Encyclopedia - Lazy Evaluation: Encyclopedia Ii - Lazy Evaluation - Delayed Evaluation
Delayed evaluation is used particularly in functional languages. When using delayed evaluation, an expression is not evaluated as soon as...   » Read the article

Encyclopedia - Fixed Point Combinator: Encyclopedia Ii - Fixed Point Combinator - Existence Of Fixed Point Combinators
In certain formalizations of mathematics, such as the untyped lambda calculus and combinatorial calculus, every expression can be conside...   » Read the article

Encyclopedia - Omega: Encyclopedia - Omega
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 me...   » Read the article

Encyclopedia - Church–turing Thesis: Encyclopedia - Church–turing Thesis
In computability theory the Church–Turing thesis, Church's thesis, Church's conjecture or Turing's thesis, named after Alonzo Church an...   » Read the article

Encyclopedia - Y: Encyclopedia - Y
Y is the twenty-fifth letter of the Latin alphabet. Its name in English is wy, sometimes spelled wye. See V. In Ancient Greek ΥψιλοÎ...   » Read the article

Encyclopedia - K: Encyclopedia - K
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 op...   » Read the article

Encyclopedia - Fixed Point Combinator: Encyclopedia Ii - Fixed Point Combinator - Other Fixed Point Combinators
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 or...   » Read the article

Encyclopedia - Fixed Point Combinator: Encyclopedia Ii - Fixed Point Combinator - Example
Consider the factorial function (under Church encoding). The usual recursive mathematical equation is fact(n) = if n=0 then 1 else n * f...   » Read the article

Encyclopedia - Fixed Point Combinator: Encyclopedia Ii - Fixed Point Combinator - Other Fixed Point Combinators
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 or...   » Read the article

Encyclopedia - List Of Paradoxes: Encyclopedia Ii - List Of Paradoxes - Logical Except Mathematical
List of paradoxes - Semantic paradoxes. These form a well-known (and well-studied) class having in common that any permissible assignme...   » Read the article

Encyclopedia - Curry-howard: Encyclopedia Ii - Curry-howard - Programs Are Proofs
A second aspect of the Curry-Howard isomorphism is that a program whose type corresponds to a logical formula is itself analogous to a pr...   » Read the article

Encyclopedia - Mathematical Logic: Encyclopedia Ii - Mathematical Logic - Technical Reference
Mathematical logic - First-order languages and structures. Definition. A first-order language is a collection of distinct typographica...   » Read the article

Encyclopedia - Church–turing Thesis: Encyclopedia Ii - Church–turing Thesis - Church–turing Thesis
The thesis can be stated as: "Every 'function which would naturally be regarded as computable' can be computed by a Turing machine." Du...   » Read the article

Encyclopedia - Parameter: Encyclopedia Ii - Parameter - Types Of Parameter
Parameter - Mathematical. In mathematics, the difference in meaning between a parameter and an argument of a function is that the param...   » Read the article

Encyclopedia - K: Encyclopedia Ii - K - Codes For Computing
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...   » Read the article

Encyclopedia - Lambda Calculus: Encyclopedia Ii - Lambda Calculus - Formal Definition
Formally, we start with a countably infinite set of identifiers, say {a, b, c, ..., x, y, z, x1, x2, ...}. The set of all lambda expressi...   » Read the article

Encyclopedia - Church–turing Thesis: Encyclopedia Ii - Church–turing Thesis - Church–turing Thesis
The thesis, in Turing's own words, can be stated as: "Every 'function which would naturally be regarded as computable' can be computed b...   » Read the article

Encyclopedia - Free Variables And Bound Variables: Encyclopedia Ii - Free Variables And Bound Variables - Examples
Before stating a precise definition of free variable and bound variable (or dummy variable), we present some examples that perhaps make t...   » Read the article

Encyclopedia - Simply Typed Lambda Calculus: Encyclopedia Ii - Simply Typed Lambda Calculus - Terms
To define the set of well typed lambda terms of a given type, we introduce typing contexts which are sequences of typing assumptions of ...   » Read the article

Encyclopedia - Y: Encyclopedia Ii - Y - Codes For Computing
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...   » Read the article

Encyclopedia - K: Encyclopedia Ii - K - Alternative Representations
Kilo represents the letter K in the NATO phonetic alphabet. In international Morse code the letter K is DahDitDah: - · - In Braille the ...   » Read the article

Encyclopedia - Raymond Smullyan: Encyclopedia Ii - Raymond Smullyan - Selected Publications
Raymond Smullyan - Popular. The Tao is Silent The Chess Mysteries of Sherlock Holmes The Chess Mysteries of the Arabian Knights The La...   » Read the article

Encyclopedia - Curry-howard: Encyclopedia Ii - Curry-howard - Types
Following the lambda calculus, we will use λx.E to denote the function with formal parameter x and body E. When applied to an argument, ...   » Read the article

Encyclopedia - Church–turing Thesis: Encyclopedia Ii - Church–turing Thesis - History
In his 1943 paper Recursive Predicates and Quantifiers (reprinted in The Undecidable, p. 255) Stephen Kleene first proposed his "THESIS I...   » Read the article

Encyclopedia - Curry-howard: Encyclopedia Ii - Curry-howard - The Type Inhabitation Problem
It's clear that λ-expressions can have quite complicated types. One might ask whether there is a λ-expression with any given type. The ...   » Read the article

Encyclopedia - Curry-howard: Encyclopedia Ii - Curry-howard - Hilbert-style Proofs
One simple way to formally characterize intuitionistic logic is as follows. It has two axiom schemas. All formulas of the form α → β...   » Read the article

Encyclopedia - Curry-howard: Encyclopedia Ii - Curry-howard - Sequent Calculus
Hilbert-style proofs can be difficult to construct. A more intuitive way to prove theorems of logic is Gentzen's 'sequent calculus'. Sequ...   » Read the article

Encyclopedia - Raymond Smullyan: Encyclopedia Ii - Raymond Smullyan - Quotations
Raymond Smullyan - Quotations About Smullyan. I now introduce Professor Smullyan, who will prove to you that either he doesn't exist or...   » Read the article

Encyclopedia - Curry-howard: Encyclopedia Ii - Curry-howard - Intuitionistic Logic
Although it is true that all inhabited types correspond to theorems of logic, the converse is not true. Even if we restrict our attention...   » Read the article

Encyclopedia - Curry-howard: Encyclopedia Ii - Curry-howard - Point Of View Of Category Theory
Starting from the point of view that functional programming supports programming languages that are typed and have higher-order functions...   » Read the article

Encyclopedia - Simply Typed Lambda Calculus: Encyclopedia Ii - Simply Typed Lambda Calculus - Types
The types of the simply typed lambda calculus are constructed from base types (or type variables) and given types σ,Ï„ we can construct...   » Read the article

Encyclopedia - Mathematical Logic: Encyclopedia Ii - Mathematical Logic - History
Mathematical logic was the name given by Giuseppe Peano to what is also known as symbolic logic. In essentials, it is still the logic of ...   » Read the article

Encyclopedia - Lambda Calculus: Encyclopedia Ii - Lambda Calculus - Arithmetic In Lambda Calculus
There are several possible ways to define the natural numbers in lambda calculus, but by far the most common are the Church numerals, whi...   » Read the article

Encyclopedia - Lambda Calculus: Encyclopedia Ii - Lambda Calculus - Logic And Predicates
By convention, the following two definitions (known as Church booleans) are used for the boolean values TRUE and FALSE: TRUE := λ ...   » Read the article

Encyclopedia - Lambda Calculus: Encyclopedia Ii - Lambda Calculus - Informal Description
In lambda calculus, every expression stands for a function with a single argument; the argument of the function is in turn a function wit...   » Read the article

Encyclopedia - Free Variables And Bound Variables: Encyclopedia Ii - Free Variables And Bound Variables - Formal Explanation
Variable-binding mechanisms occur in different contexts in mathematics, logic and computer science but in all cases they are purely synta...   » Read the article

Encyclopedia - Church–turing Thesis: Encyclopedia Ii - Church–turing Thesis - Success Of The Thesis
Since that time, many other formalisms for describing effective computability have been proposed, including recursive functions, the lamb...   » Read the article

Encyclopedia - Church–turing Thesis: Encyclopedia Ii - Church–turing Thesis - Philosophical Implications
The Church–Turing thesis has some profound implications for the philosophy of mind. There are also some important open questions which ...   » Read the article

Encyclopedia - Lambda Calculus: Encyclopedia Ii - Lambda Calculus - Recursion
Recursion is the definition of a function using the function itself; on the face of it, lambda calculus does not allow this. However, thi...   » Read the article

Encyclopedia - Lambda Calculus: Encyclopedia Ii - Lambda Calculus - Undecidability Of Equivalence
There is no algorithm which takes as input two lambda expressions and outputs TRUE or FALSE depending on whether or not the two expressio...   » Read the article

Encyclopedia - Parameter: Encyclopedia Ii - Parameter - Mathematical Analysis
In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form In this formula, t is the ar...   » Read the article

Encyclopedia - Parameter: Encyclopedia Ii - Parameter - Probability Theory
In probability theory, one may describe the distribution of a random variable as belonging to a family of probability distributions, dist...   » Read the article

Encyclopedia - Parameter: Encyclopedia Ii - Parameter - Analytic Geometry
In analytic geometry, curves are often given as the image of some function. The argument of the function is invariably called "the parame...   » Read the article

Encyclopedia - Mathematical Logic: Encyclopedia Ii - Mathematical Logic - Topics In Mathematical Logic
The main areas of mathematical logic include model theory, proof theory and recursion theory (often now referred to as computability theo...   » Read the article

Encyclopedia - Lambda Calculus: Encyclopedia Ii - Lambda Calculus - Lambda Calculus And Programming Languages
Most programming languages are equivalent to the lambda calculus extended with some additional programming language constructs. The class...   » Read the article

Encyclopedia - List Of Paradoxes: Encyclopedia Ii - List Of Paradoxes - Mathematical And Statistical
List of paradoxes - Infinity. Burali-Forti paradox: If the ordinal numbers formed a set, it would be an ordinal number which is smalle...   » Read the article

Encyclopedia - Parameter: Encyclopedia Ii - Parameter - Statistics
In statistics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on ...   » Read the article