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Empty Type
In type theory, the empty type or absurd type, typically denoted \mathbb 0 is a type with no terms. Such a type may be defined as the nullary coproduct (i.e. disjoint sum of no types). It may also be defined as the polymorphic type \forall t.t For any type P, the type \neg P is defined as P\to \mathbb 0. As the notation suggests, by the Curry–Howard correspondence, a term of type \mathbb 0 is a false proposition, and a term of type \neg P is a disproof of proposition P. A type theory need not contain an empty type. Where it exists, an empty type is not generally unique. For instance, T \to \mathbb 0 is also uninhabited for any inhabited type T. If a type system contains an empty type, the bottom type In type theory, a theory within mathematical logic, the bottom type of a type system is the type that is a subtype of all other types. Where such a type exists, it is often represented with the up tack (⊥) symbol. When the bottom type is empty ... must be uninhabited too, so n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of mathematics. Two influential type theories that were proposed as foundations are Alonzo Church's typed λ-calculus and Per Martin-Löf's intuitionistic type theory. Most computerized proof-writing systems use a type theory for their foundation. A common one is Thierry Coquand's Calculus of Inductive Constructions. History Type theory was created to avoid a paradox in a mathematical foundation based on naive set theory and formal logic. Russell's paradox, which was discovered by Bertrand Russell, existed because a set could be defined using "all possible sets", which included itself. Between 1902 and 1908, Bertrand Russell proposed various "theories of type" to fix the problem. By 1908 Russell arrived at a "ramified" theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curry–Howard Correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and the logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Inhabitation
In type theory, a branch of mathematical logic, in a given typed calculus, the type inhabitation problem for this calculus is the following problem: given a type \tau and a typing environment \Gamma, does there exist a \lambda-term M such that \Gamma \vdash M : \tau? With an empty type environment, such an M is said to be an inhabitant of \tau. Relationship to logic In the case of simply typed lambda calculus, a type has an inhabitant if and only if its corresponding proposition is a tautology of minimal implicative logic. Similarly, a System F type has an inhabitant if and only if its corresponding proposition is a tautology of intuitionistic second-order logic. Girard's paradox shows that type inhabitation is strongly related to the consistency of a type system with Curry–Howard correspondence. To be sound, such a system must have uninhabited types. Formal properties For most typed calculi, the type inhabitation problem is very hard. Richard Statman proved that for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bottom Type
In type theory, a theory within mathematical logic, the bottom type of a type system is the type that is a subtype of all other types. Where such a type exists, it is often represented with the up tack (⊥) symbol. When the bottom type is empty, a function whose return type is bottom cannot return any value, not even the lone value of a unit type. In such a language, the bottom type may therefore be known as the zero or never type. In the Curry–Howard correspondence, an empty type corresponds to falsity. Computer science applications In subtyping systems, the bottom type is a subtype of all types. It is dual to the top type, which spans all possible values in a system. If a type system is sound, the bottom type is uninhabited and a term of bottom type represents a logical contradiction. In such systems, typically no distinction is drawn between the bottom type and the empty type, and the terms may be used interchangeably. If the bottom type is inhabited, its terms typically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |