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Topic: Boolean satisfiability problem

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Boolean algebra

Boolean satisfiability problem

list of Boolean algebra topics

Boolean ring

Boolean logic

Boolean prime ideal theorem

Canonical form Boolean algebra

Monadic Boolean algebras

Boolean lattice

boolean circuit

Boolean operator

Complete Boolean algebra

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	Boolean satisfiability problem - Wikipedia, the free encyclopedia
		The Boolean satisfiability problem (SAT) is a decision problem considered in complexity theory.
		Although this problem seems easier, it has been shown that if there is a practical (randomized polynomial-time) algorithm to solve this problem, then all problems in NP can be solved just as easily.
		The maximum satisfiability problem, an FNP generalization of SAT, asks for the maximum number of clauses which can be satisfied by any assignment.
	en.wikipedia.org /wiki/Boolean_satisfiability_problem (1323 words)

	Encyclopedia :: encyclopedia : Computational complexity theory (Site not responding. Last check: 2007-11-07)
		The time complexity of a problem is the number of steps that it takes to solve an instance of the problem as a function of the size of the input (usually measured in bits), using the most efficient algorithm.
		A decision problem is a problem where the answer is always YES/NO. For example, the problem IS-PRIME is: given an integer written in binary, return whether it is a prime number or not.
		Problems that are solvable in theory, but can't be solved in practice, are called intractable.
	www.hallencyclopedia.com /Computational_complexity_theory (1121 words)

	Cook's theorem - Wikipedia, the free encyclopedia
		The Boolean satisfiability problem is in NP because a non-deterministic Turing machine can guess an assignment of truth values to the variables, determine the value of the expression under that assignment, and accept if the assignment makes the entire expression true.
		This means that if the Boolean satisfiability problem could be solved in polynomial time by a deterministic Turing machine, then all problems in NP could be solved in polynomial time, and so the complexity class NP would be equal to the complexity class P. Cook's theorem was the first proof of NP-completeness for any problem.
		For example, the problem 3-SAT (the satisfiability problem for Boolean expressions in conjunctive normal form with three variables or negations of variables per clause) can be shown to be NP-complete by showing how to reduce any instance of SAT to an equivalent instance of 3-SAT.
	en.wikipedia.org /wiki/Cook's_theorem (704 words)

	NP-complete - Wikipedia, the free encyclopedia
		At first it seems rather surprising that NP-complete problems should even exist, but in the celebrated Cook-Levin theorem (independently proved by Leonid Levin), Cook proved that the Boolean satisfiability problem is NP-complete (a simpler, but still highly technical proof of this is available).
		The Graph Isomorphism problem is suspected to be neither in P nor NP-complete, though it is obviously in NP.
		For example, the 3SAT problem, a restriction of the boolean satisfiability problem, remains NP-complete, whereas the slightly more restricted 2SAT problem is in P (specifically, NL-complete), and the slightly more general MAX 2SAT problem is again NP-complete.
	en.wikipedia.org /wiki/NP-complete (1772 words)

	Boolean satisfiability problem
		The class of satisfiable propositional formulae is NP-complete, as is that of its variant 3-satisfiability.
		The propositional satisfiability problem (SAT), which is given a propositional formula is to decide whether or not it is satisfiable, is of central importance in various areas of computer science, including theoretical computer science, algorithmics, artificial intelligence, hardware design and verification.
		Given such an algorithm it is possible to construct a polynomial-time algorithm that, given the size of the certificate, constructs a boolean circuit that is polynomially large in the certificate size and decides whether its input is a binary encoding of a valid certificate or not.
	www.guajara.com /wiki/en/wikipedia/b/bo/boolean_satisfiability_problem.html (905 words)

	NP-hard
		This intuition is supported by the fact that if we can find an algorithm A that solves one of these problems H in polynomial time then we can construct a polynomial time algorithm for every problem in NP by first executing the reduction from this problem to H and then executing the algorithm A.
		For example the boolean satisfiability problem can be reduced to the halting problem by transforming it to the description of a Turing machine that tries all truth value assignments and when it finds one that satisfies the formula it halts and otherwise it goes into an infinite loop.
		In this sense, the problem H is NP-hard if for every decision problem L in NP there is an oracle machine that has an oracle for solving H and this oracle machine can solve L in polynomial time.
	www.ebroadcast.com.au /lookup/encyclopedia/np/NP-hard.html (597 words)

	GridSAT - Boolean Satisfiability — VGrADS at Rice University
		GridSAT is a parallel and complete boolean satisfiability solver used to solve non-trivial SAT problems in a grid environment.
		It splits the problem if memory is exhausted on a node, choosing the best available resources and tailoring the split itself to make the best use of those resources.
		Satisfiability problems can be arbitrarily hard and may take long periods of time even using large resource pools.
	vgrads.rice.edu /research/applications/gridsat (738 words)

	biology - Decision problem
		In logic, a decision problem is determining whether or not there exists a decision procedure or algorithm for a class S of questions requiring a Boolean value (i.e., a true or false, or yes or no).
		For example, the decision problem for the class of questions "Does x divide y without remainder?" is decidable because there exists a mechanical procedure, namely long division, which allows us to determine for any x and any y whether the answer for "Does x divide y without remainder?" is yes or no.
		Nearly every problem can be cast as a decision problem by using reductions, often with little effect on the amount of time or space needed to solve the problem.
	www.biologydaily.com /biology/Decision_problem (413 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		We turn to the Boolean K-Satisfiability problem [3] as a mathematical tool to model large sets of interdependent variables, and then apply this model to populations of neurons in the brain as an example.
		In the presence of noise, the satisfiability of the K-SAT expression may fluctuate around the point on the phase transition curve, which corresponds to the satisfiability of the expression under this pattern of perturbation.
		The satisfiability of such an expression is an inverse of the satisfiability of the K-SAT expression in its canonical conjunctive normal form (1).
	www.genobyte.com /k-sat.doc (4627 words)

	[No title]
		I know that problem is NP-complete, but since I know that no input will ever be complemented, satisfying (finding the inputs that will make the function false in my case) is trivial: choose all inputs as false and the function is false.
		Suppose one were to reduce problem A to problem B, and problem B is known to be NP-complete.
		To shed a little light on this particular problem, the example function I gave is directly related to golomb rulers of length 11; if g evaluates false, you have a legal golomb ruler of length 11.
	www.math.niu.edu /~rusin/known-math/99/boolean (2228 words)

	Tufts University \| Computer Science Department -- Colloquia >> Colloquia
		The last few years have seen an increasing interest in the Boolean Satisfiability (SAT) problem which is concerned with identifying a variable assignment for a given Boolean formula, expressed in product-of-sums form, that evaluates the formula to true, or proving that no such assignment exists and that the formula is false.
		While the general SAT problem is NP-complete, we observe that SAT instances arising from real-world applications possess an innate structure that, once uncovered, can drastically simplify the instance.
		In this talk, we introduce the Boolean satisfiability problem and describe the commonly used algorithmic approaches for solving SAT.
	www.cs.tufts.edu /colloquia/colloquia.php?event=241 (279 words)

	CS 4804 Homework #2
		Pose the magic square puzzle as a boolean satisfiability (SAT) problem in CNF form, i.e., the boolean expression must be a conjunction of disjunctions of propositional variables.
		Pose this puzzle as a boolean satisfiability (SAT) problem in CNF form, i.e., the boolean expression must be a conjunction of disjunctions of propositional variables.
		You are not going to solve it as a SAT problem (although that is how you posed it in the previous question); instead, you are going to think of it as a traditional CSP (domains, variables, constraints, representable as a constraint graph) and use traditional CSP techniques (e.g., backtracking, maintaining arc-consistency etc.).
	courses.cs.vt.edu /~cs4804/Fall03/assignments/two.html (720 words)

	Previous sessions (1997-98) of Seminar of the Laboratory of Mathematical Logic (Site not responding. Last check: 2007-11-07)
		Problems of modeling computations by means of modal logics are considered.
		We mean that local search algorithm is an algorithm that goes from an assignment to its neighbour that satisfies more variables than the first assignment, etc. During the last years there was a comprehensive experimental research for SAT algorithms.
		NP-completeness of the problem of feasibility of a system of strict and/or non-strict linear inequalities with rational coefficients and integer and/or rational variables is established.
	logic.pdmi.ras.ru /~seminar/1998/indexall_l.html (2796 words)

	An Exploration of Physical Design Problems Via Boolean Satisfiability (SAT)
		Typically, most physical design problems can be cast as classical optimization problems with one or more objective functions that must be maximized or minimized subject to a set of constraints.
		Even though the Boolean satisfiability (SAT) problem fueled major advances in various electronic design automation (EDA) areas during the last decade, especially in logic synthesis, simulation, testing, formal functional and timing verification, the physical design domain has not been actively attacked using Boolean SAT approach except a few works [1][5][6].
		SAT problems are intrinsically "exact" methods; we create a large but atomic Boolean function whose internal structure captures the geometrical optimization we seek and respects the constraints we must obey.
	www.sigda.org /daforum/backup/99/criteria.abstract.html (846 words)

	The difficulty of schema conformance problems
		A problem can be solved in polynomial time if the number of steps required to solve the problem is bounded by a polynomial function of the size of the problem.
		Thus, the DTD conformance problem is at least as hard as the 3SAT problem.
		If we had an algorithm that solved the NRL conformance problem, where the subschemas are limited to RELAX NG schemas, then we could use this algorithm (and the algorithm that encodes instances of the 3SAT problem into RELAX NG schemas) to solve the 3SAT problem, which is NP-complete.
	www.idealliance.org /papers/extreme03/xml/2003/Lyons01/EML2003Lyons01.xml (3352 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Project Proposal: Implementing Conflict Analysis in a SAT Solver for Difference Logic The problem of checking the satisfiability of a Boolean formula, also known as SAT, is one of the fundamental and most challenging algorithmic problems.
		More than forty years of research on the problem have led to the development of powerful SAT solvers that can tackle progressively larger problems, culminating in recent solvers such as Grasp and Chaff that can handle problem of enormous size that model real-life problems in system design.
		The satisfiability problem for this ``difference logic'' is the found whether there is a truth assignment for the Boolean variables AND a real-valued assignment to the continuous variables such that the formula is satisfied.
	www.di.ens.fr /~pocchiol/STAGE03/maler.txt (622 words)

	[No title]
		The problem is how to build a voltage profile to minimize the system's power consumption without violating application's timing constraints.
		In the (boolean) satisfiability problem, we are given a formula on a set of (boolean) variables, and we are asked to assign each variable either 0 or 1 to make the formula true.
		Because these problem solvers come from very different fields and target very different type of formulas, it is difficult to compare their performance.
	www.ee.umd.edu /rite/ice02_project_descs.htm (2333 words)

	Automated Reasoning
		We should remark that this problem is non-trivial since deciding whether a finite set of equations provides a basis for Boolean algebra is undecidable, that is, it does not permit an algorithmic representation; also, the problem was attacked by Robbins, Huntington, Tarski and many of his students with no success.
		Problems stated in terms of objects or structures that involve recursive definitions or some form of repetition invariably require mathematical induction for their solving.
		Problem solving in mathematics involves the interplay of deduction and calculation, with decision procedures being a reminder of the fuzzy division between the two; hence, the integration of deductive and symbolic systems, which we coin here as Deductive Computer Algebra (DCA), is bound to be a fruitful combination.
	plato.stanford.edu /entries/reasoning-automated (12208 words)

	Charles Babbage Institute: RESEARCH PROGRAM> Current research (Site not responding. Last check: 2007-11-07)
		According to computational complexity, a research area in computer science which attempts to assess the computer resource requirements of computable problems, NP (nondeterministic polynomial time) problems include a subset of so-called “infeasible problems” that have identified solutions which are subject to verification in polynomial time, as well as all feasibly computable problems.
		NP-complete problems form a subset of particularly difficult NP problems, the solution of any one of which in polynomial time would result in the solution of them all.
		Examples of other NP-complete problems include the Hamiltonian path problem, the rural postman problem (qv), the bin-packing problem (qv), and the traveling salesman problem.
	www.cbi.umn.edu /shp/entries/npcompleteness.html (331 words)

	Forced Satisfiable CSP and SAT Benchmarks of Model RB - A Simple Way for Generating Hard Satisfiable CSP and SAT ... (Site not responding. Last check: 2007-11-07)
		Finding challenging benchmarks for the satisfiability problem is not only of significance for the experimental evaluation of SAT algorithms but also of interest to the theoretical computer science community.
		A Constraint Satisfaction Problem, or CSP for short, consists of a set of variables, a set of possible values for each variable (its domain) and a set of constraints defining the allowed tuples of values for the variables (a well-studied special case of it is SAT).
		The CSP is a fundamental problem in Artificial Intelligence, with a distinguished history and many applications, such as in knowledge representation, scheduling and pattern recognition.
	www.nlsde.buaa.edu.cn /~kexu/benchmarks/benchmarks.htm (2054 words)

	2.5.3 Planning as Satisfiability
		This means that the planning problem of Formulation 2.4 can be solved by determining whether some assignment of variables is possible for a Boolean expression that leads to a
		Generic methods for determining satisfiability can be directly applied to the Boolean expression that encodes the planning problem.
		The full problem is encoded by combining all of the given expressions into an enormous conjunction.
	msl.cs.uiuc.edu /planning/node68.html (570 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		It is not clear that hardest problems in NP should exist.
		If H is a hardest problem in NP, then it’s relatively easy to show that another problem J is too — just show that J is as hard as H, and is also in NP.
		Since it’s easier to show that a problem is easy than to show that it’s hard, most researchers believe that all NP-complete problems are hard.
	www.mathcs.sjsu.edu /faculty/smithj/oldclass/146f02/slides/np/NP.doc (359 words)

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