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Entscheidungsproblem

In mathematics and computer science, the Entscheidungsproblem is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statement is universally valid, i.e. valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic. In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notion of "effectively calculable" is captured by the functions computable by a Turing machine; this assumption is now known as the Church–Turing thesis. The origin of the Entscheidungsproblem goes back to Gottfried Leibniz, who in the seventeenth century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements.

He realized that the first step would have to be a clean formal language, much of his subsequent work was directed towards that goal. In 1928, David Hilbert and Wilhelm Ackermann posed the question in the form outlined above. In continuation of his "program", Hilbert posed three questions at an international conference in 1928, the third of which became known as "Hilbert's Entscheidungsproblem." In 1929, Moses Schönfinkel published one paper on special cases of the decision problem, prepared by Paul Bernays. As late as 1930, Hilbert believed. Before the question could be answered, the notion of "algorithm" had to be formally defined; this was done by Alonzo Church in 1935 with the concept of "effective calculability" based on his λ-calculus and by Alan Turing the next year with his concept of Turing machines. Turing recognized that these are equivalent models of computation; the negative answer to the Entscheidungsproblem was given by Alonzo Church in 1935–36 and independently shortly thereafter by Alan Turing in 1936.

Church proved that there is no computable function which decides for two given λ-calculus expressions whether they are equivalent or not. He relied on earlier work by Stephen Kleene. Turing reduced the question of the existence of a'general method' which decides whether any given Turing Machine halts or not to the question of the existence of an'algorithm' or'general method' able to solve the Entscheidungsproblem. If'Algorithm' is understood as being equivalent to a Turing Machine, with the answer to the latter question negative, the question about the existence of an Algorithm for the Entscheidungsproblem must be negative. In his 1936 paper, Turing says: "Corresponding to each computing machine'it' we construct a formula'Un' and we show that, if there is a general method for determining whether'Un' is provable there is a general method for determining whether'it' prints 0"; the work of both Church and Turing was influenced by Kurt Gödel's earlier work on his incompleteness theorem by the method of assigning numbers to logical formulas in order to reduce logic to arithmetic.

The Entscheidungsproblem is related to Hilbert's tenth problem, which asks for an algorithm to decide whether Diophantine equations have a solution. The non-existence of such an algorithm, established by Yuri Matiyasevich in 1970 implies a negative answer to the Entscheidungsproblem; some first-order theories are algorithmically decidable. The general first-order theory of the natural numbers expressed in Peano's axioms cannot be decided with an algorithm, however. Having practical decision procedures for classes of logical formulas is of considerable interest for program verification and circuit verification. Pure Boolean logical formulas are decided using SAT-solving techniques based on the DPLL algorithm. Conjunctive formulas over linear real or rational arithmetic can be decided using the simplex algorithm, formulas in linear integer arithmetic can be decided using Cooper's algorithm or William Pugh's Omega test. Formulas with negations and disjunctions combine the difficulties of satisfiability testing with that of decision of conjunctions.

Real polynomial arithmetic known as the theory of real closed fields, is decidable. Automated theorem proving Hilbert's second problem Oracle machine Turing's proof David Hilbert and Wilhelm Ackermann. Grundzüge der theoretischen Logik. Springer-Verlag, ISBN 0-8218-2024-9. Alonzo Church, "An unsolvable problem of elementary number theory", American Journal of Mathematics, 58, pp 345–363 Alonzo Church, "A note on the Entscheidungsproblem", Journal of Symbolic Logic, 1, pp 40–41. Martin Davis, 2000, Engines of Logic, W. W. Norton & Company, London, ISBN 0-393-32229-7 pbk. Alan Turing, "On Computable Numbers, with an Appli

Atlantykron

Atlantykron is an annual Summer Academy of Learning in Romania, sponsored by David Lewis Anderson's World Genesis Foundation, with support from the Romanian National Commission for UNESCO. Atlantykron is held on an island on the Danube River near the village and ancient Roman ruins of Capidava in Romania; the first Atlantykron summer academy was held in the summer of 1989. Since Atlantykron has attracted hundreds of youth and teachers from all around the world, providing them the opportunity to meet and learn together with famous scientists, artists and other professionals in a learning experience held in a wilderness setting. In 2001 Atlantykron was the host of Eurocon, the annual European science fiction convention coordinated by the European Science Fiction Society. 1989: Alexandru Mironov, Aurel Carasel, Sorin Repanovici, Gabriel Grosu, Simona Vladareanu, Aida Gusan Cristian Panfilov 1990: Valentin M. Ionescu, Alexandru Mironov, Aurel Carasel, Sorin Repanovici, Ovidiu Petcu 1991: Stefan Ghidoveanu, Mihaela Muraru Mandrea, Dan Merisca, Lucian Merisca, Romulus Bărbulescu, George Anania 1992: Valentin Nicolau, Roberto Quaglia, Mihaela Muraru Mandrea 1993: Florin Munteanu, Dan Milici, Bridget Wilkeson, Aurel Manole, Victor Sutac 1994: Aurel Manole, Bridget Wilkeson 1995: Florin Munteanu, Pierre de Hillerin 1996: Roberto Quaglia, Cristian Parghie 1997: Alain Le Bussy, Dan Milici 1998: Mihnea Muraru Mandrea 1999: David Lewis Anderson, Mihai Manea, Emil Strainu 2000: Edie Statescu, Mihai Manea, Emil Strainu 2001: Joe Haldeman, Ion Hobana, Norman Spinrad, David Lewis Anderson, Mircea Nanu-Muntean, Adriana Gheorghe 2002: Randy Gordon, Mihaela Muraru Mandrea 2003: Robert Sheckley, Dan Farcas 2004: Lucian Biro, Wilhelm Frey, Iona Frey 2005: Theodor Vasile, Vilmos Zsombori, Cristian Carstoiu, Viorel Lazar, Valentin Tanase 2006: Sibylle Colberg, Wilhelm Frey, Paul Rosner, Daniela Pascu 2007: Chan Chow Wah, Dan Lucas, Gyuri Pascu, Norman Rosner 2008: Joel Castellenos, Peter Moon, Viviana Vladutescu, Casper Werner 2009: Edward Belbruno, Corneliu Chisu, Mary Ann Martini, Cristian Gretcu, Caresella Craciun The Atlantykron summer academy is coordinated by Sorin Repanovici, Vice President of David Lewis Anderson's World Genesis Foundation, which became the key organization coordinating international participation and funding for Atlantykron beginning in 1999.

The Academy was founded in 1989 by Aurel Carasel and Sorin Repanovici. In 2009 the Academy celebrated its 20th anniversary in which Sorin Repanovici received the "Lifetime of Hope" award for his personal and professional contributions in creating new hope and opportunities for youth in the world; the Lifetime of Hope award was established by the Norman Rosner family in honor of Paul Rosner, president of the U. S. Kickboxing Association and Senior Director of the World Genesis Foundation; the primary program organizers today are: Center for Complexity Studies Cygnus Scientific Society Living Through Arts Romanian Sports and Culture Association World Genesis Foundation United States Kickboxing Association Official Website of the Atlantykron Summer Academy Historical Photo Archive of the Atlantykron Summer Academy Historical Video Archive of the Atlantykron Summer Academy

List of counties of Scotland 1890–1975

This is a list of counties of Scotland created by the Local Government Act 1889 and abolished in 1975 by the Local Government Act 1973. The list includes the county town and population density. Traditionally cities were parts of larger counties. Edinburgh was in Midlothian, Aberdeen in Aberdeenshire, Glasgow in Lanarkshire. Counties of Scotland List of counties of Scotland by area in 1951 List of counties of Scotland by population in 1951 List of counties of Scotland by population in 1971 List of Scottish counties by highest point List of Scottish council areas by area List of Scottish council areas by highest point "Scotland Table 1" A Vision of Britain. Retrieved 27 December 2007. Keay, J. & Keay, J. Collins Encyclopaedia of Scotland. London. HarperCollins. "Scotland Table 1" A Vision of Britain. Retrieved 27 December 2007

Muswell Hill, Buckinghamshire

Muswell Hill is a 0.2 hectare geological Site of Special Scientific Interest north-west of Brill in Buckinghamshire. The local planning authorities are Buckinghamshire County Council and Aylesbury Vale District Council, it is listed by the Joint Nature Conservation Committee as a Geological Conservation Review site. This site has sandy ironstones, it is problematic as their precise age and the circumstances of deposition are uncertain, but they are thought to be early Cretaceous, with late Jurassic underlying layers. There is considerable potential for further research on dating the transition between the two periods

Menzoberranzan: City of Intrigue

Menzoberranzan: City of Intrigue is a system-neutral supplement to the Dungeons & Dragons role-playing game published towards the end of 4th edition. Menzoberranzan: City of Intrigue details the drow city Menzoberranzan. There are six chapters in this book: Campaign of Intrigue The Way of Lolth Drow Factions City of Spiders The Northdark Be a DrowShannon Appelcline commented that "Menzoberranzan: City of Intrigue brings together geographical details of the Realms that were available two decades earlier in FOR2: The Drow of the Underdark and the original Menzoberranzan, it contains some information being published in'Council of Spiders'. Alongside this reprinted material — covering the setting of Menzoberranzan, the houses of Menzoberranzan, the drow of the Realms — Menzoberranzan: City of Intrigue advances the timeline of the city; this includes details on how the War of the Spider Queen changed the city and new material on the Spellplague. The result turns the system-neutral Menzoberranzan supplement into an era-neutral supplement that allows players to run Menzoberranzan in any era, from its 2e origins to the 4e present-day.

Some of Menzoberranzan moves beyond the city into the Underdark that surrounds it: the Northdark. This brings together a number of classic realms, including Mithral Hall from R. A. Salvatore's novels, the ruins of Hellgate Keep from Hellgate Keep, the Dungeon of Death from Dungeon of Death. Most of the locales in the Northdark are described in just a paragraph or two, but nonetheless the section links the UnderRealms together". Menzoberranzan: City of Intrigue was published on August 21, 2012 as a hardcover book, it was made available as a PDF or a softcover book through the DMs Guild. Menzoberranzan: City of Intrigue was part of the tenth season of Encounters, Council of Spiders. Appelcline wrote that "when Menzoberranzan was first announced in early 2012 as part of the Rise of the Underdark event, it was called a'Campaign Setting'; that would have made it the fifth Campaign Setting for D&D 4e and the third setting for the Forgotten Realms, following on from Neverwinter Campaign Setting the previous year.

Wizards had announced the end of D&D 4e on January 9, 2012 when they told The New York Times that they were working on a new version of the game. Following that, just three more 4e books appeared: Heroes of the Elemental Chaos, Halls of Undermountain, Into the Unknown: The Dungeon Survival Handbook. Menzoberranzan: City of Intrigue still featured the 4e trade dress, but it was instead published as an edition-neutral supplement, marking the end of the almost-four year reign of D&D 4e. The'Campaign Setting' became a'City of Intrigue' around the same time, to show how the new book was different from the 4e Campaign books that had preceded it. Like, the Underdark drow city of Menzoberranzan: City of Intrigue was a popular locale in the Forgotten Realms, it had first received serious attention in R. A. Salvatore's novel, Homeland, it was featured in a dense, boxed RPG set called Menzoberranzan. Menzoberranzan: City of Intrigue was part of a massive multimedia crossover called Rise of the Underdark, which had begun a few months previous with the final 4e roleplaying supplement, Into the Unknown: The Dungeon Survival Handbook.

Menzoberranzan was tied to two Rise of the Underdark releases: Encounters Season 10,'Council of Spiders' provided a drow PC adventure, while the Drow Treachery Fortune Cards offered new drow-centric Fortune Cards where players could cause troubles for each other". In 2013, Menzoberranzan: City of Intrigue was nominated for three ENnie Awards: "Product of the Year", "Best Setting", "Best Supplement"; the book was nominated for "Best Roleplaying Supplement" in the 2013 Origins Awards. John Baichtal of Wired complimented the "particularly excellent" illustrations of the noble families, stated that "this book is just the resource you need to run a campaign in the city or to inspire you to create your own evil city."John ONeill, for Black Gate, wrote "one of my favorite RPG settings of all time is Menzoberranzan, the 1992 boxed set from TSR that drew liberally from R. A. Salvatore’s best-selling Drizzt Do’Urden novels. Written by Ed Greenwood and Douglas Niles, the box detailed the famous City of Spiders, the subterranean birthplace of the drow ranger, in three thick books and a set of gorgeous maps.

Packed with 20,000 drow inhabitants, hundreds of thousands of humanoid slaves, countless secrets and simmering rivalries, the home of the drow was an ideal adventure site for intrepid players. Released nearly 20 years ago for second edition AD&D, Menzoberranzan has not seen an update since and has been out of print for over 15 years, it was featured in the popular Menzoberranzan PC game from SSI/DreamForge, part of their Forgotten Realms product line, in 1994, prominently in the six volume War of the Spider Queen novels, but it’s been far too long since my favorite underdark city-state appeared in a new edition. The wait is over. Wizards of the Coast has released an updated version in Menzoberranzan: City of Intrigue, now available in hardcover". Marshall Honorof, for The Escapist, wrote "if you're a fan of classic Dungeons & Dragons games, you have some fond memories of the Underdark. Rise of the Underdark, the latest D&D campaign, gives Dungeon Masters and players the tools and setting necessary to journey into the bowels of Faerûn in the hopes of thwarting the Spider Queen

Hans Remmer

Hans Remmer was a Luftwaffe ace and recipient of the Knight's Cross of the Iron Cross during World War II. The Knight's Cross of the Iron Cross was awarded to recognise extreme battlefield bravery or successful military leadership. Hans Rammer was killed on 2 April 1944 after he bailed out of his Bf 109, his parachute failed to open and he fell to his death, he was posthumously awarded the Knight's Cross of the Iron Cross on 9 June 1944. During his career he was credited with all against Western forces. Flugzeugführerabzeichen Front Flying Clasp of the Luftwaffe Ehrenpokal der Luftwaffe Iron Cross 2nd Class 1st Class German Cross in Gold on 31 August 1943 as Oberleutnant in the I./Jagdgeschwader 27 Knight's Cross of the Iron Cross on 9 June 1944 as Hauptmann and Staffelkapitän of the 1./Jagdgeschwader 27 World War 2 Awards.com Aces of the Luftwaffe