|
Trace Monoid
In computer science, a trace is an equivalence class of strings, wherein certain letters in the string are allowed to commute, but others are not. Traces generalize the concept of strings by relaxing the requirement for all the letters to have a definite order, instead allowing for indefinite orderings in which certain reshufflings could take place. In an opposite way, traces generalize the concept of sets with multiplicities by allowing for specifying some incomplete ordering of the letters rather than requiring complete equivalence under all reorderings. The trace monoid or free partially commutative monoid is a monoid of traces. Traces were introduced by Pierre Cartier and Dominique Foata in 1969 to give a combinatorial proof of MacMahon's master theorem. Traces are used in theories of concurrent computation, where commuting letters stand for portions of a job that can execute independently of one another, while non-commuting letters stand for locks, synchronization poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, applied disciplines (including the design and implementation of Computer architecture, hardware and Software engineering, software). Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and preventing security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D (both from C to D), then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding Equivalence class, quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. Definition The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions of congruence can be made for group (mathematics), groups, ring (mathematics), rings, vector spaces, module (mathematics), modules, semigroups, lattice (order), lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concatenation
In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalizations of concatenation theory, also called string theory, string concatenation is a primitive notion. Syntax In many programming languages, string concatenation is a binary infix operator, and in some it is written without an operator. This is implemented in different ways: * Overloading the plus sign + Example from C#: "Hello, " + "World" has the value "Hello, World". * Dedicated operator, such as . in PHP, & in Visual Basic, and , , in SQL. This has the advantage over reusing + that it allows implicit type conversion to string. * string literal concatenation, which means that adjacent strings are concatenated without any operator. Example from C: "Hello, " "World" has the value "Hello, World". In many scientific publications or standards the con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Closure
In mathematics, the transitive closure of a homogeneous binary relation on a set (mathematics), set is the smallest Relation (mathematics), relation on that contains and is Transitive relation, transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets is the unique minimal element, 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". More formally, the transitive closure of a binary relation on a set is the smallest (w.r.t. ⊆) transitive relation on such that ⊆ ; see . We have = if, and only if, itself is transitive. Conversely, transitive reduction adduces a minimal relation from a given relation such that they have the same closure, that is, ; however, many differen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Transitive Closure
In mathematics, a subset of a given set is closed under an operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest superset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set equipped with one or several methods for producing elements of from other elements of . Operations and (partial) multivariate function are examples of such methods. If is a topological space, the limit of a sequence of eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleene Star
In mathematical logic and theoretical computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation on a Set (mathematics), set to generate a set of all finite-length strings that are composed of zero or more repetitions of members from . It was named after American mathematician Stephen Cole Kleene, who first introduced and widely used it to characterize Automata theory, automata for regular expressions. In mathematics, it is more commonly known as the free monoid construction. Definition Given a set V, define :V^=\ (the set consists only of the empty string), :V^=V, and define recursively the set :V^=\ for each i>0. V^i is called the i-th power of V, it is a shorthand for the Concatenation#Concatenation of sets of strings, concatenation of V by itself i times. That is, ''V^i'' can be understood to be the set of all strings that can be represented as the concatenation of i members from V. The definition of Kleene star on V is : V^*=\bigcup_V^i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Monoid
In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thread joins) for providing rendezvous points between a set of independently executing processes or threads. History monoids occur in the theory of concurrent computation, and provide a low-level mathematical foundation for process calculi, such as CSP the language of communicating sequential processes, or CCS, the calculus of communicating systems. History monoids were first presented by M.W. Shields.M.W. Shields "Concurrent Machines", ''Computer Journal'', (1985) 28 pp. 449–465. History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a Set (mathematics), set of objects where some pairs of the objects are in some sense "related". The objects are represented by 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. 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'' means that ''A'' owes money to ''B'', then this graph is directed, because owing mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
