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Olog
The theory of ologs is an attempt to provide a rigorous mathematical framework for knowledge representation, construction of scientific models and data storage using category theory, linguistic and graphical tools. Ologs were introduced in 2010 by David Spivak, a research scientist in the Department of Mathematics, MIT. Etymology The term "olog" is short for "ontology log". "Ontology" derives from '' onto-'', from the Greek '' ὤν, ὄντος'' "being; that which is", present participle of the verb '' εἰμί'' "be", and -λογία, -logia: ''science'', ''study'', ''theory''. Mathematical formalism An olog \mathcal for a given domain is a category whose objects are boxes labeled with phrases (more specifically, singular indefinite noun phrases) relevant to the domain, and whose morphisms are directed arrows between the boxes, labeled with verb phrases also relevant to the domain. These noun and verb phrases combine to form sentences that express relationships between obj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ontology (information Science)
In computer science and information science, an ontology encompasses a representation, formal naming, and definition of the categories, properties, and relations between the concepts, data, and entities that substantiate one, many, or all domains of discourse. More simply, an ontology is a way of showing the properties of a subject area and how they are related, by defining a set of concepts and categories that represent the subject. Every academic discipline or field creates ontologies to limit complexity and organize data into information and knowledge. Each uses ontological assumptions to frame explicit theories, research and applications. New ontologies may improve problem solving within that domain. Translating research papers within every field is a problem made easier when experts from different countries maintain a controlled vocabulary of jargon between each of their languages. For instance, the definition and ontology of economics is a primary concern in Marxist econo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Language
Greek ( el, label=Modern Greek, Ελληνικά, Elliniká, ; grc, Ἑλληνική, Hellēnikḗ) is an independent branch of the Indo-European family of languages, native to Greece, Cyprus, southern Italy (Calabria and Salento), southern Albania, and other regions of the Balkans, the Black Sea coast, Asia Minor, and the Eastern Mediterranean. It has the longest documented history of any Indo-European language, spanning at least 3,400 years of written records. Its writing system is the Greek alphabet, which has been used for approximately 2,800 years; previously, Greek was recorded in writing systems such as Linear B and the Cypriot syllabary. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic, and many other writing systems. The Greek language holds a very important place in the history of the Western world. Beginning with the epics of Homer, ancient Greek literature includes many works of lasting impo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
-logia
''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Greek ending in ('). The earliest English examples were anglicizations of the French '' -logie'', which was in turn inherited from the Latin '' -logia''. The suffix became productive in English from the 18th century, allowing the formation of new terms with no Latin or Greek precedent. The English suffix has two separate main senses, reflecting two sources of the suffix in Greek: *a combining form used in the names of school or bodies of knowledge, e.g., ''theology'' (loaned from Latin in the 14th century) or ''sociology''. In words of the type ''theology'', the suffix is derived originally from (''-log-'') (a variant of , ''-leg-''), from the Greek verb (''legein'', 'to speak')."-logy." ''The Concise Oxford Dictionary of English Etymology''. Oxford University Press, 1986. retrieved 20 August 2008. The suffix has the sense of "the character or deportment of one who speaks or treats ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olog
The theory of ologs is an attempt to provide a rigorous mathematical framework for knowledge representation, construction of scientific models and data storage using category theory, linguistic and graphical tools. Ologs were introduced in 2010 by David Spivak, a research scientist in the Department of Mathematics, MIT. Etymology The term "olog" is short for "ontology log". "Ontology" derives from '' onto-'', from the Greek '' ὤν, ὄντος'' "being; that which is", present participle of the verb '' εἰμί'' "be", and -λογία, -logia: ''science'', ''study'', ''theory''. Mathematical formalism An olog \mathcal for a given domain is a category whose objects are boxes labeled with phrases (more specifically, singular indefinite noun phrases) relevant to the domain, and whose morphisms are directed arrows between the boxes, labeled with verb phrases also relevant to the domain. These noun and verb phrases combine to form sentences that express relationships between obj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ontology Language
In computer science and artificial intelligence, ontology languages are formal languages used to construct ontologies. They allow the encoding of knowledge about specific domains and often include reasoning rules that support the processing of that knowledge. Ontology languages are usually declarative languages, are almost always generalizations of frame languages, and are commonly based on either first-order logic or on description logic. Classification of ontology languages Classification by syntax Traditional syntax ontology languages * Common Logic - and its dialects * CycL * DOGMA (Developing Ontology-Grounded Methods and Applications) * F-Logic (Frame Logic) * FO-dot (First-order logic extended with types, arithmetic, aggregates and inductive definitions) * KIF (Knowledge Interchange Format) ** Ontolingua based on KIF * KL-ONE * KM programming language * LOOM (ontology) * OCML (Operational Conceptual Modelling Language) * OKBC ( Open Knowledge Base Connectivity) * PLI ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functors
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous function, continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a Linguistics, linguistic context; see function word. Definition Let ''C'' and ''D'' be category (mathematics), categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operad Theory
In mathematics, an operad is a structure that consists of abstract Operation (mathematics), operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a Group (mathematics), group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by Michael Boardman, J. Michael Boardman and Rainer M. Vogt in 1969 and by J. Peter May in 1970. The word "operad" was created by May as a portmanteau of "operations" and "monad (category theory), monad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial from an abstract simplicial complex, the former is often called a geometric simplicial complex.'', Section 4.3'' Definitions A simplicial complex \mathcal is a set of simplices that satisfies the following conditions: :1. Every face of a simplex from \mathcal is also in \mathcal. :2. The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial ''k''-complex \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modeling Language
A modeling language is any artificial language that can be used to express information or knowledge or systems in a structure that is defined by a consistent set of rules. The rules are used for interpretation of the meaning of components in the structure. Overview A modeling language can be graphical or textual. * ''Graphical'' modeling languages use a diagram technique with named symbols that represent concepts and lines that connect the symbols and represent relationships and various other graphical notation to represent constraints. * ''Textual'' modeling languages may use standardized keywords accompanied by parameters or natural language terms and phrases to make computer-interpretable expressions. An example of a graphical modeling language and a corresponding textual modeling language is EXPRESS. Not all modeling languages are executable, and for those that are, the use of them doesn't necessarily mean that programmers are no longer required. On the contrary, executab ... [...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 , 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 morphism \eta_X : F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
