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The Lambda Calculus

The Lambda Calculus was created by Alonzo Church in the 1930s as a construction in abstract logic but it has had practical application in the design of programming languages, most notably LISP. The Lambda Calculus involves the evaluation of lambda expressions, which are definitions of unnamed functions. The format of a lambda expression is:

lambda [parameter list]. [body of function]

where the body of a function is just the rules for computing the function's value. For example, the function of taking the square of a number is expressed in lambda notation as:

lambda x . x*x

Originally the notation involved the use of a "^" above the first parameter but this was typographically inconvenient and so the "^" was replaced by the Greek letter Λ which roughly resembles a "^". Of course the Λ must be placed before the first parameter rather than above it.

Students encountering the lambda notation in LISP are often puzzled as to why it is needed. The lambda notation is essential for several reasons. First there is a need for a means of defining a function without giving it a name. Naming functions is an easy task but there is an arbitrariness to choosing a name that does not fit in well with computer processing. Functions could be named serially in the order in which they arise in a program. This is feasible but awkward. Often there is not real need for a function to have a name. So the lambda notation handles the problem of creating functions without naming them. Of course functions can be given names to facilitate reference to them if that is needed.

The power of the lambda notation is in its generality. The lambda notation will handle the case in which the value of a function is a function. In many computer languages the value of a function must be an element of a set, such as a number or a string or the label of a function. In the lambda notation the value can be a function, not the name or label of a function but a function itself.

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