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Zhegalkin Normal Form
Zhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials (russian: полиномы Жегалкина), also known as algebraic normal form, are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, ''x''2 = ''x''. Hence a polynomial such as 3''x''2''y''5''z'' is congruent to, and can therefore be rewritten as, ''xyz''. __TOC__ Boolean equivalent Prior to 1927, Boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, and so on. Zhegalkin showed that all Boolean operations could be writt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Normal Form
In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), '' Zhegalkin normal form'', or '' Reed–Muller expansion'' is a way of writing logical formulas in one of three subforms: * The entire formula is purely true or false: *: 1 *: 0 * One or more variables are combined into a term by AND (\and), then one or more terms are combined by XOR (\oplus) together into ANF. Negations are not permitted: : a \oplus b \oplus \left(a \and b\right) \oplus \left(a \and b \and c\right) * The previous subform with a purely true term: : 1 \oplus a \oplus b \oplus \left(a \and b\right) \oplus \left(a \and b \and c\right) Formulas written in ANF are also known as Zhegalkin polynomials and Positive Polarity (or Parity) Reed–Muller expressions (PPRM). Common uses ANF is a normal form, which means that two equivalent formulas will convert to the same ANF, easily showing whether two formulas are equivalent for automated theorem proving. Unlike other normal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Functions
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory. A Boolean function takes the form f:\^k \to \, where \ is known as the Boolean domain and k is a non-negative integer called the arity of the function. In the case where k=0, the function is a constant element of \. A Boolean function with multiple outputs, f:\^k \to \^m with m>1 is a ''vectorial'' or ''vector-valued'' Boolean function (an S-box in symmetric cryptography). There are 2^ different Boolean functions with k arguments; equal to the number of different truth tables with 2^k entries. Every k-ary Boolean function can be expressed as a propositional formula in k variables x_1,...,x_k, and two propositional formulas a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunctive Normal Form
In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or (in philosophical logic) a ''cluster concept''. As a normal form, it is useful in automated theorem proving. Definition A logical formula is considered to be in DNF if it is a disjunction of one or more conjunctions of one or more literals. A DNF formula is in full disjunctive normal form if each of its variables appears exactly once in every conjunction. As in conjunctive normal form (CNF), the only propositional operators in DNF are and (\wedge), or (\vee), and not (\neg). The ''not'' operator can only be used as part of a literal, which means that it can only precede a propositional variable. The following is a context-free grammar for DNF: # ''DNF'' → (''Conjunction'') \vee ''DNF'' # ''DNF'' → (''Conjunction'') # ''Conjunction'' → ''Literal'' \wedge ''Conju ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Disjunctive Normal Form
In Boolean algebra, any Boolean function can be expressed in the canonical disjunctive normal form ( CDNF) or minterm canonical form and its dual canonical conjunctive normal form ( CCNF) or maxterm canonical form. Other canonical forms include the complete sum of prime implicants or Blake canonical form (and its dual), and the algebraic normal form (also called Zhegalkin or Reed–Muller). ''Minterms'' are called products because they are the logical AND of a set of variables, and ''maxterms'' are called sums because they are the logical OR of a set of variables. These concepts are dual because of their complementary-symmetry relationship as expressed by De Morgan's laws. Two dual canonical forms of ''any'' Boolean function are a "sum of minterms" and a "product of maxterms." The term "Sum of Products" (SoP or SOP) is widely used for the canonical form that is a disjunction (OR) of minterms. Its De Morgan dual is a "Product of Sums" (PoS or POS) for the canonical form that i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sierpiński Triangle
The Sierpiński triangle (sometimes spelled ''Sierpinski''), also called the Sierpiński gasket or Sierpiński sieve, is a fractal curve, fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursion, recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similarity, self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Poland, Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński. Constructions There are many different ways of constructing the Sierpinski triangle. Removing triangles The Sierpinski triangle may be constructed from an equilateral triangle by repeated removal of triangular subsets: # Start with an equilateral triangle. # Subdivide it into four smaller congruent equilateral triangles and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Matrix
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. Matrix representation of a relation If ''R'' is a binary relation between the finite indexed sets ''X'' and ''Y'' (so ), then ''R'' can be represented by the logical matrix ''M'' whose row and column indices index the elements of ''X'' and ''Y'', respectively, such that the entries of ''M'' are defined by :M_ = \begin 1 & (x_i, y_j) \in R, \\ 0 & (x_i, y_j) \not\in R. \end In order to designate the row and column numbers of the matrix, the sets ''X'' and ''Y'' are indexed with positive integers: ''i'' ranges from 1 to the cardinality (size) of ''X'', and ''j'' ranges from 1 to the cardinality of ''Y''. See the entry on indexed sets for more detail. Example The binary relation ''R'' on the set is defined so that ''aRb'' holds if and only if ''a'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Using A Karnaugh Map
Use may refer to: * Use (law), an obligation on a person to whom property has been conveyed * Use (liturgy), a special form of Roman Catholic ritual adopted for use in a particular diocese * Use–mention distinction, the distinction between using a word and mentioning it * Consumption (economics) ** Resource depletion, use to the point of lack of supply ** Psychological manipulation, in a form that treats a person is as a means to an end * Rental utilization, quantification of the use of assets to be continuously let See also * Use case * User story * USE (other) * Used (other) Used may refer to: Common meanings *Used good, goods of any type that have been used before or pre-owned *Used to, English auxiliary verb Places *Used, Huesca, a village in Huesca, Aragon, Spain *Used, Zaragoza, a town in Zaragoza, Aragon, Spain ... * User (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Summation Method
''The'' () is a grammatical article in English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with pronouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of pronoun ''thee'') when followed by a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Pascal Method
''The'' () is a grammatical article in English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with pronouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of pronoun ''thee'') when followed by a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Using Tables
Use may refer to: * Use (law), an obligation on a person to whom property has been conveyed * Use (liturgy), a special form of Roman Catholic ritual adopted for use in a particular diocese * Use–mention distinction, the distinction between using a word and mentioning it * Consumption (economics) ** Resource depletion, use to the point of lack of supply ** Psychological manipulation, in a form that treats a person is as a means to an end * Rental utilization, quantification of the use of assets to be continuously let See also * Use case * User story * USE (other) * Used (other) Used may refer to: Common meanings *Used good, goods of any type that have been used before or pre-owned *Used to, English auxiliary verb Places *Used, Huesca, a village in Huesca, Aragon, Spain *Used, Zaragoza, a town in Zaragoza, Aragon, Spain ... * User (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Using The Canonical Disjunctive Normal Form
Use may refer to: * Use (law), an obligation on a person to whom property has been conveyed * Use (liturgy), a special form of Roman Catholic ritual adopted for use in a particular diocese * Use–mention distinction, the distinction between using a word and mentioning it * Consumption (economics) ** Resource depletion, use to the point of lack of supply ** Psychological manipulation, in a form that treats a person is as a means to an end * Rental utilization, quantification of the use of assets to be continuously let See also * Use case * User story * USE (other) * Used (other) Used may refer to: Common meanings *Used good, goods of any type that have been used before or pre-owned *Used to, English auxiliary verb Places *Used, Huesca, a village in Huesca, Aragon, Spain *Used, Zaragoza, a town in Zaragoza, Aragon, Spain ... * User (other) {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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