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DPLL(T)
In computer science, DPLL(T) is a framework for determining the satisfiability of SMT problems. The algorithm extends the original SAT-solving DPLL algorithm with the ability to reason about an arbitrary theory ''T''. At a high level, the algorithm works by transforming an SMT problem into a SAT formula where atoms are replaced with Boolean variables. The algorithm repeatedly finds a satisfying valuation for the SAT problem, consults a theory solver to check consistency under the domain-specific theory, and then (if a contradiction is found) refines the SAT formula with this information. Many modern SMT solvers, such as Microsoft's Z3 Theorem Prover Z3, also known as the Z3 Theorem Prover, is a cross-platform satisfiability modulo theories (SMT) solver by Microsoft. Overview Z3 was developed in the ''Research in Software Engineering'' (RiSE) group at Microsoft Research and is targeted at sol ..., use DPLL(T) to power their core solving capabilities. References Automated t ... [...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] |
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] |
Logical Theory
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a deductively closed theory T is then called a theorem of the theory. In many deductive systems there is usually a subset \Sigma \subseteq T that is called "the set of axioms" of the theory T, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms. General theories (as expressed in formal language) When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate. The construction of a theory begins by specifying a definite non-empty ''conceptual class'' \mathcal, the element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solver
A solver is a piece of mathematical software, possibly in the form of a stand-alone computer program or as a software library, that 'solves' a mathematical problem. A solver takes problem descriptions in some sort of generic form and calculates their solution. In a solver, the emphasis is on creating a program or library that can easily be applied to other problems of similar type. Solver types Types of problems with existing dedicated solvers include: * Linear and non-linear equations. In the case of a single equation, the "solver" is more appropriately called a root-finding algorithm. * Systems of linear equations. * Nonlinear systems. * Systems of polynomial equations, which are a special case of non linear systems, better solved by specific solvers. * Linear and non-linear optimisation problems * Systems of ordinary differential equations * Systems of differential algebraic equations * Boolean satisfiability problems, including SAT solvers * Quantified boolean formula s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microsoft
Microsoft Corporation is an American multinational technology corporation producing computer software, consumer electronics, personal computers, and related services headquartered at the Microsoft Redmond campus located in Redmond, Washington, United States. Its best-known software products are the Windows line of operating systems, the Microsoft Office suite, and the Internet Explorer and Edge web browsers. Its flagship hardware products are the Xbox video game consoles and the Microsoft Surface lineup of touchscreen personal computers. Microsoft ranked No. 21 in the 2020 Fortune 500 rankings of the largest United States corporations by total revenue; it was the world's largest software maker by revenue as of 2019. It is one of the Big Five American information technology companies, alongside Alphabet, Amazon, Apple, and Meta. Microsoft was founded by Bill Gates and Paul Allen on April 4, 1975, to develop and sell BASIC interpreters for the Altair 8800. It rose to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z3 Theorem Prover
Z3, also known as the Z3 Theorem Prover, is a cross-platform satisfiability modulo theories (SMT) solver by Microsoft. Overview Z3 was developed in the ''Research in Software Engineering'' (RiSE) group at Microsoft Research and is targeted at solving problems that arise in software verification and program analysis. Z3 supports arithmetic, fixed-size bit-vectors, extensional arrays, datatypes, uninterpreted functions, and quantifiers. Its main applications are extended static checking, test case generation, and predicate abstraction. In 2015, it received the ''Programming Languages Software Award'' from ACM SIGPLAN. In 2018, Z3 received the ''Test of Time Award'' from the European Joint Conferences on Theory and Practice of Software (ETAPS). Microsoft researchers Nikolaj Bjørner and Leonardo de Moura received the 2019 Herbrand Award for Distinguished Contributions to Automated Reasoning in recognition of their work in advancing theorem proving with Z3. Z3 was open sourced in ... [...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] |
SAT Solvers
The SAT ( ) is a standardized test widely used for college admissions in the United States. Since its debut in 1926, its name and scoring have changed several times; originally called the Scholastic Aptitude Test, it was later called the Scholastic Assessment Test, then the SAT I: Reasoning Test, then the SAT Reasoning Test, then simply the SAT. The SAT is wholly owned, developed, and published by the College Board, a private, not-for-profit organization in the United States. It is administered on behalf of the College Board by the Educational Testing Service, which until recently developed the SAT as well. The test is intended to assess students' readiness for college. The SAT was originally designed not to be aligned with high school curricula, but several adjustments were made for the version of the SAT introduced in 2016, and College Board president David Coleman has said that he also wanted to make the test reflect more closely what students learn in high school with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
