A lambda term is a variable x, a lambda abstraction x. The syntax of basic \\ lambda\ calculus is quite sparse, making it an elegant, focused notation for representing functions. A lambda expression that cannot be reduced further, by either. The lambda calculus stanford encyclopedia of philosophy. Put the following expressions into beta normal form use reduction as far as possible, conversion as needed.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Lambda calculus is based on three basic building blocks. Lambda calculus for mortal developers the programming corner. In lambda calculus, function application is regarded as leftassociative, so that st. Already, our factorial example above is shorter than equivalent code in many highlevel languages. Lesson2 lambda calculus basics university of chicago. In the following, we rename x to z, or any other fresh variable. In lambda calculus, there are 3 substitution rules for expression to be reduced. An alphacorecursion principle for the infinitary lambda calculus. Although the lambdas are not explicit, they exist hidden in the definition of. Functional languages are just the lambda calculus with a more palatable syntax.
The lambda calculus department of computer science, columbia. Some compilers include an alpha conversion stage to rename all program variables such that variable names become unique. A brief and informal introduction to the lambda calculus. Callbyvalue free variable argument application reduction. Lambda abstraction the only other thing in the lambda calculus is lambda abstraction. Lambda calculus with lambda calculus mogensen describes a delightful encoding of lambda terms with lambda terms. Spare a thought for students struggling to make turing machines do simple tasks.
Lambda calculus examples in this handout, we look at several examples of lambda terms in order to provide a. An abstract black box machine that takes an input and produces an output. Lambda calculus is turing complete, that is, it is a universal model of computation that can be used to simulate any turing machine. You dont need to write down the alpha conversions now.
Then reduce e1 to normal form using applicative order reduction. This reduction erroneously captures the free occurrence of y. 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. The previous exams at the \resources link might be helpful. As we will see, each of these alternatives has dramatically different properties from the others. Finally, we could have a look at an example of beta reduction with a name clash, to see the how alpha equivalence works in practice. It is the most reducible form of all functional programming languages. Or, more concretely, a redex is always of the form. These functions can be of without names and single arguments. A term is in normal form if it is stuck in normal order semantics. 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. In this chapter we take a brief but careful look at the lambda calculus. Rule 5 says that equal terms can be substituted on either side of an application, or as the body of a function, taking care not to mess up free variables.
An alphacorecursion principle for the infinitary lambda. Some compilers include an alphaconversion stage to rename all program variables such that variable names become unique. Lambda calculus is a formal system in mathematical logic for expressing computation based on. Remember were assuming leftassociation as shown in part a. March 18, 2004 abstract the lambda calculus can appear arcane on. As we will see, each of these alternatives has dramatically different properties from.
The lambda calculus is fundamental to denotational semantics, the mathematical theory of what computer programs mean. Apply a function fon an expression nis written as fn. Further reductions could be applied to reduce 2 7 7 to 98. The lambda calculus is an abstract mathematical theory of computation, involving. A mathematical object set of ordered pairs from domain and range, and. Most functional programming languages have a syntactic form for lambda abstractions. Leftmostinnermost reduction which doesnt reduce under lambda is callbyvalue reduction sec. Functional programming lambda calculus tutorialspoint. These names will be either written in bold such as name or underlines such as name. Lambda calculus 6 beta reduction computation in the lambda calculus takes the form of beta reduction. A correct reduction first renames y to z, or any other fresh variable e. Church developed the lambda calculus in the 1930s as a theory of functions.
It is a universal model of computation that can be used to simulate any turing machine. Modern processors are just overblown turing machines. Alpha renaming is used to prevent capturing free occurrences of variables when reducing a lambda calculus expression, e. Thanks for contributing an answer to mathematics stack exchange. Also see lambda calculus and the wikipedia lambda calculus article. Heres how to multiply two numbers in lambda calculus. If e1 is a lambda abstraction, do beta reduction, and reduce the result to normal form using applicative order reduction. Two lambda expressions are equivalent if they can be beta reduced to the same expresion, subject to alpha conversion. In the notation of lambda calculus this is denoted as x.
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. 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. Introduction to the lambda calculus bu computer science. 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. An expression with no redex is said to be in normal form. In lambda calculus, lambda expressions bound variables can be substituted with different name. Csc173 lambda calculus exercises 1 practice and practical.
E to denote a function in which x is a formal argument and e is the functional body. Lambda calculus 5 scope, free and bound occurences lx. Alpha renaming is used to prevent capturing free occurrences of variables when betareducing a lambda calculus expression. Alpha equivalence edit a basic form of equivalence, definable on lambda terms, is alpha equivalence. In the lambda calculus, this means in e1 e2, reduce e2 to normal form using applicative order reduction.
There are basically two and a half processes in lambda calculus. It is the basis for functional programming like turing machines are the foundation of imperative. Assume that, in contradiction to the corollary, y and z are two different normal forms of x. Doing alpha conversions and beta reductions, lambda calculus. If my understanding is correct, a redex is a reducible expression, that is, the application of a lambda term to a lambda abstraction. Csc173 lambda calculus exercises 1 practice and practical help. In this interpretation, if the expression never reduces to normal form then the program never terminates, and. If one term can be alpha reduced to another we say the terms are. No free variables can be captured become bound in the process. What i assumed from your original posting was that you were originally doing 1, and that in trying to do 3 you were in fact still reducing under lambda, which would explain how you manage to. A short introduction to the lambda calculus achim jung. For purposes of modeling computation, we typically.
Alpha reductions are usually called alpha equivalences or alpha rewrite rules. In typed lambda calculus, functions can be applied only if they are capable of. Lambda calculus sounds like an arcane term that only functional programming wizards can understand. Deductive lambda calculus considers what happens when lambda terms are regarded as mathematical expressions. To complete the lambda calculus, we need to be able to rename a bound variable formal parameter. The main ideas are applying a function to an argument and forming functions by abstraction. Programming language features programming languages. The lambda calculus has turned out to capture two aspects of a function. Lambda calculus alpharenaming, beta reduction, applicative and normal evaluation orders, churchrosser theorem. The lambda calculus can be thought of as the theoretical foundation of functional programming. The calculi of lambdaconversion by alonzo church pdf.
With a little syntax sugar, lambda calculus becomes a practical programming language. Analyse the following lambda expression to clarify its structure. Recall that corollary 2 says that if lambdaterm x has normal form y then y is unique up to alphareduction. The \\ lambda\ calculus is, at heart, a simple notation for functions and application. This is an intermediate situation, where we may specify, for instance, that a term has a type of the form x x for all x, without actually specifying x. 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. The key rule in expression evaluation is the reduction.
Lambda calculus 6 beta reduction computation in the lambda calculus takes the form of betareduction. Through lambda calculus, is pointed at from both \readings and \resources links and the course schedule. Rule 4 says that if one lambdaexpression alpha or betareduces to another, then the two are equivalent. Lambda calculus lambda calculus stanford university. In the simplest form of lambda calculus, terms are built using only the. The lambda calculus cmu school of computer science. In the lambdacalculus, this means in e1 e2, reduce e2 to normal form using applicative order reduction. Recall that corollary 2 says that if lambda term x has normal form y then y is unique up to alpha reduction. Reduction consists of replacing a part pof eby another expression p0 according to the given rewrite rules. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. 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.
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