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The distinction between the two is subtle: "higher-order" describes a mathematical concept of functions that operate on other functions, while "first-class" is a computer science term for programming language entities that have no restriction on their use (thus first-class functions can appear anywhere in the program that other first-class ...
A function (which in mathematics is generally defined as mapping the elements of one set A to elements of another B) is called "A onto B" (instead of "A to B" or "A into B") only if it is surjective; it may even be said that "f is onto" (i. e. surjective). Not translatable (without circumlocutions) to some languages other than English.
The fixed-point combinator may be defined in mathematics and then implemented in other languages. General mathematics defines a function based on its extensional properties. [3] That is, two functions are equal if they perform the same mapping. Lambda calculus and programming languages regard function identity as an intensional property. A ...
Function application applied to this form should give another expression in the same form. In this way any expression on functions of multiple values may be treated as if it had one value. It is not sufficient for the form to represent only the set of values. Each value must have a condition that determines when the expression takes the value.
A built-in function, or builtin function, or intrinsic function, is a function for which the compiler generates code at compile time or provides in a way other than for other functions. [23] A built-in function does not need to be defined like other functions since it is built in to the programming language. [24]
It is usually instantiated with, or borrowed from, models of computation such as lambda calculus which make heavy use of higher-order functions. A programming language can be considered higher-order if components, such as procedures or labels, can be used just like data. For example, these elements could be used in the same way as arguments or ...
The above definition of a function is essentially that of the founders of calculus, Leibniz, Newton and Euler. However, it cannot be formalized, since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory.
Say i := 1 and A[i] := 2, so every time swap is referenced it'll return the other combination of the values ([1,2], [2,1], [1,2] and so on). A similar situation occurs with a random function passed as actual argument. Call-by-name is known by many compiler designers for the interesting "thunks" that are used to implement it.