his AUTOMATH for the analysis and checking of mathematical texts. Chapter 4. Though type theory clearly served as a method to prevent certain logical paradoxes, the logical system stood apart from the type system until the 1950s. In Chapter 4 we study the ways in which logic can be included in a type system. The various methods are all based on the idea that the proof of a logical implication can be seen as a function. More precisely: A proof of the proposition is implemented as a function that takes a proof of A as argument, and returns a proof of B. In this way, the proposition can be seen as the type of all functions from (proofs of) A to (proofs of) B. Similarly, a proof of becomes a term of type One calls this principle: propositions as types, or: proofs as terms. Both expressions are abbreviated by PAT. PAT was

13 xii Preface discovered independently by different people. In Section 4a we give a historical sketch. Moreover, in this book we try to present, in a uniform framework, various important type systems that were proposed during this century. An important part of this framework is formed by the so-called Pure Type Systems (PTSs). Therefore, a short introduction to typed lambda calculus and PTSs is essential for the understanding of this book. Hence, in Section 4b we present the basic definitions and properties of the Then in section 4c we introduce a general framework of type systems which includes the most influential type theories developed since the 1960s including the polymorphic type theory, the dependent type theory and the Pure Type Systems (PTSs). We also introduce the so-called Barendregt cube [3] which is a collection of eight influential type systems. PTSs in general and the cube in particular, will be used to represent in the modern setting, some of the historical systems discussed so far. Moreover, PTSs in general and the cube in particular will be subject to various extensions in Part III of this book. Chapter 5. In Chapter 5 we describe how the two most important type systems of the pre-pat-era, RTT and STT, can be described in a PAT style. This gives insight in the various ways in which PAT-implementations can be made. Chapter 6. In Chapter 6, we return to Russell s orders which we use to place RTT in a context with a modern system of computer mathematics (Constable s Nuprl). Nuprl is based on a modern type theory which differs from Pure Type Systems: Martin-Löf s type theory. As another illustration of the generality of PTSs, we present a complex type system (again Nuprl) as a simple and compact PTS. Chapter 7. One of the important applications of PAT is the mechanical verification of mathematical proofs. The first tool for such a verification was AUTOMATH. It was developed in the late 1960s. The languages of the various AUTOMATH systems have been studied intensively. In [3], a description of two of the most important systems within the framework of Pure Type Systems is given, but without explanation. In Chapter 7 we study the original language of AUTOMATH and translate it to a PTS format. In doing so, we obtain descriptions similar to those of [3]. Part III The third part which consists of three chapters, deals with some extensions of type theory, especially those related to functions and parameters (arguments). Chapter 8. In many type theories and lambda calculi, there is no formal possibility to use abbreviations or definitions, i.e., to introduce names for large expressions which can be used several times in a program or a proof. This possibility is essential for practical use, and indeed implementations of Pure Type Systems such as Coq [44] and Nuprl [35] do provide this possibility. Moreover, most implementations of programming languages (Haskell, ML, CAML, etc.) use names for large

14 Acknowledgements xiii expressions via a well-known programming language concept: the so-called let expressions or definitions. This chapter introduces definitions to Pure Type Systems in two different ways: definitions can be added only to the context or can be added to both the contexts and the syntax of terms. We compare the two notions of definitions studied in this chapter with the definitions of AUTOM ATH. Chapter 9. Our study of the function concept leads to a natural refinement of the Barendregt cube with what we call parameters. This leads to the division of the cube into eight smaller cubes which will enable more accurate presentations of various systems. The refinement of this chapter will be used in the next chapter together with the concept of definitions discussed in Chapter 8, to give a powerful refinement of Pure Type Systems in general and of the Barendregt cube in particular. Chapter 10. The description of AUTOMATH given in Chapter 7 is precise, but does not take into account two of its most important mechanisms: the definition mechanism and the parameter mechanism. Many other type systems use these mechanisms as well. This motivates us to extend the framework of PTSs with definitions and parameters in Chapter 10. Our extension results in a refinement of the framework of PTSs. In this refined framework, various modern type systems (like LF and ML) can be described in a more precise way than in the PTS framework without definitions and parameters. Acknowledgements This book is based on various projects by the authors in the past few years. Of these projects we acknowledge: The project De ontwikkeling van het Typebegrip ( the evolution of the type concept ), which was financially supported by the Co-operation Centre Tilburg and Eindhoven Universities (SOBU) and which resulted in the PhD thesis by Twan Laan [98]. NWO, Eindhoven University of Technology, the Royal Society and the British Council provided funding for the trips of Twan Laan to Scotland in spring 1995 and winter 1996 and of Fairouz Kamareddine to the Netherlands between 1993 and The project Advantages of a new lambda-notation, which was financially supported by the Engineering and Physical Sciences Research Council (EP- SRC). That project studied amongst other things, the correspondence between Martin-Löf type theory, Russell s ramified type theory and Pure Type Systems [83], the correspondence between Kripke s theories of truth and Russell s orders [82], and also led to the formalisation of definitions in typed lambda calculi by the authors [14, 81] (in collaboration with Roel Bloo).

15 xiv Preface The international exchange project(s) supported by Eindhoven University of Technology which allowed Fairouz Kamareddine to make various short and long visits to Eindhoven University of Technology during which she collaborated on various topics elaborated in this book [84, 83, 14, 82, 13, 86, 85, 87]. Many people have influenced the work in this book and we cannot mention them all. In particular, we would like to express our gratitude to Harrie de Swart, and Jos Baeten for making this project possible, for the useful discussions and invaluable support, and for facilitating our exchange visits. Henk Barendregt, Dirk van Dalen, Roger Hindley, Randall Holmes, Jonathan Seldin, and Joe Wells provided much valuable feedback. Joe Wells also provided much valued help with for which we are extremely grateful. Last but not least, we are very grateful for Jane Spurr who handled this book in her very prompt, efficient, friendly and professional manner. Producing this manuscript would have been much harder without you Jane. Thank you. Fairouz Kamareddine, Twan Laan and Rob Nederpelt Edinburgh and Eindhoven, March 2004