In theoretical settings and programming languages where functions are defined in curried form, such as the simply typed lambda calculus, a function type depends on exactly two types, the set-theoretic functions mapping " A " to " B ".
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The parameter and return values determine a function type; thus, is the type of functions that take a and a 32-bit signed integer, and return a signed integer ( of default width ) and a value of the built-in interface type.
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Consider the simply typed lambda calculus, where types ? can be basic types ( ? ), function types ( ?! ), or products ( ?), given by the following Backus Naur form grammar ( ?! associating to the right, as usual ):
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Starting with a polytype \ forall \ alpha . \ alpha, the specialization could either replace the body by another quantified variable, actually a rename or by some type constant ( including the function type ) which may or may not have parameters filled either with monotypes or quantified type variables.
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This technology formed the basis of SSL'A Series'consoles starting with Axiom and followed by Aysis, Avant, Aysis Air and Axiom MT . These consoles offered knob per function type control surfaces that were familiar to a generation of engineers who had grown up with SSL's analogue consoles.
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Intersection types are useful for describing overloaded function types : For example, if " ?! " is the type of functions taking an integer argument and returning an integer, and " ?! " is the type of functions taking a float argument and returning a float, then the intersection of these two types can be used to describe functions that do one or the other, based on what type of input they are given.
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