__NOTOC__ In mathematical logic and type theory, the λ-cube is a framework for exploring the axes of refinement in Coquand's calculus of constructions, starting from the simply typed lambda calculus (written as \lambda\rightarrow in the cube diagram to the right) as the vertex of a cube placed at the origin, and the calculus of constructions (higher order dependently-typed polymorphic lambda calculus; written as λPω in the diagram) as its diametrically opposite vertex. Each axis of the cube represents a new form of abstraction: * Terms depending on types, or polymorphism. System F, aka se...

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). Whereas simply typed lambda calculus has variables r...

In computer science and logic, a dependent type is a type that depends on a value. It is an overlapping of feature of math-encoding type theory and bug-stopping 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 ATS, Agda and Epigram, dependent types prevent bugs by allowing very expressive types. Two common examples of dependent types are dependent functions and dependent pairs. A dependent function's return type may depend on the value (not just type) of an argu...

In programming languages and type theory, parametric polymorphism is a way to make a language more expressive, while still maintaining full static type-safety. Using parametric polymorphism, a function or a data type can be written generically so that it can handle values identically without depending on their type. Such functions and data types are called generic functions and generic datatypes respectively and form the basis of generic programming. For example, a function append that joins two lists can be constructed so that it does not care about the type of elements: it can append li...

Intuitionistic logic, sometimes more generally called constructive logic, is a system of symbolic logic that differs from classical logic by replacing the traditional concept of truth with the concept of constructive provability. For example, in classical logic, propositional formulae are always assigned a truth value from the two element set of trivial propositions \{\top, \bot\} ("true" and "false" respectively) regardless of whether we have direct evidence for either case. In contrast, propositional formulae in intuitionistic logic are not assigned any definite truth value at all and ins...

In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus, and was introduced by M. Parigot. It introduces two new operators: the mu operator (which is completely different both from the mu operator found in computability theory and from the μ operator of modal μ-calculus) and the bracket operator. Proof-theoretically, it provides a well-behaved formulation of classical natural deduction. One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic. According to the Curry&...