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To Mock A Mockingbird
''To Mock a Mockingbird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic'' (1985, {{isbn, 0-19-280142-2) is a book by the mathematician and logician Raymond Smullyan. It contains many nontrivial recreational puzzles of the sort for which Smullyan is well known. It is also a gentle and humorous introduction to combinatory logic and the associated metamathematics, built on an elaborate ornithological metaphor. Combinatory logic, functionally equivalent to the lambda calculus, is a branch of symbolic logic having the expressive power of set theory, and with deep connections to questions of computability and provability. Smullyan's exposition takes the form of an imaginary account of two men going into a forest and discussing the unusual "birds" (combinators) they find there (bird watching was a hobby of one of the founders of combinatory logic, Haskell Curry, and another founder Moses Schönfinkel's name means beautiful bird). Each species of bird ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mock A Mocking Bird
Mock is an imitation, usually of lesser quality Mock may refer to: Names *Mock (surname) *Mock, or Duncan Stump, a member of the band Mock & Toof *Mock, a character in the Japanese anime series ''Mock & Sweet'' Places *Mock, Washington, a ghost town Imitations *Mockery, imitation to express ridiculing derision *Mock object, a programming object that mimics the behavior of real objects in controlled ways *Mock trial, an act or imitation trial See also *''Mock - 1'', a 1998 album by Mocking Shadows *"Mock", a 2015 song by The Story So Far from ''The Story So Far (The Story So Far album), The Story So Far'' *''Mock the Week'', a British topical comedy panel show broadcast on BBC Two Synonyms *Fake (other) *Imaginary (other) *Insult *Parody *Pretending (other) *Simulation Derived terms *Mockup *Mocker (other) *Mock orange (other) *Mockery (other) *Mock olive Possible misspellings *Mack (other) *Meck (other) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinator
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 logic t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Puzzle Books
A puzzle is a game, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together ( or take them apart) in a logical way, in order to arrive at the correct or fun solution of the puzzle. There are different genres of puzzles, such as crossword puzzles, word-search puzzles, number puzzles, relational puzzles, and logic puzzles. The academic study of puzzles is called enigmatology. Puzzles are often created to be a form of entertainment but they can also arise from serious mathematical or logical problems. In such cases, their solution may be a significant contribution to mathematical research. Etymology The ''Oxford English Dictionary'' dates the word ''puzzle'' (as a verb) to the end of the 16th century. Its earliest use documented in the ''OED'' was in a book titled ''The Voyage of Robert Dudley...to the West Indies, 1594–95, narrated by Capt. Wyatt, by himself, and by Abram Kendall, master'' (published circa 1595). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1985 Non-fiction Books
The year 1985 was designated as the International Youth Year by the United Nations. Events January * January 1 ** The Internet's Domain Name System is created. ** Greenland withdraws from the European Economic Community as a result of a new agreement on fishing rights. * January 7 – Japan Aerospace Exploration Agency launches ''Sakigake'', Japan's first interplanetary spacecraft and the first deep space probe to be launched by any country other than the United States or the Soviet Union. * January 15 – Tancredo Neves is elected president of Brazil by the Congress, ending the 21-year military rule. * January 20 – Ronald Reagan is privately sworn in for a second term as President of the United States. * January 27 – The Economic Cooperation Organization (ECO) is formed, in Tehran. * January 28 – The charity single record "We Are the World" is recorded by USA for Africa. February * February 4 – The border between Gibraltar and Spain reopen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites". In logic, many paradoxes exist that are known to be invalid arguments, yet are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself, and showed that attempts to found set theory on the identification ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brain Teaser
A brain teaser is a form of puzzle that requires thought to solve. It often requires thinking in unconventional ways with given constraints in mind; sometimes it also involves lateral thinking. Logic puzzles and riddles are specific types of brain teasers. One of the earliest known brain teaser enthusiasts was the Greek mathematician Archimedes. He devised mathematical problems for his contemporaries to solve. Example :Q: ''If three hens lay three eggs in three days, how many eggs does a (statistical) hen lay in one day?'' :A1: ''One third.'' (Note: 3 hens = 3 eggs / 3 days → 3 hens = (3 / 3) (eggs / days) → 1 hen = (1 / 3) (egg / days)) :A2: ''Zero or one'' (it's hard to lay a third of an egg). One can argue about the answers of many brain teasers; in the given example with hens, one might claim that all the eggs in the question were laid in the first day, so the answer would be three. :Q: ''Mary's father has five daughters: 1. Nana, 2. Nene, 3. Nini, 4. Nono. What is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Puzzle
A logic puzzle is a puzzle deriving from the mathematics, mathematical field of deductive reasoning, deduction. History The logic puzzle was first produced by Charles Lutwidge Dodgson, who is better known under his pen name Lewis Carroll, the author of ''Alice's Adventures in Wonderland''. In his book ''The Game of Logic'' he introduced a game to solve problems such as confirming the conclusion "Some greyhounds are not fat" from the statements "No fat creatures run well" and "Some greyhounds run well". Puzzles like this, where we are given a list of premises and asked what can be deduced from them, are known as syllogisms. Dodgson goes on to construct much more complex puzzles consisting of up to 8 premises. In the second half of the 20th century mathematician Raymond Smullyan, Raymond M. Smullyan continued and expanded the branch of logic puzzles with books such as ''The Lady or the Tiger?'', ''To Mock a Mockingbird'' and ''Alice in Puzzle-Land''. He popularized the "knights an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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(\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Combinator
In mathematics and computer science in general, a '' fixed point'' of a function is a value that is mapped to itself by the function. In combinatory logic for computer science, a fixed-point combinator (or fixpoint combinator) is a higher-order function \textsf that returns some fixed point of its argument function, if one exists. Formally, if the function ''f'' has one or more fixed points, then : \textsf\ f = f\ (\textsf\ f)\ , and hence, by repeated application, : \textsf\ f = f\ (f\ ( \ldots f\ (\textsf\ f) \ldots))\ . Y combinator In the classical untyped lambda calculus, every function has a fixed point. A particular implementation of fix is Curry's paradoxical combinator Y, represented by : \textsf = \lambda f. \ (\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))\ .Throughout this article, the syntax rules given in Lambda calculus#Notation are used, to save parentheses.For an arbitrary lambda term ''f'', the fixed-point property can be validated by beta reducing the left- and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
B, C, K, W System
The B, C, K, W system is a variant of combinatory logic that takes as primitive the combinators B, C, K, and W. This system was discovered by Haskell Curry in his doctoral thesis ''Grundlagen der kombinatorischen Logik'', whose results are set out in Curry (1930). Definition The combinators are defined as follows: * B ''x y z'' = ''x'' (''y z'') * C ''x y z'' = ''x z y'' * K ''x y'' = ''x'' * W ''x y'' = ''x y y'' Intuitively, * B ''x y'' is the composition of ''x'' and ''y''; * C ''x'' is ''x'' with the flipped arguments order; * K ''x'' is the "constant ''x''" function, which discards the next argument; * W duplicates its second argument for the doubled application to the first. Thus, it "joins" its first argument's two expectations for input into one. Connection to other combinators In recent decades, the SKI combinator calculus, with only two primitive combinators, K and S, has become the canonical approach to combinatory logic. B, C, and W can be expressed in terms of S an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gödel's Incompleteness Theorems
Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic is the study of logic, 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 for ... that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible. The first incompleteness theorem states that no consistency, consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
