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Conflict-driven Clause Learning
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] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (including the design and implementation of hardware and software). Computer science is generally considered an area of academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing security vulnerabilities. Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of repositories o ... [...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-complet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DPLL Algorithm
In logic and computer science, the Davis–Putnam–Logemann–Loveland (DPLL) algorithm is a complete, backtracking-based search algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i.e. for solving the CNF-SAT problem. It was introduced in 1961 by Martin Davis, George Logemann and Donald W. Loveland and is a refinement of the earlier Davis–Putnam algorithm, which is a resolution-based procedure developed by Davis and Hilary Putnam in 1960. Especially in older publications, the Davis–Logemann–Loveland algorithm is often referred to as the "Davis–Putnam method" or the "DP algorithm". Other common names that maintain the distinction are DLL and DPLL. Implementations and applications The SAT problem is important both from theoretical and practical points of view. In complexity theory it was the first problem proved to be NP-complete, and can appear in a broad variety of applications such as ''model checking'', automat ... [...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-exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false when P is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes ''truth'' to ''falsity'' (and vice versa). In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P is the proposition whose proofs are the refutations of P. Definition ''Classical negation'' is an operation on one logical value, typically the value of a proposition, that produces a value of ''true'' when its operand is false, and a value of ''false'' when its operand is true. Thus if statement is true, then \neg P ... [...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 wel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this operator is typically written as \wedge or . A \land B is true if and only if A is true and B is true, otherwise it is false. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English "and". * In programming languages, the short-circuit and control structure. * In set theory, intersection. * In lattice theory, logical conjunction (greatest lower bound). * In predicate logic, universal quantification. Notation And is usually denoted by an infix operator: in mathematics and logic, it is denoted by \wedge, or ; in electronics, ; and in programming languages, &, &&, or and. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Postfix Notation
Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in which operators ''precede'' their operands. It does not need any parentheses as long as each operator has a fixed number of operands. The description "Polish" refers to the nationality of logician Jan Łukasiewicz, who invented Polish notation in 1924. The first computer to use postfix notation, though it long remained essentially unknown outside of Germany, was Konrad Zuse's Z3 in 1941 as well as his Z4 in 1945. The reverse Polish scheme was again proposed in 1954 by Arthur Burks, Don Warren, and Jesse Wright and was independently reinvented by Friedrich L. Bauer and Edsger W. Dijkstra in the early 1960s to reduce computer memory access and use the stack to evaluate expressions. The algorithms and notation for this scheme were exten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brute-force Search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's statement. A brute-force algorithm that finds the divisors of a natural number ''n'' would enumerate all integers from 1 to n, and check whether each of them divides ''n'' without remainder. A brute-force approach for the eight queens puzzle would examine all possible arrangements of 8 pieces on the 64-square chessboard and for each arrangement, check whether each (queen) piece can attack any other. While a brute-force search is simple to implement and will always find a solution if it exists, implementation costs are proportional to the number of candidate solutionswhich in many practical problems tends to grow very quickly as the size of the problem increases ( §Combinato ... [...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 In computer s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Davis–Putnam Algorithm
The Davis–Putnam algorithm was developed by Martin Davis and Hilary Putnam for checking the validity of a first-order logic formula using a resolution-based decision procedure for propositional logic. Since the set of valid first-order formulas is recursively enumerable but not recursive, there exists no general algorithm to solve this problem. Therefore, the Davis–Putnam algorithm only terminates on valid formulas. Today, the term "Davis–Putnam algorithm" is often used synonymously with the resolution-based propositional decision procedure (Davis–Putnam procedure) that is actually only one of the steps of the original algorithm. Overview The procedure is based on Herbrand's theorem, which implies that an unsatisfiable formula has an unsatisfiable ground instance, and on the fact that a formula is valid if and only if its negation is unsatisfiable. Taken together, these facts imply that to prove the validity of ''φ'' it is enough to prove that a ground instance of ''¬� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
State Space Search
State space search is a process used in the field of computer science, including artificial intelligence (AI), in which successive configurations or ''states'' of an instance are considered, with the intention of finding a ''goal state'' with the desired property. Problems are often modelled as a state space, a set of ''states'' that a problem can be in. The set of states forms a graph where two states are connected if there is an ''operation'' that can be performed to transform the first state into the second. State space search often differs from traditional computer science search methods because the state space is ''implicit'': the typical state space graph is much too large to generate and store in memory. Instead, nodes are generated as they are explored, and typically discarded thereafter. A solution to a combinatorial search instance may consist of the goal state itself, or of a path from some ''initial state'' to the goal state. Representation In state space searc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
