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Knuth–Bendix Completion Algorithm
The Knuth–Bendix completion algorithm (named after Donald Knuth and Peter Bendix) is a semi-decision algorithm for transforming a set of equations (over terms) into a confluent term rewriting system. When the algorithm succeeds, it effectively solves the word problem for the specified algebra. Buchberger's algorithm for computing Gröbner bases is a very similar algorithm. Although developed independently, it may also be seen as the instantiation of Knuth–Bendix algorithm in the theory of polynomial rings. Introduction For a set ''E'' of equations, its deductive closure () is the set of all equations that can be derived by applying equations from ''E'' in any order. Formally, ''E'' is considered a binary relation, () is its rewrite closure, and () is the equivalence closure of (). For a set ''R'' of rewrite rules, its deductive closure ( ∘ ) is the set of all equations that can be confirmed by applying rules from ''R'' left-to-right to both sides until they are literal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald E
Donald is a masculine given name derived from the Gaelic name ''Dòmhnall''.. This comes from the Proto-Celtic *''Dumno-ualos'' ("world-ruler" or "world-wielder"). The final -''d'' in ''Donald'' is partly derived from a misinterpretation of the Gaelic pronunciation by English speakers, and partly associated with the spelling of similar-sounding Germanic names, such as ''Ronald''. A short form of ''Donald'' is ''Don''. Pet forms of ''Donald'' include ''Donnie'' and ''Donny''. The feminine given name ''Donella'' is derived from ''Donald''. ''Donald'' has cognates in other Celtic languages: Modern Irish ''Dónal'' (anglicised as ''Donal'' and ''Donall'');. Scottish Gaelic ''Dòmhnall'', ''Domhnull'' and ''Dòmhnull''; Welsh '' Dyfnwal'' and Cumbric ''Dumnagual''. Although the feminine given name ''Donna'' is sometimes used as a feminine form of ''Donald'', the names are not etymologically related. Variations Kings and noblemen Domnall or Domhnall is the name of many ancie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Transitive Closure
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 elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Presented Group
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set ''R'' of relations among those generators. We then say ''G'' has presentation :\langle S \mid R\rangle. Informally, ''G'' has the above presentation if it is the "freest group" generated by ''S'' subject only to the relations ''R''. Formally, the group ''G'' is said to have the above presentation if it is isomorphic to the quotient of a free group on ''S'' by the normal subgroup generated by the relations ''R''. As a simple example, the cyclic group of order ''n'' has the presentation :\langle a \mid a^n = 1\rangle, where 1 is the group identity. This may be written equivalently as :\langle a \mid a^n\rangle, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) have the same number of elements (cardinality) as does . Furthermore, itself is both a left coset and a right coset. The number of left cosets of in is equal to the number of right cosets of in . This common value is called the index of in and is usually denoted by . Cosets are a basic tool in the study of groups; for example, they play a central role in Lagrange's theorem that states that for any finite group , the number of elements of every subgroup of divides the number of elements of . Cosets of a particular type of subgroup (a normal subgroup) can be used as the elements of another group called a quotient group or factor group. Cosets also appear in other areas of mathematics such as vector spaces and error-correcting codes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Group Theory
In mathematics, computational group theory is the study of group (mathematics), groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracted interest because for many interesting groups (including most of the sporadic groups) it is impractical to perform calculations by hand. Important algorithms in computational group theory include: * the Schreier–Sims algorithm for finding the order (group theory), order of a permutation group * the Todd–Coxeter algorithm and Knuth–Bendix algorithm for coset enumeration * the product-replacement algorithm for finding random elements of a group Two important computer algebra systems (CAS) used for group theory are GAP computer algebra system, GAP and Magma computer algebra system, Magma. Historically, other systems such as CAS (for character theory) and Cayley computer algebra system, Cayley (a predecessor of Magma) were important. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Word Problem (mathematics)
In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. A deep result of computational theory is that answering this question is in many important cases undecidable. Background and motivation In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a ''solution to the word problem''. For example, imagine that x,y,z are symbols representing real numbers - then a relevant solution to the word problem would, given the input (x \cdot y)/z \mathrel (x/z)\cdot y, produce the output EQUAL, and similarly pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolution (logic)
In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation complete theorem-proving technique for sentences in propositional logic and first-order logic. For propositional logic, systematically applying the resolution rule acts as a decision procedure for formula unsatisfiability, solving the (complement of the) Boolean satisfiability problem. For first-order logic, resolution can be used as the basis for a semi-algorithm for the unsatisfiability problem of first-order logic, providing a more practical method than one following from Gödel's completeness theorem. The resolution rule can be traced back to Davis and Putnam (1960); however, their algorithm required trying all ground instances of the given formula. This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson's syntactical unification algorithm, which allowed one to instantiate the formula during the proof "on demand" just as far as needed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E Theorem Prover
E is a high-performance theorem prover for full first-order logic with equality. It is based on the equational superposition calculus and uses a purely equational paradigm. It has been integrated into other theorem provers and it has been among the best-placed systems in several theorem proving competitions. E is developed by Stephan Schulz, originally in the ''Automated Reasoning Group'' at TU Munich, now at Baden-Württemberg Cooperative State University Stuttgart. System The system is based on the equational superposition calculus. In contrast to most other current provers, the implementation actually uses a purely equational paradigm, and simulates non-equational inferences via appropriate equality inferences. Significant innovations include shared term rewriting (where many possible equational simplifications are carried out in a single operation), several efficient term indexing data structures for speeding up inferences, advanced inference literal selection strategies, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Critical Pair (logic)
A critical pair arises in a term rewriting system when two rewrite rules overlap to yield two different terms. In more detail, (''t''1, ''t''2) is a critical pair if there is a term ''t'' for which two different applications of a rewrite rule (either the same rule applied differently, or two different rules) yield the terms ''t''1 and ''t''2. Definitions The actual definition of a critical pair is slightly more involved as it excludes pairs that can be obtained from critical pairs by substitution and orients the pair based on the overlap. Specifically, for a pair of overlapping rules \rho_0 : l_0 \to r_0 and \rho_1 : l_1 \to r_1, with the overlap being that l_0 = D /math> for some non-empty context D ;/math>, and the term s (that is not a variable) matches l_1 under some substitutions s \sigma = l_1 \tau that are most general, the critical pair is (D \sigma _1 \tau r_0 \sigma). When both sides of the critical pair can reduce to the same term, the critical pair is called ''con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variant (logic)
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first-order term is recursively constructed from constant symbols, variables and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, is a term built from the constant 1, the variable , and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of . Besides in logic, terms play important roles in universal algebra, and rewriting systems. Formal definition Given a set ''V'' of variable symbols, a set ''C'' of constant symbols and sets ''F''''n'' of ''n''-ary fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encompassment Ordering
In theoretical computer science, in particular in automated theorem proving and term rewriting, the containment, or encompassment, preorder (≤) on the set of term (logic), terms, is defined by :''s'' ≤ ''t'' if a Term (logic)#Operations with terms, subterm of ''t'' is a substitution instance of ''s''. It is used e.g. in the Knuth–Bendix completion algorithm. Properties * Encompassment is a preorder, i.e. reflexive relation, reflexive and transitive relation, transitive, but not anti-symmetric relation, anti-symmetric,since both ''f''(''x'') ≤ ''f''(''y'') and ''f''(''y'') ≤ ''f''(''x'') for Term (logic)#Formal definition, variable symbols ''x'', ''y'' and a Term (logic)#Formal definition, function symbol ''f'' nor total order, totalsince neither ''a'' ≤ ''b'' nor ''b'' ≤ ''a'' for distinct Term (logic)#Formal definition, constant symbols ''a'', ''b'' * The preorder#Constructions, corresponding equivalence relation, def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent Term Rewrite System
In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviated ARS) is a formalism that captures the quintessential notion and properties of rewriting systems. In its simplest form, an ARS is simply a set (of "objects") together with a binary relation, traditionally denoted with \rightarrow; this definition can be further refined if we index (label) subsets of the binary relation. Despite its simplicity, an ARS is sufficient to describe important properties of rewriting systems like normal forms, termination, and various notions of confluence. Historically, there have been several formalizations of rewriting in an abstract setting, each with its idiosyncrasies. This is due in part to the fact that some notions are equivalent, see below in this article. The formalization that is most commonly encountered in monographs and textbooks, and which is generally followed here, is due to Gérard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
