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14
Counting models using connected components
 In Proceedings of the AAAI National Conference
, 2000
"... Recent work by Birnbaum & Lozinskii [1999] demonstrated that a clever yet simple extension of the wellknown DavisPutnam procedure for solving instances of propositional satisfiability yields an efficient scheme for counting the number of satisfying assignments (models). We present a new extension, ..."
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Cited by 47 (0 self)
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Recent work by Birnbaum & Lozinskii [1999] demonstrated that a clever yet simple extension of the wellknown DavisPutnam procedure for solving instances of propositional satisfiability yields an efficient scheme for counting the number of satisfying assignments (models). We present a new extension, based on recursively identifying connected constraintgraph components, that substantially improves counting performance on random 3SAT instances as well as benchmark instances from the SATLIB and Beijing suites. In addition, from a structurebased perspective of worstcase complexity, while polynomial time satisfiability checking is known to require only a backtrack search algorithm enhanced with nogood learning, we show that polynomial time counting using backtrack search requires an additional enhancement: good learning.
Approximate Resolution of Hard Numbering Problems
, 1996
"... We present a new method for estimating the number of solutions of constraint satisfaction problems 1 . We use a stochastic forward checking algorithm for drawing a sample of paths from a search tree. With this sample, we compute two values related to the number of solutions of a CSP instance. ..."
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Cited by 6 (2 self)
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We present a new method for estimating the number of solutions of constraint satisfaction problems 1 . We use a stochastic forward checking algorithm for drawing a sample of paths from a search tree. With this sample, we compute two values related to the number of solutions of a CSP instance. First, an unbiased estimate, second, a lower bound with an arbitrary low error probability. We will describe applications to the Boolean Satisfiability problem and the Queens problem. We shall give some experimental results for these problems. Introduction The class NP is the set of decision problems whose instances are assertions that can be proved in polynomial time. The NPComplete problems are the hardest problems in NP. These can not be solved in polynomial time under the assumption P 6= NP . All these problems have the same expression power in the sense that every NPComplete problem can be polynomialy reduced to each another (Garey & Johnson 1979). Some of them, such as CSP 2...
Improved InclusionExclusion Identities and Bonferroni Inequalities with Applications to Reliability Analysis of Coherent Systems
, 2000
"... ..."
InclusionExclusion for kCNF Formulas
 Inf. Process. Lett
, 2002
"... We show that the number of satisfying assignments of a kCNF formula is determined uniquely from the numbers of unsatisfying assignments for clausesets of size up to k#+ 2. The information of this size is also shown to be necessary. key words: combinatorial problems; SAT; kCNF formula; counting ..."
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We show that the number of satisfying assignments of a kCNF formula is determined uniquely from the numbers of unsatisfying assignments for clausesets of size up to k#+ 2. The information of this size is also shown to be necessary. key words: combinatorial problems; SAT; kCNF formula; counting; inclusionexclusion 1
#2,2SAT is solvable in lineartime
, 1997
"... A lineartime algorithm, with respect to the size of the instance Boolean formula, is presented for the #SAT problem restricted to formulae of the form #2; 2CF, i.e. formulae whose clauses have just two literals and in which each variable appears at most twice. Keyword: Satisfiability, #SAT problem ..."
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A lineartime algorithm, with respect to the size of the instance Boolean formula, is presented for the #SAT problem restricted to formulae of the form #2; 2CF, i.e. formulae whose clauses have just two literals and in which each variable appears at most twice. Keyword: Satisfiability, #SAT problem, counting problem, lineartime algorithm. 1 Introduction 1.1. The satisfiability problem for Boolean formulae, SAT, is NPcomplete, as was shown by Cook [2]. Its restriction, 2SAT, to formulae whose clauses have just two literals can be solved in linear time [5], but the corresponding "optimization" and "counting" problems Max2SAT (given a 2formula find an assignment which maximizes the number of satisfied clauses), and #2SAT (given a 2formula count the number of assignments satisfying all the clauses), are NPhard. 1.2. From the standpoint of NPcompleteness, one basic approach to compare the problem classes P and NP is to consider for a NPcomplete problem a restriction of it in P, a...
Algorithm to solve the problem 2SAT
"... An algorithm to solve the problem #2SAT is shown. We show that the computational time of the presented algorithm is bounded above by a function of order O , where n is the number of the Boolean variables and m is the number of clauses in the instance conjunctive form. Nevertheless, this ..."
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An algorithm to solve the problem #2SAT is shown. We show that the computational time of the presented algorithm is bounded above by a function of order O , where n is the number of the Boolean variables and m is the number of clauses in the instance conjunctive form. Nevertheless, this is a strict upper bound, which can be improved.
Sat Is Solvable In LinearTime
, 1998
"... A lineartime algorithm, with respect to the size of the instance Boolean formula, is presented for the #SAT problem restricted to formulae of the form #2; 2CF, i.e. formulae whose clauses have just two literals and in which each variable appears at most twice. ..."
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A lineartime algorithm, with respect to the size of the instance Boolean formula, is presented for the #SAT problem restricted to formulae of the form #2; 2CF, i.e. formulae whose clauses have just two literals and in which each variable appears at most twice.
A Branch And Bound Algorithm For Sat
"... A branch and bound algorithm to solve the problem #2SAT is shown. The computational time of the proposed algorithm is O (r nm), where n is the number of the Boolean variables, m is the number of clauses in the instance conjunctive form and r < 3 is the unique positive real root of the X 1. ..."
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A branch and bound algorithm to solve the problem #2SAT is shown. The computational time of the proposed algorithm is O (r nm), where n is the number of the Boolean variables, m is the number of clauses in the instance conjunctive form and r < 3 is the unique positive real root of the X 1. Basic notation X = fx 1 ; : : : ; xn g: Set of Boolean variables Lit(X) = X [ fxjx 2 Xg: Set of literals kclauses: c = f` 1 ; : : : ; ` k g 2 Lit(X) Conjunctive form: Any set of clauses.
Algorithms for Propositional . . .
, 2005
"... A large number of practical applications rely on effective algorithms for propositional model enumeration and counting. Examples include knowledge compilation, model checking and hybrid solvers. Besides practical applications, the problem of counting propositional models is of key relevancy in comp ..."
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A large number of practical applications rely on effective algorithms for propositional model enumeration and counting. Examples include knowledge compilation, model checking and hybrid solvers. Besides practical applications, the problem of counting propositional models is of key relevancy in computational complexity. In recent years a number of algorithms have been proposed for propositional model counting and enumeration. However, the algorithms for model counting and model enumeration have been developed in different contexts, and so are based on fairly different techniques. This paper surveys algorithms for both model enumeration and model counting, proposes optimizations to model enumeration algorithms, and addresses open topics in model counting. Experimental results, obtained on practical and representative problem instances, indicate that the proposed techniques are effective for model enumeration.
The Good Old DavisPutnam . . .
 JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
, 1999
"... As was shown recently, many important AI problems require counting the number of models of propositional formulas. The problem of counting models of such formulas is, according to present knowledge, computationally intractable in a worst case. Based on the DavisPutnam procedure, we present an al ..."
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As was shown recently, many important AI problems require counting the number of models of propositional formulas. The problem of counting models of such formulas is, according to present knowledge, computationally intractable in a worst case. Based on the DavisPutnam procedure, we present an algorithm, CDP, that computes the exact number of models of a propositional CNF or DNF formula F . Let m and n be the number of clauses and variables of F , respectively, and let p denote the probability that a literal l of F occurs in a clause C of F , then the average running time of CDP is shown to be O(m n), where d = d e. The practical performance of CDP has been estimated in a series of experiments on a wide variety of CNF formulas.