|
Newman's Lemma
In mathematics, in the theory of rewriting systems, Newman's lemma, also commonly called the diamond lemma, states that a terminating (or strongly normalizing) abstract rewriting system (ARS), that is, one in which there are no infinite reduction sequences, is confluent if it is locally confluent. In fact a terminating ARS is confluent precisely when it is locally confluent. Equivalently, for every binary relation with no decreasing infinite chains and satisfying a weak version of the diamond property, there is a unique minimal element in every connected component of the relation considered as a graph. Today, this is seen as a purely combinatorial result based on well-foundedness due to a proof of Gérard Huet in 1980. Newman's original proof was considerably more complicated. Diamond lemma In general, Newman's lemma can be seen as a combinatorial result about binary relations → on a set ''A'' (written backwards, so that ''a'' → ''b'' means that ''b'' is below ''a'') wit ... [...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] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wellfoundedness
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s R m'' (for instance, "''s'' is not smaller than ''m''") for any ''s'' ∈ ''S''. In other words, a relation is well founded if :(\forall S \subseteq X)\; \neq \emptyset \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel m) Some authors include an extra condition that ''R'' is set-like, i.e., that the elements less than any given element form a set. Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, which can be proved when there is no infinite sequence ''x''0, ''x''1, ''x''2, ... of elements of ''X'' such that ''x''''n''+1 ''R'' ''x''n for every natural number ''n''. In order theory, a partial order is called well-founded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Cohn
Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutative rings.Independent Ancestry and early life He was the only child of Jewish parents, James (or Jakob) Cohn, owner of an import business, and Julia (''née'' Cohen), a schoolteacher.Autobiography Both of his parents were born in Hamburg, as were three of his grandparents. His ancestors came from various parts of Germany. His father fought in the German army in World War I; he was wounded several times and awarded the Iron Cross. A street in Hamburg is named in memory of his mother.De Morgan When he was born, his parents were living with his mother's mother in Isestraße. After her death in October 1925, the family moved to a rented flat in a new building in Lattenkamp, in the Winterhude quarter. He attended a kindergarten then, in Ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Closure
In mathematics, the transitive closure of a binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets it is the unique minimal transitive superset of . For example, if is a set of airports and means "there is a direct flight from airport to airport " (for and in ), then the transitive closure of on is the relation such that means "it is possible to fly from to in one or more flights". Informally, the ''transitive closure'' gives you the set of all places you can get to from any starting place. More formally, the transitive closure of a binary relation on a set is the transitive relation on set such that contains and is minimal; see . If the binary relation itself is transitive, then the transitive closure is that same binary relation; otherwise, the transitive closure is a different relation. Conversely, transitive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Closure
In mathematics, the reflexive closure of a binary relation ''R'' on a set ''X'' is the smallest reflexive relation on ''X'' that contains ''R''. For example, if ''X'' is a set of distinct numbers and ''x R y'' means "''x'' is less than ''y''", then the reflexive closure of ''R'' is the relation "''x'' is less than or equal to ''y''". Definition The reflexive closure ''S'' of a relation ''R'' on a set ''X'' is given by :S = R \cup \left\ In English, the reflexive closure of ''R'' is the union of ''R'' with the identity relation on ''X''. Example As an example, if :X = \left\ :R = \left\ then the relation R is already reflexive by itself, so it does not differ from its reflexive closure. However, if any of the pairs in R was absent, it would be inserted for the reflexive closure. For example, if on the same set X :R = \left\ then the reflexive closure is :S = R \cup \left\ = \left\ . See also * Transitive closure * Symmetric closure References * Franz Baader and Tobi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-founded Relation
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s R m'' (for instance, "''s'' is not smaller than ''m''") for any ''s'' ∈ ''S''. In other words, a relation is well founded if :(\forall S \subseteq X)\; \neq \emptyset \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel m) Some authors include an extra condition that ''R'' is set-like, i.e., that the elements less than any given element form a set. Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, which can be proved when there is no infinite sequence ''x''0, ''x''1, ''x''2, ... of elements of ''X'' such that ''x''''n''+1 ''R'' ''x''n for every natural number ''n''. In order theory, a partial order is called well-founded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemme Newman Demonstration
The Lemme is a 35 km torrent, a right tributary of the Orba, which flows through the Province of Alessandria in northern Italy. Geography Its source is near Monte Calvo; from there it passes through the communes of Fraconalto, Voltaggio, Carrosio, Gavi, San Cristoforo, Francavilla Bisio, Basaluzzo and finally Predosa where it enters the Orba. Main tributaries left hand tributaries * rio delle Acque Striate; * rio Morsone: it starts near Monte Tobbio and meets the Lemme in Voltaggio; * rio Ardana: its upper valley includes Bosio, from which flows northwards meeting the Lemme in Gavi; * rio Riolo: from San Cristoforo it runs not faraway from the Lemme meeting it near its confluence with the Orba.''Carta Tecnica Regionale'' raster 1:10.000 (vers.3.0); publisher:Regione Piemonte - 2007 right hand tributaries * rio Carbonasca: from Fraconalto goes till Voltaggio where it meets the Lemme; * torrente Neirone: * torrente Riasco: it collects the waters flowing in the area of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
JACM
The ''Journal of the ACM'' is a Peer review, peer-reviewed scientific journal covering computer science in general, especially theoretical aspects. It is an official journal of the Association for Computing Machinery. Its current editor-in-chief is Venkatesan Guruswami. The journal was established in 1954 and "computer scientists universally hold the ''Journal of the ACM'' in high esteem". See also * ''Communications of the ACM'' References External links * Publications established in 1954 Computer science journals Association for Computing Machinery academic journals Bimonthly journals English-language journals {{compu-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gérard Huet
Gérard Pierre Huet (; born 7 July 1947) is a French computer scientist, linguist and mathematician. He is senior research director at INRIA and mostly known for his major and seminal contributions to type theory, programming language theory and to the theory of computation. Biography Gérard Huet graduated from the Université Denis Diderot (Paris VII), Case Western Reserve University, and the Université de Paris. He is senior research director at INRIA, a member of the French Academy of Sciences, and a member of Academia Europaea. Formerly he was a visiting professor at Asian Institute of Technology in Bangkok, a visiting professor at Carnegie Mellon University, and a guest researcher at SRI International. He is the author of a unification algorithm for simply typed lambda calculus, and of a complete proof method for Church's theory of types ( constrained resolution). He worked on the Mentor program editor in 1974–1977 with Gilles Kahn. He worked on the Knuth†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-foundedness
In mathematics, a binary relation ''R'' is called well-founded (or wellfounded) on a class ''X'' if every non-empty subset ''S'' ⊆ ''X'' has a minimal element with respect to ''R'', that is, an element ''m'' not related by ''s R m'' (for instance, "''s'' is not smaller than ''m''") for any ''s'' ∈ ''S''. In other words, a relation is well founded if :(\forall S \subseteq X)\; \neq \emptyset \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel m) Some authors include an extra condition that ''R'' is set-like, i.e., that the elements less than any given element form a set. Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, which can be proved when there is no infinite sequence ''x''0, ''x''1, ''x''2, ... of elements of ''X'' such that ''x''''n''+1 ''R'' ''x''n for every natural number ''n''. In order theory, a partial order is called well-founded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
