mathematics

Function type

Function type is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Function type rather than just read about it. In short: In computer science and mathematical logic, a function type (or arrow type or exponential) is the type of a variable or parameter to which a function has or can be assigned, or an argument or result type of a higher-order function taking or 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…

Key takeaways

Function type belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Function type to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Function type from memory before moving on to harder problems.

Reference excerpt

In computer science and mathematical logic, a function type (or arrow type or exponential) is the type of a variable or parameter to which a function has or can be assigned, or an argument or result type of a higher-order function taking or 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 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 domain A and the range B. Here a function type is often denoted A → B, following mathematical convention, or BA, based on there existing exactly BA (exponentially many) set-theoretic functions mappings A to B in the category of sets. The class of such maps or functions is called the exponential object. The act of currying makes the function type adjoint to the product type; this is explored in detail in the article on currying. The function type can be considered to be a special case of the dependent product type, which among other properties, encompasses the idea of a polymorphic function.

Programming languages The syntax used for function types in several programming languages can be summarized, including an example type signature for the higher-order function composition function:

When looking at the example type signature of, for example C#, the type of the function compose is actually Func<Func<A,B>,Func<B,C>,Func<A,C>>. Due to type erasure in C++11's std::function, it is more common to use templates for higher order function parameters and type inference (auto) for closures.

Denotational semantics The function type in programming languages does not correspond to the space of all set-theoretic functions. Given 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 that can be defined 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 expression to the set of computable functions is not sufficient either if the programming language allows writing non-terminating computations (which is the case if the programming language is Turing complete). Expression must be restricted 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. (2002). Types and Programming Languages. The MIT Press. pp. 99–100. ISBN 9780262162098. Mitchell, John C. Foundations for Programming Languages. The MIT Press. function type at the nLab Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study. See section 1.2.

Worked examples

Example 1 — a first encounter with Function type

Start with the simplest possible case. Write down what Function type claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Function type before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Function type ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Function type

In research: Function type appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Function type in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Function type is common in secondary-school and first-year university syllabi. It links to neighbouring topics Data types, Subroutines, Type theory, so understanding it makes those chapters shorter.
In everyday life: Look for Function type outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Function type in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Function type means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Function type out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Function type in simple terms?

Why does Function type matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Function type?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Function type.