§ 3.9. Function application syntax

powerUncurry is just like a C function taking a pair of arguments. Unlike C and other procedural languages, ML also allows the programmer to write a function that takes multiple arguments in a sequence like powerCurry. For proper understanding of this feature, let us review function application syntax. In mathematics, function application is written as f(x), but in ML it is written as f x, simply juxtaposing function and its argument.

Let expr be the expressions consisting of constants, variables and function applications. Its syntax is given by the following grammar.

This grammar is ambiguous in the sequence of function applications. To understand this problem, let us examine familiar arithmetic expressions. Suppose we introduce expr+expr, expr-expr, and expr*expr for integer addition, subtraction and multiplication. Then we can write 10 - 6 - 2 but this expression itself does not determine the order of subtractions. As a consequence, two different results are possible: (10 - 6) - 2 = 2 or 10 - (6 - 2) = 4. As in elementary school arithmetic, ML interprets this as (10 - 6) - 2 = 2. We say that the subtraction associates to the left or is left associative. For anther example, ML interprets 10 + 6 * 2 as 10 + (6 * 2). We say that the multiplication * associates stronger than addition +.

In ML and other lambda calculus based functional languages, there is the following important rule:

function application associates to the left and its associatibity is the strongest

e1 e2 e3 ⋯ en is interpreted as (⋯ ((e1 e2) e3) ⋯ en). For example, powerCurry 2 3 is interpreted as ((powerCurry 2) 3).