The idea is due to sch on nkel 1924 but is often called currying, after h. An introduction to functional programming through lambda. An introduction to functional programming through lambda calculus greg michaelson department of computing and electrical engineering heriotwatt university. The lambda calculus is a good medium to represent mathematics on a computer with the aim to exchange and store reliable mathematical knowledge. A model of the untyped lambda calculus univocally induces a lambda theory i. Lambda calculus lambda calculus stanford university. A semantics of lambda calculus is equationally incomplete if there exists a lambda theory that is not induced by any model in the semantics. I think the reference on the subject of lambda calculus itself still is barendregts book.
These diagrammatics create a twoway street between the topology of knots and links in three dimensional space and key. This wellrespected text offers an accessible introduction to functional programming concepts and techniques for students of mathematics and computer science. Beyond that it pretty much depends on what part of lambda calculus you are interested in. Every partial order is the specialization order of a space. The typed lambda calculus also has a special type called the unit type, which we denote as 1.
Lecture notes on the lambda calculus by peter selinger. Lambdacalculus is thus effectively a programming language, from which other languages can be built. The purpose of the talk is to show how this result generalizes to the partial case. The scott topology was first defined by dana scott for complete lattices and later defined for arbitrary partially ordered sets. The tree topology on the set of lambda terms is the smallest topology that makes the mapping from lambda terms to bo.
Topological incompleteness and order incompleteness of the. There is a class of problems of elementary number theory which can be stated in the form that it is required to find an effectively calculable function f of n positive integers, such that f x, x. In this chapter we indicate some of the connections between topology and calculus, as examples of sort of unexpected relationships and applications between seemingly unrelated fields of mathematics. The scott topology is the most frequently used in this. Algebra and topology in lambda calculus request pdf.
In this paper we will look at another property of these calculi, namely continuity. Its unclear how we can similarly tame turing machines. A longstanding open problem is whether there exists a nonsyntactical model of untyped lambdacalculus whose theory is exactly the least equational lambdatheory lb. Algebra and topology in lambda calculus antonino salibra. However, the graphic beta move can be applied outside the sector of untyped lambda calculus, and the graphic lambda calculus can be used for other purposes than the one of visual representing lambda calculus. Occasionally, some elementary notions of topology 64 are needed. The lambda calculus is a language of pure functions. Lambda calculus lambda calculus was introduced around 1930 by a. Lecture notes on the lambda calculus pdf 106p download book. We present an extension of the lambda calculus with differential constructions. Jean galliers home page university of pennsylvania.
Functional programming is rooted in lambda calculus, which constitutes the worlds smallest programming language. However, the graphic beta move can be applied outside the sector of untyped lambda calculus, and the graphic lambda calculus can be used for other purposes than that. Stochastic lambdacalculus university of california, berkeley. Although the lambda calculus has the power to represent all computable functions, its uncomplicated syntax and semantics provide an excellent vehicle for studying the meaning of programming language. This formalism was developed by alonzo church as a tool for studying the mathematical properties of e. Connectingtopologyandlogicwithcategorytheory nicolaikraus.
From the viewpoint of category theory, the need for this. I think the reference on the subject of lambdacalculus itself still is barendregts book. The partial lambda calculus joint work with pieter hofstra it is a classic result for the lambda calculus that a lambda algebra gives rise to a cmonoid and whence a cartesian closed category. The process of application and evaluation reflects the computational behavior of many modern functional programming languages, which explains the interest in the lambda calculus among computer scientists. The mission of the center of excellence in computation, algebra and topology cecat is to provide a forum for world class interdisciplinary research where computation meets algebra, topology and related mathematical fields. A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus. More generally, typed lambda calculus turns out to be deeply connected to the foundations of mathematics. A longstanding open problem is whether there exists a nonsyntactical model of untyped lambda calculus whose theory is exactly the least equational lambda theory lb. The visser topology of lambda calculus chantal berline giulio manzonetto antonino salibra cnrs, laboratoire pps dipartimento di informatica universite paris 7 universit. The lambda calculus, its syntax and semantics studies in logic and the foundations of mathematics, volume 103. There may, indeed, be other applications of the system than its use as a logic. Buy the lambda calculus, its syntax and semantics studies in logic and the foundations of mathematics, volume 103. An unsolvable problem of elementary number theory alonzo. In this paper we make use of the visser topology for investigating the more general question of whether the equational resp.
This formalism was developed by alonzo church as a tool for studying the mathematical properties of e ectively computable functions. 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. Lambda calculus between algebra and topology topology re nes partial orderings through the separation axioms. A tutorial introduction to the lambda calculus arxiv. Although lambda calculus is a very basic language, it is sufficient to express all the computable functions. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. The process of application and evaluation reflects the computational behavior of many modern functional programming languages, which explains the interest in. Lambda calculus between algebra and topology antonino. A tutorial introduction to the lambda calculus raul rojas freie universit at berlin version 2.
Full intersection types and topologies in lambda calculus. The lambda calculus, its syntax and semantics studies in. Algebra and topology in lambda calculus antonino salibra department of computer science university of venice, italy. 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. Lambda calculus and types oxford department of computer. A short introduction to the lambda calculus achim jung. What are some resources for learning lambda calculus. Since topology refines partial orderings through separation axioms, salibra 35 has introduced other topological semantics of lambda calculus. The final section introduces a topology on the set of lambda terms which is meant to explain an illusory contradiction. 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. An introduction pdf aspects of harmonic analysis and representation theory html a gentle introduction to homology, cohomology, and sheaf cohomology html algebra, topology, differential calculus, and optimization theory for computer science and machine learning html aspects of convex geometry.
Sufficiently advanced types make bugs impossible to express, that. Department of philosophy, carnegie mellon university. The existence of two join gates, with the same topology, is justified by the fact that they appear in different moves. Students too often see mathematics as compartmentalized. 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. Each of these is a field in itself, and i dont know of any book that covers it all. The course is entitled lambda calculus and types and it appears in various forms. Its main move is called the graphic beta move for its relation to the beta reduction in lambda calculus. Center of excellence in computation, algebra, and topology. Lambda calculus princeton university computer science. In simply typed lambda calculus, each variable and term has a type, and we can only form the application f t f t if t t is of some type a a while f f is of a function type a b b a a \to b ba whose domain is a a. An introduction to functional programming through lambda calculus greg michaelson department of computing and electrical engineering heriotwatt university riccarton campus edinburgh eh14 4as 2 preface overview this book aims to provide a gentle introduction to functional programming. Lambda calculus between algebra and topology antonino salibra.
The lambda calculus is algebraic dalhousie university. Topology is an important instrument not only to obtain models of lambda calculus, but also to investigate syntactical properties of. Revised edition on free shipping on qualified orders. Given a lambda term, is it possible to create an automatic analyzer that decides, yes or no, whether or not a lambda term will ever get stuck. Nh topology and its applications elsevier topology and its applications 89 1998 359 topology, domain theory and theoretical computer science michael w. Lecture notes on the lambda calculus department of mathematics. In the 1930s, while turing was developing what are now called turing machines as a model for computation, church and his student kleene were developing a different model, called the lambda calculus 29, 63. Scottcontinuous functions show up in the study of models for lambda calculi and the denotational semantics of computer programs. Baez department of mathematics, university of california. Beyond that it pretty much depends on what part of lambdacalculus you are interested in. Suppose tm is a lambda term that simulates a turing machine consider. The cecat has a track record of supporting its mission by hosting workshops that attract national and international. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a. There is a single term of this type, which we denote as.
A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the or the the least sensible theory h generated by. For this reason when considering the topology of computation it is common to focus on the topology of. Lecture notes on the lambda calculus pdf 106p download. Topological representation of the lambdacalculus carnegie. The partial lambda calculus joint work with pieter hofstra it is a classic result for the lambdacalculus that a lambdaalgebra gives rise to a cmonoid and whence a cartesian closed category. First, we define a theory of untyped extensional lambda calculus. Graphic lambda calculus, a visual language that can be used for representing untyped lambda calculus, is introduced and studied. Its main move is called graphic beta move for its relation to the beta reduction in lambda calculus.
Topology, domain theory and theoretical computer science. The known topologies in the lambda calculus are the tree topology and the filter topology. March 18, 2004 abstract the lambda calculus can appear arcane on. Mislove 1 department of mathematics, tulane university, new orleans, ga 70118, usa received 30 august 1996. Church as part of an investigation in the formal foundations of math. Note that this is not necessarily a complete description of the topology of computation, since functions which are equivalent in. By adding types, we can ensure lambda calculus programs always halt.
In this article, we introduce a new technique to prove in a uniform way the incompleteness of all denotational semantics of lambda calculus that have been proposed so far, including the strongly stable. We conclude that the lambda calculus is algebraic, in the sense that its canonical class of models is the class of lambda algebras. The lambda calculus, its syntax and semantics this is a must. The visser topology of lambda calculus request pdf. We present an extension of the lambdacalculus with differential constructions. 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 lambdacalculus.
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