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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. 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] |
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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (''and'') denoted as ∧, disjunction (''or'') denoted as ∨, and the negation (''not'') denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. So Boolean algebra is 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 Huntington, the term "Boolean alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Morgan's Laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. The rules can be expressed in English as: * The negation of a disjunction is the conjunction of the negations * The negation of a conjunction is the disjunction of the negations or * The complement of the union of two sets is the same as the intersection of their complements * The complement of the intersection of two sets is the same as the union of their complements or * not (A or B) = (not A) and (not B) * not (A and B) = (not A) or (not B) where "A or B" is an " inclusive or" meaning ''at least'' one of A or B rather than an " exclusive or" that means ''exactly'' one of A or B. In set theory and Boolean algebra, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" ( conjunction), "or" (disjunction), "not" (negation) and "if" ... [...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 function has been reached. It is sometimes referred to as the tabulation method. The method involves two steps: # Finding all prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blake Canonical Form
In Boolean logic, a 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 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 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 presented his canonical form at a meeting of the American Mathematical Society in 1932, and in his 1937 dissertation. He called it the "simplified canonical form"; it was named the "B ... [...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 f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analysis Of Algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that relates the size of an algorithm's input to the number of steps it takes (its time complexity) or the number of storage locations it uses (its space complexity). An algorithm is said to be efficient when this function's values are small, or grow slowly compared to a growth in the size of the input. Different inputs of the same size may cause the algorithm to have different behavior, so best, worst and average case descriptions might all be of practical interest. When not otherwise specified, the function describing the performance of an algorithm is usually an upper bound, determined from the worst case inputs to the algorithm. The term "analysis of algorithms" was coined by Donald Knuth. Algorithm analysis is an important part of a br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautology (logic)
In mathematical logic, a tautology (from el, ταυτολογία) is a formula or assertion that is true in every possible interpretation. An example is "x=y or x≠y". Similarly, "either the ball is green, or the ball is not green" is always true, regardless of the colour of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, borrowing from rhetoric, where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. It cannot be untrue. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables. The d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Co-NP-hard
In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that any problem in co-NP can be reformulated as a special case of any co-NP-complete problem with only polynomial overhead. If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there exists a way to solve a co-NP-complete problem quickly, then that algorithm can be used to solve all co-NP problems quickly. Each co-NP-complete problem is the complement of an NP-complete problem. There are some problems in both NP and co-NP, for example all problems in P or integer factorization. However, it is not known if the sets are equal, although inequality is thought more likely. See co-NP and NP-complete for more details. Fortune showed in 1979 that if any sparse language is co-NP-complete (or even just co-NP-hard), then , a critical foundation for Mahaney's theorem. Formal definition A decision ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality Principle (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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (''and'') denoted as ∧, disjunction (''or'') denoted as ∨, and the negation (''not'') denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. So Boolean algebra is 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 Huntington, the term "Boolean alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hardness
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem. A more precise specification is: a problem ''H'' is NP-hard when every problem ''L'' in NP can be reduced in polynomial time to ''H''; that is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve any NP-hard problem would give polynomial time algorithms for all the problems in NP. As it is suspected that P≠NP, it is unlikely that such an algorithm exists. It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven. Moreover, the class P, in which all problems can be solved in polynomial time, is contained in the NP class. Def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Satisfiability Problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called ''satisfiable''. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. SAT is the first problem that was proved to be NP-compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
