DOCSLIB.ORG

A Type System for Certified Runtime Type Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Type System for Certified Runtime Type Analysis A Type System For Certified Runtime Type Analysis A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Bratin Saha Dissertation Director: Professor Zhong Shao December 2002 Abstract A Type System For Certified Runtime Type Analysis Bratin Saha 2002 Modern programming paradigms are increasingly giving rise to applications that require type information at runtime. For example, services like garbage collection, marshalling, and serializa- tion need to analyze type information at runtime. When compiling code which uses runtime type inspections, most existing compilers use untyped intermediate languages and discard type informa- tion at an early stage. Unfortunately, such an approach is incompatible with type-based certifying compilation. A certifying compiler generates not only the code but also a proof that the code obeys a security policy. Therefore, one need not trust the correctness of a compiler generating certified code, instead one can verify the correctness of the generated code. This allows a code consumer to accept code from untrusted sources which is specially advantageous in a networked environment. In practice, most certifying compilers use a type system for generating and encoding the proofs. These systems are called type-based certifying compilers. This dissertation describes a type system that can be used for supporting runtime type analysis in type-based certifying compilers. The type system has two novel features. First, type analysis can be applied to the type of any runtime value. In particular quantified types such as polymorphic and existential types can also be analyzed, yet type-checking remains decidable. This allows the system to be used for applications such as a copying garbage collector. Type analysis plays the key role in formalizing the contract between the mutator and the collector. Second, the system integrates runtime type analysis with the explicit representation of proofs and propositions. Essentially, it integrates an entire proof language into the type system for a com- piler intermediate language. Existing certifying compilers have focussed only on simple memory and control-flow safety. This system can be used for certifying more advanced properties while still retaining decidable type-checking. Copyright c 2002 by Bratin Saha All rights reserved. i Acknowledgements First and foremost, I want to thank my advisor Professor Zhong Shao for his encouragement and support during my graduate study. He was a constant source of ideas, sound advice, and enthu- siasm. He exposed me to a wealth of knowledge in programming languages, and has taught me by example the skills of a good researcher. In my first summer at Yale, he put in a huge effort in helping me get up to speed with the SML/NJ compiler. I am particularly grateful for his efforts on our “Optimal Type Lifting” paper to improve my writing. I am indebted to him for all the other things that he has taught me: giving talks, writing research proposals, and much more. This work would not have been possible without the tremendous effort that he put into guiding me. I was privileged to be his student. I would like to thank my collaborators in the FLINT group. Valery Trifonov helped out a great deal with the work on intensional type analysis. I worked with Stefan Monnier on the application of intensional type analysis to type-safe garbage collectors. Stefan was always there to lend a hand with system administration issues. I would like to thank Chris League for many useful discussions and specially for Latex related help. I would like to thank Carsten Schurmann, Arvind Krishnamurthy, John Peterson, and Dana Angluin in the Yale Computer Science department. All of them have helped me in various ways during my graduate study and have taken an active interest in my work. Arvind and John served on my thesis committee. Carsten helped me refine my thoughts through stimulating discussions. I would also like to thank all the people in the administrative offices at Yale – specially Judy Smith and Linda Dobb for helping me out with all the paperwork. I want to thank Andrew Appel and Robert Harper for their help in various stages of my graduate study. Andrew served on my thesis committee and also helped me in other ways. Robert Harper shaped my thoughts through several stimulating discussions. His papers are an education in type- theory. ii Finally, my deepest thanks to my parents without whom I would not be here. They have provided me with loving support throughout my childhood and adulthood, and have been a great source of inspiration. I am very grateful to my brother-in-law Prasantada, my sister Gargi, Buni and Bublu for their constant love and support. I am indebted to Atrayee for her love and encouragement. Graduating during an economic depression can be particularly depressing. She was always there to cheer me up. The later part of graduate study can get particularly stressful specially in trying to tie up all the loose ends; she was always there to lend me a hand. Without Atrayee’s cheerful and constant support this work would have been impossible. I also wish to thank all my friends in Yale and outside – specially Narayan and Kishnan. This work was sponsored in part by the Defense Advanced Research Projects Agency under contract F30602-99-1-0519 and by the National Science Foundation under contracts CCR 9633390, CCR 9901011, CCR 0081590. I am grateful to both these organizations. iii Contents 1 Introduction 1 1.1 Runtime type analysis . 1 1.2 Certifying compilation . 2 1.3 Fully reflexive type analysis . 3 1.4 Integrating type analysis with a proof system . 4 1.5 Summary of contributions . 4 1.6 Outline of this dissertation . 5 2 Overview Of The Typed Lambda Calculi 7 2.1 The simply typed lambda calculus . 7 2.2 The polymorphically typed lambda calculus . 10 2.3 The third order lambda calculus . 12 2.4 The ! order lambda calculus . 13 2.5 Pure type systems . 14 3 A Language For Runtime Type Analysis 18 3.1 Introduction . 18 3.2 Background . 19 3.3 Analyzing quantified types . 24 3.4 Type erasure semantics . 36 ! ! 3.5 Translation from λi to λR . 44 3.6 The untyped language . 48 3.7 Related work . 50 iv 4 Applying Runtime Type Analysis: Garbage Collection 52 4.1 Introduction and motivation . 52 4.2 Source language λCLOS . 55 4.3 Target language λGC . 56 4.4 Translating λCLOS to λGC . 63 4.5 Summary and related work . 66 5 Integrating Runtime Type Analysis With a Proof System 67 5.1 Introduction . 67 5.2 Approach . 68 i 5.3 The type language λCC . 72 i 5.4 Formalization of λCC . 76 5.5 The computation language λH . 81 5.6 Summary and related work . 89 6 Implementation 91 6.1 Implementing a pure type system . 94 6.2 Rationale for design decisions . 98 6.3 Implementation Details . 99 6.4 Representing the base kind . 102 6.5 Performance measurement . 106 6.6 Related Work . 109 7 Conclusions and Future Work 110 7.1 Summary of contributions . 111 7.2 Future work . 112 Appendix 113 ! A Formal Properties of λi 114 ! A.1 Soundness of λi . 114 A.2 Strong normalization . 120 A.3 Confluence . 137 v B Formal Properties Of λGC 145 i C Formal Properties of λCC 154 C.1 Subject reduction . 154 C.2 Strong normalization . 168 C.3 Church-Rosser property . 201 C.4 Consistency . 204 Bibliography 204 vi Chapter 1 Introduction This dissertation describes a type system for supporting runtime type analysis in a certifying com- piler. Why is this important ? Runtime type analysis plays a crucial role in many applications, and modern language implementations are progressively targeting the generation of certified code. The following sections show how runtime type analysis plays a crucial role, and why it is important to support it in a certifying framework. 1.1 Runtime type analysis Modern programming paradigms are increasingly giving rise to applications that rely critically on type information at runtime. For example: • Java adopts dynamic linking as a key feature, and to ensure safe linking, an external module must be dynamically verified to satisfy the expected interface type. • A garbage collector must keep track of all live heap objects, and for that type information must be kept at runtime to allow traversal of data structures. • In a distributed computing environment, code and data on one machine may need to be pickled for transmission to a different machine, where the unpickler reconstructs the data structures from the bit stream. If the type of the data is not statically known at the destination (as is the case for the environment components of function closures), the unpickler must use type information, encoded in the bit stream, to correctly interpret the encoded value. 1 • Type-safe persistent programming requires language support for dynamic typing: the pro- gram must ensure that data read from a persistent store is of the expected type. • Finally, in polymorphic languages like ML, the type of a value may not be known statically; therefore, compilers have traditionally used inefficient, uniformly boxed data representation. To avoid this, several modern compilers use runtime type information to support unboxed data representation. Existing compilers can not type-check the code that involves runtime type inspections. When compiling such code they use untyped intermediate languages, and reify runtime types into values at some early stage. However, discarding the type information during compilation makes this approach incompatible with certifying compilation. 1.2 Certifying compilation A certifying compiler [Nec98] generates not only the object code, but also a machine-checkable proof that the code satisfies a given security policy.
Load more

Recommended publications
  • Introduction to the Calculus of Inductive Constructions Christine Paulin-Mohring
    Introduction to the Calculus of Inductive Constructions Christine Paulin-Mohring To cite this version: Christine Paulin-Mohring. Introduction to the Calculus of Inductive Constructions. Bruno Woltzen- logel Paleo; David Delahaye. All about Proofs, Proofs for All, 55, College Publications, 2015, Studies in Logic (Mathematical logic and foundations), 978-1-84890-166-7. hal-01094195 HAL Id: hal-01094195 https://hal.inria.fr/hal-01094195 Submitted on 11 Dec 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Introduction to the Calculus of Inductive Constructions Christine Paulin-Mohring1 LRI, Univ Paris-Sud, CNRS and INRIA Saclay - ˆIle-de-France, Toccata, Orsay F-91405 [email protected] 1 Introduction The Calculus of Inductive Constructions (CIC) is the formalism behind the interactive proof assis- tant Coq [24, 5]. It is a powerful language which aims at representing both functional programs in the style of the ML language and proofs in higher-order logic. Many data-structures can be repre- sented in this language: usual data-types like lists and binary trees (possibly polymorphic) but also infinitely branching trees. At the logical level, inductive definitions give a natural representation of notions like reachability and operational semantics defined using inference rules.
    [Show full text]
  • Proof-Assistants Using Dependent Type Systems
    CHAPTER 18 Proof-Assistants Using Dependent Type Systems Henk Barendregt Herman Geuvers Contents I Proof checking 1151 2 Type-theoretic notions for proof checking 1153 2.1 Proof checking mathematical statements 1153 2.2 Propositions as types 1156 2.3 Examples of proofs as terms 1157 2.4 Intermezzo: Logical frameworks. 1160 2.5 Functions: algorithms versus graphs 1164 2.6 Subject Reduction . 1166 2.7 Conversion and Computation 1166 2.8 Equality . 1168 2.9 Connection between logic and type theory 1175 3 Type systems for proof checking 1180 3. l Higher order predicate logic . 1181 3.2 Higher order typed A-calculus . 1185 3.3 Pure Type Systems 1196 3.4 Properties of P ure Type Systems . 1199 3.5 Extensions of Pure Type Systems 1202 3.6 Products and Sums 1202 3.7 E-typcs 1204 3.8 Inductive Types 1206 4 Proof-development in type systems 1211 4.1 Tactics 1212 4.2 Examples of Proof Development 1214 4.3 Autarkic Computations 1220 5 P roof assistants 1223 5.1 Comparing proof-assistants . 1224 5.2 Applications of proof-assistants 1228 Bibliography 1230 Index 1235 Name index 1238 HANDBOOK OF AUTOMAT8D REASONING Edited by Alan Robinson and Andrei Voronkov © 2001 Elsevier Science Publishers 8.V. All rights reserved PROOF-ASSISTANTS USING DEPENDENT TYPE SYSTEMS 1151 I. Proof checking Proof checking consists of the automated verification of mathematical theories by first fully formalizing the underlying primitive notions, the definitions, the axioms and the proofs. Then the definitions are checked for their well-formedness and the proofs for their correctness, all this within a given logic.
    [Show full text]
  • A Calculus of Constructions with Explicit Subtyping Ali Assaf
    A calculus of constructions with explicit subtyping Ali Assaf To cite this version: Ali Assaf. A calculus of constructions with explicit subtyping. 2014. hal-01097401v1 HAL Id: hal-01097401 https://hal.archives-ouvertes.fr/hal-01097401v1 Preprint submitted on 19 Dec 2014 (v1), last revised 14 Jan 2016 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A calculus of constructions with explicit subtyping Ali Assaf September 16, 2014 Abstract The calculus of constructions can be extended with an infinite hierar- chy of universes and cumulative subtyping. In this hierarchy, each uni- verse is contained in a higher universe. Subtyping is usually left implicit in the typing rules. We present an alternative version of the calculus of constructions where subtyping is explicit. This new system avoids prob- lems related to coercions and dependent types by using the Tarski style of universes and by introducing additional equations to reflect equality. 1 Introduction The predicative calculus of inductive constructions (PCIC), the theory behind the Coq proof system [15], contains an infinite hierarchy of predicative universes Type0 : Type1 : Type2 : ... and an impredicative universe Prop : Type1 for propositions, together with a cumulativity relation: Prop ⊆ Type0 ⊆ Type1 ⊆ Type2 ⊆ ...
    [Show full text]
  • Type Theory and Applications
    Type Theory and Applications Harley Eades [email protected] 1 Introduction There are two major problems growing in two areas. The first is in Computer Science, in particular software engineering. Software is becoming more and more complex, and hence more susceptible to software defects. Software bugs have two critical repercussions: they cost companies lots of money and time to fix, and they have the potential to cause harm. The National Institute of Standards and Technology estimated that software errors cost the United State's economy approximately sixty billion dollars annually, while the Federal Bureau of Investigations estimated in a 2005 report that software bugs cost U.S. companies approximately sixty-seven billion a year [90, 108]. Software bugs have the potential to cause harm. In 2010 there were a approximately a hundred reports made to the National Highway Traffic Safety Administration of potential problems with the braking system of the 2010 Toyota Prius [17]. The problem was that the anti-lock braking system would experience a \short delay" when the brakes where pressed by the driver of the vehicle [106]. This actually caused some crashes. Toyota found that this short delay was the result of a software bug, and was able to repair the the vehicles using a software update [91]. Another incident where substantial harm was caused was in 2002 where two planes collided over Uberlingen¨ in Germany. A cargo plane operated by DHL collided with a passenger flight holding fifty-one passengers. Air-traffic control did not notice the intersecting traffic until less than a minute before the collision occurred.
    [Show full text]
  • Equality Is Typable in Semi-Full Pure Type Systems
    Equality is typable in Semi-Full Pure Type Systems Vincent Siles Hugo Herbelin Ecole Polytechnique / INRIA / Laboratoire PPS INRIA / Laboratoire PPS Equipe πr2 Equipe πr2 23 avenue d’Italie, 75013 Paris, France 23 avenue d’Italie, 75013 Paris, France [email protected] [email protected] Abstract—There are two usual ways to describe equality in Surprisingly, showing the equivalence between those two a dependent typing system, one that uses an external notion definitions is difficult. Geuvers and Werner [7] early noticed of computation like beta-reduction, and one that introduces a that being able to lift an untyped equality to a typed one, typed judgement of beta-equality directly in the typing system. After being an open problem for some time, the general i.e. to turn a system with β-conversion into a system with equivalence between both approaches has been solved by judgmental equality requires to show Subject Reduction in Adams for a class of pure type systems (PTSs) called functional. the latter system: In this paper, we relax the functionality constraint and prove the equivalence for all semi-full PTSs by combining the ideas If Γ `e M : A and M β N then Γ `e M = N : A. of Adams with a study of the general shape of types in PTSs. As one application, an extension of this result to systems Subject Reduction requires the injectivity of dependent A with sub-typing would be a first step toward bringing closer products Πx :B : the theory behind a proof assistant such as Coq to its A C implementation.
    [Show full text]
  • Investigation on the Typing of Equality in Type Systems Vincent Siles
    Investigation on the typing of equality in type systems Vincent Siles To cite this version: Vincent Siles. Investigation on the typing of equality in type systems. Logic in Computer Science [cs.LO]. Ecole Polytechnique X, 2010. English. pastel-00556578 HAL Id: pastel-00556578 https://pastel.archives-ouvertes.fr/pastel-00556578 Submitted on 17 Jan 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Ecole´ Polytechnique Th`ese de Doctorat Sp´ecialit´eInformatique Investigation on the typing of equality in type systems Pr´esent´ee et soutenue publiquement par Vincent Siles le 25 Novembre 2010 devant le jury compos´ede Bruno Barras Co-directeur de th`ese Gilles Barthe Rapporteur Roberto Di Cosmo Herman Geuvers Rapporteur Hugo Herbelin Co-directeur de th`ese Alexandre Miquel Rapporteur Randy Pollack Benjamin Werner Abstract Pure Type Systems are a good way to factorize the questions of meta-theory about a large family of type systems. They have been introduced as a gener- alization of Barendregt’s λ-cube, an abstraction of several type systems like the Simply Typed λ-Calculus, System F or the Calculus of Constructions. One critical detail of the Pure Type Systems is their conversion rule that allows to do computation at the level of types.
    [Show full text]
  • Notes: Barendregt's Cube and Programming with Dependent Types
    Notes: Barendregt's cube and programming with dependent types Eric Lu I. ANNOTATED BIBLIOGRAPHY A. Lambda Calculi with Types @incollection{barendregt:1992 author = {Barendregt, Henk}, title = {Lambda Calculi with Types}, booktitle = {Handbook of Logic in Computer Science}, publisher = {Oxford University Press}, year = {1992}, editor = {Abramsky, S. and Gabbay, D.M. and Maibaum, T.S.E.}, volume = {2}, pages = {117--309}, address = {New York, New York} } Summary: As setup for introducing the lambda cube, Barendregt introduces the type-free lambda calculus and reviews some of its properties. Subsequently he presents accounts of typed lambda calculi from Curry and Church, including some Curry-style systems not belonging to the lambda cube (λU and λµ). He then describes the lambda cube construction that was first noted by Barendregt in 1991. The lambda cube describes an inclusion relation amongst eight typed lambda calculi. Moreover it explains a fine structure for the calculus of constructions arising from the presence or absence of three axiomatic additions to the simply-typed lambda calculus. The axes of the lambda cube are intuitively justified by the notion of dependency. After introducing the lambda cube, Barendregt gives properties of the languages belonging to the lambda cube, as well as some properties of pure type systems, which generalize the means of description of the axioms of the systems in the lambda cube. Evaluation: This work originates as a reference from the logical/mathematical community, but its content was important in providing a firm foundation for types in programming languages. The lambda cube explains prior work on a number of type systems by observing a taxonomy and relating them to logical systems via the Curry-Howard isomorphism.
    [Show full text]
  • Basics of Type Theory and Coq
    Basics of type theory and Coq Michael Shulman January 31, 2012 1 / 77 Type-theoretic foundations Set theory Type theory Logic Types ^; _; ); :; 8; 9 ×; +; !; Q; P Sets Logic ×; +; !; Q; P ^; _; ); :; 8; 9 x 2 A is a proposition x : A is a typing judgment 2 / 77 Type theory is programming For now, think of type theory as a programming language. • Closely related to functional programming languages like ML, Haskell, Lisp, Scheme. • More expressive and powerful. • Can manipulate “mathematical objects”. 4 / 77 Typing judgments Type theory consists of rules for manipulating judgments. The most important judgment is a typing judgment: x1 : A1; x2 : A2;::: xn : An ` b : B The turnstile ` binds most loosely, followed by commas. This should be read as: In the context of variables x1 of type A1, x2 of type A2,..., and xn of type An, the expression b has type B. Examples ` 0: N x : N; y : N ` x + y : N f : R ! R; x : R ` f (x): R 1 (n) 1 f : C (R; R); n: N ` f : C (R; R) 5 / 77 Type constructors The basic rules tell us how to construct valid typing judgments, i.e. how to write programs with given input and output types. This includes: 1 How to construct new types (judgments Γ ` A: Type). 2 How to construct terms of these types. 3 How to use such terms to construct terms of other types. Example (Function types) 1 If A: Type and B : Type, then A ! B : Type. 2 If x : A ` b : B, then λxA:b : A ! B.
    [Show full text]
  • Interactive Theorem Proving in Coq and the Curry-Howard Isomorphism
    Interactive Theorem Proving in Coq and the Curry-Howard Isomorphism Abhishek Kr Singh January 7, 2015 Abstract There seems to be a general consensus among mathematicians about the notion of a correct proof. Still, in mathematical literature, many invalid proofs remain accepted over a long period of time. It happens mostly because proofs are incomplete, and it is easy to make mistake while verifying an incomplete proof. However, writing a complete proof on paper consumes a lot of time and effort. Proof Assistants, such as Coq [8, 2], can minimize these overheads. It provides the user with lots of tools and tactics to interactively develop a proof. Most of the routine jobs, such as creation of proof-terms and type verification, are done internally by the proof assistant. Realization of such a computer based tool for representing proofs and propositions, is made possible because of the Curry- Howard Isomorphism [6]. It says that, proving and programming are essentially equivalent tasks. More precisely, we can use a programming paradigm, such as typed lambda calculus [1], to encode propositions and their proofs. This report tries to discuss curry-howard isomorphism at different levels. In particular, it discusses different features of Calculus of Inductive Constructions [3, 9], that makes it possible to encode different logics in Coq. Contents 1 Introduction 1 1.1 Proof Assistants . 1 1.2 The Coq Proof assistant . 2 1.3 An overview of the report . 2 2 Terms, Computations and Proofs in Coq 3 2.1 Terms and Types . 3 2.2 Computations . 5 2.3 Propositions and Proofs .
    [Show full text]
  • Explicit Convertibility Proofs in Pure Type Systems
    Explicit Convertibility Proofs in Pure Type Systems Floris van Doorn Herman Geuvers Freek Wiedijk Utrecht University Radboud University Nijmegen Radboud University Nijmegen fl[email protected] [email protected] [email protected] Abstract In dependent type systems, terms occur in types, so one can We define type theory with explicit conversions. When type check- do (computational) reduction steps inside types. This includes ing a term in normal type theory, the system searches for convert- β-reduction steps, but, in case one has inductive types, also ι- ibility paths between types. The results of these searches are not reduction and, in case one has definitions, also δ-reduction. A type stored in the term, and need to be reconstructed every time again. B that is obtained from A by a (computational) reduction step is In our system, this information is also represented in the term. considered to be “equal” to A. A common approach to deal with The system we define has the property that the type derivation this equality between types is to use an externally defined notion of of a term has exactly the same structure as the term itself. This has conversion. In case one has only β-reduction, this is β-conversion the consequence that there exists a natural LF encoding of such a which is denoted by A 'β B. This is the least congruence con- system in which the encoded type is a dependent parameter of the taining the β-reduction step. Then there is a conversion rule of the type of the encoded term.
    [Show full text]
  • Logics and Type Systems
    Logics and Typ e Systems een wetenschapp elijke pro eve op het gebied van de wiskunde en informatica proefschrift ter verkrijging van de graad van do ctor aan de Katholieke Universiteit Nijmegen volgens b esluit van het College van Decanen in het op enbaar te verdedigen op dinsdag septemb er des namiddags te uur precies do or Jan Herman Geuvers geb oren mei te Deventer druk Universiteitsdrukkerij Nijmegen Promotor Professor dr H P Barendregt Logics and Typ e Systems Herman Geuvers Cover design Jean Bernard Ko eman cipgegevens Koninklijke Bibliotheek Den Haag Geuvers Jan Herman Logics and typ e systems Jan Herman Geuvers Sl sn Nijmegen Universiteitsdrukkerij Nijmegen Pro efschrift Nijmegen Met lit opgreg ISBN Trefw logica vo or de informatica iv Contents Intro duction Natural Deduction Systems of Logic Intro duction The Logics Extensionality Some useful variants of the systems Some easy conservativity results Conservativity b etween the logics Truth table semantics for classical prop ositional logics Algebraic semantics for intuitionistic prop ositional logics Kripke semantics for intuitionistic prop ositional logics Formulasastyp es Intro duction The formulasastyp es notion a la Howard Completeness of the emb edding Comparison with other emb eddings Reduction of derivations and extensions
    [Show full text]
  • Pure Type Systems Without Explicit Contexts
    Pure Type Systems without Explicit Contexts Herman Geuvers Radboud University Nijmegen Technical University Eindhoven The Netherlands Robbert Krebbers James McKinna Freek Wiedijk Radboud University Nijmegen Institute for Computing and Information Sciences Heyendaalseweg 135, 6525 AJ Nijmegen The Netherlands We present an approach to type theory in which the typing judgments do not have explicit contexts. Instead of judgments of shape Γ ⊢ A : B, our systems just have judgments of shape A : B. A key feature is that we distinguish free and bound variables even in pseudo-terms. Specifically we give the rules of the ‘Pure Type System’ class of type theories in this style. We prove that the typing judgments of these systems correspond in a natural way with those of Pure Type Systems as traditionally formulated. I.e., our systems have exactly the same well-typed terms as traditional presentations of type theory. Our system can be seen as a type theory in which all type judgments share an identical, infinite, typing context that has infinitely many variables for each possible type. For this reason we call our system Γ∞. This name means to suggest that our type judgment A : B should be read as Γ∞ ⊢ A : B, with a fixed infinite type context called Γ∞. 1 Introduction 1.1 Problem One of the important insights type theory gives us is the need to be aware of the context in which one works. This was already stressed by de Bruijn in his 1979 paper Wees contextbewust in WOT [6], Dutch for “Be context aware in the mathematical vernacular”. In type theory a term always is considered with respect to a context Γ, which gives the types of the variables occurring free in the term.
    [Show full text]