#LAMBDA CALCULUS BETA REDUCTION EXAMPLES FREE#

Now we perform the substitution in the body of the expression and throw away the head, since all the bound variables are substituted, leaving only: x) - an annotation inside the parameter list showing the substitution that will be performed inside the body (arguments have already been removed) Two types of annotations are commonly seen, you can use either (or both!): x - an annotation on the right of the lambda body showing the substitution that will be applied to the expression on the left We can reduce this expression to a simpler form by a substitution, indicated by a bit of intermediate notation. in (λx.x)y, the brackets make it clear that y is not part of the lambda λx.x, but rather the lambda is being applied to y. Brackets are required to delineate the start and end of an expression, e.g. Note that while in JavaScript application of a function (x=>x) to an argument y requires brackets around the argument: (x=>x)(y), in the Lambda Calculus application of some expression f to some other expression x is indicated simply fx. Here, we apply the lambda (λx.x) to the variable y: (λx.x)y

What can we do with such a lambda expression? Well we can apply it to another expression (The same way we can apply anonymous functions to an argument in JavaScript). The K combinator which we wrote as x=>y=>x in JavaScript, is written λxy.x. The expression λx.xy is not a combinator, because y is not bound to any parameter, it is free.

Thus, the expression λx.x is a combinator because the variable x is bound to the parameter.

#LAMBDA CALCULUS BETA REDUCTION EXAMPLES FREE#

A combinator is a lambda expression (function) with no free variables.

We have already discussed combinators in JavaScript, now we can give them a more formal definition: a sequence of nested univariate functions). x y, but they are implicitly curried (e.g.

Lambda functions can have multiple parameters in the parameter list, e.g.: λxy.

Thus, λx.x is semantically equivalent (or alpha equivalent) to λy.y or any other possible renaming of the variable.

The names of variables bound to parameters in a lambda expression are only meaningful within the context of that expression.

See the discussion of Church Encodings, below, to see how this is done. It’s lambdas all the way down! However, to actually model and perform useful computations we say that certain expressions represent values.

The only values that Lambda Calculus variables can take on is other functions (i.e.

Note that anonymous functions in languages like JavaScript and Python are also frequently called lambda expressions, or just lambdas.

A lambda expression has no name, it is anonymous.

Some things to note about such lambda expressions: When we discussed combinators in JavaScript, we gave this function a name. Here is a simple Lambda Abstraction of a function: It is worth looking at this notation before studying haskell-like languages because it was the inspiration for Haskell syntax. Lambda Calculus expressions are written with a standard system of notation. The operations we can apply to Lambda Calculus expressions to simplify (or reduce) them, or to prove equivalence, can also be applied to pure functions in a programming language that supports higher-order functions. The Lambda Calculus is also important to study as it is the basis of functional programming. It has been proven that, as a model of computation, the Lambda Calculus is just as powerful as Turing Machines, that is, any computation that can be modelled with a Turing Machine can also be modeled with the Lambda Calculus. However, while the Turing Machine is based on a hypothetical physical machine (involving tapes from which instructions are read and written) the Lambda Calculus was conceived as a set of rules and operations for function abstraction and application. You are probably aware of the more famous model for computation developed around the same time by Alan Turing: the Turing Machine. The Lambda Calculus is a model of computation developed in the 1930s by the mathematician Alonzo Church. Apply conversion and reduction rules to simplify lambda expressions.Relate the lambda calculus to functional programming.

Understand that the lambda calculus provides a complete model of computation.

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