In type theory a typing environment (or typing context) represents the association between variable names and data types.
More formally an environment is a set or ordered list of pairs , usually written as , where is a variable and its type.
The judgement
is read as " has type in context ". [1]
In statically typed programming languages these environments are used and maintained by typing rules to type check a given program or expression. [1]
In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a type to every "term". Usually the terms are various constructs of a computer program, such as variables, expressions, functions, or modules. A type system dictates the operations that can be performed on a term. For variables, the type system determines the allowed values of that term. Type systems formalize and enforce the otherwise implicit categories the programmer uses for algebraic data types, data structures, or other components.
In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
In type theory, a typing rule is an inference rule that describes how a type system assigns a type to a syntactic construction. These rules may be applied by the type system to determine if a program is well-typed and what type expressions have. A prototypical example of the use of typing rules is in defining type inference in the simply typed lambda calculus, which is the internal language of Cartesian closed categories.
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces quadratic form, and solitons. As Grassmann algebras, they appear in quantum field theory.
Bosonic string theory is the original version of string theory, developed in the late 1960s named after Satyendra Nath Bose. It is so called because it contains only bosons in the spectrum.
The Hamiltonian constraint arises from any theory that admits a Hamiltonian formulation and is reparametrisation-invariant. The Hamiltonian constraint of general relativity is an important non-trivial example.
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.
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 scaled inverse chi-squared distribution is the distribution for x = 1/s2, where s2 is a sample mean of the squares of ν independent normal random variables that have mean 0 and inverse variance 1/σ2 = τ2. The distribution is therefore parametrised by the two quantities ν and τ2, referred to as the number of chi-squared degrees of freedom and the scaling parameter, respectively.
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.
In computer science, Programming Computable Functions (PCF) is a typed functional language introduced by Gordon Plotkin in 1977, based on previous unpublished material by Dana Scott. It can be considered to be an extended version of the typed lambda calculus or a simplified version of modern typed functional languages such as ML or Haskell.
In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance. A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming.
In programming language semantics, normalisation by evaluation (NBE) is a style of obtaining the normal form of terms in the λ-calculus by appealing to their denotational semantics. A term is first interpreted into a denotational model of the λ-term structure, and then a canonical representative is extracted by reifying the denotation. Such an essentially semantic approach differs from the more traditional syntactic description of normalisation as a reductions in a term rewrite system where β-reductions are allowed deep inside λ-terms.
Quantile regression is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional mean of the response variable across values of the predictor variables, quantile regression estimates the conditional median of the response variable. Quantile regression is an extension of linear regression used when the conditions of linear regression are not met.
A Hindley–Milner (HM) type system is a classical type system for the lambda calculus with parametric polymorphism. It is also known as Damas–Milner or Damas–Hindley–Milner. It was first described by J. Roger Hindley and later rediscovered by Robin Milner. Luis Damas contributed a close formal analysis and proof of the method in his PhD thesis.
In mathematical logic, category theory, and computer science, kappa calculus is a formal system for defining first-order functions.
In mathematical logic, the intersection type discipline is a branch of type theory encompassing type systems that use the intersection type constructor to assign multiple types to a single term. In particular, if a term can be assigned both' the type and the type , then can be assigned the intersection type . Therefore, the intersection type constructor can be used to express finite heterogeneous ad hoc polymorphism . For example, the λ-term can be assigned the type in most intersection type systems, assuming for the term variable both the function type and the corresponding argument type .
This programming language theory or type theory-related article is a stub. You can help Wikipedia by expanding it. |
