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Encompassment Ordering
In theoretical computer science, in particular in automated theorem proving and term rewriting, the containment, or encompassment, preorder (≤) on the set of terms, is defined by :''s'' ≤ ''t'' if a 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 and transitive, but not anti-symmetric,since both ''f''(''x'') ≤ ''f''(''y'') and ''f''(''y'') ≤ ''f''(''x'') for variable symbols ''x'', ''y'' and a function symbol ''f'' nor totalsince neither ''a'' ≤ ''b'' nor ''b'' ≤ ''a'' for distinct constant symbols ''a'', ''b'' * The corresponding equivalence relation, defined by ''s'' ~ ''t'' if ''s'' ≤ ''t'' ≤ ''s'', is equality modulo renaming. * ''s'' ≤ ''t'' whenever ''s'' is a subterm of ''t''. * ''s'' ≤ ''t'' whenever ''t'' is a substitution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Relation
In mathematics, a binary relation on a set (mathematics), set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Every partial order and every equivalence relation is transitive. For example, less than and equality (mathematics), equality among real numbers are both transitive: If and then ; and if and then . Definition A homogeneous relation on the set is a ''transitive relation'' if, :for all , if and , then . Or in terms of first-order logic: :\forall a,b,c \in X: (aRb \wedge bRc) \Rightarrow aRc, where is the infix notation for . Examples As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy is also an ancestor of Carrie. On the other hand, "is the birth mother of" is not a transitive relation, because if Alice is the birth mother of Brenda, and Brenda is the birth mother of Claire, then it does ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreflexive Kernel
In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. Etymology The word ''reflexive'' is originally derived from the Medieval Latin ''reflexivus'' ('recoiling' reflex.html" ;"title="f. ''reflex">f. ''reflex'' or 'directed upon itself') (c. 1250 AD) from the classical Latin ''reflexus-'' ('turn away', 'reflection') + ''-īvus'' (suffix). The word entered Early Modern English in the 1580s. The sense of the word meaning 'directed upon itself', as now used in mathematics, surviving mostly by its use in philosophy and grammar (cf. ''Reflexive verb'' and ''Reflexive pronoun''). The first exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-founded
In mathematics, a binary relation is called well-founded (or wellfounded or foundational) on a set (mathematics), set or, more generally, a Class (set theory), class if every non-empty subset has a minimal element with respect to ; that is, there exists an such that, for every , one does not have . In other words, a relation is well-founded if: (\forall S \subseteq X)\; [S \neq \varnothing \implies (\exists m \in S) (\forall s \in S) \lnot(s \mathrel m)]. Some authors include an extra condition that is Set-like relation, set-like, i.e., that the elements less than any given element form a set. Equivalently, assuming the axiom of dependent choice, a relation is well-founded when it contains no infinite descending chains, meaning there is no infinite sequence of elements of such that for every natural number . In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution (logic)
A substitution is a syntactic transformation on formal expressions. To ''apply'' a substitution to an expression means to consistently replace its variable, or placeholder, symbols with other expressions. The resulting expression is called a ''substitution instance'', or ''instance'' for short, of the original expression. Propositional logic Definition Where ''ψ'' and ''φ'' represent formulas of propositional logic, ''ψ'' is a ''substitution instance'' of ''φ'' if and only if ''ψ'' may be obtained from ''φ'' by substituting formulas for propositional variables in ''φ'', replacing each occurrence of the same variable by an occurrence of the same formula. For example: ::''ψ:'' (R → S) & (T → S) is a substitution instance of ::''φ:'' P & Q That is, ''ψ'' can be obtained by replacing P and Q in ''φ'' with (R → S) and (T → S) respectively. Similarly: ::''ψ:'' (A ↔ A) ↔ (A ↔ A) is a substitution instance of: ::''φ:'' (A ↔ A) since ''ψ'' can be obta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rewrite Order
In theoretical computer science, in particular in automated reasoning about formal equations, reduction orderings are used to prevent endless loops. Rewrite orders, and, in turn, rewrite relations, are generalizations of this concept that have turned out to be useful in theoretical investigations. Motivation Intuitively, a reduction order ''R'' relates two terms ''s'' and ''t'' if ''t'' is properly "simpler" than ''s'' in some sense. For example, simplification of terms may be a part of a computer algebra program, and may be using the rule set . In order to prove impossibility of endless loops when simplifying a term using these rules, the reduction order defined by "''sRt'' if term ''t'' is properly shorter than term ''s''" can be used; applying any rule from the set will always properly shorten the term. In contrast, to establish termination of "distributing-out" using the rule ''x''*(''y''+''z'') → ''x''*''y''+''x''*''z'', a more elaborate reduction order will be needed, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anti-symmetric Relation
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Antisymmetric or skew-symmetric may refer to: * Antisymmetry in linguistics * Antisymmetry in physics * Antisymmetric relation in mathematics * Skew-symmetric graph * Self-complementary graph In mathematics, especially linear algebra, and in theoretical physics, the adjective antisymmetric (or skew-symmetric) is used for matrices, tensors, and other objects that change sign if an appropriate operation (e.g. matrix transposition) is performed. See: * Skew-symmetric matrix (a matrix ''A'' for which ) * Skew-symmetric bilinear form is a bilinear form ''B'' such that for all ''x'' and ''y''. * Antisymmetric tensor in matrices and index subsets. * "antisymmetric function" – odd function See also * Symmetry in mathematics Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. Given a structured obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Relation
In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. Etymology The word ''reflexive'' is originally derived from the Medieval Latin ''reflexivus'' ('recoiling' reflex.html" ;"title="f. ''reflex">f. ''reflex'' or 'directed upon itself') (c. 1250 AD) from the classical Latin ''reflexus-'' ('turn away', 'reflection') + ''-īvus'' (suffix). The word entered Early Modern English in the 1580s. The sense of the word meaning 'directed upon itself', as now used in mathematics, surviving mostly by its use in philosophy and grammar (cf. ''Reflexive verb'' and ''Reflexive pronoun''). The first e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 lite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution Instance
A substitution is a syntactic transformation on formal expressions. To ''apply'' a substitution to an expression means to consistently replace its variable, or placeholder, symbols with other expressions. The resulting expression is called a ''substitution instance'', or ''instance'' for short, of the original expression. Propositional logic Definition Where ''ψ'' and ''φ'' represent formulas of propositional logic, ''ψ'' is a ''substitution instance'' of ''φ'' if and only if ''ψ'' may be obtained from ''φ'' by substituting formulas for propositional variables in ''φ'', replacing each occurrence of the same variable by an occurrence of the same formula. For example: ::''ψ:'' (R → S) & (T → S) is a substitution instance of ::''φ:'' P & Q That is, ''ψ'' can be obtained by replacing P and Q in ''φ'' with (R → S) and (T → S) respectively. Similarly: ::''ψ:'' (A ↔ A) ↔ (A ↔ A) is a substitution instance of: ::''φ:'' (A ↔ A) since ''ψ'' can be obtain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
