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Implication Graph
In mathematical logic and graph theory, an implication graph is a skew-symmetric, directed graph composed of vertex set and directed edge set . Each vertex in represents the truth status of a Boolean literal, and each directed edge from vertex to vertex represents the material implication "If the literal is true then the literal is also true". Implication graphs were originally used for analyzing complex Boolean expressions. Applications A 2-satisfiability instance in conjunctive normal form can be transformed into an implication graph by replacing each of its disjunctions by a pair of implications. For example, the statement (x_0\lor x_1) can be rewritten as the pair (\neg x_0 \rightarrow x_1), (\neg x_1 \rightarrow x_0). An instance is satisfiable if and only if no literal and its negation belong to the same strongly connected component of its implication graph; this characterization can be used to solve 2-satisfiability instances in linear time. In CDCL SAT-solvers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implication Graph
In mathematical logic and graph theory, an implication graph is a skew-symmetric, directed graph composed of vertex set and directed edge set . Each vertex in represents the truth status of a Boolean literal, and each directed edge from vertex to vertex represents the material implication "If the literal is true then the literal is also true". Implication graphs were originally used for analyzing complex Boolean expressions. Applications A 2-satisfiability instance in conjunctive normal form can be transformed into an implication graph by replacing each of its disjunctions by a pair of implications. For example, the statement (x_0\lor x_1) can be rewritten as the pair (\neg x_0 \rightarrow x_1), (\neg x_1 \rightarrow x_0). An instance is satisfiable if and only if no literal and its negation belong to the same strongly connected component of its implication graph; this characterization can be used to solve 2-satisfiability instances in linear time. In CDCL SAT-solvers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S , assuming that R abbreviates "it is raining" and S abbreviates "it is snowing". In classical logic, disjunction is given a truth functional semantics according to which a formula \phi \lor \psi is true unless both \phi and \psi are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an ''inclusive'' interpretation of disjunction, in contrast with exclusive disjunction. Classical proof theoretical treatments are often given in terms of rules such as disjunction introduction and disjunction elimination. Disjunction has also been given numerous non-classical treatments, motivated by problems including Aristotle's sea battle argument, Heisenberg's uncertainty principle, as well t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 algebra" wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Propagation
Unit propagation (UP) or Boolean Constraint propagation (BCP) or the one-literal rule (OLR) is a procedure of automated theorem proving that can simplify a set of (usually propositional) clauses. Definition The procedure is based on unit clauses, i.e. clauses that are composed of a single literal, in conjunctive normal form. Because each clause needs to be satisfied, we know that this literal must be true. If a set of clauses contains the unit clause l, the other clauses are simplified by the application of the two following rules: # every clause (other than the unit clause itself) containing l is removed (the clause is satisfied if l is); # in every clause that contains \neg l this literal is deleted (\neg l can not contribute to it being satisfied). The application of these two rules lead to a new set of clauses that is equivalent to the old one. For example, the following set of clauses can be simplified by unit propagation because it contains the unit clause a. : \ Sinc ... [...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-complete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CDCL
In computer science, conflict-driven clause learning (CDCL) is an algorithm for solving the Boolean satisfiability problem (SAT). Given a Boolean formula, the SAT problem asks for an assignment of variables so that the entire formula evaluates to true. The internal workings of CDCL SAT solvers were inspired by DPLL solvers. The main difference between CDCL and DPLL is that CDCL's back jumping is non-chronological. Conflict-driven clause learning was proposed by Marques-Silva and Sakallah (1996, 1999) and Bayardo and Schrag (1997). Background Boolean satisfiability problem The satisfiability problem consists in finding a satisfying assignment for a given formula in conjunctive normal form (CNF). An example of such a formula is: :( ( not ''A'') or (not ''C'') ) and (''B'' or ''C''), or, using a common notation:In the pictures below, "+" is used to denote "or", multiplication to denote "and", and a postfix "'" to denote "not". :(\lnot A \lor \lnot C) \land (B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Tarjan
Robert Endre Tarjan (born April 30, 1948) is an American computer scientist and mathematician. He is the discoverer of several graph algorithms, including Tarjan's off-line lowest common ancestors algorithm, and co-inventor of both splay trees and Fibonacci heaps. Tarjan is currently the James S. McDonnell Distinguished University Professor of Computer Science at Princeton University, and the Chief Scientist at Intertrust Technologies Corporation. Early life and education He was born in Pomona, California. His father, raised in Hungary, was a child psychiatrist, specializing in mental retardation, and ran a state hospital. As a child, Tarjan read a lot of science fiction, and wanted to be an astronomer. He became interested in mathematics after reading Martin Gardner's mathematical games column in Scientific American. He became seriously interested in math in the eighth grade, thanks to a "very stimulating" teacher. While he was in high school, Tarjan got a job, where he work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Plass
Michael may refer to: People * Michael (given name), a given name * Michael (surname), including a list of people with the surname Michael Given name "Michael" * Michael (archangel), ''first'' of God's archangels in the Jewish, Christian and Islamic religions * Michael (bishop elect), English 13th-century Bishop of Hereford elect * Michael (Khoroshy) (1885–1977), cleric of the Ukrainian Orthodox Church of Canada * Michael Donnellan (1915–1985), Irish-born London fashion designer, often referred to simply as "Michael" * Michael (footballer, born 1982), Brazilian footballer * Michael (footballer, born 1983), Brazilian footballer * Michael (footballer, born 1993), Brazilian footballer * Michael (footballer, born February 1996), Brazilian footballer * Michael (footballer, born March 1996), Brazilian footballer * Michael (footballer, born 1999), Brazilian footballer Rulers =Byzantine emperors= *Michael I Rangabe (d. 844), married the daughter of Emperor Nikephoros I *Mic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Connected Component
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(''V'' + ''E'')). Definitions A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first. In a directed graph ''G'' that may not itself be strongly connected, a pair of vertices ''u'' and ''v'' are said to be strongly connected to each other if there is a path in each direction between them. The binary relation of being strongly connected is an equivalence relation, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjunctive Normal Form
In Boolean logic, 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. As a canonical normal form, it is useful in automated theorem proving and 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 propositional connectives a formula in CNF can contain are and, or, and not. The not operator can only be used as part of a literal, which means that it can only precede a propositional variable or a predicate symbol. 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. Examples and non-examples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
