|
Horn Clause
In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form that gives it useful properties for use in logic programming, formal specification, universal algebra and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951. Definition A Horn clause is a disjunctive clause (a disjunction of literals) with at most one positive, i.e. unnegated, literal. Conversely, a disjunction of literals with at most one negated literal is called a dual-Horn clause. A Horn clause with exactly one positive literal is a definite clause or a strict Horn clause; a definite clause with no negative literals is a unit clause, and a unit clause without variables is a fact; a Horn clause without a positive literal is a goal clause. The empty clause, consisting of no literals (which is equivalent to ''false''), is a goal clause. These three kinds of Horn clauses are illustrated in the follo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Computational Logic
Computational logic is the use of logic to perform or reason about computation. It bears a similar relationship to computer science and engineering as mathematical logic bears to mathematics and as philosophical logic bears to philosophy. It is an alternative term for "logic in computer science". Computational logic has also come to be associated with logic programming, because much of the early work in logic programming in the early 1970s also took place in the Department of Computational Logic in Edinburgh. It was reused in the early 1990s to describe work on extensions of logic programming in the EU Basic Research Project "Compulog" and in the associated Network of Excellence. Krzysztof Apt, who was the co-ordinator of the Basic Research Project Compulog-II, reused and generalized the term when he founded the ACM Transactions on Computational Logic in 2000 and became its first Editor-in-Chief. The term “computational logic” came to prominence with the founding of the ACM ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Reduced Product
In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let be a nonempty family of structures of the same signature σ indexed by a set ''I'', and let ''U'' be a proper filter on ''I''. The domain of the reduced product is the quotient of the Cartesian product :\prod_ S_i by a certain equivalence relation ~: two elements (''ai'') and (''bi'') of the Cartesian product are equivalent if :\left\\in U If ''U'' only contains ''I'' as an element, the equivalence relation is trivial, and the reduced product is just the direct product. If ''U'' is an ultrafilter, the reduced product is an ultraproduct. Operations from σ are interpreted on the reduced product by applying the operation pointwise. Relations are interpreted by :R((a^1_i)/,\dots,(a^n_i)/) \iff \\in U. For example, if each structure is a vector space In mathematics and physics, a vector space (also cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Variety (universal Algebra)
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called ''finitary algebraic categories''. A ''covariety'' is the class of all coalgebraic structures of a given signature. Terminology A variety of algebras should not be confused with an algebraic variety, which means a set of solutions to a system of polynomial equations. They are formally quite distinct and their theories have little in common. The term "variety of algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quasivariety
In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class. __TOC__ Definition A ''trivial algebra'' contains just one element. A quasivariety is a class ''K'' of algebras with a specified signature satisfying any of the following equivalent conditions: # ''K'' is a pseudoelementary class closed under subalgebras and direct products. # ''K'' is the class of all models of a set of quasi-identities, that is, implications of the form s_1 \approx t_1 \land \ldots \land s_n \approx t_n \rightarrow s \approx t, where s, s_1, \ldots, s_n,t, t_1, \ldots, t_n are terms built up from variables using the operation symbols of the specified signature. # ''K'' contains a trivial algebra and is closed under isomorphisms, subalgebras, and reduced products. # ''K'' contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and ultraproducts. Examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quasi-identity
In universal algebra, a quasi-identity is an implication of the form :''s''1 = ''t''1 ∧ … ∧ ''s''''n'' = ''t''''n'' → ''s'' = ''t'' where ''s''1, ..., ''s''''n'', ''t''1, ..., ''t''''n'', ''s'', and ''t'' are terms built up from variables using the operation symbols of the specified signature. A quasi-identity amounts to a conditional equation for which the conditions themselves are equations. Alternatively, it can be seen as a disjunction of inequations and one equation ''s''1 ≠ ''t''1 ∨ ... ∨ ''s''''n'' ≠ ''t''''n'' ∨ ''s'' = ''t''—that is, as a definite Horn clause. A quasi-identity with ''n'' = 0 is an ordinary identity or equation, so quasi-identities are a generalization of identities. See also * Quasivariety In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class. __TOC__ Definition A ''trivial algebra'' contains just one element. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
NP-complete
In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any input to the problem, the output is either "yes" or "no". # When the answer is "yes", this can be demonstrated through the existence of a short (polynomial length) ''solution''. # The correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # The problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. Hence, if we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Boolean Satisfiability Problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) asks whether there exists an Interpretation (logic), interpretation that Satisfiability, satisfies a given Boolean logic, Boolean Formula (mathematical logic), formula. In other words, it asks whether the formula's variables can be consistently replaced by the values TRUE or FALSE to make the formula evaluate to TRUE. If this is the case, the formula is called ''satisfiable'', else ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem. This means that all problems in the complexity class NP (complexity), NP, which includes a wide range of natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Time
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P-complete
In computational complexity theory, a decision problem is P-complete ( complete for the complexity class P) if it is in P and every problem in P can be reduced to it by an appropriate reduction. The notion of P-complete decision problems is useful in the analysis of: * which problems are difficult to parallelize effectively, * which problems are difficult to solve in limited space. specifically when stronger notions of reducibility than polytime-reducibility are considered. The specific type of reduction used varies and may affect the exact set of problems. Generically, reductions stronger than polynomial-time reductions are used, since all languages in P (except the empty language and the language of all strings) are P-complete under polynomial-time reductions. If we use NC reductions, that is, reductions which can operate in polylogarithmic time on a parallel computer with a polynomial number of processors, then all P-complete problems lie outside NC and so cannot be effec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Horn-satisfiability
In formal logic, Horn-satisfiability, or HORNSAT, is the problem of deciding whether a given conjunction of propositional Horn clauses is satisfiable or not. Horn-satisfiability and Horn clauses are named after Alfred Horn. A Horn clause is a clause with at most one positive literal, called the ''head'' of the clause, and any number of negative literals, forming the ''body'' of the clause. A Horn formula is a propositional formula formed by conjunction of Horn clauses. Horn satisfiability is actually one of the "hardest" or "most expressive" problems which is known to be computable in polynomial time, in the sense that it is a P-complete problem.Author's 2008 draft version see p.213f) The extension of the problem for quantified Horn formulae can be also solved in polynomial time. The Horn satisfiability problem can also be asked for propositional many-valued logics. The algorithms are not usually linear, but some are polynomial; see Hähnle (2001 or 2003) for a survey. Algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
