Lambda calculus between algebra and topology antonino salibra. Algebra and topology in lambda calculus antonino salibra. The idea is due to sch on nkel 1924 but is often called currying, after h. 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. Lambda calculus between algebra and topology antonino. The process of application and evaluation reflects the computational behavior of many modern functional programming languages, which explains the interest in. The visser topology of lambda calculus chantal berline giulio manzonetto antonino salibra cnrs, laboratoire pps dipartimento di informatica universite paris 7 universit. Scottcontinuous functions show up in the study of models for lambda calculi and the denotational semantics of computer programs. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. We conclude that the lambda calculus is algebraic, in the sense that its canonical class of models is the class of lambda algebras. Since topology refines partial orderings through separation axioms, salibra 35 has introduced other topological semantics of lambda calculus.
By adding types, we can ensure lambda calculus programs always halt. The lambda calculus is a good medium to represent mathematics on a computer with the aim to exchange and store reliable mathematical knowledge. These diagrammatics create a twoway street between the topology of knots and links in three dimensional space and key. Students too often see mathematics as compartmentalized. Beyond that it pretty much depends on what part of lambdacalculus you are interested in. More generally, typed lambda calculus turns out to be deeply connected to the foundations of mathematics. A tutorial introduction to the lambda calculus raul rojas freie universit at berlin version 2.
Graphic lambda calculus, a visual language that can be used for representing untyped lambda calculus, is introduced and studied. Every partial order is the specialization order of a space. Lecture notes on the lambda calculus by peter selinger. Church as part of an investigation in the formal foundations of math. 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.
We present an extension of the lambdacalculus with differential constructions. 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. In this paper we will look at another property of these calculi, namely continuity. The cecat has a track record of supporting its mission by hosting workshops that attract national and international. Lambda calculus lambda calculus was introduced around 1930 by a. Lecture notes on the lambda calculus department of mathematics. Baez department of mathematics, university of california. In this paper we make use of the visser topology for investigating the more general question of whether the equational resp. Department of philosophy, carnegie mellon university. Its main move is called graphic beta move for its relation to the beta reduction in lambda calculus. 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 scott topology was first defined by dana scott for complete lattices and later defined for arbitrary partially ordered sets. The lambda calculus is algebraic dalhousie university.
Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a. 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. Topological incompleteness and order incompleteness of the. 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 lambda calculus, its syntax and semantics studies in. There is a single term of this type, which we denote as. While a turing machine can be seen as an idealized, simplified model of computer hardware, the lambda calculus is more like a simple model of software. A short introduction to the lambda calculus achim jung.
An unsolvable problem of elementary number theory alonzo. Impact of lambda calculus on logic and computer science. From the viewpoint of category theory, the need for this. The scott topology is the most frequently used in this.
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. Topological representation of the lambdacalculus carnegie. This formalism was developed by alonzo church as a tool for studying the mathematical properties of e ectively computable functions. 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. The lambda calculus, its syntax and semantics studies in logic and the foundations of mathematics, volume 103. Each of these is a field in itself, and i dont know of any book that covers it all. Lecture notes on the lambda calculus pdf 106p download. Topology is an important instrument not only to obtain models of lambda calculus, but also to investigate syntactical properties of. 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. For this reason when considering the topology of computation it is common to focus on the topology of. Its unclear how we can similarly tame turing machines. 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. 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. 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.
Lambda calculus between algebra and topology topology re nes partial orderings through the separation axioms. Full intersection types and topologies in lambda calculus. The existence of two join gates, with the same topology, is justified by the fact that they appear in different moves. A semantics of lambda calculus is equationally incomplete if there exists a lambda theory that is not induced by any model in the semantics. The course is entitled lambda calculus and types and it appears in various forms. A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus. Lambda calculus and types oxford department of computer. Connectingtopologyandlogicwithcategorytheory nicolaikraus. Lambdacalculus is thus effectively a programming language, from which other languages can be built. Lambda calculus princeton university computer science.
Stochastic lambdacalculus university of california, berkeley. Topology, domain theory and theoretical computer science. 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. Its main move is called the graphic beta move for its relation to the beta reduction in lambda calculus. 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. Occasionally, some elementary notions of topology 64 are needed. Nh topology and its applications elsevier topology and its applications 89 1998 359 topology, domain theory and theoretical computer science michael w.
A longstanding open problem is whether there exists a nonsyntactical model of untyped lambdacalculus whose theory is exactly the least equational lambdatheory lb. Lecture notes on the lambda calculus pdf 106p this notes contains the details about the untyped lambda calculus, the churchrosser theorem, combinatory algebras, the curryhoward isomorphism, polymorphism, weak and strong normalization, denotational semantics of pcf. 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. I think the reference on the subject of lambdacalculus itself still is barendregts book. Note that this is not necessarily a complete description of the topology of computation, since functions which are equivalent in. I think the reference on the subject of lambda calculus itself still is barendregts book. A tutorial introduction to the lambda calculus arxiv. 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. The known topologies in the lambda calculus are the tree topology and the filter topology.
The purpose of the talk is to show how this result generalizes to the partial case. Suppose tm is a lambda term that simulates a turing machine consider. Revised edition on free shipping on qualified orders. Lambda calculus lambda calculus stanford university. What are some resources for learning lambda calculus. An introduction to functional programming through lambda. Mislove 1 department of mathematics, tulane university, new orleans, ga 70118, usa received 30 august 1996. We present an extension of the lambda calculus with differential constructions. 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.
An introduction to functional programming through lambda calculus greg michaelson department of computing and electrical engineering heriotwatt university. 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. Beyond that it pretty much depends on what part of lambda calculus you are interested in. This wellrespected text offers an accessible introduction to functional programming concepts and techniques for students of mathematics and computer science. Di erential lambdacalculus a di erential simple term is a variable x or an abstraction x s where x is a variable and s is a simple. 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. Center of excellence in computation, algebra, and topology. Buy the lambda calculus, its syntax and semantics studies in logic and the foundations of mathematics, volume 103. This formalism was developed by alonzo church as a tool for studying the mathematical properties of e. The tree topology on the set of lambda terms is the smallest topology that makes the mapping from lambda terms to bo. The final section introduces a topology on the set of lambda terms which is meant to explain an illusory contradiction.
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. There may, indeed, be other applications of the system than its use as a logic. March 18, 2004 abstract the lambda calculus can appear arcane on. The lambda calculus is a language of pure functions. First, we define a theory of untyped extensional lambda calculus. Algebra and topology in lambda calculus request pdf. The visser topology of lambda calculus request pdf.
Sufficiently advanced types make bugs impossible to express, that. Jean galliers home page university of pennsylvania. Algebra and topology in lambda calculus antonino salibra department of computer science university of venice, italy. A model of the untyped lambda calculus univocally induces a lambda theory i. 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. The typed lambda calculus also has a special type called the unit type, which we denote as 1. The lambda calculus, its syntax and semantics this is a must. Although lambda calculus is a very basic language, it is sufficient to express all the computable functions. Lecture notes on the lambda calculus pdf 106p download book. Functional programming is rooted in lambda calculus, which constitutes the worlds smallest programming language. 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.