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Armstrong's Axioms
Armstrong's axioms are a set of references (or, more precisely, inference rules) used to infer all the functional dependencies on a relational database. They were developed by William W. Armstrong in his 1974 paper. The axioms are sound in generating only functional dependencies in the closure of a set of functional dependencies (denoted as F^) when applied to that set (denoted as F). They are also complete in that repeated application of these rules will generate all functional dependencies in the closure F^+. More formally, let \langle R(U), F \rangle denote a relational scheme over the set of attributes U with a set of functional dependencies F. We say that a functional dependency f is logically implied by F, and denote it with F \models f if and only if for every instance r of R that satisfies the functional dependencies in F, r also satisfies f. We denote by F^ the set of all functional dependencies that are logically implied by F. Furthermore, with respect to a set of infe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inference Rule
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of inference called ''modus ponens'' takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion. Typically, a rule of inference preserves truth, a semantic property. In many-valued logic, it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. rules suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Dependency
In relational database theory, a functional dependency is a constraint between two sets of attributes in a relation from a database. In other words, a functional dependency is a constraint between two attributes in a relation. Given a relation ''R'' and sets of attributes X,Y \subseteq R, ''X'' is said to functionally determine ''Y'' (written ''X'' → ''Y'') if and only if each ''X'' value in ''R'' is associated with precisely one ''Y'' value in ''R''; ''R'' is then said to ''satisfy'' the functional dependency ''X'' → ''Y''. Equivalently, the projection \Pi_R is a function, i.e. ''Y'' is a function of ''X''. In simple words, if the values for the ''X'' attributes are known (say they are ''x''), then the values for the ''Y'' attributes corresponding to ''x'' can be determined by looking them up in ''any'' tuple of ''R'' containing ''x''. Customarily ''X'' is called the ''determinant'' set and ''Y'' the ''dependent'' set. A functional dependency FD: ''X'' → ''Y'' is called ''tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relational Database
A relational database is a (most commonly digital) database based on the relational model of data, as proposed by E. F. Codd in 1970. A system used to maintain relational databases is a relational database management system (RDBMS). Many relational database systems are equipped with the option of using the SQL (Structured Query Language) for querying and maintaining the database. History The term "relational database" was first defined by E. F. Codd at IBM in 1970. Codd introduced the term in his research paper "A Relational Model of Data for Large Shared Data Banks". In this paper and later papers, he defined what he meant by "relational". One well-known definition of what constitutes a relational database system is composed of Codd's 12 rules. However, no commercial implementations of the relational model conform to all of Codd's rules, so the term has gradually come to describe a broader class of database systems, which at a minimum: # Present the data to the user as relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Ward Armstrong
William Ward Armstrong is a Canadian mathematician and computer scientist. He earned his Ph.D. from the University of British Columbia in 1966 and is most known as the originator Armstrong's axioms of dependency in a Relational database A relational database is a (most commonly digital) database based on the relational model of data, as proposed by E. F. Codd in 1970. A system used to maintain relational databases is a relational database management system (RDBMS). Many relatio .... Works * William W. Armstrong, Yatsuka Nakamura, Piotr Rudnicki, "Armstrong's Axioms", Formalized Mathematics V. 11 No. 1, pp. 39 – 51, University of Bialystok, 2003. * Guoliang Qu, John R. Feddes, Richard N. Coleman, William W. Armstrong, Jerry L. Leonard, Method and Apparatus for estimating odor concentration using an electronic nose, US patent 6,411,905 B1, June 25, 2002. * W. W. Armstrong, D. O. Gorodnichy, Breaking Hyperplanes to fit Data with Applications to 3D World Modeling and Oil Sand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soundness
In logic or, more precisely, deductive reasoning, an argument is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical systems are sound if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system. Definition In deductive reasoning, a sound argument is an argument that is valid and all of its premises are true (and as a consequence its conclusion is true as well). An argument is valid if, assuming its premises are true, the conclusion ''must'' be true. An example of a sound argument is the following well-known syllogism: : ''(premises)'' : All men are mortal. : Socrates is a man. : ''(conclusion)'' : Therefore, Socrates is mortal. Because of the logical necessity of the conclusion, this argument is valid; and because the argument is valid and its premises are true, the argument is sound. However, an argument can be valid without ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set is closed under an operation of 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 subset 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 element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is expressively complete if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Database Theory
Database theory encapsulates a broad range of topics related to the study and research of the theoretical realm of databases and database management systems. Theoretical aspects of data management include, among other areas, the foundations of query languages, computational complexity and expressive power of queries, finite model theory, database design theory, dependency theory, foundations of concurrency control and database recovery, deductive databases, temporal and spatial databases, real-time databases, managing uncertain data and probabilistic databases, and Web data. Most research work has traditionally been based on the relational model, since this model is usually considered the simplest and most foundational model of interest. Corresponding results for other data models, such as object-oriented or semi-structured models, or, more recently, graph data models and XML, are often derivable from those for the relational model. A central focus of database theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
