We use a specific equality in order to translate the extensional conversion relation into an intensional system. Some results on extensionality in lambda calculus sciencedirect. It converts any table of derivatives into a table of integrals and vice versa. Lambda calculi with types, handbook of logic in comp.
Heres an example of a simple lambda expression that defines the plus one function. When we write an equation e e0, we shall implicitly refer to the principal typing giving eand e0the same type a. The lambda calculus is algebraic dalhousie university. The lambdacalculus was invented in the early 1930s, by a. The chain rule and the second fundamental theorem of calculus1 problem 1. Find the derivative of the function gx z v x 0 sin t2 dt, x 0. The most interesting aspect is a system of substitution primitives and an accompanying. This is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of ottawa in 2001 and at dalhousie university in 2007 and 20. Request pdf some results on extensionality in lambda calculus in this paper we consider the problem due to a. Also it forms a compact language to denote mathematical proofs logic provides a formal language in which mathematical statements can be formulated and provides deductive power to derive these.
The chain rule and the second fundamental theorem of calculus. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics university of ottawa abstract this is a set of lecture notes for the course mathematical foundations of computation, which i taught at the university of ottawa in fall 2001. Introduction to the lambda calculus iowa state university. In haskell inf inf definition of the value inf alwayseven x 7 definition of the function alwayseven alwayseven inf expression to be evaluated two possible choices in the evaluation of alwayseven inf. We conclude that the lambda calculus is algebraic, in the sense that its canonical class of models is the class of lambda algebras. The chain rule and the second fundamental theorem of. Abstraction algorithm, combinatory completeness, translations to and from untyped lambda calculus, mismatches between combinary logic and lambda calculus, basis. The lambda calculus is a simple programming language a model of computation akin to turing machines and recursive functions, through which we can study the computability and complexity of functions and predicates. Topics covered in these notes include the untyped lambda calculus, the. While we have concentrated on models of the untyped lambda calculus, similar considerations apply to the algebraic modeling of any language with variables and binders.
Lambdacalculus and combinators, an introduction combinatory logic and. One problem we found was that, in contrast to combinators, for hequality a n explicit definition of model did not exist. Some results on extensionality in lambda calculus request pdf. Abstraction algorithm, combinatory completeness, translations to and from untyped lambdacalculus, mismatches between combinary logic and lambdacalculus, basis. Church 1936 invented the lambda calculus, written calculus. This formalism was developed by alonzo church as a tool for studying the mathematical properties of e ectively computable functions. Lambda calculus introduces a direct notation for functions. Lambda calculus as a formal system free and bound variables. The syntax of basic \\ lambda\ calculus is quite sparse, making it an elegant, focused notation for representing functions. Calculus models and extensionality hindley, r longo, g.
Lambda calculus and recursion theory preliminary version dana scott oxford university, oxford, england 0. Lambda calculus as a programming language computability background. The \\ lambda\ calculus is, at heart, a simple notation for functions and application. Extensionality in acalculus 7a extensional equality 7b 377reduction in acalculus. Lambdacalculus and combinators, an introduction book. The most interesting aspect is a system of substitution primitives and an accompanying equational theory providing for algebraic proofs. This paper presents a method to translate a proof in an extensional version of the calculus of constructions into a proof in the calculus of inductive constructions extended with a few axioms. Some results on extensionality in lambda calculus core. In the pure lambda calculus, the only values are functions \x. What are the axioms, inference rules, and formal semantics. Extensionality in the calculus of constructions springerlink. Church, and has been considerably developed since then. Lambda calculus and types oxford department of computer.
Lecture notes on the lambda calculus nanjing university. The authors previous book served as the main reference for introductory courses on lambda calculus for over 20 years. Typed lambda calculi are weaker than the untyped lambda calculus, which is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can, but on the other hand typed lambda calculi allow more things to be proved. Second recursion theorem, scottcurry theorem, undecideability of equality in lambda beta. A lambda expression that cannot be reduced further, by either. Lambda calculus is a language with clear operational and denotational semantics capable of expressing algorithms. Combinatory logic and lambdacalculus, originally devised in the 1920s, have since developed into linguistic tools, especially useful in programming languages. Course notes in typed lambda calculus page has been moved. They have since evolved into important tools for the development and study of programming languages. A calculus is a method of moving symbols, not the undergrad course. The authors previous book served as the main reference for introductory courses on lambdacalculus for over 20 years. Combinatory logic and lambda calculus, originally devised in the 1920s, have since developed into linguistic tools, especially useful in programming languages.
In the untyped lambda calculus, any term can be applied to another term. Terms which can be connected by a zigzag of beta reductions in either direction are said to be betaequivalent another basic operation often assumed in the lambda calculus is eta reductionexpansion, which consists of identifying a function, f f with the lambda abstraction. Equality in the untyped lambda calculus without extensionality. Appendix a studies the intermediatecoercion calculus used in section 4. These two settheoretical postulates are analogous to the postulates of beta and eta conversion in the lambda calculus, where the role of set abstraction is played by functional abstraction, and the role of having a member. Topics covered in these notes include the untyped lambda calculus, the churchrosser theorem, combinatory algebras. Also, maybe the discussion is lambda calculus a logic. If we wish to take an extensional viewpoint then the. J p seldin in this uptodate account of the subject, the grammar and basic properties of both combinatory logic and lambdacalculus are discussed, followed by an introduction to typetheory. A short introduction to the lambda calculus achim jung. This book is an introduction to some aspects of the theory today. The lambda calculus stanford encyclopedia of philosophy. The course is entitled lambda calculus and types and it appears in various forms. Lambdacalculus and combinators, an introduction book, 2008.
Lambda calculus and recursion theory preliminary version. Calculus models and extensionality, mathematical logic. He is i think talking about models of the untyped lambda calculus without the following axiom of extensionality. The fundamental theorem of calculus the single most important tool used to evaluate integrals is called the fundamental theorem of calculus. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. The lambda calculus can be thought of as the theoretical foundation of functional programming. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics dalhousie university, halifax, canada abstract this is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of ottawa in 2001 and at dalhousie university in 2007. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics dalhousie university,halifax, canada abstract this is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of. The main ideas are applying a function to an argument and forming functions by abstraction. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics dalhousie university, halifax, canada abstract this is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of. So from now on, all equations in rare written without type, the types. Lambda calculus and turing machines are equivalent, in the sense that any function that can be defined using one can be defined using the other. Selfreferential sentences and the recursion theorem 146 6.
Modern processors are just overblown turing machines. Lecture notes on the lambda calculus department of mathematics. A tutorial introduction to the lambda calculus raul rojas freie universit at berlin version 2. It was introduced by the mathematician alonzo church in the 1930s as part of his research into the. Introduction a latticetheoretic and topological method for defining a class of models of the churchcurry acalculus was discovered by the author in 1969 cf. A tutorial introduction to the lambda calculus arxiv. Curry models the lambda calculus using a basic set of combinators, namely sand k, together with a collectionof equations to explaintheirbehavior. This formalism was developed by alonzo church as a tool for studying the mathematical properties of e. Functional languages are just the lambda calculus with a more palatable syntax. Pdf in this work we provide alternative formulations of the concepts of lambda theory and extensional theory without introducing the notion of.
It is a universal model of computation that can be used to simulate any turing machine. However, sand kare so basic that complex equations are needed to capture the rules expressing a weak extensionality of functions in the lambda calculus and especially. March 18, 2004 abstract the lambda calculus can appear arcane on. Pages in category lambda calculus the following 52 pages are in this category, out of 52 total. Lambda calculus is a model of computation, invented by church in the early 1930s.
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