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SKI Combinator Calculus
The SKI combinator calculus is a combinatory logic system and a computational system. It can be thought of as a computer programming language, though it is not convenient for writing software. Instead, it is important in the mathematical theory of algorithms because it is an extremely simple Turing complete language. It can be likened to a reduced version of the untyped lambda calculus. It was introduced by Moses Schönfinkel and Haskell Curry. All operations in lambda calculus can be encoded via abstraction elimination into the SKI calculus as binary trees whose leaves are one of the three symbols S, K, and I (called ''combinators''). Notation Although the most formal representation of the objects in this system requires binary trees, for simpler typesetting they are often represented as parenthesized expressions, as a shorthand for the tree they represent. Any subtrees may be parenthesized, but often only the right-side subtrees are parenthesized, with left associativity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 log ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fixed-point Combinator
In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator) is a higher-order function (i.e., a function which takes a function as argument) that returns some '' fixed point'' (a value that is mapped to itself) of its argument function, if one exists. Formally, if \mathrm is a fixed-point combinator and the function f has one or more fixed points, then \mathrm\ f is one of these fixed points, i.e., : \mathrm\ f\ = f\ (\mathrm\ f) . Fixed-point combinators can be defined in the lambda calculus and in functional programming languages, and provide a means to allow for recursive definitions. ''Y'' combinator in lambda calculus In the classical untyped lambda calculus, every function has a fixed point. A particular implementation of \mathrm is Haskell Curry's paradoxical combinator ''Y'', given byThroughout this article, the syntax rules given in Lambda calculus#Notation are used, to save parentheses.According to Barendregt p.132, the name origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Peirce's Law
In logic, Peirce's law is named after the philosopher and logician Charles Sanders Peirce. It was taken as an Axiom#Mathematics, axiom in his first axiomatisation of propositional logic. It can be thought of as the law of excluded middle written in a form that involves only one sort of connective, namely implication. In propositional calculus, Peirce's law says that ((''P''→''Q'')→''P'')→''P''. Written out, this means that ''P'' must be true if there is a proposition ''Q'' such that the truth of ''P'' Logical consequence, follows from the truth of "if ''P'' then ''Q''". Peirce's law does not hold in intuitionistic logic or intermediate logics and cannot be deduced from the deduction theorem alone. Under the Curry–Howard isomorphism, Peirce's law is the type of continuation operators, e.g. call/cc in Scheme (programming language), Scheme. v \end \end and where Peirce's law as a formula can be simplified to: : \begin ((u \mathrel v) \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Law Of Excluded Middle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens or De Morgan's laws. The law is also known as the law/principle of the excluded third, in Latin ''principium tertii exclusi''. Another Latin designation for this law is ''tertium non datur'' or "no third ossibilityis given". In classical logic, the law is a tautology. In contemporary logic the principle is distinguished from the semantical principle of bivalence, which states that every proposition is either true or false. The principle of bivalence always implies the law of excluded middle, while the converse is not always true. A commonly cited counterexample uses statements unprovable n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Classical Logic
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class shares characteristic properties: Gabbay, Dov, (1994). 'Classical vs non-classical logic'. In D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (Eds), ''Handbook of Logic in Artificial Intelligence and Logic Programming'', volume 2, chapter 2.6. Oxford University Press. # Law of excluded middle and double negation elimination # Law of noncontradiction, and the principle of explosion # Monotonicity of entailment and idempotency of entailment # Commutativity of conjunction # De Morgan duality: every logical operator is dual to another While not entailed by the preceding conditions, contemporary discussions of classical logic normally only include propositional and first-order logics. Shapiro, Stewart (2000). Classical Logic. In St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Implicational Propositional Calculus
In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus that uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ..., then ...", "→", "\rightarrow ", etc.. Functional (in)completeness Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it. For example, the two-place truth function that always returns '' false'' is not definable from → and arbitrary propositional variables: any formula constructed from → and propositional variables must receive the value ''true'' when all of its variables are evaluated to true. It follows that is not functionally complete. However, if one adds a nullary connective ⊥ for falsity, then one can define all other truth functions. Formulas over the resulting set of connectives are called f-implicational. If ''P'' and ''Q'' are pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Modus Ponens
In propositional logic, (; MP), also known as (), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P'' implies ''Q.'' ''P'' is true. Therefore, ''Q'' must also be true." ''Modus ponens'' is a mixed hypothetical syllogism and is closely related to another valid form of argument, '' modus tollens''. Both have apparently similar but invalid forms: affirming the consequent and denying the antecedent. Constructive dilemma is the disjunctive version of ''modus ponens''. The history of ''modus ponens'' goes back to antiquity. The first to explicitly describe the argument form ''modus ponens'' was Theophrastus. It, along with '' modus tollens'', is one of the standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal. Explanation The form of a ''modus ponens'' argument is a mixed hypothetical syllogism, with two premises and a con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Sentential Logic
The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contrast it with System F, but it should not be confused with first-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation. Some sources include other connectives, as in the table below. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Boolean Logic
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as Logical conjunction, conjunction (''and'') denoted as , disjunction (''or'') denoted as , and negation (''not'') denoted as . Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his ''An Investigation of the Laws of Thought'' (1854). According to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Strict Programming Language
A strict programming language is a programming language that only allows strict functions (functions whose parameters must be evaluated completely before they may be called) to be defined by the user. A non-strict programming language allows the user to define non-strict functions, and hence may allow lazy evaluation. In most non-strict languages, the non-strictness extends to data constructors. Description A strict programming language is a programming language which employs a strict programming paradigm, allowing only strict functions (functions whose parameters must be evaluated completely before they may be called) to be defined by the user. A non-strict programming language allows the user to define non-strict functions, and hence may allow lazy evaluation. Non-strictness has several disadvantages which have prevented widespread adoption: * Because of the uncertainty regarding if and when expressions will be evaluated, non-strict languages generally must be purely fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
