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First-order Query
In descriptive complexity, a query is a mapping from structures of one signature to structures of another vocabulary. Neil Immerman, in his book Descriptive Complexity, "use the concept of query as the fundamental paradigm of computation" (p. 17). Given signatures \sigma and \tau, we define the set of structures on each language, \mboxsigma/math> and \mboxtau/math>. A query is then any mapping I : \mboxsigma\to \mboxtau/math> Computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ... can then be phrased in terms of the power of the mathematical logic necessary to express a given query. Order-independent queries A query is order-independent if the ordering of objects in the structure does not affect the results of the query. In databases, these queri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Complexity
Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the formal language, languages in them. For example, PH (complexity), PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic. This connection between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods and providing additional evidence that the main complexity classes are somehow "natural" and not tied to the specific abstract machines used to define them. Specifically, each logical system produces a set of query (complexity), queries expressible in it. The queries – when restricted to finite structures – correspond to the computational problems of traditional complexity theory. The first main result of descriptive com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic. Definition Formally, a (single-sorted) signature can be defined as a 4-tuple \sigma = \left(S_, S_, S_, \operatorname\right), where S_ and S_ are disjoint sets not containing any other basic logical symbols, called respectively * '' function symbols'' (examples: +, \times), * ''s'' or '' predicates'' (examples: \,\leq, \, \in), * '' constant symbols'' (examples: 0, 1), and a function \operatorname : S_ \cup S_ \to \N which assigns a natural number called ''arity'' to every function or relation symbol. A function or relation symbol is called n-ary if its arity is n. Some authors define a nullary (0-ary) function symbol as ''constant symbol'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neil Immerman
Neil Immerman (born 24 November 1953, Manhasset, New York) is an American theoretical computer science, theoretical computer scientist, a professor of computer science at the University of Massachusetts Amherst.Faculty directory: Neil Immerman Computer Science Department, University of Massachusetts Amherst, retrieved 2010-01-23. He is one of the key developers of descriptive complexity, an approach he is currently applying to research in model checking, database theory, and computational complexity theory. Professor Immerman is an editor of the ''SIAM Journal on Computing'' and of ''Logical Methods in Computer Science''. He received B.S. and M.S. degrees from Yale University in 1974 and his Ph.D. from Cornell University in 1980 under the supervision of Juris Hartmanis, a Turing Award winner at Cornell. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structure (mathematical Logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures of first-order theories with no relation symbols. Model theory has a different scope that encompasses more arbitrary first-order theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a '' semantic model'' when one discusses the notion in the more general setting of mathematical models. Log ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Query
Generic or generics may refer to: In business * Generic term, a common name used for a range or class of similar things not protected by trademark * Generic brand, a brand for a product that does not have an associated brand or trademark, other than the trading name of the business providing the product * Generic trademark, a trademark that sometimes or usually replaces a common term in colloquial usage * Generic drug, a drug identified by its chemical name rather than its brand name In computer programming * Generic function, a computer programming entity made up of all methods having the same name * Generic programming, a computer programming paradigm based on method/functions or classes defined irrespective of the concrete data types used upon instantiation ** Generics in Java In linguistics *A pronoun or other word used with a less specific meaning, such as: ** generic ''you'' ** generic ''he'' or generic ''she'' ** generic ''they'' * Generic mood, a grammatical mood used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
