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12
Improvements To Propositional Satisfiability Search Algorithms
, 1995
"... ... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable ..."
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Cited by 161 (0 self)
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... quickly across a wide range of hard SAT problems than any other SAT tester in the literature on comparable platforms. On a Sun SPARCStation 10 running SunOS 4.1.3 U1, POSIT can solve hard random 400variable 3SAT problems in about 2 hours on the average. In general, it can solve hard nvariable random 3SAT problems with search trees of size O(2 n=18:7 ). In addition to justifying these claims, this dissertation describes the most significant achievements of other researchers in this area, and discusses all of the widely known general techniques for speeding up SAT search algorithms. It should be useful to anyone interested in NPcomplete problems or combinatorial optimization in general, and it should be particularly useful to researchers in either Artificial Intelligence or Operations Research.
Tractable Reasoning via Approximation
 Artificial Intelligence
, 1995
"... Problems in logic are wellknown to be hard to solve in the worst case. Two different strategies for dealing with this aspect are known from the literature: language restriction and theory approximation. In this paper we are concerned with the second strategy. Our main goal is to define a semantical ..."
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Cited by 92 (0 self)
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Problems in logic are wellknown to be hard to solve in the worst case. Two different strategies for dealing with this aspect are known from the literature: language restriction and theory approximation. In this paper we are concerned with the second strategy. Our main goal is to define a semantically wellfounded logic for approximate reasoning, which is justifiable from the intuitive point of view, and to provide fast algorithms for dealing with it even when using expressive languages. We also want our logic to be useful to perform approximate reasoning in different contexts. We define a method for the approximation of decision reasoning problems based on multivalued logics. Our work expands and generalizes in several directions ideas presented by other researchers. The major features of our technique are: 1) approximate answers give semantically clear information about the problem at hand; 2) approximate answers are easier to compute than answers to the original problem; 3) approxim...
New methods for 3SAT decision and worstcase analysis
 THEORETICAL COMPUTER SCIENCE
, 1999
"... We prove the worstcase upper bound 1:5045 n for the time complexity of 3SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2 and 3clauses, called "blocked clauses", generalizing the e ..."
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Cited by 66 (12 self)
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We prove the worstcase upper bound 1:5045 n for the time complexity of 3SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2 and 3clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3clausesets and can be applied also to arbitrary trees. Keywords: 3SAT, worstcase upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3SAT decision and prove the worstcase upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results...
Satisfiability Testing with More Reasoning and Less Guessing
, 1995
"... A new algorithm for testing satisfiability of propositional formulas in conjunctive normal form (CNF) is described. It applies reasoning in the form of certain resolution operations, and identification of equivalent literals. Resolution produces growth in the size of the formula, but within a global ..."
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Cited by 51 (10 self)
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A new algorithm for testing satisfiability of propositional formulas in conjunctive normal form (CNF) is described. It applies reasoning in the form of certain resolution operations, and identification of equivalent literals. Resolution produces growth in the size of the formula, but within a global quadratic bound; most previous methods avoid operations that produce any growth, and generally do not identify equivalent literals. Computational experience indicates that the method does substantially less "guessing" than previously reported algorithms, while keeping a polynomial time bound on the work done between guesses. Experiments indicate that, for larger problems, the time investment in reasoning returns a profit in reduced searching, and the profit increases with increasing problem size. Experimental data compares six branching strategies of the proposed algorithm on a variety of problems, including several Dimacs benchmarks. These branching strategies were shown to perform differently with statistical signi cance. A new scheme based on Johnson's maximum satisfiability approximation algorithm was found to be the best overall. Both satisfiable and unsatifi able random 3CNF formulas with 50283 variables and 4.27 ratio of clauses to variables have been tested; this class is generally acknowledged to be relatively "hard" and
Evidence for a Satisfiability Threshold for Random 3CNF Formulas
, 1993
"... This paper presents empirical evidence of a satisfiability threshold in random 3CNF formulas. The paper also expands on and supports the conjecture of Mitchell, Selman, and Levesque [13] that hard randomly generated CNF formulas will be hard for any reasonable satisfiability algorithm. We report sta ..."
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Cited by 46 (1 self)
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This paper presents empirical evidence of a satisfiability threshold in random 3CNF formulas. The paper also expands on and supports the conjecture of Mitchell, Selman, and Levesque [13] that hard randomly generated CNF formulas will be hard for any reasonable satisfiability algorithm. We report statistics for a much larger set of variables than have been previously reported; in particular, we show that for each clause to variable ratio less than 4.2, the percentage of satisfiable formulas increases, and for each clause to variable ratio greater than 4.2, the percentage of satisfiable formulas decreases as a function of number of variables. We found that several algorithms behaved qualitatively in the same fashion. We report on the relative performance of each algorithm. Keywords: backtracking, logic, reasoning, satisfiability, threshold behavior
Deciding propositional tautologies: Algorithms and their complexity
, 1997
"... We investigate polynomial reductions and efficient branching rules for algorithms deciding propositional tautologies for DNF and coNPcomplete subclasses. Upper bounds on the time complexity are given with exponential part 2 ff\Delta(F ) where (F ) is one of the measures n(F ) = #f variables g, ` ..."
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Cited by 38 (8 self)
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We investigate polynomial reductions and efficient branching rules for algorithms deciding propositional tautologies for DNF and coNPcomplete subclasses. Upper bounds on the time complexity are given with exponential part 2 ff\Delta(F ) where (F ) is one of the measures n(F ) = #f variables g, `(F ) = #f literal occurrences g and k(F ) = #f clauses g. We start with a discussion of variants of the algorithms from [Monien/Speckenmeyer85] and [Luckhardt84] with the known upper bound 2 0:695\Deltan for 3DNF and (roughly) (2 \Delta (1 \Gamma 2 \Gammap )) n for pDNF, p 3, where p is the maximal clause length, giving now an uniform treatment for all pDNF including the easy decidable case p 2. Recently for 3DNF the bound has been lowered to 2 0:5892\Deltan ([K2]; see also [Sch2], [K3]). In this article further improvements are achieved by studying two additional characteristic groups of parameters. The first group differentiates according to the maximal numbers (a; b) of occ...
New WorstCase Upper Bounds for SAT
 Journal of Automated Reasoning
, 2000
"... In 1980 Monien and Speckenmeyer proved that satisfiability of a propositional formula consisting of K clauses (of arbitrary length) can be checked in time of the order 2^{K/3}. Recently Kullmann and Luckhardt proved the worstcase upper bound 2^{L/9}, where L is the length of the input formula. The ..."
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Cited by 35 (8 self)
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In 1980 Monien and Speckenmeyer proved that satisfiability of a propositional formula consisting of K clauses (of arbitrary length) can be checked in time of the order 2^{K/3}. Recently Kullmann and Luckhardt proved the worstcase upper bound 2^{L/9}, where L is the length of the input formula. The algorithms leading to these bounds are based on the splitting method which goes back to the Davis{Putnam procedure. Transformation rules (pure literal elimination, unit propagation etc.) constitute a substantial part of this method. In this paper we present a new transformation rule and two algorithms using this rule. We prove that these algorithms have the worstcase upper bounds 2^{0.30897K} and 2^{0.10299L}, respectively.
Dynamic Algebras as a wellbehaved fragment of Relation Algebras
 In Algebraic Logic and Universal Algebra in Computer Science, LNCS 425
, 1990
"... The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect ..."
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Cited by 35 (5 self)
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The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras. Dept. of Computer Science, Stanford, CA 94305. This paper is based on a talk given at the conference Algebra and Computer Science, Ames, Iowa, June 24, 1988. It will appear in the proceedings of that conference, to be published by SpringerVerlag in the Lecture Notes in Computer Science series. This work was supported by the National Science Foundation under grant number CCR8814921 ...
Using Neural Networks and Genetic Algorithms as Heuristics for NPComplete Problems
, 1983
"... Paradigms for using neural networks (NNs) and genetic algorithms (GAs) to heuristically solve boolean satisfiability (SAT) problems are presented. Since SAT is NPComplete, any other NPComplete problem can be transformed into an equivalent SAT problem in polynomial time, and solved via either parad ..."
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Cited by 15 (8 self)
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Paradigms for using neural networks (NNs) and genetic algorithms (GAs) to heuristically solve boolean satisfiability (SAT) problems are presented. Since SAT is NPComplete, any other NPComplete problem can be transformed into an equivalent SAT problem in polynomial time, and solved via either paradigm. This technique is illustrated for hamiltonian circuit (HC) problems. INTRODUCTION NPComplete problems are problems that are not currently solvable in polynomial time. However, they are polynomially equivalent in the sense that any NPComplete problem can be transformed into any other in polynomial time. Thus, if any NPComplete problem can be solved in polynomial time, they all can [Garey]. The canonical example of an NPComplete problem is the boolean satisfiability (SAT) problem: Given an arbitrary boolean expression of n variables, does there exist an assignment to those variables such that the expression is true? Other familiar examples include job shop scheduling, bin packing, a...
Algorithms for SAT/TAUT decision based on various measures
 Information and Computation
, 1999
"... We investigate algorithms deciding propositional tautologies for DNF and coNPcomplete subclasses given by restrictions on the number of occurrences of literals. Especially polynomial use of resolution for reductions in combination with a new combinatorial principle called "Generalized Sign Princip ..."
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Cited by 11 (8 self)
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We investigate algorithms deciding propositional tautologies for DNF and coNPcomplete subclasses given by restrictions on the number of occurrences of literals. Especially polynomial use of resolution for reductions in combination with a new combinatorial principle called "Generalized Sign Principle" is studied. Upper bounds on time complexity are given with exponential part 2 ff\Delta(F ) where the measure (F ) for a clause set F either is the number n(F ) of variables, the number `(F ) of literal occurrences or the number k(F ) of clauses. ff is called a "power coefficient" for the class of formulas under consideration w.r.t. measure . Power coefficients are derived with the help of a method estimating the size of trees, which is also used to find "good" branching rules. Under natural conditions power coefficients ff; fi; fl for n; k; ` respectively fulfill ff fi fl. We obtain the following power coefficients.  0:1112 for DNF w.r.t. `  0:3334 for DNF w.r.t. k These result...