Differential calculus is extensively applied in many fields of mathematics, in particular in geometry. We present an extension of the lambda calculus with differential constructions. What are some resources for learning lambda calculus. E to denote a function in which x is a formal argument and e is the functional body. A function is a mapping from the elements of a domain set to the elements of a codomain set given by a rulefor example, cube.
An introduction to functional programming through lambda. Lambda calculus is a language with clear operational and denotational semantics capable of expressing algorithms. The boolean differential calculus introduction and examples. We present an extension of the lambdacalculus with differential constructions. Intuitionistic differential nets and lambdacalculus. Math 221 1st semester calculus lecture notes version 2. Try writing a lambda calculus interpetter, ideally in a functional language using the build in syntax of the language rather than via a parser.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. The quote business is a special feature that will be explained later. The differential lambdacalculus introduction citeseerx. Differential calculus makes it possible to compute the limits of a function in many cases when this is not feasible by the simplest limit theorems cf. Moggi, kripkestyle models for typed lambda calculus, annals of pure and applied logic 51 1991 99124. Lambda calculus is a framework developed by alonzo church in 1930s to study computations with functions. Lambda calculus 5 scope, free and bound occurences lx.
Christian posthoff the university of the west indies st. Lambda calculus is a notation for describing mathematical functions and programs. This wellrespected text offers an accessible introduction to functional programming concepts and techniques for students of mathematics and computer science. A basic understanding of calculus is required to undertake a study of differential equations. Modern processors are just overblown turing machines. It is a universal model of computation that can be used to simulate any turing machine. The main result is that every model of the untyped differential.
Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a strong theoretical foundation. What is a categorical model of the differential and the resource. March 18, 2004 abstract the lambda calculus can appear arcane on. Abstractwe present an extension of the lambda calculus with differential constructions. Publication date 1962 topics natural sciences, mathematics, analysis publisher. Tranquillitheoreticalcomputerscience412201119791997 1983 fig. Free differential calculus books download ebooks online.
Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. Differential and integral calculus lecture notes pdf 143p. The present volume is essentially a supplement to book 3, placing more emphasis on mathematics as a human activity and on the people who made it in the course of many centuries and in many parts of the world. The lambda calculus, its syntax and semantics studies in logic and the foundations of mathematics, volume 103. These functions can be of without names and single arguments. The majority of functional programming languages at all do not require you to learn lambda calculus, whatever that would mean, lambda calculus is insanely minimal, you can learn its axioms in an under an hour. The common root of the two calculi is a further calculus based on linear logic. Limits and continuity, differentiation rules, applications of differentiation, curve sketching, mean value theorem, antiderivatives and differential equations, parametric equations and polar coordinates, true or false and multiple choice problems. A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus.
Functional programming lambda calculus tutorialspoint. Differential equations department of mathematics, hkust. Thus our interpreter actually runs more than plain lambda calculus. Functional programming is rooted in lambda calculus, which constitutes the worlds smallest programming language. Backpropagation in the simply typed lambdacalculus with. The lambda calculus, its syntax and semantics studies in. The notes were written by sigurd angenent, starting from an extensive collection of notes and problems compiled by joel robbin.
The differential lambdacalculus is a syntactical out come of the first authors work on the k. Introduction to the lambda calculus iowa state university. 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. We state and prove some basic results confluence, strong normalization in the typed case, and also a theorem relating the usual taylor series of analysis to the linear head reduction of lambda calculus. Resource lambda calculus, relational semantics, full abstraction, differential linear logic. Functional languages are just the lambda calculus with a more palatable syntax.
The theorems for differential operators can be specialized for vectorial derivatives, too. As it turns out, it is possible to define models of the typed lambdacalculus or equivalently of full linear logic where types are interpreted as vector spaces and. Haskell similarly permits unrestricted recursion via let expressions. Lambda calculus and probabilistic computation extended. Lambda calculus 6 beta reduction computation in the lambda calculus takes the form of betareduction. A short introduction to the lambda calculus achim jung. Learn differential calculus for freelimits, continuity, derivatives, and derivative applications. It is a mathematical system for studying the interaction of functional abstraction and functional application. Tranquillitheoreticalcomputerscience412201119791997 1981 fig. This book itself is purely theoretical and principally aimed for researchersstudents of its field. Differentiable programming, lambdacalculus, linear logic. We define pure intuitionistic differential proof nets, extending ehrhard and regniers differential interaction nets with the exponential box of linea. We state and prove some basic results confluence, strong.
It captures some of the essential, common features of a wide variety of programming languages. For other visual, diagrammatic representations of lambda calculus see the vex language 2, or keenans website 3. 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 and 20. Differential interaction nets thomas ehrhard and laurent.
The motivation for introducing graphic lambda calculus comes from the study of emergent algebras. It is the basis for functional programming like turing machines are the foundation of imperative. The paper shows how the scottkoymans theorem for the untyped. This is surprisingly easy and a good way to improve your feel for it. We introduce interaction nets for the differential lambdacalculus and exhibit in this framework a new symmetry between the of course and the why not modalities. Lambda calculus lambda calculus stanford university.
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