An Introduction to Functional Programming Through Lambda Calculus

- 191 -(> 9 8 7 6 5) ->tThe primitive:numberpreturns true if its argument is a number. For example,(numberp 42) ->t10.6. Lambda functionsLISP uses a notation like that for the λ calculus to define nameless functions.It is important to note that LISP functions do not have all the properties we might expect from the λ calculus. Inparticular, special techniques are needed to pass functions as arguments, to return functions as values and to applyfunctions returned as values to new arguments.Functions are defined as forms with the primitive:lambdafollowed by a flat list of bound variables and the body form:where:(lambda () ) ::= | For example, to square a number:(lambda (x) (* x x))or to find the sum of the squares of two numbers:(lambda (x y) (+ (* x x) (* y y)))or to find the value of the quadratic:ax 2 +bx+cgiven a, b, c and x:(lambda (a b c x) (+ (* a (* x x)) (* b x) c)))Note that functions are normally uncurried in LISP.Note that LISP systems will reject attempts to present lambda functions directly as values other than in the functionposition in a form. lambda is not a primitive which denotes a system function. Instead it acts as a marker to indicatea function form. However, if a LISP system sees a naked lambda function form it will try to find a function associatedwith lambda and fail. The special techniques needed to manipulate function values are discussed below.A function is applied to arguments in a form with the function followed by the arguments. The function’s body isevaluated with the bound variables associated initially with the corresponding arguments: