|
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'', automate ... [...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] |
Model Checking
In computer science, model checking or property checking is a method for checking whether a finite-state model of a system meets a given specification (also known as correctness). This is typically associated with hardware or software systems, where the specification contains liveness requirements (such as avoidance of livelock) as well as safety requirements (such as avoidance of states representing a system crash). In order to solve such a problem algorithmically, both the model of the system and its specification are formulated in some precise mathematical language. To this end, the problem is formulated as a task in logic, namely to check whether a structure satisfies a given logical formula. This general concept applies to many kinds of logic and many kinds of structures. A simple model-checking problem consists of verifying whether a formula in the propositional logic is satisfied by a given structure. Overview Property checking is used for verification when two desc ... [...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] |
Polarity (mathematical Logic)
In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive normal form and the method of resolution. Literals can be divided into two types: * A positive literal is just an atom (e.g., x). * A negative literal is the negation of an atom (e.g., \lnot x). The polarity of a literal is positive or negative depending on whether it is a positive or negative literal. In logics with double negation elimination (where \lnot \lnot x \equiv x) the complementary literal or complement of a literal l can be defined as the literal corresponding to the negation of l. We can write \bar to denote the complementary literal of l. More precisely, if l\equiv x then \bar is \lnot x and if l\equiv \lnot x then \bar is x. Double negation elimination occurs in classical logics but not in intuitionistic logic. In the context of a formula in the conjunctive normal form ... [...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] |
Truth Value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false". Classical logic In classical logic, with its intended semantics, the truth values are ''true'' (denoted by ''1'' or the verum ⊤), and '' untrue'' or '' false'' (denoted by ''0'' or the falsum ⊥); that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantics of l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Theory
A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms. It can also simultaneously be a body of knowledge (e.g., based on known axioms and definitions), and so in this sense can refer to an area of mathematical research within the established framework. Explanatory depth is one of the most significant theoretical virtues in mathematics. For example, set theory has the ability to systematize and explain number theory and geometry/analysis. Despite the widely logical necessity (and self-evidence) of arithmetic truths such as 1<3, 2+2=4, 6-1=5, and so on, a theory that just postulates an infinite blizzard of such truths would be inadequate. Rather an adequate theory is one in which such truths are derived from explanatorily prior axioms, such as the Peano Axioms or set theoretic axioms, which lie at the foundation of ZFC axiomatic set theory. The singular accomplishment of axiomatic set theory is its ability to give a foundation for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Variable
In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics. Uses Formulas in logic are typically built up recursively from some propositional variables, some number of logical connectives, and some logical quantifiers. Propositional variables are the atomic formulas of propositional logic, and are often denoted using capital roman letters such as P, Q and R. ;Example In a given propositional logic, a formula can be defined as follows: * Every propositional variable is a formula. * Given a formula ''X'', the negation ''¬X'' is a formula. * Given two formulas ''X'' and ''Y'', and a binary connective ''b'' (such as the logical conjunction ∧),the expression ''(X b Y)'' is a formula. (Note the parentheses.) Through this const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Satisfiability Modulo Theories
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings. The name is derived from the fact that these expressions are interpreted within ("modulo") a certain formal theory in first-order logic with equality (often disallowing quantifiers). SMT solvers are tools which aim to solve the SMT problem for a practical subset of inputs. SMT solvers such as Z3 and cvc5 have been used as a building block for a wide range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing. Since Boolean satisfiability is already NP-complete, the SMT problem is typically NP-hard, and for many theories it is undecidable. Resea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automated Theorem Proving
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science. Logical foundations While the roots of formalised logic go back to Aristotle, the end of the 19th and early 20th centuries saw the development of modern logic and formalised mathematics. Frege's ''Begriffsschrift'' (1879) introduced both a complete propositional calculus and what is essentially modern predicate logic. His ''Foundations of Arithmetic'', published 1884, expressed (parts of) mathematics in formal logic. This approach was continued by Russell and Whitehead in their influential ''Principia Mathematica'', first published 1910–1913, and with a revised second edition in 1927. Russell and Whitehead thought they could derive all mathematical truth using axioms and inference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ZChaff
Chaff is an algorithm for solving instances of the Boolean satisfiability problem in programming. It was designed by researchers at Princeton University. The algorithm is an instance of the DPLL algorithm with a number of enhancements for efficient implementation. Implementations Some available implementations of the algorithm in software are mChaff and zChaff, the latter one being the most widely known and used. zChaff was originally written by Dr. Lintao Zhang, at Microsoft Research, hence the “z”. It is now maintained by researchers at Princeton University and available for download as both source code and binaries on Linux Linux ( or ) is a family of open-source Unix-like operating systems based on the Linux kernel, an operating system kernel first released on September 17, 1991, by Linus Torvalds. Linux is typically packaged as a Linux distribution, w .... zChaff is free for non-commercial use. References * M. Moskewicz, C. Madigan, Y. Zhao, L. Zhan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GRASP (SAT Solver)
GRASP is a well known SAT instance solver. It was developed by João Marques Silva, a Portuguese Portuguese may refer to: * anything of, from, or related to the country and nation of Portugal ** Portuguese cuisine, traditional foods ** Portuguese language, a Romance language *** Portuguese dialects, variants of the Portuguese language ** Portu ... computer science researcher. It stands for ''Generic seaRch Algorithm for the Satisfiability Problem''. External links GRASP home page References {{Reflist SAT solvers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> Model Checking")