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49
Binary decision diagrams in theory and practice
, 2001
"... Decision diagrams (DDs) are the stateoftheart data structure in VLSI CAD and have been successfully applied in many other fields.DDs are widely used and are also integrated in commercial tools.This special section comprises six contributed articles on various aspects of the theory and application ..."
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Cited by 31 (7 self)
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Decision diagrams (DDs) are the stateoftheart data structure in VLSI CAD and have been successfully applied in many other fields.DDs are widely used and are also integrated in commercial tools.This special section comprises six contributed articles on various aspects of the theory and application of DDs.As preparation for these contributions, the present article reviews the basic definitions of binary decision diagrams (BDDs). We provide a brief overview and study theoretical and practical aspects.Basic properties of BDDs are discussed and manipulation algorithms are described.Extensions of BDDs are investigated and by this we give a deeper insight into the basic data structure.Finally we outline several applications of BDDs and their extensions and suggest a number of articles and books for those who wish to pursue the topic in more depth.
Symbolic Decision Procedures for QBF
 Proceedings of 10th Int. Conf. on Principles and Practice of Constraint Programming (CP 2004
, 2004
"... Much recent work has gone into adapting techniques that were originally developed for SAT solving to QBF solving. In particular, QBF solvers are often based on SAT solvers. Most competitive QBF solvers are searchbased. In this work we explore an alternative approach to QBF solving, based on symb ..."
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Cited by 28 (1 self)
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Much recent work has gone into adapting techniques that were originally developed for SAT solving to QBF solving. In particular, QBF solvers are often based on SAT solvers. Most competitive QBF solvers are searchbased. In this work we explore an alternative approach to QBF solving, based on symbolic quantifier elimination. We extend some recent symbolic approaches for SAT solving to symbolic QBF solving, using various decisiondiagram formalisms such as OBDDs and ZDDs. In both approaches, QBF formulas are solved by eliminating all their quantifiers. Our first solver, QMRES, maintains a set of clauses represented by a ZDD and eliminates quantifiers via multiresolution. Our second solver, QBDD, maintains a set of OBDDs, and eliminate quantifier by applying them to the underlying OBDDs. We compare our symbolic solvers to several competitive searchbased solvers. We show that QBDD is not competitive, but QMRES compares favorably with searchbased solvers on various benchmarks consisting of nonrandom formulas.
Search vs. symbolic techniques in satisfiability solving
 in Proceedings 7th International Conference on Theory and Applications of Satisfiability Testing
, 2004
"... Abstract. Recent work has shown how to use OBDDs for satisfiability solving. The idea of this approach, which we call symbolic quantifier elimination, is to view an instance of propositional satisfiability as an existentially quantified propositional formula. Satisfiability solving then amounts to q ..."
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Cited by 21 (3 self)
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Abstract. Recent work has shown how to use OBDDs for satisfiability solving. The idea of this approach, which we call symbolic quantifier elimination, is to view an instance of propositional satisfiability as an existentially quantified propositional formula. Satisfiability solving then amounts to quantifier elimination; once all quantifiers have been eliminated we are left with either 1 or 0. Our goal in this work is to study the effectiveness of symbolic quantifier elimination as an approach to satisfiability solving. To that end, we conduct a direct comparison with the DPLLbased ZChaff, as well as evaluate a variety of optimization techniques for the symbolic approach. In comparing the symbolic approach to ZChaff, we evaluate scalability across a variety of classes of formulas. We find that no approach dominates across all classes. While ZChaff dominates for many classes of formulas, the symbolic approach is superior for other classes of formulas. Once we have demonstrated the viability of the symbolic approach, we focus on optimization techniques for this approach. We study techniques from constraint satisfaction for finding a good plan for performing the symbolic operations of conjunction and of existential quantification. We also study various variableordering heuristics, finding that while no heuristic seems to dominate across all classes of formulas, the maximumcardinality search heuristic seems to offer the best overall performance. 1
Discovering coherent biclusters from gene expression data using zerosuppressed binary decision diagrams
 Computational Biology and (C) International Journal of Engineering Sciences & Research Technology
"... Abstract—The biclustering method can be a very useful analysis tool when some genes have multiple functions and experimental conditions are diverse in gene expression measurement. This is because the biclustering approach, in contrast to the conventional clustering techniques, focuses on finding a s ..."
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Cited by 16 (5 self)
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Abstract—The biclustering method can be a very useful analysis tool when some genes have multiple functions and experimental conditions are diverse in gene expression measurement. This is because the biclustering approach, in contrast to the conventional clustering techniques, focuses on finding a subset of the genes and a subset of the experimental conditions that together exhibit coherent behavior. However, the biclustering problem is inherently intractable, and it is often computationally costly to find biclusters with high levels of coherence. In this work, we propose a novel biclustering algorithm that exploits the zerosuppressed binary decision diagrams (ZBDDs) data structure to cope with the computational challenges. Our method can find all biclusters that satisfy specific input conditions, and it is scalable to practical gene expression data. We also present experimental results confirming the effectiveness of our approach. Index Terms—Clustering, life and medical sciences, bioinformatics (genome or protein) databases, logic design. 1
Bisimulation Minimization in an AutomataTheoretic Verification Framework
 In Formal Methods in ComputerAided Design (FMCAD
, 1998
"... Bisimulation is a seemingly attractive statespace minimization technique because it can be computed automatically and yields the smallest model preserving all ¯calculus formulas. It is considered impractical for symbolic model checking, however, because the required BDDs are prohibitively large fo ..."
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Bisimulation is a seemingly attractive statespace minimization technique because it can be computed automatically and yields the smallest model preserving all ¯calculus formulas. It is considered impractical for symbolic model checking, however, because the required BDDs are prohibitively large for most designs. We revisit bisimulation minimization, this time in an automatatheoretic framework. Bisimulation has potential in this framework because after intersecting the design with the negation of the property, minimization can ignore most of the atomic propositions. We compute bisimulation using an algorithm due to Lee and Yannakakis that represents bisimulation relations by their equivalence classes and only explores reachable classes. This greatly improves on the time and memory usage of naive algorithms. We demonstrate that bisimulation is practical for many designs within the automatatheoretic framework. In most cases, however, the cost of performing this reduction still outweigh...
SNF: A Special Normal Form for ESOPs
 Mississipi State University, Starkville (Mississipi) USA
, 2001
"... This paper introduces a new normal form for Exclusive SumsofProducts (ESOPs) of completely specified Boolean functions. We study the properties of the SNF and show its special place among canonical ReedMuller representations. We propose to use the SNF in a number of applications related to the ex ..."
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Cited by 13 (10 self)
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This paper introduces a new normal form for Exclusive SumsofProducts (ESOPs) of completely specified Boolean functions. We study the properties of the SNF and show its special place among canonical ReedMuller representations. We propose to use the SNF in a number of applications related to the exact ESOP minimization. We describe an efficient way to compute the SNF with the complexity proportional to the number of nodes in the BDD of the given function. Experimental results speak for the potential usefulness of the SNF. 1
An Introduction to ZeroSuppressed Binary Decision Diagrams. http://www.ee.pdx.edu/~alanmi/research
, 2001
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Deduction in ManyValued Logics: a Survey
 Mathware & Soft Computing, iv(2):6997
, 1997
"... this article, there is considerable activity in MVL deduction which is why we felt justified in writing this survey. Needless to say, we cannot give a general introduction to MVL in the present context. For this, we have to refer to general treatments such as [153, 53, 93]. 2 A classification of man ..."
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Cited by 10 (1 self)
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this article, there is considerable activity in MVL deduction which is why we felt justified in writing this survey. Needless to say, we cannot give a general introduction to MVL in the present context. For this, we have to refer to general treatments such as [153, 53, 93]. 2 A classification of manyvalued logics according to their intended application
Correctness Proof of a BDD Manager in the Context of Satisfiability Checking
 Department of Computer Sciences
, 2000
"... We present a compositional proof of correctness for a binary decision diagram (BDD) manager used in the context of a propositional satisfiability checker implemented using SingleThreaded Objects (stobjs) in ACL2. The use of stobjs affords the definition of an efficient BDD manager which ensures uniq ..."
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Cited by 10 (2 self)
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We present a compositional proof of correctness for a binary decision diagram (BDD) manager used in the context of a propositional satisfiability checker implemented using SingleThreaded Objects (stobjs) in ACL2. The use of stobjs affords the definition of an efficient BDD manager which ensures unique construction, allows constanttime comparison, and caches previously computed results. The use of ACL2 means we can prove that the BDD manager implements the prescribed task of building a normalform representation of a boolean formula. We divide the proof requirements into (1) showing that a simpler set of BDD functions is correct, and (2) showing that the stobjbased BDD functions return values consistent with these simpler functions. We conclude the paper with a discussion of future extensions and refinements to the BDD manager presented.
Complexity Theoretical Results for Randomized Branching Programs
, 1998
"... This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straigh ..."
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Cited by 8 (7 self)
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This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps readonce branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...