Delayed evaluation is used particularly in functional languages. When using delayed evaluation, an expression is not evaluated as soon as it gets bound to a variable, but when the evaluator is forced to produce the expression's value. Some programming languages delay evaluation of expressions by default, and some others provide functions or special syntax to delay evaluation. In Miranda and Haskell, evaluation of function arguments is delayed by default. In many other languages, evaluation can be delayed by explicitly suspending the computation using special syntax (as with Scheme's "delay" and "force") or, ...

See also:

Lazy evaluation, Lazy evaluation - Delayed evaluation

Read more here: » Lazy evaluation: Encyclopedia II - Lazy evaluation - Delayed evaluation

In certain formalizations of mathematics, such as the untyped lambda calculus and combinatorial calculus, every expression can be considered a higher-order function. In these formalizations, the existence of a fixed-point combinator means that every function has at least one fixed point; a function may have more than one distinct fixed point. In some other systems, for example the simply typed lambda calculus, a well-typed fixed-point combinator cannot be written -- in those systems any support for recursion must be explicitly ...

See also:

Fixed point combinator, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Example, Fixed point combinator - Other fixed point combinators

Read more here: » Fixed point combinator: Encyclopedia II - Fixed point combinator - Existence of fixed point combinators

Omega (Ω ω) is the 24th and last letter of the Greek alphabet. In the Greek numeric system it had a value of 800. The word literally means "big O" (ō mega, mega meaning 'big'), as opposed to Omicron, which means "little O" (o mikron, micron meaning 'little').[1]. This name is Byzantine; in Classical Greek, the letter was called ō (ὦ), whereas the Omicron was called ou (Including:

• Omega - The symbol Ω upper-case letter
• Omega - The symbol ω lower-case letter

Read more here: » Omega: Encyclopedia - Omega

In computability theory the Church–Turing thesis, Church's thesis, Church's conjecture or Turing's thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. The thesis claims that any calculation that is possible can be performed by an algorithm running on a computer, provided that sufficient time and storage space are available. It is generally assumed that an algorithm must satisfy the following requirements: ...

Including:

• Church–Turing thesis - Church–Turing thesis
• Church–Turing thesis - History
• Church–Turing thesis - Success of the thesis
• Church–Turing thesis - Philosophical implications
• Church–Turing thesis - Reference

Read more here: » Church–Turing thesis: Encyclopedia - Church–Turing thesis

Y is the twenty-fifth letter of the Latin alphabet. Its name in English is wy, sometimes spelled wye. See V. In Ancient Greek Υψιλον (Ypsilon) was pronounced IPA [u], later on [y], now [i]. The Romans borrowed Y directly from the Greek, because they felt that V no longer adequately represented Greek [y]. The letter Y was used in Old English, as in Latin, with the value [y]; however, some think that this use was an independent invention in England created by stacking a V and an I, unrelated to the Lati ...

Including:

• Y - Codes for computing
• Y - Meanings of Y

Read more here: » Y: Encyclopedia - Y

The eleventh letter of the Latin alphabet, K, or k comes from the Greek Κ or κ (Kappa) developed from the Semitic Kap, symbol for an open hand. The Semitic soundish value /k/ was maintained in most Classic as well as Modern Languages, although Latin abandoned K almost completely, preferring C. Therefore, the Romance languages have K only in foreign words. Its name in English is kay. In the International phonetic alphabet, [k] is the symbol for the voiceless velar plosive. K - Alternative re ...

Including:

• K - Alternative representations
• K - Computing
• K - Meanings for K

Read more here: » K: Encyclopedia - K

A version of the Y combinator that can be used in call-by-value (applicative-order) evaluation is given by η-expansion of part of the ordinary Y combinator: Z = λf. (λx. f (λy. x x y)) (λx. f (λy. x x y)) The Y combinator can be expressed in the SKI-calculus as Y = S (K (S I I)) (S (S (K S) K) (K (S I I))) The simplest fixed point combinator in the SK-calculus, found by John Tr ...

See also:

Fixed point combinator, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Example, Fixed point combinator - Other fixed point combinators

Read more here: » Fixed point combinator: Encyclopedia II - Fixed point combinator - Other fixed point combinators

Consider the factorial function (under Church encoding). The usual recursive mathematical equation is fact(n) = if n=0 then 1 else n * fact(n-1) We can express a "single step" of this recursion in lambda calculus as F = λf. λx. (ISZERO x) 1 (MULT x (f (PRED x))), where "f" is a place-holder argument for the factorial function to be passed to itself. The function F performs a single step in the evaluation of the recursive formula. A ...

See also:

Fixed point combinator, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Example, Fixed point combinator - Other fixed point combinators

Read more here: » Fixed point combinator: Encyclopedia II - Fixed point combinator - Example

A version of the Y combinator that can be used in call-by-value (applicative-order) evaluation is given by η-expansion of part of the ordinary Y combinator: Z = λf. (λx. f (λy. x x y)) (λx. f (λy. x x y)) The Y combinator can be expressed in the SKI-calculus as Y = S (K (S I I)) (S (S (K S) K) (K (S I I))) The simplest fixed point combinator in the SK-calculus, found by John Tromp, is Y = S S K (S (K (S S (S (S S K)))) K) which corresponds to the lambda expres ...

See also:

Fixed point combinator, Fixed point combinator - Existence of fixed point combinators, Fixed point combinator - Example, Fixed point combinator - Other fixed point combinators

Read more here: » Fixed point combinator: Encyclopedia II - Fixed point combinator - Other fixed point combinators

List of paradoxes - Semantic paradoxes. These form a well-known (and well-studied) class having in common that any permissible assignment of semantic value (truth, reference) to an expression immediately implies the assignment of a different value. Berry paradox: What is "The first number not nameable in under ten words"? (And has it not just been named in nine?) Curry's paradox: "If this sentence is true, the world will end in a week." Epimenides paradox: A Cretan says "All Cretans ...

See also:

List of paradoxes, List of paradoxes - Logical except mathematical, List of paradoxes - Semantic paradoxes, List of paradoxes - Vagueness, List of paradoxes - Mathematical and statistical, List of paradoxes - Infinity, List of paradoxes - Geometry and topology, List of paradoxes - Psychological and rational, List of paradoxes - Physical, List of paradoxes - Philosophical, List of paradoxes - Economic

Read more here: » List of paradoxes: Encyclopedia II - List of paradoxes - Logical except mathematical

A second aspect of the Curry-Howard isomorphism is that a program whose type corresponds to a logical formula is itself analogous to a proof of that formula. Consider the two following functions of λ-calculus: K: λxy.x S: λxyz. (x z (y z)) It can be shown that any function can be created by suitable applications of K and S to each other. (See the combinatory logic article for a proof.) For example, the function B ...

See also:

Curry-Howard, Curry-Howard - Types, Curry-Howard - The type inhabitation problem, Curry-Howard - Intuitionistic logic, Curry-Howard - Hilbert-style proofs, Curry-Howard - Programs are proofs, Curry-Howard - Proof of α → α, Curry-Howard - Proof of β → α → γ → β → γ → α, Curry-Howard - Sequent calculus, Curry-Howard - Point of view of category theory

Read more here: » Curry-Howard: Encyclopedia II - Curry-Howard - Programs are proofs

Mathematical logic - First-order languages and structures. Definition. A first-order language is a collection of distinct typographical symbols classified as follows: The equality symbol ; the connectives , ; the universal quantifier and the parentheses , . A countable set of variable symbols . A set of constant symbols . A set of function symbol ...

See also:

Mathematical logic, Mathematical logic - History, Mathematical logic - Topics in mathematical logic, Mathematical logic - Some fundamental results, Mathematical logic - Technical reference, Mathematical logic - First-order languages and structures, Mathematical logic - Terms formulas and sentences, Mathematical logic - Assignment functions, Mathematical logic - Logical satisfaction, Mathematical logic - Logical implication and truth, Mathematical logic - Variable substitution, Mathematical logic - Substitutability

Read more here: » Mathematical logic: Encyclopedia II - Mathematical logic - Technical reference

The thesis can be stated as: "Every 'function which would naturally be regarded as computable' can be computed by a Turing machine." Due to the vagueness of the concept of a "function which would naturally be regarded as computable", the thesis cannot formally be proven. Disproof would be possible only if humanity found ways of building hypercomputers whose results should "naturally be regarded as computable". Any computer program can be translated into a Turing machine, and any Turing machine can be ...

See also:

Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

Read more here: » Church–Turing thesis: Encyclopedia II - Church–Turing thesis - Church–Turing thesis

Parameter - Mathematical. In mathematics, the difference in meaning between a parameter and an argument of a function is that the parameters are the symbols that are part of the function's definition, while arguments are the symbols that are supplied to the function when it is used. The value or objects assigned to the parameters by the corresponding arguments of a function or system are not reassigned during the function's evaluation. So, parameters are effectively constants during th ...

See also:

Parameter, Parameter - Types of parameter, Parameter - Mathematical, Parameter - Computer science, Parameter - Logic, Parameter - Engineering, Parameter - Analytic geometry, Parameter - Mathematical analysis, Parameter - Probability theory, Parameter - Statistics

Read more here: » Parameter: Encyclopedia II - Parameter - Types of parameter

In Unicode the capital K is codepoint U+004B and the lowercase k is U+006B. The ASCII code for capital K is 75 and for lowercase k is 107; or in binary 01001011 and 01101011, correspondingly. The EBCDIC code for capital K is 210, and for lowercase k, 146. The numeric character references in HTML and XML are "K" and "k" for upper and lower case respectively. ...

See also:

K, K - Codes for computing, K - Meanings for K

Read more here: » K: Encyclopedia II - K - Codes for computing

Formally, we start with a countably infinite set of identifiers, say {a, b, c, ..., x, y, z, x1, x2, ...}. The set of all lambda expressions can then be described by the following context-free grammar in BNF: <expr> ::= <identifier> <expr> ::= (λ <identifier>. <expr>) <expr ...

See also:

Lambda calculus, Lambda calculus - History, Lambda calculus - Informal description, Lambda calculus - Formal definition, Lambda calculus - α-conversion, Lambda calculus - β-reduction, Lambda calculus - η-conversion, Lambda calculus - Arithmetic in lambda calculus, Lambda calculus - Logic and predicates, Lambda calculus - Recursion, Lambda calculus - Computable functions and lambda calculus, Lambda calculus - Undecidability of equivalence, Lambda calculus - Lambda calculus and programming languages

Read more here: » Lambda calculus: Encyclopedia II - Lambda calculus - Formal definition

The thesis, in Turing's own words, can be stated as: "Every 'function which would naturally be regarded as computable' can be computed by a Turing machine." Due to the vagueness of the concept of a "function which would naturally be regarded as computable", the thesis cannot formally be proven. Disproof would be possible only if humanity found ways of building hypercomp ...

See also:

Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

Read more here: » Church–Turing thesis: Encyclopedia II - Church–Turing thesis - Church–Turing thesis

Before stating a precise definition of free variable and bound variable (or dummy variable), we present some examples that perhaps make these two concepts clearer than the definition would (unfortunately the term dummy variable is used by many statisticians to mean an indicator variable or some variant thereof; the name is really not apt for that purpose, but magnificently conveys the intuition behind the definition of t ...

See also:

Free variables and bound variables, Free variables and bound variables - Examples, Free variables and bound variables - Variable-binding operators, Free variables and bound variables - Formal explanation

Read more here: » Free variables and bound variables: Encyclopedia II - Free variables and bound variables - Examples

To define the set of well typed lambda terms of a given type, we introduce typing contexts which are sequences of typing assumptions of the form x:σ where x is a variable. We introduce the judgment which means that t is a term of type σ in context Γ which is given by the following typing rules: Examples of closed terms are: (I), (K), and (S). These are the typed lambda calculus represen ...

See also:

Simply typed lambda calculus, Simply typed lambda calculus - Types, Simply typed lambda calculus - Terms, Simply typed lambda calculus - Important results

Read more here: » Simply typed lambda calculus: Encyclopedia II - Simply typed lambda calculus - Terms

In Unicode the capital Y is codepoint U+0059 and the lowercase y is U+0079. The ASCII code for capital Y is 89 and for lowercase y is 121; or in binary 01011001 and 01111001, correspondingly. The EBCDIC code for capital Y is 232 and for lowercase y is 168. The numeric character references in HTML and XML are "&#89;" and "&#121;" for upper and lower case respectively. ...

See also:

Y, Y - Codes for computing, Y - Meanings of Y

Read more here: » Y: Encyclopedia II - Y - Codes for computing