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Boyce Codd normal form





Encyclopedia results for Boyce Codd normal form

  1. Boyce?Codd normal form

    Original research date April 2011 Boyce Codd normal form or BCNF or 3.5NF is a Database normalization Normal forms normal form used in database normalization . It is a slightly stronger version of the third normal form 3NF . BCNF was developed in 1974 by Raymond F. Boyce and Edgar F. Codd to address certain types of anomaly not dealt with by 3NF as originally defined. ref name Codd Codd, E. F. Recent Investigations into Relational Data Base Systems. IBM Research Report RJ1385 April 23, 1974 . Republished in Proc. 1974 Congress Stockholm, Sweden, 1974 . New York, N.Y. North Holland 1974 . ref Chris Date has pointed out that a definition of what we now know as BCNF appeared in a paper by Ian Heath in 1971. ref name Heath Heath, I. Unacceptable File Operations in a Relational Database. Proc. 1971 ACM SIGFIDET Workshop on Data Description, Access, and Control , San Diego, Calif. November 11th&ndash 12th, 1971 . ref Date writes blockquote Since that definition predated Boyce and Codd s own definition by some three years, it seems to me that BCNF ought by rights to be called Heath normal form ... schema R is in Boyce Codd normal form if and only if for every one of its Functional dependency ... of Texas. Database normalization Category Database normalization BCNF cs Boyce Coddova norm ln forma de Normalisierung Datenbank Boyce Codd Normalform BCNF es Forma normal de Boyce Codd kk ... be a candidate key. Recall that Second normal form 2NF prohibits partial functional dependencies of non ... normal form 3NF prohibits transitive dependency transitive functional dependencies of non ... problems related to the design of normal form relational schemas. ACM Transactions on Database Systems 4 1 , March 1979, p. 50. ref Thus, unlike the first three normal forms, BCNF is not always achievable ... to BCNF is possible. ref name Zaniolo Zaniolo, Carlo. A New Normal Form for the Design of Relational ... made by members PREMIUM B, for Court 2 bookings made by non members Note that, Court 1 normal ...   more details



  1. Normal form

    Normal form may refer to Normal form abstract rewriting Database normalization Normal forms Normal form databases Normal form game Normal form game theory Normal form mathematics In formal language theory Beta normal form Chomsky normal form Greibach normal form Kuroda normal form Normal form abstract rewriting , an element of a rewrite system which cannot be further rewritten In logic Algebraic normal form Clausal normal form Conjunctive normal form Negation normal form Prenex normal form Skolem normal form In lambda calculus Beta normal form See also Normalization disambiguation Normalization property Set music Musical set theory mathdab Category Mathematical terminology de Normalform fr Forme normale nl Normaalvorm pl Posta normalna ru sv Normalform uk ...   more details



  1. Third normal form

    serve as a definition of the slightly stronger Boyce Codd normal form Each attribute must represent ... by such anomalies these are tables which either fall short of Boyce Codd normal form BCNF or, if they meet BCNF, fall short of the higher normal forms Fourth normal form 4NF or Fifth normal form 5NF . See also Attribute value system Boyce Codd normal form First normal form Second normal form Fourth normal form Fifth normal form References Reflist Further reading Refbegin Date, C. J. 1999 , http www.aw ...In computer science , the third normal form 3NF is a Database normalization Normal forms normal form used in database normalization . 3NF was originally defined by E.F. Codd in 1971. ref name Codd Codd ... Symposia Series 6 . Prentice Hall, 1972. ref Codd s definition states that a table is in 3NF if and only if both of the following conditions hold The relation R table is in second normal form 2NF .... A New Normal Form for the Design of Relational Database Schemata. ACM Transactions on Database ... of the key ensures that the table is in First normal form 1NF requiring that non key attributes be dependent on the whole key ensures Second normal form 2NF further requiring that non key attributes .... ref name Codd2 Codd, p. 43. ref A transitive dependency is a functional dependency in which X Z X determines Z indirectly, by virtue of X Y and Y Z where it is not the case that Y X . ref Codd, p. 45&ndash 46. ref A 3NF definition that is equivalent to Codd s, but expressed differently, was given by Carlo ... Zaniolo s definition gives a clear sense of the difference between 3NF and the more stringent Boyce Codd normal form BCNF . BCNF simply eliminates the third alternative A is a prime attribute . Nothing but the key A memorable statement of Codd s definition of 3NF, paralleling the traditional sworn ..., William. http www.bkent.net Doc simple5.htm A Simple Guide to Five Normal Forms in Relational Database ... this definition with the oath so help me Edgar F. Codd Codd . ref name Diehr The author of a 1989 ...   more details



  1. Fourth normal form

    Fourth normal form 4NF is a Database normalization Normal forms normal form used in database normalization . Introduced by Ronald Fagin in 1977, 4NF is the next level of normalization after Boyce Codd normal form BCNF . Whereas the Second normal form second , Third normal form third , and Boyce Codd normal form Boyce Codd normal forms are concerned with functional dependency functional dependencies , 4NF is concerned with a more general type of dependency known as a multivalued dependency . A Table database Table is in 4NF if and only if , for every one of its non trivial multivalued dependencies X math twoheadrightarrow math Y , X is a superkey that is, X is either a candidate key or a superset thereof. ref name Fagin A relation schema R is in fourth normal form 4NF if, whenever a nontrivial multivalued dependency X math twoheadrightarrow math Y holds for R , then so does the functional dependency X A for every column name A of R . Intuitively all dependencies are the result of keys. cite journal first Ronald last Fagin title Multivalued Dependencies and a New Normal Form for Relational Databases journal ACM Transactions on Database Systems volume 2 issue 1 month September year 1977 ... one or more tables that violated 4NF while meeting all lower normal forms. ref name Wu cite journal first Margaret S. last Wu title The Practical Need for Fourth Normal Form journal ACM SIGCSE Bulletin ... value system Third normal form Fifth normal form Sixth normal form References Reflist Further ... together form the whole set of attributes of the relation. A functional dependency is a special case ... it meets all normal forms up to BCNF. If we assume, however, that pizza varieties offered by a restaurant ... all lower normal forms are rarely encountered in business applications. This belief may not be accurate ..., W. 1983 http www.bkent.net Doc simple5.htm A Simple Guide to Five Normal Forms in Relational Database ... Database normalization 4NF de Normalisierung Datenbank Vierte Normalform 4NF es Cuarta forma normal ...   more details



  1. Domain/key normal form

    Domain key normal form DKNF is a Database normalization Normal forms normal form used in database normalization which requires that the database contains no constraints other than Data domain domain constraints and key constraints. A domain constraint specifies the permissible values for a given attribute, while a key constraint specifies the attributes that uniquely identify a row in a given table. The domain key normal form is achieved when every constraint on the relation is a logical consequence of the definition of keys and domains, and enforcing key and domain restraints and conditions causes ... key normal form is to avoid having general constraints in the database that are not clear domain or key ... normal form than it is to convert lesser databases which may contain numerous anomalies. However, successfully building a domain key normal form database remains a difficult task, even for experienced database programmers. Thus, while the domain key normal form eliminates the problems found in most databases, it tends to be the most costly normal form to achieve. However, failing to achieve the domain key normal form may carry long term, hidden costs due to anomalies which appear in databases adhering only to lower normal forms over time. The third normal form , Boyce Codd normal form , fourth normal form and fifth normal form are special cases of the domain key normal form. All have either ... normal forms were unconstrained so all domain constraints are satisfied. However, transforming a higher normal form into domain key normal form is not always a dependency preserving transformation ... cs people fagin tods81.pdf title A Normal Form for Relational Databases That Is Based on Domains ... however would normally require special database programming in the form of stored procedures .... http phlonx.com resources nf3 A tutorial on the first 3 normal forms by Fred Coulson http support.microsoft.com ... Category Database normalization DKNF es Forma normal de dominio clave ru ...   more details



  1. First normal form

    First normal form 1NF or Minimal Form is a Database normalization Normal forms normal form used in database .... J. http www.dbdebunk.com page page 629796.htm What First Normal Form Really Means in Date on Database ... name Kent First normal form excludes variable repeating fields and groups. Kent, William. http www.bkent.net Doc simple5.htm A Simple Guide to Five Normal Forms in Relational Database Theory , Communications ... First Normal Form Really Means , pp. 127 8 ref name Date5Cr Date, C. J. http www.dbdebunk.com page page 629796.htm What First Normal Form Really Means pp. 127 128. ref Violation of any of these conditions ... name DateRepg Date, C. J. http www.dbdebunk.com page page 629796.htm What First Normal Form Really ... for second normal form second and third normal form third normal form 3NF . Atomicity Some definitions ... Attributes or, Will the Real First Normal Form Please Stand Up? , in C. J. Date and Hugh Darwen, Relational ... page 629796.htm What First Normal Form Really Means in Date on Database Writings 2000 2006 Springer ... name DateRVA Date, C. J. http www.dbdebunk.com page page 629796.htm What First Normal Form Really Means pp. 121 126. ref Normalization beyond 1NF Any table that is in second normal form 2NF or higher is, by definition, also in 1NF each normal form has more stringent criteria than its predecessor . On the other ... normal form 3NF , and so on. Normal forms higher than 1NF are intended to deal with situations ... as the foreign key . The tables would conform to second normal form 2NF in addition to 1NF. See also Attribute value system Entity attribute value model Second normal form Third normal form For other .... 315. ref following the precedent established by Edgar F. Codd excludes relation valued attributes tables ... of condition 4 is controversial. It marks an important departure from Edgar F. Codd Codd s later vision of the relational model , ref name DateNullsLater Codd first defined the relational model ... , Appendix A.2. ref which made explicit provision for nulls. ref name CoddRule The third of Codd ...   more details



  1. Second normal form

    Second normal form 2NF is a Database normalization Normal forms normal form used in database normalization . 2NF was originally defined by E.F. Codd in 1971. ref name Codd Codd, E.F. Further Normalization of the Data Base Relational Model. Presented at Courant Computer Science Symposia Series 6, Data Base Systems, New York City, May 24th 25th, 1971. IBM Research Report RJ909 August 31st, 1971 . Republished in Randall J. Rustin ed. , Data Base Systems Courant Computer Science Symposia Series 6 . Prentice Hall, 1972. ref A table that is in first normal form 1NF must meet additional criteria if it is to qualify for second normal form. Specifically a table is in 2NF if and only if , it is in 1NF and no non prime attribute is dependent on any proper subset of any candidate key of the table. A non prime attribute of a table is an attribute that is not a part of any candidate key of the table. Put simply, a table is in 2NF if and only if, it is in 1NF and every non prime attribute of the table is either dependent on the whole of a candidate key, or on another non prime attribute. Note that when a 1NF table has no composite candidate keys candidate keys consisting of more than one attribute , the table is automatically in 2NF. Example Consider a table describing employees skills class wikitable Employees Skills u Employee u u Skill u Current Work Location Jones Typing 114 Main Street Jones ... in turn depends on the key Tournament Year. This problem is addressed by third normal form 3NF . 2NF ... Toothmaster Hoch X Prime Hoch X Prime See also Attribute value system First normal form Third normal form References Reflist Further reading Refbegin http www.troubleshooters.com littstip ltnorm.html ..., W. 1983 http www.bkent.net Doc simple5.htm A Simple Guide to Five Normal Forms in Relational Database ... Hillyer. http phlonx.com resources nf3 A tutorial on the first 3 normal forms by Fred Coulson ... 2NF es Segunda forma normal ko 2 ru uk vi D ng ...   more details



  1. Normal form (natural deduction)

    Unreferenced stub auto yes date December 2009 Orphan date December 2009 An inference of natural deduction is a normal form , according to Dag Prawitz , if no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule. DEFAULTSORT Normal Form Natural Deduction Category Logic Logic stub ...   more details



  1. Kuroda normal form

    In formal language theory , a formal grammar grammar is in Kuroda normal form if, and only if, all production rules are of the form AB &rarr CD or A &rarr BC or A &rarr B A &rarr where A, B, C and D are nonterminal symbols and is a terminal symbol . Every grammar in Kuroda normal form is noncontracting grammar monotonic , and therefore, generates a context sensitive language . Conversely, every context sensitive language which does not generate the empty string can be generated by a grammar in Kuroda normal form. It is named for linguistics linguist S. Y. Kuroda . See also Backus Naur form Chomsky normal form Greibach normal form References S. Y. Kuroda, Classes of languages and linear bounded automata , Information and Control , 7 2 207&ndash 223, June 1964. Category Formal languages formalmethods stub bs Kurodin normalni oblik de Kuroda Normalform hr Kurodin normalni oblik it Forma normale di Kuroda ja pl Posta normalna Kurody zh ...   more details



  1. Negation normal form

    Negation normal form is an elementary canonical form in mathematical logic. There are similar requirements for negation normal form in different logic fragments. In predicate logic , a logical formula is in negation normal form if negation occurs only immediately above elementary propositions, and math lnot, lor, land math are the only allowed Boolean connectives. In classical logic each formula can be brought into this form by replacing implications and equivalences by their definitions, using De Morgan s laws to push negation inside, and eliminating double negations. This process can be represented using the following rewrite rule s math lnot forall x. G to exists x. lnot G math math lnot exists x. G to forall x. lnot G math math lnot lnot G to G math math lnot G 1 land G 2 to lnot G 1 lor lnot G 2 math math lnot G 1 lor G 2 to lnot G 1 land lnot G 2 math A formula in negation normal form can be put into the stronger conjunctive normal form or disjunctive normal form by applying the distributivity laws. External links http www.izyt.com BooleanLogic applet.php Java applet for converting logical formula to Negation Normal Form, showing laws used Category Propositional calculus Category Normal forms logic logic stub de Negationsnormalform es Forma normal negativa ja pt Forma normal da nega o sr ...   more details



  1. Beta normal form

    Unreferenced date December 2009 In the lambda calculus , a term is in beta normal form if no lambda calculus reduction beta reduction is possible. A term is in beta eta normal form if neither a beta reduction nor an lambda calculus conversion eta reduction is possible. A term is in head normal form if there is no beta redex in head position . Beta reduction In the lambda calculus, a beta redex is a term of the form math mathbf lambda x . A x t math where math A x math is a term possibly involving variable math x math . A beta reduction is an application of the following rewrite rule to a beta redex math mathbf lambda x . A x t rightarrow A t math where math A t math is the result of substituting the term math t math for the variable math x math in the term math A x math . A beta reduction is in head position if it is of the following form math lambda x 0 ldots lambda x i 1 . lambda x i . A x i M 1 M 2 ldots M n rightarrow lambda x 0 ldots lambda x i 1 . A M 1 M 2 ldots M n math , where math i geq 0, n geq 1 math . Any reduction not in this form is an internal beta reduction. Reduction strategies In general, there can be several different beta reductions possible for a given term. Normal order reduction is the evaluation strategy in which one continually applies the rule for beta ... term is in head normal form . In contrast, in applicative order reduction , one applies the internal .... Normal order reduction is complete, in the sense that if a term has a head normal form, then normal ..., even when the term has a normal form. For example, using applicative order reduction, the following .... See also Lambda calculus Normal form DEFAULTSORT Beta Normal Form Category Lambda calculus zh Beta pt Forma normal beta ... math ldots math But using normal order reduction, the same starting point reduces quickly to normal form math mathbf lambda x . z lambda w. w w w lambda w. w w w math math rightarrow z math Sinot s director ...   more details



  1. Clausal normal form

    merge to Conjunctive normal form date January 2011 The clausal normal form or clause normal form , conjunctive normal form , CNF of a logical formula is used in logic programming and many automated theorem proving theorem proving systems. A formula in clause normal form is a set of clauses, interpreted as a conjunction. A Clause logic clause is an implicitly universally quantified set of literals, interpreted as a disjunction. ref cite book last Loveland first Donald W. title Automated Theorem Proving A Logical Basis publisher North Holland date 1978 ref Conversion to clausal normal form The procedure to convert a formula into clausal form can destroy the structure of the formula, and naive translations often causes exponential growth exponential blowup in the size of the resulting formula. The procedure begins with any formula of classical First order predicate calculus first order logic Put the formula into negation normal form . Standardize variables math forall x , P x vee exists x , P x math becomes math forall x , P x vee exists c , P c math , where math c math is new Skolem normal form Skolemize replace Existential quantification existential variables with Skolem constants or Skolem functions of Universal quantification universal variables, from the outside inward. Make the following replacements math forall x exists y , P y math becomes math , forall x , P f c x math , where math f c math is new Discard the universal quantifier s which are implicit in CNF . Put the formula into conjunctive normal form . Replace math C1 wedge cdots wedge Cn math with math C1 , dots , Cn math . Each conjunct is of the form math neg A1 vee cdots vee neg Am vee B1 vee cdots vee Bn math ... Clause Normal Forms url http www.mpi inf.mpg.de weidenb publications handbook99small.ps.gz accessdate 2009 07 05 ref References references Category Logic programming Category Normal forms logic ja pt Forma normal clausal ...   more details



  1. Disjunctive normal form

    refimprove date November 2010 In boolean logic , a disjunctive normal form DNF is a standardization or normalization of a logical formula which is a disjunction of conjunctive clause logic clauses . As a normal form , it is useful in automated theorem proving . A logical formula is considered to be in DNF iff if and only if it is a logical disjunction disjunction of one or more logical conjunction conjunctions of one or more literal mathematical logic literals . A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every clause. As in conjunctive normal form CNF , the only propositional operators in DNF are logical conjunction and , logical disjunction or , and logical negation not . The not operator can only be used as part of a literal, which means that it can only precede a propositional variable . For example, all of the following formulas are in DNF math A and B math math A math math A and B or C math math A and neg B and neg C or neg D and E ... and only one full disjunctive normal form, one of the two canonical form Boolean algebra canonical form s. The following is a formal grammar for DNF disjunct conjunct disjunct disjunct conjunct conjunct ... as any variable. See also Algebraic normal form Boolean function Boolean valued function Conjunctive normal form Horn clause Logical graph Propositional logic Quine McCluskey algorithm Truth table External ... distributive law . All logical formulas can be converted into disjunctive normal form. However, in some cases conversion to DNF can lead to an exponential explosion of the formula. For example, in DNF, logical formulas of the following form have 2 sup n sup terms math X 1 or Y ... boolean logic expressions to CNF and DNF, showing the laws used Category Normal forms logic de Disjunktive Normalform es Forma normal disyuntiva fr Forme normale disjonctive it Forma normale ... normalform pl Dysjunkcyjna posta normalna pt Forma normal disjuntiva ru ...   more details



  1. Greibach normal form

    In computer science and formal language theory, a context free grammar is in Greibach normal form if the right hand sides of all productions start with a terminal symbol, optionally followed by some variables. A non strict form allows one exception to this format restriction for allowing the empty word epsilon, to be a member of the described language. The normal form bears the name of Sheila Greibach . More precisely, a context free grammar is in Greibach normal form, if all production rules are of the form math A to alpha A 1 A 2 cdots A n math or math S to varepsilon math where A is a nonterminal symbol , is a terminal symbol , math A 1, A 2, ldots, A n math is a possibly empty sequence of nonterminal symbols not including the start symbol, S is the start symbol, and is the empty word . Observe that the grammar must be without left recursion s. Every context free grammar can be transformed into an equivalent grammar in Greibach normal form. Some definitions do not consider the second form of rule to be permitted, in which case a context free grammar that can generate the empty word cannot be so transformed. In particular, there is a construction ensuring that the resulting normal form grammar is of size at most O n sup 4 sup , where n is the size of the original grammar. ref Blum and Koch 1999 ref This conversion can be used to prove that every context free language can be accepted by a non deterministic pushdown automaton . Given a grammar in GNF and a derivable string in the grammar with length n , any top down parsing top down parser will halt at depth n . See also Backus Naur form Chomsky normal form Kuroda normal form Notes references References John E. Hopcroft ... Koch Greibach Normal Form Transformation Revisited. Information and Computation 150 1 , 1999, pp ... Normalform es Forma normal de Greibach hr Greibachov normalni oblik it Forma normale di Greibach nl Greibach normaalvorm ja pl Posta normalna Greibach pt Forma normal de Greibach ro Forma normal ...   more details



  1. Normal form (abstract rewriting)

    Cleanup date August 2009 In abstract rewriting , a normal form is an element of the system which cannot be rewritten any further. Stated formally, for some reduction relation &sdot   &rarr   &sdot over X a term t in X is a normal form if there does not exist a term t &prime in X such that t &rarr t &prime . Consider the basic term rewriting system with reduction rule g x , y x . The term g g 4, 2 , g 3, 1 has the following reduction sequence, according to the usual outermost strategy term rewriting strategy , that is, if the reduction rule is applied to each outermost occurrence of g math g g 4,2 , g 3,1 rightarrow rho g 4,2 rightarrow rho 4. math There is no rule that permits us to rewrite 4, so 4 is a normal form for this term rewriting system. Related concepts refer to the possibility of rewriting an element into normal form. Weak normalization means that some element can be rewritten into a normal form. Strong normalization means that any reduction sequence starting from some element terminates. We say that the system is weakly normalizing or strongly normalizing if all elements are weakly normalizing resp. strongly normalizing . Newman s lemma states that if an abstract reduction system A is strongly normalizing and is confluence term rewriting weakly confluent , then A is in fact confluent. The result enables us to further generalize the critical pair lemma . References cite book first1 Franz last1 Baader authorlink1 Franz Baader first2 Tobias last2 Nipkow authorlink2 Tobias Nipkow title Term Rewriting and All That year 1998 publisher Cambridge University Press ref harv DEFAULTSORT Normal Form Term Rewriting Category Computability theory Category Formal languages Category Rewriting systems comp sci theory stub pt Forma normal ...   more details



  1. Sixth normal form

    Sixth normal form 6NF is a term in relational database theory, used in two different ways. 6NF C. Date s definition A book by Christopher J. Date and others on temporal database s, ref da da lo Date et al., 2003 ref defined sixth normal form as a Database normalization Normal forms normal form for databases based on an extension of the relational algebra. In this work, the relational operators, such as join , are generalized to support a natural treatment of interval data, such as sequences of dates or moments in time. ref da da lo op. cit. , chapter 9 Generalizing the relational operators ref Sixth normal form is then based on this generalized join, as follows blockquote A relvar R table is in sixth normal form abbreviated 6NF if and only if it satisfies no nontrivial join dependencies at all where, as before, a join dependency is trivial if and only if at least one of the projections possibly U projections involved is taken over the set of all attributes of the relvar table concerned. Date et al. ref da da lo op. cit. , section 10.4, p. 176 ref blockquote Any relation in 6NF is also in fifth normal form 5NF . Sixth normal form is intended to decompose relation variables to irreducible components. Though this may be relatively unimportant for non temporal relation variables, it can be important when dealing with temporal variables or other interval data. For instance, if a relation ... sixth normal form differently, namely, as a synonym for Domain key normal form DKNF . citation ... on this topic ref Usage The sixth normal form is currently being used in some data warehouse ... website for a website that describes a data warehouse modelling method based on the sixth normal form ... queries in databases modelled in the Third normal form . Note this is not what the original text nor ... attributes. See also First normal form Fifth normal form References reflist 2 Further reading Refbegin ... NF NORMAL FORM url http www.dbdebunk.com page page 621935.htm NOTE redundant reference see footnotes ...   more details



  1. Hesse normal form

    Noref date August 2009 Image Hesse normal form.png thumb Drawing of the normal and the distance calculated with the Hesse normal form The Hesse normal form named after Otto Hesse , is an equation used in analytic geometry , and describes a line in math mathbb R 2 math or a plane in Euclidean space math mathbb R 3 math or a hyperplane in higher dimensions. It is primarily used for calculating distances, and is written in vector notation as math vec r cdot vec n 0 d 0. , math This equation is satisfied by all points P described by the location vector math vec r math , which lie precisely in the plane E or in 2D, on the line g . The vector math vec n 0 math represents the unit normal vector of E or g , that points from the origin of the coordinate system to the plane or line, in 2D . The distance math d ge 0 math is the distance from the origin to the plane or line . The dot math cdot math indicates the scalar product or dot product. Derivation Calculation from the normal form Note For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D. In the normal form, math vec r vec a cdot vec n 0 , math a plane is given by a normal vector math vec n math as well as an arbitrary position vector math vec a math of a point math A in E math . The direction of math vec n math is chosen to satisfy the following inequality math vec a cdot vec n geq 0 , math By dividing the normal vector math vec n math by its Euclidean vector Length of a vector Magnitude math vec n math , we obtain the unit or normalized normal vector math vec n 0 vec n over vec n , math and the above equation can be rewritten as math vec r vec a cdot vec n 0 0. , math Substituting math d vec a cdot vec n 0 geq 0 , math we obtain the Hesse normal form math vec r cdot vec n 0 d 0. , math center File Ebene Hessesche Normalform.PNG center In this diagram, d is the distance from the origin. Because math vec r cdot vec n 0 d math holds for every point in the plane, it is also true ...   more details



  1. Negational normal form

    merge Negation normal form date February 2012 Negative normal form is way to represent the formula by keeping the negation symbol only to the literals. If and are boolean formulas, then logical conjunction math wedge math , logical disjunction math vee math , negation math neg math , implication math rightarrow math are all boolean formulas. The boolean formulas are made up of literals which are propositional atoms. A formula can be converted to the NNF by simply moving the negations across braces either using De Morgan s laws or the property that math neg neg math is itself. External links http aracne.usal.es congress PDF AntonioMoreno.pdf Math logic tutor Category Mathematical logic ...   more details



  1. Conjunctive normal form

    merge from Clausal normal form date January 2011 In Boolean logic , a formula is in conjunctive normal form CNF if it is a logical conjunction conjunction of clause logic clauses , where a clause is a logical disjunction disjunction of literal mathematical logic literal s. As a Normal form mathematics Rewriting systems normal form , it is useful in automated theorem proving . It is similar to the Canonical form Boolean algebra product of sums form used in circuit theory . All conjunctions of literals and all disjunctions of literals are in CNF, as they can be seen as conjunctions of one literal clauses and conjunctions of a single clause, respectively. As in the disjunctive normal form DNF , the only ..., this formula is also in disjunctive normal form . The following formulae are not in CNF ... all logical formulae can be converted into an equivalent formula in conjunctive normal form, proofs ... In first order logic, conjunctive normal form can be taken further to yield the clausal normal form ... of a boolean formula expressed in Conjunctive Normal Form, such that the formula is true. The k ... in polynomial time . Typical problems in this case involve formulas in 3CNF form conjunctive normal form with no more than three variables per conjunct. Examples of such formulas encountered ... A modern Approach 1995... Russel and Norvig ref Convert to negation normal form . Eliminate implications ... between CNF and DNF for different De Morgan Triples DEFAULTSORT Conjunctive Normal Form Category Normal forms logic cs Konjunktivn norm ln forma de Konjunktive Normalform es Forma normal conjuntiva ... details Drop universal quantifiers Distribute ORs over ANDs. Notes references See also Algebraic normal form Disjunctive normal form Horn clause Quine&ndash McCluskey algorithm References Paul Jackson, Daniel Sheridan Clause Form Conversions for Boolean Circuits. In Holger H. Hoos, David G. Mitchell ... nl Conjunctieve normaalvorm ja pl Koniunkcyjna posta normalna pt Forma normal conjuntiva ...   more details



  1. Frobenius normal form

    In linear algebra , the Frobenius normal form , Turner binormal projective form or rational canonical form of a square matrix A is a canonical form for Matrix mathematics matrices that reflects the structure ... example The Frobenius normal form M of a matrix A with entries in a field F can be obtained ... matrix can be transformed by a similarity transformation to its Frobenius normal form. For example, let ... expect the other two blocks comprising the normal form of the matrix to be identical to A sub 1 sub . So, we can simply write down the Frobenius normal form math M A 1 oplus A 2 oplus A 3 begin pmatrix ... used in the construction of the Jordan normal form do not exist over F x , so the invariant factors ... polynomial, the Frobenius normal form is the companion matrix of the characteristic polynomial ... matrices A and B are similar if and only if they have the same rational canonical form. A rational normal form generalizing the Jordan normal form Primary rational canonical form The Frobenius normal ... normal form rather different than other normal forms that do depend on factoring the characteristic ... the Jordan normal form if the characteristic polynomial splits into linear factors . For instance, the Frobenius normal form of a diagonal matrix with distinct diagonal entries is just the companion matrix of its characteristic polynomial. There is another way to define a normal form, that like the Frobenius normal form is always defined over the same field F as A , but that does reflect a possible ... factors over F , and which reduces to the Jordan normal form in case this factorization only ... ref is sometimes called the generalized Jordan normal form , or primary rational canonical form ..., as a direct sum of cyclic F x modules like is done for the Frobenius normal form above , where the characteristic ... abs storjohann.html An O n sup 3 sup Algorithm for Frobenius Normal Form http portal.acm.org ft gateway.cfm?id 281570&type pdf An Algorithm for the Frobenius Normal Form pdf http www.numbertheory.org ...   more details



  1. Chomsky normal form

    In formal language theory, a context free grammar is said to be in Chomsky normal form if all of its production rules are of the form math A rightarrow BC math or math A rightarrow alpha math or math S rightarrow varepsilon math where math A math , math B math and math C math are nonterminal symbols, is a terminal symbol a symbol that represents a constant value , math S math is the start symbol, and is the empty string . Also, neither math B math nor math C math may be the start symbol. Every grammar in Chomsky normal form is context free grammar context free , and conversely, every context free grammar can be transformed into an equivalent one which is in Chomsky normal form. Several algorithms for performing such a transformation are known. Transformations are described in most textbooks on automata theory, such as Hopcroft and Ullman, 1979. ref John E. Hopcroft, Rajeev Motwani, and Jeffrey ... way to define Chomsky normal form e.g., Hopcroft and Ullman 1979, and Hopcroft et al. 2006 is A formal grammar is in Chomsky reduced form if all of its production rules are of the form math A rightarrow ... string, can be transformed into Chomsky reduced form. Converting a grammar to Chomsky Normal Form Introduce ... that are not in Chomsky normal form. dt Replace math A rightarrow u 1 u 2 dotso u k, k ge 3, u 1 ... . See also CYK algorithm Backus Naur form Greibach normal form Kuroda normal form Footnotes references ... isbn 978 0201029550 Pages 103 106. Cole, Richard. Converting CFGs to CNF Chomsky Normal Form , October ... dt math epsilon math rules are rules of the form math A rightarrow epsilon math where math A not S 0 ... 240 of section 6.6 simplified forms and normal forms. cite book authorlink Michael A. Harrison author ... normaal vorm ar bs Chomskyjev normalni oblik ca Forma normal de Chomsky cs Chomsk ho norm ln forma de Chomsky Normalform es Forma normal de Chomsky fr Forme normale de Chomsky ... pl Posta normalna Chomsky ego pt Forma Normal de Chomsky ru fi Chomskyn ...   more details



  1. Jordan normal form

    In linear algebra , a Jordan normal form often called Jordan canonical form ref Shilov defines the term Jordan canonical form and in a footnote says that Jordan normal form is synonymous. These terms are sometimes ... is the field of complex number s. The diagonal entries of the normal form are the eigenvalues of the operator ... is originally given by a square matrix M , then its Jordan normal form is also called the Jordan normal form of M . Any square matrix has a Jordan normal form if the field of coefficients ... is particularly simple on a basis on which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrix normal matrices , is a special case of the Jordan normal form. The Jordan normal form is named after Camille Jordan . Motivation ... . math The matrix J is almost diagonal. This is the Jordan normal form of A . The section Example ... and the superdiagonal. J is called the Jordan normal form of A . Each J sub i sub is called .... The Jordan normal form is obtained by some similarity transformation P sup &minus 1 sup AP J , i.e. ... that every square matrix A can be put in Jordan normal form is equivalent to the claim that there exists ... normal form. Next consider the subspace Ker A &minus I . If math mathrm Ran A lambda I cap mathrm ... chains from the Jordan normal form of A . We can extend the chains by taking the preimages of these lead ... eigenvectors of A , and this shows A can be put in Jordan normal form. Uniqueness It can be shown that the Jordan normal form of a given matrix A is unique up to the order of the Jordan blocks ... the Jordan normal form of A . Assuming the algebraic multiplicity m of an eigenvalue is known ... to show the uniqueness of the Jordan form. Let J sub 1 sub and J sub 2 sub be two Jordan normal ... in the new basis. Consequences One can see that the Jordan normal form is essentially a classification ... as its consequences. Spectral mapping theorem Using the Jordan normal form, direct calculation gives ...   more details



  1. Smith normal form

    In mathematics, the Smith normal form is a Canonical form normal form that can be defined for any matrix not necessarily square with entries in a principal ideal domain PID . The Smith normal form of a matrix ... always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working ... i mid alpha i 1 forall 1 le i r math . This is the Smith normal form of the matrix A . The elements ... normal form. Phrased more abstractly, the goal is to show that, thinking of A as a map from math R ... s identity . To put a matrix into Smith normal form, one can repeatedly apply the following, where t loops ... that the product S A T satisfies the definition of a Smith normal form. In particular, this shows that the Smith normal form exists, which was assumed without proof in the definition. Applications The Smith normal form is useful for computing the homology mathematics homology of a chain complex when ..., we will find the Smith normal form of the following matrix over the integers. math begin pmatrix ... 2 & 0 & 0 0 & 6 & 0 0 & 0 & 12 end pmatrix math So the Smith normal form is math begin pmatrix ... The Smith normal form can be used to determine whether or not matrices with entries over a common ... the Smith normal form of their characteristic matrices match, but are not similar to C because the Smith normal form of the characteristic matrices do not match. References cite journal last Smith ... normal form Hermite normal form Invariant factor Henry John Stephen Smith 1826 1883 , eponym of the Smith normal form Structure theorem for finitely generated modules over a principal ideal domain External ... getobj Smith normal form article at PlanetMath GFDL http planetmath.org encyclopedia ExampleOfSmithNormalForm.html Example of Smith normal form at PlanetMath Category Matrix ... S cdot A cdot T math has the simple form of a diagonal matrix . The matrices S and T can be found ... ring theory ideals generated by the elements at position t , j sub t sub form an ascending ...   more details



  1. Prenex normal form

    rho x math is logically equivalent but not in prenex normal form. Conversion to prenex form Every ... formula in prenex normal form. There are several conversion rules that can be recursively applied to convert a formula to prenex normal form. The rules depend on which logical connective s appear in the formula ... x does not appear as a free variable of math , phi math in 2 and 4 . Use of prenex form Some proof calculus proof calculi will only deal with a theory whose formulae are written in prenex normal form ... that all formulae have been recast in prenex normal form. See also Wiktionary prenex Herbrandization ... x forall z phi lor psi rightarrow rho math . This is not the only prenex form equivalent to the original ..., the prenex form math forall z forall x phi lor psi rightarrow rho math can be obtained math forall ... rho math . Intuitionistic logic The rules for converting a formula to prenex form make heavy ... why some formulas have no intuitionistically equivalent prenex form. In this interpretation, a proof ... 1 will not be equivalent to formula 2 . The rules for converting a formula to prenex form ... Normal forms logic de Pr nexform es Forma prenexa fr Forme pr nexe it Forma prenessa hu Prenex formula nl Prenex normaalvorm ja pl Forma preneksowa pt Forma normal prenex zh ...   more details



  1. Hermite normal form

    In linear algebra , the Hermite normal form is an analogue of reduced echelon form for matrix mathematics matrices over the integer s Z . Nonsingular square matrices A nonsingular matrix nonsingular square matrix M     m sub ij sub with integer entries is in Hermite normal form HNF if M is upper triangular , ref Some authors prefer using lower triangular matrices suitable adjustments must be made to the rest of the definition ref its diagonal entries, m sub ii sub , are positive, for j > i , m sub ii sub > m sub ij sub 0, i.e. in a given row, the entries to the right of the diagonal are less than the diagonal, and at least zero. Example The matrix math begin pmatrix 5 & 3 & 1 & 4 0 & 1 & 0 & 0 0 & 0 & 19 & 16 0 & 0 & 0 & 3 end pmatrix math is in HNF. General matrices More generally, an m × n matrix with integer entries is in HNF if there exists r with 0 r n , a strictly increasing function f r 1, n 1, m , such that the first r columns of M are zero, and for r 1 j n m sub f j j sub > 0, m sub ij sub 0 when i > f j , m sub f j j sub > m sub f j k sub 0 when k > j . Example math begin pmatrix 0&0&5 & 0 & 1 & 4 0&0&0 & 1 & 4 & 99 0&0&0 & 20 & 19 & 16 0&0&0 & 0 & 2 & 1 0&0&0 & 0 & 0 & 3 0&0&0 & 0 & 0 & 0 end pmatrix math Here we have r 2 f 3 1, f 4 3, f 5 4, f 6 5. f j gives the row of the lowest nonzero entry in column j . Uniqueness of the Hermite normal form Given any m × n matrix M with integer entries, there is a unique m × n matrix H , in HNF, with integer entries such that math H MU math with U GL sub n sub Z i.e. U is unimodular matrix unimodular . The matrix formed by the nonzero columns of H is called the Hermite normal form of M . See also Hermite ring Notes references References Section 2.4.2 of Citation unused data Henri Cohen last Cohen first Henri author link Henri Cohen number theorist title A Course in Computational Algebraic Number Theory publisher Springer Verlag location Berlin, New York series Graduate Texts in Mathematics ...   more details




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