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Simply Typed Lambda Calculus
The simply typed lambda calculus (\lambda^\to), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor (\to) 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 use of the untyped lambda calculus. The term ''simple type'' is also used to refer extensions of the simply typed lambda calculus such as products, coproducts or natural numbers ( System T) or even full recursion (like PCF). In contrast, systems which introduce polymorphic types (like System F) or dependent types (like the Logical Framework) are not considered ''simply typed''. The simple types, except for full recursion, are still considered ''simple'' because the Church encodings of such structures can be done using only \to and suitable type variables, while polymorphism and dependency cannot. Synta ... [...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" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Church Encoding
In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way. Terms that are usually considered primitive in other notations (such as integers, booleans, pairs, lists, and tagged unions) are mapped to higher-order functions under Church encoding. The Church-Turing thesis asserts that any computable operator (and its operands) can be represented under Church encoding. In the untyped lambda calculus the only primitive data type is the function. Use A straightforward implementation of Church encoding slows some access operations from O(1) to O(n), where n is the size of the data structure, making Church encoding impractical. Research has shown that this can be addressed by targeted optimizations, but most functional programming languages instead expand their in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Call By Value
In a programming language, an evaluation strategy is a set of rules for evaluating expressions. The term is often used to refer to the more specific notion of a ''parameter-passing strategy'' that defines the kind of value that is passed to the function for each parameter (the ''binding strategy'') and whether to evaluate the parameters of a function call, and if so in what order (the ''evaluation order''). The notion of reduction strategy is distinct, although some authors conflate the two terms and the definition of each term is not widely agreed upon. To illustrate, executing a function call f(a,b) may first evaluate the arguments a and b, store the results in references or memory locations ref_a and ref_b, then evaluate the function's body with those references passed in. This gives the function the ability to look up the argument values, to modify them via assignment as if they were local variables, and to return values via the references. This is the call-by-reference eval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Call By Name
In a programming language, an evaluation strategy is a set of rules for evaluating expressions. The term is often used to refer to the more specific notion of a ''parameter-passing strategy'' that defines the kind of value that is passed to the function for each parameter (the ''binding strategy'') and whether to evaluate the parameters of a function call, and if so in what order (the ''evaluation order''). The notion of reduction strategy is distinct, although some authors conflate the two terms and the definition of each term is not widely agreed upon. To illustrate, executing a function call f(a,b) may first evaluate the arguments a and b, store the results in references or memory locations ref_a and ref_b, then evaluate the function's body with those references passed in. This gives the function the ability to look up the argument values, to modify them via assignment as if they were local variables, and to return values via the references. This is the call-by-reference eval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operational Semantics
Operational semantics is a category of formal programming language semantics in which certain desired properties of a program, such as correctness, safety or security, are verified by constructing proofs from logical statements about its execution and procedures, rather than by attaching mathematical meanings to its terms (denotational semantics). Operational semantics are classified in two categories: structural operational semantics (or small-step semantics) formally describe how the ''individual steps'' of a computation take place in a computer-based system; by opposition natural semantics (or big-step semantics) describe how the ''overall results'' of the executions are obtained. Other approaches to providing a formal semantics of programming languages include axiomatic semantics and denotational semantics. The operational semantics for a programming language describes how a valid program is interpreted as sequences of computational steps. These sequences then ''are'' the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eta Reduction
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Reduction
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Untyped Lambda Calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equational Theory
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study. Basic idea In universal algebra, an algebra (or algebraic structure) is a set ''A'' together with a collection of operations on ''A''. An ''n''- ary operation on ''A'' is a function that takes ''n'' elements of ''A'' and returns a single element of ''A''. Thus, a 0-ary operation (or ''nullary operation'') can be represented simply as an element of ''A'', or a '' constant'', often denoted by a letter like ''a''. A 1-ary operation (or ''unary operation'') is simply a function from ''A'' to ''A'', often denoted by a symbol placed in front of its argument, like ~''x''. A 2-ary operation (or '' binary operation'') is often denoted by a symbol placed between it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Inference
Type inference 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 Types in a most general view can be associated to a designated use suggesting and restricting the activities possible for an object of that type. Many nouns in language specify such uses. For instance, the word leash indicates a different use than the word line. Calling something a table indicates another designation than calling it firewood, though it might be materially the same thing. While their material properties make things usable for some purposes, they are also subject of particular designations. This is especially the case in abstract fields, namely mathematics and computer science, where the material is finally only bits or formulas. To exclude unwanted, but materially possible uses, the concept ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Erasure
In programming languages, type erasure is the load-time process by which explicit type annotations are removed from a program, before it is executed at run-time. Operational semantics that do not require programs to be accompanied by types are called ''type-erasure semantics'', to be contrasted with ''type-passing semantics''. The possibility of giving type-erasure semantics is a kind of abstraction principle, ensuring that the run-time execution of a program does not depend on type information. In the context of generic programming, the opposite of type erasure is called reification. Type inference The reverse operation is called type inference. Though type erasure can be used as an easy way to define typing over implicitly typed languages (an implicitly typed term is well-typed if and only if it is the erasure of a well-typed explicitly typed lambda term), it does not always lead to an algorithm to check implicitly typed terms. See also * Template (C++) * Problems wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatory Logic
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators, which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions—and to remove any mention of variables—particularly in predicate logic. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments. In mathematics Combinatory logic was originally intended as a 'pre-logic' that would clarify the role of quantified variables in logic, essentially by eliminating them. Another way of eliminating quantified variables is Quine's predicate functor logic. While the expressive power of combinatory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |