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Word Problem (mathematics)
In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. A deep result of computational theory is that answering this question is in many important cases undecidable. Background and motivation In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a ''solution to the word problem''. For example, imagine that x,y,z are symbols representing real numbers - then a relevant solution to the word problem would, given the input (x \cdot y)/z \mathrel (x/z)\cdot y, produce the output EQUAL, and similarly pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Mathematics
Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006. Retrieved April 2007. A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics itself. This includes mathematical experimentation for establishing conjectures (particularly in number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants. Areas of computational mathematics Computational mathematics emerged as a distinct part of applied ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Markov Jr
Andrey, Andrej or Andrei (in Cyrillic script: Андрей, Андреј or Андрэй) is a form of Andreas/Ἀνδρέας in Slavic languages and Romanian. People with the name include: *Andrei of Polotsk ( – 1399), Lithuanian nobleman *Andrei Alexandrescu, Romanian computer programmer *Andrey Amador, Costa Rican cyclist *Andrei Arlovski, Belarusian mixed martial artist *Andrey Arshavin, Russian football player * Andrej Babiš, Czech prime minister *Andrey Belousov (born 1959), Russian politician *Andrey Bolotov, Russian agriculturalist and memoirist *Andrey Borodin, Russian financial expert and businessman *Andrei Chikatilo, prolific and cannibalistic Russian serial killer and rapist *Andrei Denisov (weightlifter) (born 1963), Israeli Olympic weightlifter *Andrey Ershov, Russian computer scientist *Andrey Esionov, Russian painter *Andrei Glavina, Istro-Romanian writer and politician *Andrei Gromyko (1909–1989), Belarusian Soviet politician and diplomat * Andrey Ivanov, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory. Life Post was born in Augustów, Suwałki Governorate, Congress Poland, Russian Empire (now Poland) into a Polish-Jewish family that immigrated to New York City in May 1904. His parents were Arnold and Pearl Post. Post had been interested in astronomy, but at the age of twelve lost his left arm in a car accident. This loss was a significant obstacle to being a professional astronomer, leading to his decision to pursue mathematics rather than astronomy. Post attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B.S. in Mathematics. After completing his Ph.D. in mathematics in 1920 at Columbia University, supervised by Cassius Jackson Keyser, he did a post-doctorate at Princeton University in the 1920–1921 academic year. Pos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systems Of Logic Based On Ordinals
''Systems of Logic Based on Ordinals'' was the PhD dissertation of the mathematician Alan Turing. Turing's thesis is not about a new type of formal logic, nor was he interested in so-called ‘ranked logic’ systems derived from ordinal or relative numbering, in which comparisons can be made between truth-states on the basis of relative veracity. Instead, Turing investigated the possibility of resolving the Godelian incompleteness condition using Cantor's method of infinites. This condition can be stated thus—in all systems with finite sets of axioms, an exclusive-or condition applies to expressive power and provability; i.e. one can have power and no proof, or proof and no power, but not both. The thesis is an exploration of formal mathematical systems after Gödel's theorem. Gödel showed that for any formal system S powerful enough to represent arithmetic, there is a theorem G which is true but the system is unable to prove. G could be added as an additional axiom to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
