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MU Puzzle
The MU puzzle is a puzzle stated by Douglas Hofstadter and found in '' Gödel, Escher, Bach'' involving a simple formal system called "MIU". Hofstadter's motivation is to contrast reasoning within a formal system (ie., deriving theorems) against reasoning about the formal system itself. MIU is an example of a Post canonical system and can be reformulated as a string rewriting system. The puzzle Suppose there are the symbols , , and which can be combined to produce strings of symbols. The MU puzzle asks one to start with the "axiomatic" string and transform it into the string using in each step one of the following transformation rules: : Solution The puzzle cannot be solved: it is impossible to change the string into by repeatedly applying the given rules. In other words, MU is not a theorem of the MIU formal system. To prove this, one must step "outside" the formal system itself. In order to prove assertions like this, it is often beneficial to look for an invariant; that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Douglas Hofstadter
Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics. His 1979 book '' Gödel, Escher, Bach: An Eternal Golden Braid'' won both the Pulitzer Prize for general nonfiction"General Nonfiction" . ''Past winners and finalists by category''. The Pulitzer Prizes. Retrieved March 17, 2012. and a (at that time called The American Book Award) for Science. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independence Results
Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the status of a dependent territory. The commemoration of the independence day of a country or nation celebrates when a country is free from all forms of foreign colonialism; free to build a country or nation without any interference from other nations. Definition of independence Whether the attainment of independence is different from revolution has long been contested, and has often been debated over the question of violence as legitimate means to achieving sovereignty. In general, revolutions aim only to redistribute power with or without an element of emancipation,such as in democratization ''within'' a state, which as such may remain unaltered. For example, the Mexican Revolution (1910) chiefly refers to a multi-factional conflict that even ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unsolvable Puzzles
"Unsolvable" is the twenty-first episode of the first season of the American television police sitcom series ''Brooklyn Nine-Nine''. Written by co-executive producer Prentice Penny and directed by Ken Whittingham, it aired on Fox in the United States on March 18, 2014. In this episode, Jake decides to take on an 8-year-old case that is deemed "unsolvable" and seeks Terry's help in solving it; Amy, planning a romantic vacation with her boyfriend Teddy, tries to obscure her true intentions from Holt; and Boyle, downcast after the end of his relationship, is told of a great secret. The episode was seen by an estimated 2.50 million household viewers and gained a 1.1/3 ratings share among adults aged 18–49, according to Nielsen Media Research. The episode received mostly positive reviews from critics, who praised Andy Samberg's performance. Plot When Jake is allowed the weekend off because of a hot streak in solving cases, he decides to take on an 8-year old cold case that everyon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic Puzzles
A logic puzzle is a puzzle deriving from the mathematical field of 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 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 and knaves" puzzles, which involve knights, who a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unrestricted Grammar
In automata theory, the class of unrestricted grammars (also called semi-Thue, type-0 or phrase structure grammars) is the most general class of grammars in the Chomsky hierarchy. No restrictions are made on the productions of an unrestricted grammar, other than each of their left-hand sides being non-empty. This grammar class can generate arbitrary recursively enumerable languages. Formal definition An unrestricted grammar is a formal grammar G = (N, T, P, S), where * N is a finite set of nonterminal symbols, * T is a finite set of terminal symbols with N and T disjoint,Actually, T\cap N=\emptyset is not strictly necessary since unrestricted grammars make no real distinction between the two. The designation exists purely so that one knows when to stop generating sentential forms of the grammar; more precisely, the language L(G) recognized by G is restricted to strings of terminal symbols. * P is a finite set of production rules of the form \alpha \to \beta , where \alpha and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursive Definition
In mathematics and computer science, a recursive definition, or inductive definition, is used to define the elements in a set in terms of other elements in the set ( Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set. A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function ''n''! is defined by the rules :0! = 1. :(''n'' + 1)! = (''n'' + 1)·''n''!. This definition is valid for each natural number ''n'', because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure for computing the value of the function ''n''!, starting from ''n'' = 0 and proceeding onwards with ''n'' = 1, ''n'' = 2, ''n'' = 3 etc. The recursion theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Susanna S
Susanna may refer to: People * Susanna (Book of Daniel), a portion of the Book of Daniel and its protagonist * Susanna (disciple), a disciple of Jesus * Susanna (given name), a feminine given name (including a list of people with the name) Film and TV * ''Suzanna'' (film), a 1923 American film directed by F. Richard Jones * ''Suzanne'' (1932 film), a French film directed by Léo Joannon and Raymond Rouleau * ''Susanna'' (1967 film), Hong Kong film directed by Ho Meng Hua * ''Suzanne'' (1980 film), Canadian drama film directed by Robin Spry * ''Susanna'' (2000 film), Indian Malayalam film directed by T. V. Chandran Music * ''Susanna'' (Stradella), an oratorio by Alessandro Stradella * ''Susanna'' (Handel), an oratorio by George Frideric Handel * "Susanna" (The Art Company song), English version of their song "Suzanne" Other * ''Susanna'' - plant genus, currently relegated to ''Amellus'' and ''Felicia'' * Susanna, Missouri, a community in the United States See also * Sus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rules Of Inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called ''modus ponens'' takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion. Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. rules suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined abstraction, system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive Symbol (formal), symbols (which collectively form an Alphabet (computer science), alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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