In the area of mathematical logic and computer science known as type theory, a type constructor is a feature of a typed formal language that builds new types from old ones. Basic types are considered to be built using nullary type constructors. Some type constructors take another type as an argument, e.g., the constructors for product types, function types, power types and list types. New types can be defined by recursively composing type constructors.
For example, simply typed lambda calculus can be seen as a language with a single type constructor—the function type constructor. Product types can generally be considered "built-in" in typed lambda calculi via currying.
Abstractly, a type constructor is an n-ary type operator taking as argument zero or more types, and returning another type. Making use of currying, n-ary type operators can be (re)written as a sequence of applications of unary type operators. Therefore, we can view the type operators as a simply typed lambda calculus, which has only one basic type, usually denoted , and pronounced "type", which is the type of all types in the underlying language, which are now called proper types in order to distinguish them from the types of the type operators in their own calculus, which are called kinds .
Type operators may bind type variables. For example, giving the structure of the simply-typed λ-calculus at the type level requires binding, or higher-order, type operators. These binding type operators correspond to the 2nd axis of the λ-cube, and type theories such as the simply-typed λ-calculus with type operators, λω. Combining type operators with the polymorphic λ-calculus (System F) yields System Fω.
In mathematics and computer science, currying is the technique of converting a function that takes multiple arguments into a sequence of functions that each take a single argument. For example, currying a function that takes three arguments creates three functions:
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that each return a value, rather than a sequence of imperative statements which change the state of the program.
Lambda calculus is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.
In mathematical logic and computer science, a general recursive function or μ-recursive function, is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense. In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines. The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every μ-recursive function is a primitive recursive function—the most famous example is the Ackermann function.
In mathematics, logic, and computer science, a type system is a formal system in which every term has a "type" which defines its meaning and the operations that may be performed on it. Type theory is the academic study of type systems.
In mathematics and computer science in general, a fixed point of a function is a value that is mapped to itself by the function. In combinatory logic for computer science, a fixed-point combinator is a higher-order function that returns some fixed point of its argument function, if one exists.
In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.
In computer programming, especially functional programming and type theory, an algebraic data type is a kind of composite type, i.e., a type formed by combining other types.
A typed lambda calculus is a typed formalism that uses the lambda-symbol to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered. From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus, but from another point of view, they can also be considered the more fundamental theory and untyped lambda calculus a special case with only one type.
In computer science and mathematical logic, a function type 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.
In mathematical logic and computer science, the Calculus of Constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants.
System F, also known as the (Girard–Reynolds) polymorphic lambda calculus or the second-order lambda calculus, is a typed lambda calculus that differs from the simply typed lambda calculus by the introduction of a mechanism of universal quantification over types. System F thus formalizes the notion of parametric polymorphism in programming languages, and forms a theoretical basis for languages such as Haskell and ML. System F was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds (1974).
In mathematical logic and type theory, the λ-cube is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to bind a term or type. The respective dimensions of the λ-cube correspond to:
The SKI combinator calculus is a combinatory logic, a computational system that may be perceived as a reduced version of the untyped lambda calculus. It can be thought of as a computer programming language, though it is not convenient for writing software. Instead, it is important in the mathematical theory of algorithms because it is an extremely simple Turing complete language. It was introduced by Moses Schönfinkel and Haskell Curry.
In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Coq, F*, Epigram, and Idris, dependent types may help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations.
The simply typed lambda calculus, a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor 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 uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties.
In mathematics and computer science, apply is a function that applies a function to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in the study of the denotational semantics of computer programs, because it is a continuous function on complete partial orders. Apply is also a continuous function in homotopy theory, and, indeed underpins the entire theory: it allows a homotopy deformation to be viewed as a continuous path in the space of functions. Likewise, valid mutations (refactorings) of computer programs can be seen as those that are "continuous" in the Scott topology.
In computer science, Scott encoding is a way to represent (recursive) data types in the lambda calculus. Church encoding performs a similar function. The data and operators form a mathematical structure which is embedded in the lambda calculus.
In the area of mathematical logic and computer science known as type theory, a kind is the type of a type constructor or, less commonly, the type of a higher-order type operator. A kind system is essentially a simply typed lambda calculus "one level up", endowed with a primitive type, denoted and called "type", which is the kind of any data type which does not need any type parameters.
In computer science, partial application refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given a function , we might fix the first argument, producing a function of type . Evaluation of this function might be represented as . Note that the result of partial function application in this case is a function that takes two arguments. Partial application is sometimes incorrectly called currying, which is a related, but distinct concept.