Lecture Notes on Types for Part II of the Computer Science Tripos
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Lecture Notes on the Lambda Calculus 2.4 Introduction to Β-Reduction
2.2 Free and bound variables, α-equivalence. 8 2.3 Substitution ............................. 10 Lecture Notes on the Lambda Calculus 2.4 Introduction to β-reduction..................... 12 2.5 Formal definitions of β-reduction and β-equivalence . 13 Peter Selinger 3 Programming in the untyped lambda calculus 14 3.1 Booleans .............................. 14 Department of Mathematics and Statistics 3.2 Naturalnumbers........................... 15 Dalhousie University, Halifax, Canada 3.3 Fixedpointsandrecursivefunctions . 17 3.4 Other data types: pairs, tuples, lists, trees, etc. ....... 20 Abstract This is a set of lecture notes that developed out of courses on the lambda 4 The Church-Rosser Theorem 22 calculus that I taught at the University of Ottawa in 2001 and at Dalhousie 4.1 Extensionality, η-equivalence, and η-reduction. 22 University in 2007 and 2013. Topics covered in these notes include the un- typed lambda calculus, the Church-Rosser theorem, combinatory algebras, 4.2 Statement of the Church-Rosser Theorem, and some consequences 23 the simply-typed lambda calculus, the Curry-Howard isomorphism, weak 4.3 Preliminary remarks on the proof of the Church-Rosser Theorem . 25 and strong normalization, polymorphism, type inference, denotational se- mantics, complete partial orders, and the language PCF. 4.4 ProofoftheChurch-RosserTheorem. 27 4.5 Exercises .............................. 32 Contents 5 Combinatory algebras 34 5.1 Applicativestructures. 34 1 Introduction 1 5.2 Combinatorycompleteness . 35 1.1 Extensionalvs. intensionalviewoffunctions . ... 1 5.3 Combinatoryalgebras. 37 1.2 Thelambdacalculus ........................ 2 5.4 The failure of soundnessforcombinatoryalgebras . .... 38 1.3 Untypedvs.typedlambda-calculi . 3 5.5 Lambdaalgebras .......................... 40 1.4 Lambdacalculusandcomputability . 4 5.6 Extensionalcombinatoryalgebras . 44 1.5 Connectionstocomputerscience . -
Proving Termination of Evaluation for System F with Control Operators
Proving termination of evaluation for System F with control operators Małgorzata Biernacka Dariusz Biernacki Sergue¨ıLenglet Institute of Computer Science Institute of Computer Science LORIA University of Wrocław University of Wrocław Universit´ede Lorraine [email protected] [email protected] [email protected] Marek Materzok Institute of Computer Science University of Wrocław [email protected] We present new proofs of termination of evaluation in reduction semantics (i.e., a small-step opera- tional semantics with explicit representation of evaluation contexts) for System F with control oper- ators. We introduce a modified version of Girard’s proof method based on reducibility candidates, where the reducibility predicates are defined on values and on evaluation contexts as prescribed by the reduction semantics format. We address both abortive control operators (callcc) and delimited- control operators (shift and reset) for which we introduce novel polymorphic type systems, and we consider both the call-by-value and call-by-name evaluation strategies. 1 Introduction Termination of reductions is one of the crucial properties of typed λ-calculi. When considering a λ- calculus as a deterministic programming language, one is usually interested in termination of reductions according to a given evaluation strategy, such as call by value or call by name, rather than in more general normalization properties. A convenient format to specify such strategies is reduction semantics, i.e., a form of operational semantics with explicit representation of evaluation (reduction) contexts [14], where the evaluation contexts represent continuations [10]. Reduction semantics is particularly convenient for expressing non-local control effects and has been most successfully used to express the semantics of control operators such as callcc [14], or shift and reset [5]. -
Open C++ Tutorial∗
Open C++ Tutorial∗ Shigeru Chiba Institute of Information Science and Electronics University of Tsukuba [email protected] Copyright c 1998 by Shigeru Chiba. All Rights Reserved. 1 Introduction OpenC++ is an extensible language based on C++. The extended features of OpenC++ are specified by a meta-level program given at compile time. For dis- tinction, regular programs written in OpenC++ are called base-level programs. If no meta-level program is given, OpenC++ is identical to regular C++. The meta-level program extends OpenC++ through the interface called the OpenC++ MOP. The OpenC++ compiler consists of three stages: preprocessor, source-to-source translator from OpenC++ to C++, and the back-end C++ com- piler. The OpenC++ MOP is an interface to control the translator at the second stage. It allows to specify how an extended feature of OpenC++ is translated into regular C++ code. An extended feature of OpenC++ is supplied as an add-on software for the compiler. The add-on software consists of not only the meta-level program but also runtime support code. The runtime support code provides classes and functions used by the base-level program translated into C++. The base-level program in OpenC++ is first translated into C++ according to the meta-level program. Then it is linked with the runtime support code to be executable code. This flow is illustrated by Figure 1. The meta-level program is also written in OpenC++ since OpenC++ is a self- reflective language. It defines new metaobjects to control source-to-source transla- tion. The metaobjects are the meta-level representation of the base-level program and they perform the translation. -
The System F of Variable Types, Fifteen Years Later
Theoretical Computer Science 45 (1986) 159-192 159 North-Holland THE SYSTEM F OF VARIABLE TYPES, FIFTEEN YEARS LATER Jean-Yves GIRARD Equipe de Logique Mathdmatique, UA 753 du CNRS, 75251 Paris Cedex 05, France Communicated by M. Nivat Received December 1985 Revised March 1986 Abstract. The semantic study of system F stumbles on the problem of variable types for which there was no convincing interpretation; we develop here a semantics based on the category-theoretic idea of direct limit, so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones, thus getting rid of the circularity of variable types. To do so, one has first to simplify somehow the extant semantic ideas, replacing Scott domains by the simpler and more finitary qualitative domains. The interpretation obtained is extremely compact, as shown on simple examples. The paper also contains the definitions of a very small 'universal model' of lambda-calculus, and investigates the concept totality. Contents Introduction ................................... 159 1. Qualitative domains and A-structures ........................ 162 2. Semantics of variable types ............................ 168 3. The system F .................................. 174 3.1. The semantics of F: Discussion ........................ 177 3.2. Case of irrt ................................. 182 4. The intrinsic model of A-calculus .......................... 183 4.1. Discussion about t* ............................. 183 4.2. Final remarks ............................... -
Let's Get Functional
5 LET’S GET FUNCTIONAL I’ve mentioned several times that F# is a functional language, but as you’ve learned from previous chapters you can build rich applications in F# without using any functional techniques. Does that mean that F# isn’t really a functional language? No. F# is a general-purpose, multi paradigm language that allows you to program in the style most suited to your task. It is considered a functional-first lan- guage, meaning that its constructs encourage a functional style. In other words, when developing in F# you should favor functional approaches whenever possible and switch to other styles as appropriate. In this chapter, we’ll see what functional programming really is and how functions in F# differ from those in other languages. Once we’ve estab- lished that foundation, we’ll explore several data types commonly used with functional programming and take a brief side trip into lazy evaluation. The Book of F# © 2014 by Dave Fancher What Is Functional Programming? Functional programming takes a fundamentally different approach toward developing software than object-oriented programming. While object-oriented programming is primarily concerned with managing an ever-changing system state, functional programming emphasizes immutability and the application of deterministic functions. This difference drastically changes the way you build software, because in object-oriented programming you’re mostly concerned with defining classes (or structs), whereas in functional programming your focus is on defining functions with particular emphasis on their input and output. F# is an impure functional language where data is immutable by default, though you can still define mutable data or cause other side effects in your functions. -
An Introduction to Logical Relations Proving Program Properties Using Logical Relations
An Introduction to Logical Relations Proving Program Properties Using Logical Relations Lau Skorstengaard [email protected] Contents 1 Introduction 2 1.1 Simply Typed Lambda Calculus (STLC) . .2 1.2 Logical Relations . .3 1.3 Categories of Logical Relations . .5 2 Normalization of the Simply Typed Lambda Calculus 5 2.1 Strong Normalization of STLC . .5 2.2 Exercises . 10 3 Type Safety for STLC 11 3.1 Type safety - the classical treatment . 11 3.2 Type safety - using logical predicate . 12 3.3 Exercises . 15 4 Universal Types and Relational Substitutions 15 4.1 System F (STLC with universal types) . 16 4.2 Contextual Equivalence . 19 4.3 A Logical Relation for System F . 20 4.4 Exercises . 28 5 Existential types 29 6 Recursive Types and Step Indexing 34 6.1 A motivating introduction to recursive types . 34 6.2 Simply typed lambda calculus extended with µ ............ 36 6.3 Step-indexing, logical relations for recursive types . 37 6.4 Exercises . 41 1 1 Introduction The term logical relations stems from Gordon Plotkin’s memorandum Lambda- definability and logical relations written in 1973. However, the spirit of the proof method can be traced back to Wiliam W. Tait who used it to show strong nor- malization of System T in 1967. Names are a curious thing. When I say “chair”, you immediately get a picture of a chair in your head. If I say “table”, then you picture a table. The reason you do this is because we denote a chair by “chair” and a table by “table”, but we might as well have said “giraffe” for chair and “Buddha” for table. -
Category Theory and Lambda Calculus
Category Theory and Lambda Calculus Mario Román García Trabajo Fin de Grado Doble Grado en Ingeniería Informática y Matemáticas Tutores Pedro A. García-Sánchez Manuel Bullejos Lorenzo Facultad de Ciencias E.T.S. Ingenierías Informática y de Telecomunicación Granada, a 18 de junio de 2018 Contents 1 Lambda calculus 13 1.1 Untyped λ-calculus . 13 1.1.1 Untyped λ-calculus . 14 1.1.2 Free and bound variables, substitution . 14 1.1.3 Alpha equivalence . 16 1.1.4 Beta reduction . 16 1.1.5 Eta reduction . 17 1.1.6 Confluence . 17 1.1.7 The Church-Rosser theorem . 18 1.1.8 Normalization . 20 1.1.9 Standardization and evaluation strategies . 21 1.1.10 SKI combinators . 22 1.1.11 Turing completeness . 24 1.2 Simply typed λ-calculus . 24 1.2.1 Simple types . 25 1.2.2 Typing rules for simply typed λ-calculus . 25 1.2.3 Curry-style types . 26 1.2.4 Unification and type inference . 27 1.2.5 Subject reduction and normalization . 29 1.3 The Curry-Howard correspondence . 31 1.3.1 Extending the simply typed λ-calculus . 31 1.3.2 Natural deduction . 32 1.3.3 Propositions as types . 34 1.4 Other type systems . 35 1.4.1 λ-cube..................................... 35 2 Mikrokosmos 38 2.1 Implementation of λ-expressions . 38 2.1.1 The Haskell programming language . 38 2.1.2 De Bruijn indexes . 40 2.1.3 Substitution . 41 2.1.4 De Bruijn-terms and λ-terms . 42 2.1.5 Evaluation . -
Lightweight Linear Types in System F°
University of Pennsylvania ScholarlyCommons Departmental Papers (CIS) Department of Computer & Information Science 1-23-2010 Lightweight linear types in System F° Karl Mazurak University of Pennsylvania Jianzhou Zhao University of Pennsylvania Stephan A. Zdancewic University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/cis_papers Part of the Computer Sciences Commons Recommended Citation Karl Mazurak, Jianzhou Zhao, and Stephan A. Zdancewic, "Lightweight linear types in System F°", . January 2010. Karl Mazurak, Jianzhou Zhao, and Steve Zdancewic. Lightweight linear types in System Fo. In ACM SIGPLAN International Workshop on Types in Languages Design and Implementation (TLDI), pages 77-88, 2010. doi>10.1145/1708016.1708027 © ACM, 2010. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM SIGPLAN International Workshop on Types in Languages Design and Implementation, {(2010)} http://doi.acm.org/10.1145/1708016.1708027" Email [email protected] This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_papers/591 For more information, please contact [email protected]. Lightweight linear types in System F° Abstract We present Fo, an extension of System F that uses kinds to distinguish between linear and unrestricted types, simplifying the use of linearity for general-purpose programming. We demonstrate through examples how Fo can elegantly express many useful protocols, and we prove that any protocol representable as a DFA can be encoded as an Fo type. We supply mechanized proofs of Fo's soundness and parametricity properties, along with a nonstandard operational semantics that formalizes common intuitions about linearity and aids in reasoning about protocols. -
Run-Time Type Information and Incremental Loading in C++
Run-time Type Information and Incremental Loading in C++ Murali Vemulapati Sriram Duvvuru Amar Gupta WP #3772 September 1993 PROFIT #93-11 Productivity From Information Technology "PROFIT" Research Initiative Sloan School of Management Massachusetts Institute of Technology Cambridge, MA 02139 USA (617)253-8584 Fax: (617)258-7579 Copyright Massachusetts Institute of Technology 1993. The research described herein has been supported (in whole or in part) by the Productivity From Information Technology (PROFIT) Research Initiative at MIT. This copy is for the exclusive use of PROFIT sponsor firms. Productivity From Information Technology (PROFIT) The Productivity From Information Technology (PROFIT) Initiative was established on October 23, 1992 by MIT President Charles Vest and Provost Mark Wrighton "to study the use of information technology in both the private and public sectors and to enhance productivity in areas ranging from finance to transportation, and from manufacturing to telecommunications." At the time of its inception, PROFIT took over the Composite Information Systems Laboratory and Handwritten Character Recognition Laboratory. These two laboratories are now involved in research re- lated to context mediation and imaging respectively. LABORATORY FOR ELEC RONICS A CENTERCFORENTER COORDINATION LABORATORY ) INFORMATION S RESEARCH RESE CH E53-310, MITAL"Room Cambridge, MA 02M142-1247 Tel: (617) 253-8584 E-Mail: [email protected] Run-time Type Information and Incremental Loading in C++ September 13, 1993 Abstract We present the design and implementation strategy for an integrated programming environment which facilitates specification, implementation and execution of persistent C++ programs. Our system is implemented in E, a persistent programming language based on C++. The environment provides type identity and type persistence, i.e., each user-defined class has a unique identity and persistence across compilations. -
CS 6110 S18 Lecture 32 Dependent Types 1 Typing Lists with Lengths
CS 6110 S18 Lecture 32 Dependent Types When we added kinding last time, part of the motivation was to complement the functions from types to terms that we had in System F with functions from types to types. Of course, all of these languages have plain functions from terms to terms. So it's natural to wonder what would happen if we added functions from values to types. The result is a language with \compile-time" types that can depend on \run-time" values. While this arrangement might seem paradoxical, it is a very powerful way to express the correctness of programs; and via the propositions-as-types principle, it also serves as the foundation for a modern crop of interactive theorem provers. The language feature is called dependent types (i.e., types depend on terms). Prominent dependently-typed languages include Coq, Nuprl, Agda, Lean, F*, and Idris. Some of these languages, like Coq and Nuprl, are more oriented toward proving things using propositions as types, and others, like F* and Idris, are more oriented toward writing \normal programs" with strong correctness guarantees. 1 Typing Lists with Lengths Dependent types can help avoid out-of-bounds errors by encoding the lengths of arrays as part of their type. Consider a plain recursive IList type that represents a list of any length. Using type operators, we might use a general type List that can be instantiated as List int, so its kind would be type ) type. But let's use a fixed element type for now. With dependent types, however, we can make IList a type constructor that takes a natural number as an argument, so IList n is a list of length n. -
Kodak Color Control Patches Kodak Gray
Kodak Color Control Patches 0 - - _,...,...,. Blue Green Yellow Red White 3/Color . Black Kodak Gray Scale A 1 2 a 4 s -- - --- - -- - -- - -- -- - A Study of Compile-time Metaobject Protocol :J/1\'-(Jt.-S~..)(:$/;:t::;i':_;·I'i f-.· 7 •[:]f-. :::JJl. (:~9 ~~Jf'?E By Shigeru Chiba chiba~is.s.u-tokyo.ac.jp November 1996 A Dissertation Submitted to Department of Information Science Graduate School of Science The Un iversity of Tokyo In Partial Fulfillment of the Requirements For the Degree of Doctor of Science. Copyright @1996 by Shigeru Chiba. All Rights Reserved. - - - ---- - ~-- -- -~ - - - ii A Metaobject Protocol for Enabling Better C++ Libraries Shigeru Chiba Department of Information Science The University of Tokyo chiba~is.s.u-tokyo.ac.jp Copyrighl @ 1996 by Shi geru Chiba. All Ri ghts Reserved. iii iv Abstract C++ cannot be used to implement control/data abstractions as a library if their implementations require specialized code for each user code. This problem limits programmers to write libraries in two ways: it makes some kinds of useful abstractions that inherently require such facilities impossi ble to implement, and it makes other abstractions difficult to implement efficiently. T he OpenC++ MOP addresses this problem by providing li braries the ability to pre-process a program in a context-sensitive and non-local way. That is, libraries can instantiate specialized code depending on how the library is used and, if needed, substitute it for the original user code. The basic protocol structure of the Open C++ MOP is based on that of the CLOS MOP, but the Open C++ MOP runs metaobjects only at compile time. -
1 Simply-Typed Lambda Calculus
CS 4110 – Programming Languages and Logics Lecture #18: Simply Typed λ-calculus A type is a collection of computational entities that share some common property. For example, the type int represents all expressions that evaluate to an integer, and the type int ! int represents all functions from integers to integers. The Pascal subrange type [1..100] represents all integers between 1 and 100. You can see types as a static approximation of the dynamic behaviors of terms and programs. Type systems are a lightweight formal method for reasoning about behavior of a program. Uses of type systems include: naming and organizing useful concepts; providing information (to the compiler or programmer) about data manipulated by a program; and ensuring that the run-time behavior of programs meet certain criteria. In this lecture, we’ll consider a type system for the lambda calculus that ensures that values are used correctly; for example, that a program never tries to add an integer to a function. The resulting language (lambda calculus plus the type system) is called the simply-typed lambda calculus (STLC). 1 Simply-typed lambda calculus The syntax of the simply-typed lambda calculus is similar to that of untyped lambda calculus, with the exception of abstractions. Since abstractions define functions tht take an argument, in the simply-typed lambda calculus, we explicitly state what the type of the argument is. That is, in an abstraction λx : τ: e, the τ is the expected type of the argument. The syntax of the simply-typed lambda calculus is as follows. It includes integer literals n, addition e1 + e2, and the unit value ¹º.