The lambda calculus appendix on recursive functions. Barendregt, the lambda calculus its syntax and semantics, northholland, 1984. Functions a function may be viewed as a set of ordered pairs where. Effectively computable functions of natural numbers can be identified with those definable by. To show that all recursive functions can be represented in the \\ lambda \ calculus, one reproduces the definition of recursive functions in the \\ lambda \ calculus. 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. The course is entitled lambda calculus and types and it appears in various forms. A tutorial introduction to the lambda calculus raul rojas freie universit at berlin version 2. These notes have nothing to say about scotts subsequent work on the typed lambda calculus, and the general notion of types.
Steven syrek lambda calculus for people who cant be bothered to learn it part 1 of 2. There are three kinds of expressions also called terms in the pure lambda calculus. A tutorial introduction to the lambda calculus arxiv. A lambda calculus function or term is an implementation of a mathematical function. Although all partial recursive functions can be represented in the untyped lambdacalculus, not all such functions.
The formalism became popular and has provided a strong theoretical foundation for the family of functional programming languages. Introduction to the lambda calculus iowa state university. Lambdacalculus, combinatory logic, mu recursive functions, turing machines, and others, represent the strongest class of models of computation. These constructs are constructorconstrained recursive types, and a lifting. Like loop, we can encode rec in lambda calculus too. To show that all recursive functions can be represented in the \\lambda\calculus, one reproduces the definition of recursive functions in the \\lambda\calculus.
The formalism has since been extended with types and used in functional programming haskell, clean and proof assistants coq, isabelle, hol, used in. The fixedpoint function is generally referred to as y, and must by definition satisfy yffyf. Still, there is a way to implement recursion, discovered by haskell curry, by using a particular lambda term, called fixedpoint combinator if you are interested in. The lambdacalculus schemata of this paper are natural extensions of recursive schemes, obtained through the addition of full lambda abstraction. Good surveys of the state of schematology in 1973 are provided by chandra 3 and manna 18. Lambda calculus examples in this handout, we look at several examples of lambda terms in order to provide a. Still, there is a way to implement recursion, discovered by haskell curry, by using a particular lambdaterm, called fixedpoint combinator if you are interested in. We would like to be able to define recursive functions like factorial in the lambdacalculus. The lambda calculus is a formalism of highorder functions and it is a logical basis of some important classical functional programming languages, such as lisp, scheme, ml and haskell. Churchturing thesis theyre powerful enough to feature undecidability, i.
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. Lambda calculus formulations of recursive function theory from the 1920s and before tended to be based on making explicit definitions like those in the note above. Church introduced his formalism to give a functional foundation for mathematics but in the end mathematicians preferred axiomatic set theory. Lambda calculus and recursion theory preliminary version dana scott oxford university, oxford, england 0. Pdf principles of alphainduction and recursion for the. Lecture notes on the lambda calculus dalhousie university. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a.
I cant figure out how to do it with a lambda function though for the simple reason that the lambda function has no. Undecidability results and the second recursion theorem. Intuitively, abstractions represent functions, and applications represent the application of a function to its argument. This formalism was developed by alonzo church as a tool for studying the mathematical properties of e ectively computable functions. Evaluation of lambda calculus involves a single operation. It is not exactly easy to write large or interesting programs in the pure lambda calculus. Lambda calculus is turing complete, that is, it is a universal model of computation that can be used to simulate any turing machine. It was introduced by the mathematician alonzo church in the 1930s as part of his research into the. This foundational theory consisted of a part dealing with logical symbols and rules and a part dealing with algorithms operating on these symbols. I would like to create a lambda calculus function p such that p x y z gives x yx pp z.
The lambda calculus language pure lambda calculus supports only a single type. We formulate principles of induction and recursion for a variant of lambda calculus in its original syntax i. To achieve recursion in lambda calculus, a fixedpoint function is required. Subsequently, i was a postgraduate student at st andrews university from 1974 to 1977 where i learnt about functional language design. In the lambda calculus there are a number of combinator implementations that satisfy the mathematical definition of a fixedpoint combinator. Intuitively, abstractions represent functions, and applications represent the application of a. The research on functional quantum programming started with an attempt to define a quantum extension of lambda calculus made by maymin 165 and van tonder 212. In typed lambda calculus, functions can be applied only if they are capable of. A regular function can contain a call to itself in its definition, no problem. Modeling basic programming constructs references h. Lambda calculus university of colorado colorado springs. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a strong theoretical foundation.
Introduction a latticetheoretic and topological method for defining a class of models of the churchcurry acalculus was discovered by the author in 1969 cf. Can a lambda function call itself recursively in python. In this entry, we provide an account of the class of. This paper presents an alternative called the calculus of dependent lambda eliminations, based on pure lambda encodings with no auxiliary datatype system. I have tried using variants of the ycombinatorturing combinator, i. The lambda calculus was introduced in 19321933 by a. Church 19031995 as part of a theory intended as a foundation for mathematics. This formalism was developed by alonzo church as a tool for studying the mathematical properties of e. The recursive functions, which form a class of computable functions, take their name from the process of recurrence or recursion. Lambda calculus as a programming language computability background. The general recursion function can easily become any recursive function because f can be whatever you like. Definition the class of recursive functions is the smallest class of functions from natural numbers to. It is a universal model of computation that can be used to simulate any turing machine.
A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus. Recursion lecture 8 thursday, february 21, 20 1 lambda calculus encodings the pure lambda calculus contains only functions as values. 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. Combinatory logic is a higherorder functions theory. This tutorial shows how to perform arithmetical and. 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. Introduction to the lambda calculus overview what is computability. We can however encode objects, such as booleans, and integers. A particular implementation of fix is currys paradoxical combinator y, represented by. The course is an introductory overview of the foundations of computer science with particular reference to the lambdacalculus. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to lisp, and was treated in the first authors classic the lambda calculus 1984. A brief and informal introduction to the lambda calculus.
These names will be either written in bold such as name or underlines such as name. 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. Lecture notes on the lambda calculus nanjing university. The idea is due to sch on nkel 1924 but is often called currying, after h. In the classical untyped lambda calculus, every function has a fixed point.
Partial recursive functions are built from a set of basic functions projection, zero and successor which apply to tuples of integers using the operations of composition. 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. I cant figure out how to do it with a lambda function though for the simple reason that the lambda function has no n. Recursion lecture 9 tuesday, february 23, 2010 1 lambda calculus encodings the pure lambda calculus contains only functions as values. The course is an introductory overview of the foundations of computer science with particular reference to the lambda calculus. An introduction to functional programming through lambda calculus greg michaelson. Pdf a tutorial introduction to the lambda calculus. In its most general numerical form the process of recursion consists in defining the value of a function by using other values of the same function. Lambda calculus and recursion theory preliminary version.
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