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Allen's Interval Algebra
''For the type of boolean algebra called interval algebra, see Boolean algebra (structure)'' Allen's interval algebra is a calculus for temporal reasoning that was introduced by James F. Allen in 1983. The calculus defines possible relations between time intervals and provides a composition table that can be used as a basis for reasoning about temporal descriptions of events. Formal description Relations The following 13 base relations capture the possible relations between two intervals. Using this calculus, given facts can be formalized and then used for automatic reasoning. Relations between intervals are formalized as sets of base relations. The sentence : ''During dinner, Peter reads the newspaper. Afterwards, he goes to bed.'' is formalized in Allen's Interval Algebra as follows: \mbox \mathbf \mbox \mbox \mathbf \mbox In general, the number of different relations between ''n'' intervals, starting with ''n'' = 0, is 1, 1, 13, 409, 23917, 2244361..OEIS A0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse Relation
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if X and Y are sets and L \subseteq X \times Y is a relation from X to Y, then L^ is the relation defined so that yL^x if and only if xLy. In set-builder notation, :L^ = \. The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations as detailed below. As a unary operation, taking the converse (sometimes called conversion or transposition) commutes with the order-relate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commonsense Reasoning
In artificial intelligence (AI), commonsense reasoning is a human-like ability to make presumptions about the type and essence of ordinary situations humans encounter every day. These assumptions include judgments about the nature of physical objects, taxonomic properties, and peoples' intentions. A device that exhibits commonsense reasoning might be capable of drawing conclusions that are similar to humans' folk psychology (humans' innate ability to reason about people's behavior and intentions) and naive physics (humans' natural understanding of the physical world). Definitions and characterizations Some definitions and characterizations of common sense from different authors include: * "Commonsense knowledge includes the basic facts about events (including actions) and their effects, facts about knowledge and how it is obtained, facts about beliefs and desires. It also includes the basic facts about material objects and their properties." * "Commonsense knowledge differs from e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spatial Relation
A spatial relationD. M. Mark and M. J. Egenhofer (1994), "Modeling Spatial Relations Between Lines and Regions: Combining Formal Mathematical Models and Human Subjects Testing"PDF/ref> specifies how some object is located in space in relation to some reference object. When the reference object is much bigger than the object to locate, the latter is often represented by a point. The reference object is often represented by a bounding box. In Anatomy it might be the case that a spatial relation is not fully applicable. Thus, the degree of applicability is defined which specifies from 0 till 100% how strongly a spatial relation holds. Often researchers concentrate on defining the applicability function for various spatial relations. In spatial databases and geospatial topology the ''spatial relations'' are used for spatial analysis and constraint specifications. In cognitive development for walk and for catch objects, or for understand objects-behaviour; in robotic Natural Feature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Region Connection Calculus
The region connection calculus (RCC) is intended to serve for qualitative spatial representation and reasoning. RCC abstractly describes regions (in Euclidean space, or in a topological space) by their possible relations to each other. RCC8 consists of 8 basic relations that are possible between two regions: * disconnected (DC) * externally connected (EC) * equal (EQ) * partially overlapping (PO) * tangential proper part (TPP) * tangential proper part inverse (TPPi) * non-tangential proper part (NTPP) * non-tangential proper part inverse (NTPPi) From these basic relations, combinations can be built. For example, proper part (PP) is the union of TPP and NTPP. Axioms RCC is governed by two axioms. * for any region x, x connects with itself * for any region x, y, if x connects with y, y will connects with x Remark on the axioms The two axioms describe two features of the connection relation, but not the characteristic feature of the connect relation.Dong 2008 For example, we can say ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temporal Logic
In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I eat something"). It is sometimes also used to refer to tense logic, a modal logic-based system of temporal logic introduced by Arthur Prior in the late 1950s, with important contributions by Hans Kamp. It has been further developed by computer scientists, notably Amir Pnueli, and logicians. Temporal logic has found an important application in formal verification, where it is used to state requirements of hardware or software systems. For instance, one may wish to say that ''whenever'' a request is made, access to a resource is ''eventually'' granted, but it is ''never'' granted to two requestors simultaneously. Such a statement can conveniently be expressed in a temporal logic. Motivation Consider the statement "I am hungry". Though its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Tree
In computer science, an interval tree is a tree data structure to hold intervals. Specifically, it allows one to efficiently find all intervals that overlap with any given interval or point. It is often used for windowing queries, for instance, to find all roads on a computerized map inside a rectangular viewport, or to find all visible elements inside a three-dimensional scene. A similar data structure is the segment tree. The trivial solution is to visit each interval and test whether it intersects the given point or interval, which requires O(n) time, where n is the number of intervals in the collection. Since a query may return all intervals, for example if the query is a large interval intersecting all intervals in the collection, this is asymptotically optimal; however, we can do better by considering output-sensitive algorithms, where the runtime is expressed in terms of m, the number of intervals produced by the query. Interval trees have a query time of O(\log n + m) and an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overlapping Markup
In markup languages and the digital humanities, overlap occurs when a document has two or more structures that interact in a non-hierarchical manner. A document with overlapping markup cannot be represented as a tree. This is also known as concurrent markup. Overlap happens, for instance, in poetry, where there may be a metrical structure of feet and lines; a linguistic structure of sentences and quotations; and a physical structure of volumes and pages and editorial annotations. History The problem of non-hierarchical structures in documents has been recognised since 1988; resolving it against the dominant paradigm of text as a single hierarchy (an ''ordered hierarchy of content objects'' or ''OHCO'') was initially thought to be merely a technical issue, but has, in fact, proven much more difficult. In 2008, Jeni Tennison identified markup overlap as "the main remaining problem area for markup technologists". Markup overlap continues to be a primary issue in the digital study ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation Algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2''X''² of all binary relations on a set ''X'', that is, subsets of the cartesian square ''X''2, with ''R''•''S'' interpreted as the usual composition of binary relations ''R'' and ''S'', and with the converse of ''R'' as the converse relation. Relation algebra emerged in the 19th-century work of Augustus De Morgan and Charles Peirce, which culminated in the algebraic logic of Ernst Schröder. The equational form of relation algebra treated here was developed by Alfred Tarski and his students, starting in the 1940s. Tarski and Givant (1987) applied relation algebra to a variable-free treatment of axiomatic set theory, with the implication that mathematics founded on set theory could itself be conducted without variables. Definition A relation algebra is an algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relation Composition
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions. The word uncle indicates a compound relation: for a person to be an uncle, he must be the brother of a parent. In algebraic logic it is said that the relation of Uncle (x U z) is the composition of relations "is a brother of" (x B y) and "is a parent of" (y P z). U = BP \quad \text \quad xByPz \text xUz. Beginning with Augustus De Morgan, the traditional form of reasoning by syllogism has been subsumed by relational logical expressions and their composition. Definition If R \subseteq X \times Y and S \subseteq Y \times Z are two binary relations, then their composition R; S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus (other)
Calculus (from Latin ''calculus'' meaning ‘pebble’, plural '' calculī'') in its most general sense is any method or system of calculation. Calculus may refer to: Biology * ''Calculus'' (spider), a genus of the family Oonopidae * ''Caseolus calculus'', a genus and species of small land snails Mathematics * Infinitesimal calculus (or simply Calculus), which investigate motion and rates of change ** Differential calculus ** Integral calculus ** Non-standard calculus, an approach to infinitesimal calculus using Robinson's infinitesimals * Calculus of sums and differences (difference operator), also called the finite-difference calculus, a discrete analogue of "calculus" * Functional calculus, a way to apply various types of functions to operators * Schubert calculus, a branch of algebraic geometry * Tensor calculus (also called tensor analysis), a generalization of vector calculus that encompasses tensor fields ** Vector calculus (also called vector analysis), comprising ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
