It is a function taking one argument xand with body e. The lambda calculus cmu school of computer science. In lambda calculus, a branch of mathematical logic concerned with the formal study of functions, a reduction strategy is how a complex expression is reduced to a simple expression by successive reduction steps. Publishers pdf, also known as version of record includes final page, issue and volume numbers. It is the basis for functional programming like turing machines are the foundation of imperative. Lambda calculus is a framework developed by alonzo church in 1930s to study computations with functions.
Although the lambdas are not explicit, they exist hidden in the definition of. To enable you to focus on the lambda calculus semantics, a parser is provided in oz and in haskell. Lambda calculus lecture 7 tuesday, february, 2018 1 syntax. Ppl lecture lambda calculus alpha,beta,eta reduction. Nov 23, 2015 we covered the relation of lambda calculus to turing machines, history of lambda calculus, syntax of lambda calculus, disambiguation rules, informal description of the semantics of lambda calculus. A lambda expression that cannot be reduced further, by either. We prefer our version of the theory in which the identifications are made on. This explains why the lambda calculus has been called a calculus. These names will be either written in bold such as name or underlines such as name. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
Notes on lambdacalculus to obtain the value of an rpal program, we carry out the following steps. Reduction consists of replacing a part pof eby another expression p0 according to the given rewrite rules. The lambda calculus notes stanford encyclopedia of philosophy. Lambda calculus and types oxford department of computer. Mn reduces under reduction to the expression m with n substituted in it. In lambda calculus, lambda expressions bound variables can be substituted with different name. Lambda is an interactive, graphical, pedagogical computer program that helps students of formal semantics practice the typed lambda calculus we discussed how the lc is used in linguistics in the past check the archives. Abstraction is used to represent a function, and application to represent the. Cs1 lambda calculus worksheet this isnt homework and is worth no credit. Functional programming lambda calculus tutorialspoint.
Mx y alpha the alpha rule is needed to establish the churchrosser theorem and the other results mentioned below, but it plays no role in reduction. Further reductions could be applied to reduce 2 7 7 to 98. It has relevant and representational problems at the end of the chapters, all with answers in the back. Lambda calculus as a workflow model by peter kelly, paul coddington, and andrew wendelborn. In the previous notes on encoding natural numbers, etc. Lambda calculus 5 scope, free and bound occurences lx. The lambda calculus is an abstract mathematical theory of computation, involving. Well also see a quick example of a simple sentence. If we denote the encoding of a term \t\ by \\lceil t\rceil\, then we can recursively encode any term with the following three rules for variables, applications, and lambda abstractions, respectively. It was introduced by the mathematician alonzo church in the 1930s as part of his research into the.
Ppl lecture lambda calculus alpha,beta,eta reduction with example hindi in. A correct reduction first renames y to z, or any other fresh variable e. The lambda calculus is a language of pure functions. The most interesting aspect is a system of substitution primitives and an accompanying.
Calculus is considered as the smallest universal programming language the. Lambda calculus with lambda calculus mogensen describes a delightful encoding of lambda terms with lambda terms. Heres how to multiply two numbers in lambda calculus. Lambda calculus is a model of computation, invented by church in the early 1930s. In the pure lambda calculus, the only values are functions \x.
These functions can be of without names and single arguments. A short introduction to the lambda calculus achim jung. In the notation of lambda calculus this is denoted as x. The course is an introductory overview of the foundations of computer science with particular reference to the lambda calculus. This post will explain some of the basics of lambda calculus, mainly alphaconversion and betareduction. This is called a betareduction,and is used to apply a function to its argument. E to denote a function in which x is a formal argument and e is the functional body. Lambda calculus lambda calculus stanford university. In lambda calculus, function application is regarded as leftassociative, so that st.
Introduction to the lambda calculus bu computer science. Lambda calculus 6 beta reduction computation in the lambda calculus takes the form of beta reduction. Browse other questions tagged lambda lambda calculus or ask. Some compilers include an alpha conversion stage to rename all program variables such that variable names become unique. It is a universal model of computation that can be used to simulate any turing machine. Also see lambda calculus and the wikipedia lambda calculus article. One interpretation of the untyped lambda calculus is as a programming language where evaluation proceeds by performing reductions on an expression until it is in normal form. When laying out the early principles of \\ lambda\calculus, church restricted \\beta\ reduction to only those cases where variable capture does not occur. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation.
Topics covered in these notes include the untyped lambda calculus, the. Spare a thought for students struggling to make turing machines do simple tasks. Fundamental to all functional languages is the most atomic notion of composition, function abstraction of a single variable. The lambda calculus consists very simply of three terms and all valid recursive combinations thereof var a variable. There may, indeed, be other applications of the system than its use as a logic. 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. If one term can be alpha reduced to another we say the terms are alpha equivalent since. There are basically two and a half processes in lambda calculus. To complete the lambda calculus, we need to be able to rename a bound variable formal parameter. Lambda calculus princeton university computer science. Csc173 lambda calculus exercises 1 practice and practical. Lambda calculus examples in this handout, we look at several examples of lambda terms in order to provide a. Functional languages are just the lambda calculus with a more palatable syntax. With a little syntax sugar, lambda calculus becomes a practical programming language.
In this interpretation, if the expression never reduces to normal form then the program never terminates, and. In lambda calculus, there are 3 substitution rules for expression to be reduced. Except for the problems in section 7, you should use equational reasoning rather than cbn or cbv reduction. Lecture notes on the lambda calculus department of mathematics. We covered the relation of lambda calculus to turing machines, history of lambda calculus, syntax of lambda calculus, disambiguation rules, informal description of the semantics of lambda calculus. As an illustration of the kinds of difficulties that can arise if one is too casual about free and bound variables, one can formulate a sequent calculus. Cs311 computational structures churchs lambda calculus. Programming language features programming languages. Already, our factorial example above is shorter than equivalent code in many highlevel languages. Through lambda calculus, is pointed at from both \readings and \resources links and the course schedule. Lambda calculus provides some of the foundational structures that functional programming is built from. This post is being written with me being slightly ill so some bits may make no sense whatsoever sorry. Lambda calculus 6 beta reduction computation in the lambda calculus takes the form of betareduction.
Other notions of computability many other notions of computability have been proposed, e. The expressions of this language are the terms which are built using abstraction and application. Alpha equivalence edit a basic form of equivalence, definable on lambda terms, is alpha equivalence. Simple programming language with while loops all have been shown equivalent to turing machines by simulation proofs 2. Values are lambda expressions that have properly finished being evaluated there is nothing more to do. Reduction consists of replacing a part p of e by another expression p accord.
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. A lambda term is a variable x, a lambda abstraction x. The previous exams at the \resources link might be helpful. More lambda calculus meeting 17, csci 5535, spring 2009 announcements work on your project probably background reading i am looking at your proposals, but come talk to me if you have concerns 2 plan last time introduce lambda calculus syntax substitution today lambda calculus in real life operational. In lambda calculus there is one computation rule called. The key operation, the one that does the heavy lifting, is called. 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 course is entitled lambda calculus and types and it appears in various forms. March 18, 2004 abstract the lambda calculus can appear arcane on. Church developed the lambda calculus in the 1930s as a theory of functions. It therefore seems fitting to start my journey through functional programming with a thorough examination of lambda calculus. Heres an example of a simple lambda expression that defines the plus one function. 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. Some compilers include an alphaconversion stage to rename all program variables such that variable names become unique.
An alphacorecursion principle for the infinitary lambda calculus. Deductive lambda calculus considers what happens when lambda terms are regarded as mathematical expressions. 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. The key rule in expression evaluation is the reduction. Apply a function fon an expression nis written as fn. Finally, in 1960s, dana scott, then a professor at oxford university, and himself a former student of alonzo church, discovered a meaning for the lambda calculus. But, if we want to make the step explicit, we call it. When laying out the early principles of \\ lambda\ calculus, church restricted \\beta\reduction to only those cases where variable capture does not occur.
The lambda calculus can be thought of as the theoretical foundation of functional programming. Note that twice is another example of a polymorphic function in the sense that it may. Notes on lambda calculus to obtain the value of an rpal program, we carry out the following steps. Alpha renaming is used to prevent capturing free occurrences of variables when reducing a lambda calculus expression, e. As an illustration of the kinds of difficulties that can arise if one is too casual about free and bound variables, one can formulate a. This first post will introduce lambda calculus with references to the book an introduction to functional programming through lambda calculus. Lambda calculus is a language with clear operational and denotational semantics capable of expressing algorithms. The lambda calculus notes stanford encyclopedia of. Symbolic ai basic lambda calculus 1 computer science.