Function type in the context of Function space

In computer science, a literal is a textual representation (notation) of a value as it is written in source code. Almost all programming languages have notations for atomic values such as integers, floating-point numbers, and strings, and usually for Booleans and characters; some also have notations for elements of enumerated types and compound values such as arrays, records, and objects. An anonymous function is a literal for the function type.

In contrast to literals, variables or constants are symbols that can take on one of a class of fixed values, the constant being constrained not to change. Literals are often used to initialize variables; for example, in the following, 1 is an integer literal and the three letter string in "cat" is a string literal:

In computer programming, an anonymous function (function literal, lambda function, or block) is a function definition that is not bound to an identifier. Anonymous functions are often arguments being passed to higher-order functions or used for constructing the result of a higher-order function that needs to return a function.If the function is only used once, or a limited number of times, an anonymous function may be syntactically lighter than using a named function. Anonymous functions are ubiquitous in functional programming languages and other languages with first-class functions, where they fulfil the same role for the function type as literals do for other data types.

Anonymous functions originate in the work of Alonzo Church in his invention of the lambda calculus, in which all functions are anonymous, in 1936, before electronic computers. In several programming languages, anonymous functions are introduced using the keyword lambda, and anonymous functions are often referred to as lambdas or lambda abstractions. Anonymous functions have been a feature of programming languages since Lisp in 1958, and a growing number of modern programming languages support anonymous functions.

The simply typed lambda calculus (⁠${\displaystyle \lambda ^{\to }}$⁠), a formof type theory, is a typed interpretation of the lambda calculus with only one type constructor (⁠${\displaystyle \to }$⁠) that builds function types. It is the canonical and simplest example of a typed lambda calculus. The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical use of the untyped lambda calculus.

The term simple type is also used to refer to extensions of the simply typed lambda calculus with constructs such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast, systems that introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered simply typed. The simple types, except for full recursion, are still considered simple because the Church encodings of such structures can be done using only ${\displaystyle \to }$ and suitable type variables, while polymorphism and dependency cannot.