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Ehrenfeucht–Fraïssé Game
In the mathematical discipline of model theory, the Ehrenfeucht–Fraïssé game (also called back-and-forth games) is a technique based on game semantics for determining whether two structures are elementarily equivalent. The main application of Ehrenfeucht–Fraïssé games is in proving the inexpressibility of certain properties in first-order logic. Indeed, Ehrenfeucht–Fraïssé games provide a complete methodology for proving inexpressibility results for first-order logic. In this role, these games are of particular importance in finite model theory and its applications in computer science (specifically computer aided verification and database theory), since Ehrenfeucht–Fraïssé games are one of the few techniques from model theory that remain valid in the context of finite models. Other widely used techniques for proving inexpressibility results, such as the compactness theorem, do not work in finite models. Ehrenfeucht–Fraïssé-like games can also be defined for othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sentence (mathematical Logic)
:''This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see Sentence (linguistics), for a less technical introductory article see Statement (logic).'' In mathematical logic, a sentence (or closed formula)Edgar Morscher, "Logical Truth and Logical Form", ''Grazer Philosophische Studien'' 82(1), pp. 77–90. of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that ''must'' be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: As the free variables of a (general) formula can range over several values, the truth value of such a formula may vary. Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic formulas by applying con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Ordering
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruno Poizat
Bruno may refer to: People and fictional characters *Bruno (name), including lists of people and fictional characters with either the given name or surname * Bruno, Duke of Saxony (died 880) * Bruno the Great (925–965), Archbishop of Cologne, Duke of Lotharingia and saint * Bruno (bishop of Verden) (920–976), German Roman Catholic bishop * Pope Gregory V (c. 972–999), born Bruno of Carinthia * Bruno of Querfurt (c. 974–1009), Christian missionary bishop, martyr and saint * Bruno of Augsburg (c. 992–1029), Bishop of Augsburg * Bruno (bishop of Würzburg) (1005–1045), German Roman Catholic bishop * Pope Leo IX (1002–1054), born Bruno of Egisheim-Dagsburg * Bruno II (1024–1057), Frisian count or margrave * Bruno the Saxon (fl. 2nd half of the 11th century), historian * Saint Bruno of Cologne (d. 1101), founder of the Carthusians * Bruno (bishop of Segni) (c. 1045–1123), Italian Roman Catholic bishop and saint * Bruno (archbishop of Trier) (died 1124), German Ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilfrid Hodges
Wilfrid Augustine Hodges, FBA (born 27 May 1941) is a British mathematician and logician known for his work in model theory. Life Hodges attended New College, Oxford (1959–65), where he received degrees in both '' Literae Humaniores'' and (Christianic) Theology. In 1970 he was awarded a doctorate for a thesis in Logic. He lectured in both Philosophy and Mathematics at Bedford College, University of London. He has held visiting appointments in the department of philosophy at the University of California and in the department of mathematics at University of Colorado. Hodges was Professor of Mathematics at Queen Mary College, University of London from 1987 to 2006 and is the author of books on logic. Honors and awards Hodges was President of the British Logic Colloquium, of the European Association for Logic, Language and Information and of the Division of Logic, Methodology, and Philosophy of Science. In 2009 he was elected a Fellow of the British Academy. Writing style Hodg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joel Spencer
Joel Spencer (born April 20, 1946) is an American mathematician. He is a combinatorialist who has worked on probabilistic methods in combinatorics and on Ramsey theory. He received his doctorate from Harvard University in 1970, under the supervision of Andrew Gleason. He is currently () a professor at the Courant Institute of Mathematical Sciences of New York University. Spencer's work was heavily influenced by Paul Erdős, with whom he coauthored many papers (giving him an Erdős number of 1). In 1963, while studying at the Massachusetts Institute of Technology, Spencer became a Putnam Fellow. In 1984 Spencer received a Lester R. Ford Award. He was an Erdős Lecturer at Hebrew University of Jerusalem in 2001. In 2012 he became a fellow of the American Mathematical Society. He was elected as a fellow of the Society for Industrial and Applied Mathematics in 2017, "for contributions to discrete mathematics and theory of computing, particularly random graphs and networks, Ramsey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrzej Ehrenfeucht
Andrzej Ehrenfeucht (, born 8 August 1932) is a Polish-American mathematician and computer scientist. Life Andrzej Ehrenfeucht formulated the Ehrenfeucht–Fraïssé game, using the back-and-forth method given in Roland Fraïssé's PhD thesis. Also named for Ehrenfeucht is the Ehrenfeucht–Mycielski sequence. In 1971 Ehrenfeucht was a founding member of the Department of Computer Science at the University of Colorado at Boulder. He currently teaches and does research at the University, where he runs a project, "breaking away", with Patricia Baggett; the project, using hands-on activities, aims at raising high-school students' interest in mathematics and technology. Two of Ehrenfeucht's students, Eugene Myers and David Haussler, contributed to the sequencing of the human genome. They, with Harold Gabow, Ross McConnell, and Grzegorz Rozenberg, spoke at a 2012 University of Colorado two-day symposium honoring Ehrenfeucht's 80th birthday. Two journal issues have come out in his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roland Fraïssé
Roland Fraïssé (; 12 March 1920 – 30 March 2008) was a French mathematical logician. Fraïssé received his doctoral degree from the University of Paris in 1953. In his thesis, Fraïssé used the back-and-forth method to determine whether two model-theoretic structures were elementarily equivalent. This method of determining elementary equivalence was later formulated as the Ehrenfeucht–Fraïssé game. Fraïssé worked primarily in relation theory. Another of his important works was the Fraïssé construction of a Fraïssé limit of finite structures. He also formulated Fraïssé's conjecture on order embeddings, and introduced the notion of compensor in the theory of posets.Petits posets : dénombrement, représentabilité par cercles et compenseurs, Roland Fraïssé and Nik Lygeros, ''Comptes Rendus de l'Académie des Sciences'', Série I 313 (1991), no. 7, 417–420 Most of his career was spent as Professor at the University of Provence in Marseille Marseille ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Back-and-forth Method
In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In particular it can be used to prove that * any two countably infinite densely ordered sets (i.e., linearly ordered in such a way that between any two members there is another) without endpoints are isomorphic. An isomorphism between linear orders is simply a strictly increasing bijection. This result implies, for example, that there exists a strictly increasing bijection between the set of all rational numbers and the set of all real algebraic numbers. * any two countably infinite atomless Boolean algebras are isomorphic to each other. * any two equivalent countable atomic models of a theory are isomorphic. * the Erdős–Rényi model of random graphs, when applied to countably infinite graphs, almost surely produces a unique graph, the Rado graph. * any two many-complete recursive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation (mathematics)
In mathematics, a relation on a set may, or may not, hold between two given set members. For example, ''"is less than"'' is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1 is an asymmetric relation, but ≥ is not. Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation defined by is neither symmetric nor antisymmetric, let alone asymmetric. ; : for all , if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. ; : for all , if then or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : for all , or . This property is sometimes called "total", which is distinct from the definitions of "total" given in the section . ; : every nonempty subset of contains a minimal element with respect to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
