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. Programming language features programming languages. 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. Introduction to the lambda calculus bu computer science. Church developed the lambda calculus in the 1930s as a theory of functions. Modern processors are just overblown turing machines. Notes on lambdacalculus to obtain the value of an rpal program, we carry out the following steps. Further reductions could be applied to reduce 2 7 7 to 98. 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. An alphacorecursion principle for the infinitary lambda calculus. If one term can be alpha reduced to another we say the terms are alpha equivalent since. When laying out the early principles of \\ lambda\calculus, church restricted \\beta\ reduction to only those cases where variable capture does not occur.
Mn reduces under reduction to the expression m with n substituted in it. 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. Cs1 lambda calculus worksheet this isnt homework and is worth no credit. Calculus is considered as the smallest universal programming language the.
In the previous notes on encoding natural numbers, etc. Also see lambda calculus and the wikipedia lambda calculus article. These names will be either written in bold such as name or underlines such as name. Although the lambdas are not explicit, they exist hidden in the definition of. 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.
In the pure lambda calculus, the only values are functions \x. This is called a betareduction,and is used to apply a function to its argument. 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. 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. Lambda calculus is a language with clear operational and denotational semantics capable of expressing algorithms. 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. Topics covered in these notes include the untyped lambda calculus, the. A short introduction to the lambda calculus achim jung. This tool may be useful even for those not interested in this angle, even though thats the intended use of the software. 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. The lambda calculus is an abstract mathematical theory of computation, involving. This explains why the lambda calculus has been called a calculus. This post is being written with me being slightly ill so some bits may make no sense whatsoever sorry.
The key operation, the one that does the heavy lifting, is called. 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. The lambda calculus can be thought of as the theoretical foundation of functional programming. Ppl lecture lambda calculus alpha,beta,eta reduction with example hindi in. Heres an example of a simple lambda expression that defines the plus one function. Csc173 lambda calculus exercises 1 practice and practical. It therefore seems fitting to start my journey through functional programming with a thorough examination of lambda calculus. Lambda calculus 6 beta reduction computation in the lambda calculus takes the form of betareduction. Values are lambda expressions that have properly finished being evaluated there is nothing more to do. Already, our factorial example above is shorter than equivalent code in many highlevel languages. 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. Lambda calculus lambda calculus stanford university. Other notions of computability many other notions of computability have been proposed, e.
It is a universal model of computation that can be used to simulate any turing machine. Some compilers include an alphaconversion stage to rename all program variables such that variable names become unique. But, if we want to make the step explicit, we call it. These functions can be of without names and single arguments. Spare a thought for students struggling to make turing machines do simple tasks. Lambda calculus provides some of the foundational structures that functional programming is built from. In lambda calculus there is one computation rule called. With a little syntax sugar, lambda calculus becomes a practical programming language. Lambda calculus princeton university computer science. To complete the lambda calculus, we need to be able to rename a bound variable formal parameter. We prefer our version of the theory in which the identifications are made on.
There may, indeed, be other applications of the system than its use as a logic. There are basically two and a half processes in lambda calculus. 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. It is a function taking one argument xand with body e. In lambda calculus, there are 3 substitution rules for expression to be reduced. 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. Publishers pdf, also known as version of record includes final page, issue and volume numbers. 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. All functional programming languages can be viewed as syntactic variations of the lambda calculus, so that both their semantics and implementation can be analysed in the context of the. March 18, 2004 abstract the lambda calculus can appear arcane on. 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. Lambda calculus is a framework developed by alonzo church in 1930s to study computations with functions. The course is entitled lambda calculus and types and it appears in various forms. For purposes of modeling computation, we typically.
This first post will introduce lambda calculus with references to the book an introduction to functional programming through lambda calculus. It is the basis for functional programming like turing machines are the foundation of imperative. Alpha equivalence edit a basic form of equivalence, definable on lambda terms, is alpha equivalence. When laying out the early principles of \\ lambda\ calculus, church restricted \\beta\reduction to only those cases where variable capture does not occur. Lambda calculus with lambda calculus mogensen describes a delightful encoding of lambda terms with lambda terms. Note that twice is another example of a polymorphic function in the sense that it may. Simple programming language with while loops all have been shown equivalent to turing machines by simulation proofs 2. Lambda calculus lecture 7 tuesday, february, 2018 1 syntax. Through lambda calculus, is pointed at from both \readings and \resources links and the course schedule. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. E to denote a function in which x is a formal argument and e is the functional body.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Lambda calculus 5 scope, free and bound occurences lx. Except for the problems in section 7, you should use equational reasoning rather than cbn or cbv reduction. In this interpretation, if the expression never reduces to normal form then the program never terminates, and. A correct reduction first renames y to z, or any other fresh variable e. Apply a function fon an expression nis written as fn. Lambda calculus and types oxford department of computer. The lambda calculus notes stanford encyclopedia of philosophy. Notes on lambda calculus to obtain the value of an rpal program, we carry out the following steps. In lambda calculus, function application is regarded as leftassociative, so that st.
This post will explain some of the basics of lambda calculus, mainly alphaconversion and betareduction. In lambda calculus, lambda expressions bound variables can be substituted with different name. The expressions of this language are the terms which are built using abstraction and application. Ppl lecture lambda calculus alpha,beta,eta reduction. Lambda calculus 6 beta reduction computation in the lambda calculus takes the form of beta reduction. The lambda calculus notes stanford encyclopedia of. Lambda calculus as a workflow model by peter kelly, paul coddington, and andrew wendelborn. Functional languages are just the lambda calculus with a more palatable syntax. 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. Symbolic ai basic lambda calculus 1 computer science. Cs311 computational structures churchs lambda calculus.
Reduction consists of replacing a part p of e by another expression p accord. A lambda expression that cannot be reduced further, by either. Fundamental to all functional languages is the most atomic notion of composition, function abstraction of a single variable. The most interesting aspect is a system of substitution primitives and an accompanying. To enable you to focus on the lambda calculus semantics, a parser is provided in oz and in haskell. Lecture notes on the lambda calculus department of mathematics. Lambda calculus examples in this handout, we look at several examples of lambda terms in order to provide a. It has relevant and representational problems at the end of the chapters, all with answers in the back. The previous exams at the \resources link might be helpful.
Functional programming lambda calculus tutorialspoint. Abstraction is used to represent a function, and application to represent the. Browse other questions tagged lambda lambda calculus or ask. Lambda calculus is a model of computation, invented by church in the early 1930s. The lambda calculus cmu school of computer science. A lambda term is a variable x, a lambda abstraction x. The course is an introductory overview of the foundations of computer science with particular reference to the lambda calculus. The lambda calculus is a language of pure functions. Alpha renaming is used to prevent capturing free occurrences of variables when reducing a lambda calculus expression, e. The key rule in expression evaluation is the reduction. Deductive lambda calculus considers what happens when lambda terms are regarded as mathematical expressions. Well also see a quick example of a simple sentence. 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. Reduction consists of replacing a part pof eby another expression p0 according to the given rewrite rules.
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