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 propositional logic 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 canonical form, which means that two logically equivalent formulas will convert to the same ANF, easily showing whether two formulas are equivalent for automated theorem prov ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Boolean Algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as Logical conjunction, conjunction (''and'') denoted as , disjunction (''or'') denoted as , and negation (''not'') denoted as . Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations. Boolean algebra was introduced by George Boole in his first book ''The Mathematical Analysis of Logic'' (1847), and set forth more fully in his ''An Investigation of the Laws of Thought'' (1854). According to ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Shift Register
A shift register is a type of digital circuit using a cascade of flip-flop (electronics), flip-flops where the output of one flip-flop is connected to the input of the next. They share a single clock signal, which causes the data stored in the system to shift from one location to the next. By connecting the last flip-flop back to the first, the data can cycle within the shifters for extended periods, and in this configuration they were used as computer memory, displacing delay-line memory systems in the late 1960s and early 1970s. In most cases, several parallel shift registers would be used to build a larger memory pool known as a "bit array". Data was stored into the array and read back out in parallel, often as a computer word, while each bit was stored serially in the shift registers. There is an inherent trade-off in the design of bit arrays; putting more flip-flops in a row allows a single shifter to store more bits, but requires more clock cycles to push the data through all ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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EETimes
''EE Times'' (''Electronic Engineering Times'') is an electronics industry magazine published in the United States since 1972. EE Times is currently owned by AspenCore, a division of Arrow Electronics since August 2016. Ownership and status ''EE Times'' was launched in 1972 by Gerard G. Leeds of CMP Publications Inc. In 1999, the Leeds family sold CMP to United Business Media for $900 million. After 2000, ''EE Times'' moved more into web publishing. The shift in advertising from print to online began to accelerate in 2007, and the periodical shed staff to adjust to the downturn in revenue. In July 2013, the digital edition migrated to UBM TechWeb's DeusM community platform. On June 3, 2016, UBM announced that ''EE Times'', along with the rest of its electronics media portfolio ( EDN, Embedded.com, TechOnline, and Datasheets.com), was being sold to AspenCore Media, a company owned by Arrow Electronics, for $23.5 million. The acquisition was completed on August 1, 2016. Av ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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University Of Duisburg-Essen
The University of Duisburg-Essen () is a public research university in North Rhine-Westphalia, Germany. In the 2019 ''Times Higher Education World University Rankings'', the university was awarded 194th place in the world. It was originally founded in 1654 and re-established on 1 January 2003, as a merger of the Gerhard Mercator University of Duisburg and the university of Essen. It is based in both the cities of Duisburg and Essen, and a part of ConRuhr, University Alliance Metropolis Ruhr. With its 12 departments and around 40,000 students, the University of Duisburg-Essen is among the 10 largest German universities. Since 2014, research income has risen by 150 percent. Natural science and engineering are ranked within the top 10 in Germany, and the humanities are within the top 20 to 30. Especially, the physics field is ranked in the top 1 in Germany. History Origins: University of Duisburg (1555) The university's origins date back to the 1555 decision of Duke ''Wilhe ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Wiley-Teubner
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East O ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Springer Science + Business Media B
Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinational academic publishing group created by the merger of Springer Science+Business Media, Nature Publishing Group, Palgrave Macmillan, and Macmillan Education * Axel Springer SE, a German publishing house, including several newspapers * Springer Publishing, an American publishing company of academic journals and books, mainly focusing on public health. Places ;United States * Springer, New Mexico * Springer, Oklahoma * Springer Mountain, southern terminus of the Appalachian Trail * Springer Opera House, Columbus, Georgia Animals * In cattle, a cow or heifer near to calving * English Springer Spaniel, a breed of dog * Welsh Springer Spaniel, a breed of dog * Springer (orca), an orca (killer whale) identified as A73 in her wild community Vehic ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Boolean Ring
In mathematics, a Boolean ring is a ring for which for all in , that is, a ring that consists of only idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet , and ring addition to exclusive disjunction or symmetric difference (not disjunction , which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole. Notation There are at least four different and incompatible systems of notation for Boolean rings and algebras: * In commutative algebra the standard notation is to use for the ring sum of and , and use for their product. * In logic, a common notation is to use for the meet (same as the ring product) and use for the join, given in terms of ring notation (given just above) by . * In set theory and logic it is also common to use f ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Karnaugh Map
A Karnaugh map (KM or K-map) is a diagram that can be used to simplify a Boolean algebra expression. Maurice Karnaugh introduced the technique in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which itself was a rediscovery of Allan Marquand's 1881 ''logical diagram'' or Marquand diagram. They are also known as Marquand–Veitch diagrams, Karnaugh–Veitch (KV) maps, and (rarely) Svoboda charts. An early advance in the history of formal logic methodology, Karnaugh maps remain relevant in the digital age, especially in the fields of logical circuit design and digital engineering. Definition A Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability. It also permits the rapid identification and elimination of potential race conditions. The required Boolean results are transferred from a truth table onto a two-dimensional grid where, in Karnaugh maps, the cells are ordered in Gray code, and each cell ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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 and each conjunction appears at most once (up to the order of variables). 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'' \, \to \, ''Conjunct'' \, \mid \, ''Conjunc ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Conjunctive Normal Form
In Boolean algebra, a formula is in conjunctive normal form (CNF) or clausal normal form if it is a conjunction of one or more clauses, where a clause is a disjunction of literals; otherwise put, it is a product of sums or an AND of ORs. In automated theorem proving, the notion "''clausal normal form''" is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals. Definition A logical formula is considered to be in CNF if it is a conjunction of one or more disjunctions of one or more literals. As in disjunctive normal form (DNF), the only propositional operators in CNF are or (\vee), and (\and), 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 CNF: : ''CNF'' \, \to \, ''Disjunct'' \, \mid \, ''Disjunct'' \, \land \, ''CNF'' : ''Disjunct'' \, \to \, ''Literal'' \, \mid\, ''Literal'' ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Logical Graph
An existential graph is a type of diagrammatic or visual notation for logical expressions, created by Charles Sanders Peirce, who wrote on graphical logic as early as 1882, and continued to develop the method until his death in 1914. They include both a separate graphical notation for logical statements and a logical calculus, a formal system of rules of inference that can be used to derive theorems. Background Peirce found the algebraic notation (i.e. symbolic notation) of logic, especially that of predicate logic, which was still very new during his lifetime and which he himself played a major role in developing, to be philosophically unsatisfactory, because the symbols had their meaning by mere convention. In contrast, he strove for a style of writing in which the signs literally carry their meaning within them – in the terminology of his theory of signs: a system of iconic signs that resemble or resemble the represented objects and relations. Thus, the development of an ico ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Boolean Function
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]   |