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Valuation Algebra
The term "information algebra" refers to mathematical techniques of Data processing, information processing. Classical information theory goes back to Claude Shannon. It is a theory of information transmission, looking at communication and storage. However, it has not been considered so far that information comes from different sources and that it is therefore usually combined. It has furthermore been neglected in classical information theory that one wants to extract those parts out of a piece of information that are relevant to specific questions. A mathematical phrasing of these operations leads to an algebra of information, describing basic modes of information processing. Such an algebra involves several formalisms of computer science, which seem to be different on the surface: relational databases, multiple systems of formal logic or numerical problems of linear algebra. It allows the development of generic procedures of information processing and thus a unification of basic m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Data Processing
Data processing is the collection and manipulation of digital data to produce meaningful information. Data processing is a form of ''information processing'', which is the modification (processing) of information in any manner detectable by an observer.Data processing is distinct from '' word processing'', which is manipulation of text specifically rather than data generally. Functions Data processing may involve various processes, including: * Validation – Ensuring that supplied data is correct and relevant. * Sorting – "arranging items in some sequence and/or in different sets." * Summarization (statistical) or (automatic) – reducing detailed data to its main points. * Aggregation – combining multiple pieces of data. * Analysis – the "collection, organization An organization or organisation (English in the Commonwealth of Nations, Commonwealth English; American and British English spelling differences#-ise, -ize (-isation, -izat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Semiring Valued Algebra
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction \lor as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers \N (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid. Terminology Some authors define semirings without the requirement for there to be a 0 or 1. This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggesting that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Probabilistic Argumentation
Probabilistic argumentation refers to different formal frameworks pertaining to probabilistic logic. All share the idea that qualitative aspects can be captured by an underlying logic, while quantitative aspects of uncertainty can be accounted for by probabilistic measures. Probabilistic argumentation labellings The framework of "probabilistic labellings" refers to probability spaces where the sample space is a set of labellings of argumentation graphs . A labelling of an argumentation graph associates any argument of the graph with a label to reflect the acceptability of the argument within the graph. For example, an argument can be associated with a label "in" (the argument is accepted), "out" (the argument is rejected), or "und" (the status of the argument is undecided — neither accepted nor rejected). Consequently, the approach of probabilistic labellings associates any argument with the probability of a label to reflect the probability of the argument to be labelled as suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Scott Information System
In domain theory, a branch of mathematics and computer science, a Scott information system is a primitive kind of logical deductive system often used as an alternative way of presenting Scott domains. Definition A Scott information system, ''A'', is an ordered triple (T, Con, \vdash) * T \mbox * Con \subseteq \mathcal_f(T) \mbox T * \subseteq (Con \setminus \lbrace \emptyset \rbrace)\times T satisfying # \mbox a \in X \in Con\mboxX \vdash a # \mbox X \vdash Y \mboxY \vdash a \mboxX \vdash a # \mboxX \vdash a \mbox X \cup \ \in Con # \forall a \in T : \ \in Con # \mboxX \in Con \mbox X^\prime\, \subseteq X \mboxX^\prime \in Con. Here X \vdash Y means \forall a \in Y, X \vdash a. Examples Natural numbers The return value of a partial recursive function, which either returns a natural number or goes into an infinite recursion, can be expressed as a simple Scott information system as follows: * T := \mathbb * Con := \ \cup \ * X \vdash a\iff a \in X. That is, the result can eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Scott Domain
In the mathematical fields of order and domain theory, a Scott domain is an algebraic, bounded-complete and directed-complete partial order (dcpo). They are named in honour of Dana S. Scott, who was the first to study these structures at the advent of domain theory. Scott domains are very closely related to algebraic lattices, being different only in possibly lacking a greatest element. They are also closely related to Scott information systems, which constitute a "syntactic" representation of Scott domains. While the term "Scott domain" is widely used with the above definition, the term "domain" does not have such a generally accepted meaning and different authors will use different definitions; Scott himself used "domain" for the structures now called "Scott domains". Additionally, Scott domains appear with other names like "algebraic semilattice" in some publications. Originally, Dana Scott demanded a complete lattice, and the Russian mathematician Yuri Yershov construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Valuation Algebra
The term "information algebra" refers to mathematical techniques of Data processing, information processing. Classical information theory goes back to Claude Shannon. It is a theory of information transmission, looking at communication and storage. However, it has not been considered so far that information comes from different sources and that it is therefore usually combined. It has furthermore been neglected in classical information theory that one wants to extract those parts out of a piece of information that are relevant to specific questions. A mathematical phrasing of these operations leads to an algebra of information, describing basic modes of information processing. Such an algebra involves several formalisms of computer science, which seem to be different on the surface: relational databases, multiple systems of formal logic or numerical problems of linear algebra. It allows the development of generic procedures of information processing and thus a unification of basic m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear System
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems. Definition A general deterministic system can be described by an operator, , that maps an input, , as a function of to an output, , a type of black box description. A system is linear if and only if it satisfies the superposition principle, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.) The superposition principle means that a linear combination of inputs to the system produces a linear com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Predicate Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Polyadic Algebra
Polyadic algebras (more recently called Halmos algebras) are algebraic structures introduced by Paul Halmos. They are related to first-order logic analogous to the relationship between Boolean algebras and propositional logic (see Lindenbaum–Tarski algebra). There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras (when equality is part of the logic) and Lawvere's functorial semantics (a categorical approach). References Further reading *Paul Halmos, ''Algebraic Logic'', Chelsea Publishing The Chelsea Publishing Company was a publisher of mathematical books, based in New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on on ..., New York (1962) Algebraic logic {{mathlogic-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cylindric Algebra
In mathematics, the notion of cylindric algebra, developed by Alfred Tarski, arises naturally in the Algebraic logic, algebraization of first-order logic with equality. This is comparable to the role Boolean algebra (structure), Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equipped with additional cylindrification operations that model Quantification (logic), quantification and equality (mathematics), equality. They differ from polyadic algebras in that the latter do not model equality. The cylindric algebra should not be confused with the measure theory, measure theoretic concept ''cylindrical algebra'' that arises in the study of cylinder set measures and the cylindrical σ-algebra. Definition of a cylindric algebra A cylindric algebra of dimension \alpha (where \alpha is any ordinal number) is an algebraic structure (A,+,\cdot,-,0,1,c_\kappa,d_)_ such that (A,+,\cdot,-,0,1) is a Boolean algebra (structure), Boolean algebra, c_\kappa a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
