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Kappa Calculus
In mathematical logic, category theory, and computer science, kappa calculus is a formal system for defining first-order functions. Unlike lambda calculus, kappa calculus has no higher-order functions; its functions are not first class objects. Kappa-calculus can be regarded as "a reformulation of the first-order fragment of typed lambda calculus". Because its functions are not first-class objects, evaluation of kappa calculus expressions does not require closures. Definition ''The definition below has been adapted from the diagrams on pages 205 and 207 of Hasegawa.'' Grammar Kappa calculus consists of ''types'' and ''expressions,'' given by the grammar below: : \tau = 1 \mid \tau\times\tau \mid \ldots : e = x \mid id_\tau \mid !_\tau \mid \operatorname_\tau(e) \mid e \circ e \mid \kappa x:1\tau . e In other words, * 1 is a type * If \tau_1 and \tau_2 are types then \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Type
In the area of mathematical logic and computer science known as type theory, a unit type is a type that allows only one value (and thus can hold no information). The carrier (underlying set) associated with a unit type can be any singleton set. There is an isomorphism between any two such sets, so it is customary to talk about ''the'' unit type and ignore the details of its value. One may also regard the unit type as the type of 0-tuples, i.e. the product of no types. The unit type is the terminal object in the category of types and typed functions. It should not be confused with the ''zero'' or bottom type, which allows ''no'' values and is the initial object in this category. Similarly, the Boolean is the type with ''two'' values. The unit type is implemented in most functional programming languages. The void type that is used in some imperative programming languages serves some of its functions, but because its carrier set is empty, it has some limitations (as detaile ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Logic
Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction. It can also be characterized as linear logic with weakening. The name "affine logic" is associated with linear logic, to which it differs by allowing the weakening rule. Jean-Yves Girard introduced the name as part of the geometry of interaction semantics of linear logic, which characterizes linear logic in terms of linear algebra; here he alludes to affine transformations on vector spaces. Affine logic predated linear logic. V. N. Grishin used this logic in 1974, after observing that Russell's paradox cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom.Cf. Frederic Fitch's demonstrably consistent set theory Likewise, the logic formed the basis of a decidable sub-theory of predicate logic, called 'Direct logic' (Ketonen & Wehrauch, 1984; Ketonen & Bellin, 1989). Affine logic can be embedded into linear logic by rewriting the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Type System
Substructural type systems are a family of type systems analogous to substructural logics where one or more of the structural rules are absent or only allowed under controlled circumstances. Such systems are useful for constraining access to system resources such as files, locks and memory by keeping track of changes of state that occur and preventing invalid states. Different substructural type systems Several type systems have emerged by discarding some of the structural rules of exchange, weakening, and contraction: *Ordered type systems (discard exchange, weakening and contraction): Every variable is used exactly once in the order it was introduced. *Linear type systems (allow exchange, but neither weakening nor contraction): Every variable is used exactly once. *Affine type systems (allow exchange and weakening, but not contraction): Every variable is used at most once. *Relevant type systems (allow exchange and contraction, but not weakening): Every variable is used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substructural Logic
In logic, a substructural logic is a logic lacking one of the usual structural rules (e.g. of classical and intuitionistic logic), such as weakening, contraction, exchange or associativity. Two of the more significant substructural logics are relevance logic and linear logic. Examples In a sequent calculus, one writes each line of a proof as :\Gamma\vdash\Sigma. Here the structural rules are rules for rewriting the LHS of the sequent, denoted Γ, initially conceived of as a string (sequence) of propositions. The standard interpretation of this string is as conjunction: we expect to read :\mathcal A,\mathcal B \vdash\mathcal C as the sequent notation for :(''A'' and ''B'') implies ''C''. Here we are taking the RHS Σ to be a single proposition ''C'' (which is the intuitionistic style of sequent); but everything applies equally to the general case, since all the manipulations are taking place to the left of the turnstile symbol \vdash. Since conjunction is a commutative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrow (computer Science)
In computer science, arrows or bolts are a type class used in programming to describe computations in a pure and declarative fashion. First proposed by computer scientist John Hughes as a generalization of monads, arrows provide a referentially transparent way of expressing relationships between ''logical'' steps in a computation. Unlike monads, arrows don't limit steps to having one and only one input. As a result, they have found use in functional reactive programming, point-free programming, and parsers among other applications. Motivation and history While arrows were in use before being recognized as a distinct class, it wasn't until 2000 that John Hughes first published research focusing on them. Until then, monads had proven sufficient for most problems requiring the combination of program logic in pure code. However, some useful libraries, such as the Fudgets library for graphical user interfaces and certain efficient parsers, defied rewriting in a monadic form. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Arity
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics. In logic and philosophy, it is also called adicity and degree. In linguistics, it is usually named valency. Examples The term "arity" is rarely employed in everyday usage. For example, rather than saying "the arity of the addition operation is 2" or "addition is an operation of arity 2" one usually says "addition is a binary operation". In general, the naming of functions or operators with a given arity follows a convention similar to the one used for ''n''-based numeral systems such as binary and hexadecimal. One combines a Latin prefix with the -ary ending; for example: * A nullary function takes no arguments. ** Example: f()=2 * A unary function takes one argument. ** Example: f(x)=2x * A binary function takes two arguments. ** E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (computer Science)
In programming languages, a closure, also lexical closure or function closure, is a technique for implementing lexically scoped name binding in a language with first-class functions. Operationally, a closure is a record storing a function together with an environment. The environment is a mapping associating each free variable of the function (variables that are used locally, but defined in an enclosing scope) with the value or reference to which the name was bound when the closure was created. Unlike a plain function, a closure allows the function to access those ''captured variables'' through the closure's copies of their values or references, even when the function is invoked outside their scope. History and etymology The concept of closures was developed in the 1960s for the mechanical evaluation of expressions in the λ-calculus and was first fully implemented in 1970 as a language feature in the PAL programming language to support lexically scoped first-class funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-class Object
In programming language design, a first-class citizen (also type, object, entity, or value) in a given programming language is an entity which supports all the operations generally available to other entities. These operations typically include being passed as an argument, returned from a function, and assigned to a variable. History The concept of first- and second-class objects was introduced by Christopher Strachey in the 1960s. He did not actually define the term strictly, but contrasted real numbers and procedures in ALGOL: First and second class objects. In ALGOL, a real number may appear in an expression or be assigned to a variable, and either of them may appear as an actual parameter in a procedure call. A procedure, on the other hand, may only appear in another procedure call either as the operator (the most common case) or as one of the actual parameters. There are no other expressions involving procedures or whose results are procedures. Thus in a sense procedures ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-order Function
In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following: * takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself a procedure), * returns a function as its result. All other functions are ''first-order functions''. In mathematics higher-order functions are also termed '' operators'' or '' functionals''. The differential operator in calculus is a common example, since it maps a function to its derivative, also a function. Higher-order functions should not be confused with other uses of the word "functor" throughout mathematics, see Functor (other). In the untyped lambda calculus, all functions are higher-order; in a typed lambda calculus, from which most functional programming languages are derived, higher-order functions that take one function as argument are values with types of the form (\tau_1\to\tau_2)\to\tau_3. General examples * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
