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Principal Type
In type theory, a type system is said to have the principal type property if, given a term and an environment, there exists a principal type for this term in this environment, i.e. a type such that all other types for this term in this environment are an instance of the principal type. The principal type property is a desirable one for a type system, as it provides a way to type expressions in a given environment with a type which encompasses all of the expressions' possible types, instead of having several incomparable possible types. Type inference for systems with the principal type property will usually attempt to infer the principal type. For instance, the ML system has the principal type property and principal types for an expression can be computed by Robinson's unification algorithm, which is used by the Hindley–Milner type inference algorithm. However, many extensions to the type system of ML, such as polymorphic recursion, can make the inference of the principal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Theory
In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. 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 have been proposed as foundations are: * Typed λ-calculus of Alonzo Church * Intuitionistic type theory of Per Martin-Löf 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 paradoxes in naive set theory and formal logic, such as Russell's paradox which demonstrates that, without proper axioms, it is possible to define the set of all sets that are not members of themselves; this set both contains itself and does not contain itself. Between 1902 and 1908, Bertrand Russell proposed various solutions to this problem. By 1908, Russell arrive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type System
In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a ''type'' (for example, integer, floating point, string) to every '' term'' (a word, phrase, or other set of symbols). Usually the terms are various language 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 data types, such as "string", "array of float", "function returning boolean". Type systems are often specified as part of programming languages and built into interpreters and compilers, although the type system of a language can be extended by optional tools that perform added checks using the language's original type synta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instance (type Theory)
Instantiation or instance may refer to: Philosophy * A modern concept similar to ''participation'' in classical Platonism; see the Theory of Forms * The instantiation principle, the idea that in order for a property to exist, it must be had by some object or substance; the instance being a specific object rather than the idea of it * Universal instantiation * An instance (predicate logic), a statement produced by applying universal instantiation to a universal statement * Existential fallacy, also called existential instantiation * A substitution instance, a formula of mathematical logic that can be produced by substituting certain strings of symbols for others in formula, also can be used as the mathematical order to represent the data in an algorithm Computing * Instance (computer science), referring to any running process or to an object as an instance of a class * Table instance (or database instance), a concept in database design; see Row (database) * Creation of an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Inference
Type inference, sometimes called type reconstruction, refers to the automatic detection of the type of an expression in a formal language. These include programming languages and mathematical type systems, but also natural languages in some branches of computer science and linguistics. Nontechnical explanation In a typed language, a term's type determines the ways it can and cannot be used in that language. For example, consider the English language and terms that could fill in the blank in the phrase "sing _." The term "a song" is of singable type, so it could be placed in the blank to form a meaningful phrase: "sing a song." On the other hand, the term "a friend" does not have the singable type, so "sing a friend" is nonsense. At best it might be metaphor; bending type rules is a feature of poetic language. A term's type can also affect the interpretation of operations involving that term. For instance, "a song" is of composable type, so we interpret it as the thing created ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ML (programming Language)
ML (Meta Language) is a general-purpose, high-level, functional programming language. It is known for its use of the polymorphic Hindley–Milner type system, which automatically assigns the data types of most expressions without requiring explicit type annotations ( type inference), and ensures type safety; there is a formal proof that a well-typed ML program does not cause runtime type errors. ML provides pattern matching for function arguments, garbage collection, imperative programming, call-by-value and currying. While a general-purpose programming language, ML is used heavily in programming language research and is one of the few languages to be completely specified and verified using formal semantics. Its types and pattern matching make it well-suited and commonly used to operate on other formal languages, such as in compiler writing, automated theorem proving, and formal verification. Overview Features of ML include a call-by-value evaluation strategy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robinson's Unification Algorithm
In logic and computer science, specifically automated reasoning, unification is an algorithmic process of solving equations between symbolic expressions, each of the form ''Left-hand side = Right-hand side''. For example, using ''x'',''y'',''z'' as variables, and taking ''f'' to be an uninterpreted function, the singleton equation set is a syntactic first-order unification problem that has the substitution as its only solution. Conventions differ on what values variables may assume and which expressions are considered equivalent. In first-order syntactic unification, variables range over first-order terms and equivalence is syntactic. This version of unification has a unique "best" answer and is used in logic programming and programming language type system implementation, especially in Hindley–Milner based type inference algorithms. In higher-order unification, possibly restricted to higher-order pattern unification, terms may include lambda expressions, and equivalence is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polymorphic Recursion
In computer science, polymorphic recursion (also referred to as Robin Milner, Milner–Alan Mycroft, Mycroft typability or the Milner–Mycroft calculus) refers to a recursion (computer science), recursive parametric polymorphism, parametrically polymorphic function (computer science), function where the type parameter changes with each recursive invocation made, instead of staying constant. Type inference for polymorphic recursion is equivalent to semi-unification and therefore undecidable problem, undecidable and requires the use of a semi-algorithm or programmer-supplied type annotations. Example Nested datatypes Consider the following nested datatype in Haskell (programming language), Haskell: data Nested a = a :<: (Nested [a]) , Epsilon infixr 5 :<: nested = 1 :<: [2,3,4] :<: 5,6],[7],[8,9 :<: Epsilon A length function defined over this datatype will be polymorphically recursive, as the type of the argument changes from Nested a to ...
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Haskell (programming Language)
Haskell () is a General-purpose programming language, general-purpose, static typing, statically typed, purely functional programming, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research, and industrial applications, Haskell pioneered several programming language #Features, features such as type classes, which enable type safety, type-safe operator overloading, and Monad (functional programming), monadic input/output (IO). It is named after logician Haskell Curry. Haskell's main implementation is the Glasgow Haskell Compiler (GHC). Haskell's Semantics (computer science), semantics are historically based on those of the Miranda (programming language), Miranda programming language, which served to focus the efforts of the initial Haskell working group. The last formal specification of the language was made in July 2010, while the development of GHC continues to expand Haskell via language extensions. Haskell is used in a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Algebraic Data Type
In functional programming, a generalized algebraic data type (GADT, also first-class phantom type, guarded recursive datatype, or equality-qualified type) is a generalization of a Parametric polymorphism, parametric algebraic data type (ADT). Overview In a GADT, the product constructors (called Algebraic data type, data constructors in Haskell) can provide an explicit instantiation of the ADT as the type instantiation of their return value. This allows defining functions with a more advanced type behaviour. For a data constructor of Haskell 2010, the return value has the type instantiation implied by the instantiation of the ADT parameters at the constructor's application. -- A parametric ADT that is not a GADT data List a = Nil , Cons a (List a) integers :: List Int integers = Cons 12 (Cons 107 Nil) strings :: List String strings = Cons "boat" (Cons "dock" Nil) -- A GADT data Expr a where EBool :: Bool -> Expr Bool EInt :: Int -> Expr Int EEqual :: E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
