In computer science, a function type (also arrow type or exponential) is the type of a variable or parameter to which a function has or can be assigned or the result type of a higher-order function returning a function.
A function type depends on the type of the parameters and the result type of the function (it, or more accurately the unapplied type constructor · → ·, is a higher-kinded type). In theoretical settings and languages where functions are defined in curried form, such as the simply typed lambda calculus, a function type depends on exactly two types, the ___domain A and the range B. Here a function type is often denoted A → B, following mathematical convention, or BA, based on the fact that there exist exactly BA (exponentially many) set-theoretic functions mapping A to B.
Programming languages
The following table summarized the syntax used for function types in several programming languages, including an example type signature for the higher-order function composition function:
| Language | Notation | Example type signature | |
|---|---|---|---|
| With first-class functions, parametric polymorphism |
C# | Func<α1,α2,...,αn,ρ> |
Func<A,C> compose(Func<A,B> f, Func<B,C> g); |
| Haskell | α -> ρ |
compose :: (a -> b) -> (b -> c) -> a -> c
| |
| Scala | (α1,α2,...,αn) => ρ |
def compose[A, B, C](f: B => C, g: A => B): A => C
| |
| Without first-class functions, parametric polymorphism |
C | ρ (*)(α1,α2,...,αn) |
int (*compose(int (*f)(int), int (*g)(int)))(int);
|
When looking at the example type signature of for example C#, one should note that the type of the function compose is actually Func<Func<A,B>,Func<B,C>,Func<A,C>>.
Denotational semantics
The function type in programming languages does not correspond to the space of all set-theoretic functions. If we take the countably infinite type of natural numbers as the ___domain and the booleans as range, then there are an uncountably infinite number (2ℵ0 = c) of set-theoretic functions between them. Clearly this space of functions is larger than the number of functions we can define in any programming language as there exist only countably many programs (a program being a finite sequence of a finite number of symbols) and one of the set-theoretic functions effectively solves the halting problem.
Denotational semantics concerns itself with finding more appropriate models (called domains) to model programming language concepts such as function types. It turns out that restricting ourselves to the set of computable functions is not sufficient either if the programming language allows us to write non-terminating computations (which is the case if the programming language is Turing complete). We need to restrict ourselves to the so-called continuous functions (corresponding to continuity in the Scott topology, not continuity in the real analytical sense). Even then, the set of continuous function contains the parallel-or function, which cannot be correctly defined in all programming languages.
See also
- Cartesian closed category
- Currying
- Exponential object, category-theoretic equivalent
- First-class function
- Function space, set-theoretic equivalent
References
- Pierce, Benjamin C. Types and Programming Languages. The MIT Press. pp. 99–100.
- Mitchell, John C. Foundations for Programming Languages. The MIT Press.