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Prime Implicant
In Boolean logic, the term implicant has either a generic or a particular meaning. In the generic use, it refers to the hypothesis of an implication (implicant). In the particular use, a product term (i.e., a conjunction of literals) ''P'' is an implicant of a Boolean function ''F'', denoted P \le F, if ''P'' implies ''F'' (i.e., whenever ''P'' takes the value 1 so does ''F''). For instance, implicants of the function :f(x,y,z,w)=xy+yz+w include the terms xy, xyz, xyzw, w, as well as some others. Prime implicant A prime implicant of a function is an implicant (in the above particular sense) that cannot be covered by a more general, (more reduced, meaning with fewer literals) implicant. W. V. Quine defined a ''prime implicant'' to be an implicant that is minimal—that is, the removal of any literal from ''P'' results in a non-implicant for ''F''. Essential prime implicants (also known as core prime implicants) are prime implicants that cover an output of the function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Logic
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] |
Implicant
In Boolean logic, the term implicant has either a generic or a particular meaning. In the generic use, it refers to the hypothesis of an implication (wiktionary:implicant, implicant). In the particular use, a product term (i.e., a conjunction of literals) ''P'' is an implicant of a Boolean function ''F'', denoted P \le F, if ''P'' implies ''F'' (i.e., whenever ''P'' takes the value 1 so does ''F''). For instance, implicants of the function :f(x,y,z,w)=xy+yz+w include the terms xy, xyz, xyzw, w, as well as some others. Prime implicant A prime implicant of a function is an implicant (in the above particular sense) that cannot be covered by a more general, (more reduced, meaning with fewer literal (computer programming), literals) implicant. W. V. Quine defined a ''prime implicant'' to be an implicant that is minimal—that is, the removal of any literal from ''P'' results in a non-implicant for ''F''. Essential prime implicants (also known as core prime implicants) are pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Term
In Boolean logic, a product term is a conjunction of literals, where each literal is either a variable or its negation. Examples Examples of product terms include: :A \wedge B :A \wedge (\neg B) \wedge (\neg C) :\neg A Origin The terminology comes from the similarity of AND to multiplication as in the ring structure of Boolean rings. Minterms For a 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 functi ... of n variables , a product term in which each of the n variables appears once (in either its complemented or uncomplemented form) is called a ''minterm''. Thus, a ''minterm'' is a logical expression of ''n'' variables that employs only the ''complement'' operator and the ''conjunction'' operator. References *Fredrick J. Hill, and Gerald R. Peterson, 1974, ''Introduction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Literal (computer Programming)
In computer science, a literal is a textual representation (notation) of a value as it is written in source code. Almost all programming languages have notations for atomic values such as integers, floating-point numbers, and strings, and usually for Booleans and characters; some also have notations for elements of enumerated types and compound values such as arrays, records, and objects. An anonymous function is a literal for the function type. In contrast to literals, variables or constants are symbols that can take on one of a class of fixed values, the constant being constrained not to change. Literals are often used to initialize variables; for example, in the following, 1 is an integer literal and the three letter string in "cat" is a string literal: int a = 1; string s = "cat"; In lexical analysis, literals of a given type are generally a token type, with a grammar rule, like "a string of digits" for an integer literal. Some literals are specific keywords, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blake Canonical Form
In Boolean logic, a Formula (mathematical logic), formula for a Boolean function ''f'' is in Blake canonical form (BCF), also called the complete sum of prime implicants, the complete sum, or the disjunctive prime form, when it is a logical disjunction, disjunction of all the prime implicants of ''f''. Relation to other forms The Blake canonical form is a special case of disjunctive normal form. The Blake canonical form is not necessarily Boolean minimization, minimal (upper diagram), however all the terms of a minimal sum are contained in the Blake canonical form. On the other hand, the Blake canonical form is a canonical form, that is, it is unique up to reordering, whereas there can be multiple minimal forms (lower diagram). Selecting a minimal sum from a Blake canonical form amounts in general to solving the set cover problem, so is NP-hard. History Archie Blake (mathematician), Archie Blake presented his canonical form at a meeting of the American Mathematical Society in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quine–McCluskey Algorithm
The Quine–McCluskey algorithm (QMC), also known as the method of prime implicants, is a method used for minimization of Boolean functions that was developed by Willard V. Quine in 1952 and extended by Edward J. McCluskey in 1956. As a general principle this approach had already been demonstrated by the logician Hugh McColl in 1878, was proved by Archie Blake in 1937, and was rediscovered by Edward W. Samson and Burton E. Mills in 1954 and by Raymond J. Nelson in 1955. Also in 1955, Paul W. Abrahams and John G. Nordahl as well as Albert A. Mullin and Wayne G. Kellner proposed a decimal variant of the method. The Quine–McCluskey algorithm is functionally identical to Karnaugh mapping, but the tabular form makes it more efficient for use in computer algorithms, and it also gives a deterministic way to check that the minimal form of a Boolean F has been reached. It is sometimes referred to as the tabulation method. The Quine-McCluskey algorithm works as follows: # Finding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Petrick's Method
In Boolean algebra, Petrick's method (also known as ''Petrick function'' or ''branch-and-bound'' method) is a technique described by Stanley R. Petrick (1931–2006) in 1956 for determining all minimum sum-of-products solutions from a prime implicant chart. Petrick's method is very tedious for large charts, but it is easy to implement on a computer. The method was improved by Insley B. Pyne and Edward Joseph McCluskey in 1962. Algorithm # Reduce the prime implicant chart by eliminating the essential prime implicant rows and the corresponding columns. # Label the rows of the reduced prime implicant chart P_1, P_2, P_3, P_4, etc. # Form a logical function P which is true when all the columns are covered. ''P'' consists of a product of sums where each sum term has the form (P_ + P_ + \cdots + P_), where each P_ represents a row covering column i. # Apply De Morgan's Laws to expand P into a sum of products and minimize by applying the absorption law In algebra, the absorption l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
