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The Quest for Efficient Boolean Satisfiability Solvers
, 2002
"... has seen much interest in not just the theoretical computer science community, but also in areas where practical solutions to this problem enable significant practical applications. Since the first development of the basic search based algorithm proposed by Davis, Putnam, Logemann and Loveland (DPLL ..."
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Cited by 128 (2 self)
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has seen much interest in not just the theoretical computer science community, but also in areas where practical solutions to this problem enable significant practical applications. Since the first development of the basic search based algorithm proposed by Davis, Putnam, Logemann and Loveland (DPLL) about forty years ago, this area has seen active research effort with many interesting contributions that have culminated in stateoftheart SAT solvers today being able to handle problem instances with thousands, and in same cases even millions, of variables. In this paper we examine some of the main ideas along this passage that have led to our current capabilities. Given the depth of the literature in this field, it is impossible to do this in any comprehensive way; rather we focus on techniques with consistent demonstrated efficiency in available solvers. For the most part, we focus on techniques within the basic DPLL search framework, but also briefly describe other approaches and look at some possible future research directions. 1.
Binary Decision Diagrams and Beyond: Enabling Technologies for Formal Verification
, 1995
"... Ordered Binary Decision Diagrams (OBDDs) have found widespread use in CAD applications such as formal verification, logic synthesis, and test generation. OBDDs represent Boolean functions in a form that is both canonical and compact for many practical cases. They can be generated and manipulated by ..."
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Cited by 105 (0 self)
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Ordered Binary Decision Diagrams (OBDDs) have found widespread use in CAD applications such as formal verification, logic synthesis, and test generation. OBDDs represent Boolean functions in a form that is both canonical and compact for many practical cases. They can be generated and manipulated by efficient graph algorithms. Researchers have found that many tasks can be expressed as series of operations on Boolean functions, making them candidates for OBDDbased methods. The success of OBDDs has inspired efforts to improve their efficiency and to expand their range of applicability. Techniques have been discovered to make the representation more compact and to represent other classes of functions. This has led to improved performance on existing OBDD applications, as well as enabled new classes of problems to be solved. This paper provides an overview of the state of the art in graphbased function representations. We focus on several recent advances of particular importance for forma...
Verification of Arithmetic Circuits with Binary Moment Diagrams
 IN PROCEEDINGS OF THE 32ND ACM/IEEE DESIGN AUTOMATION CONFERENCE
, 1995
"... Binary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear functions, we can embed arbitrary functions from Boolean variables to integer values. BMDs can thus model ..."
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Cited by 96 (11 self)
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Binary Moment Diagrams (BMDs) provide a canonical representations for linear functions similar to the way Binary Decision Diagrams (BDDs) represent Boolean functions. Within the class of linear functions, we can embed arbitrary functions from Boolean variables to integer values. BMDs can thus model the functionality of data path circuits operating over wordlevel data. Many important functions, including integer multiplication, that cannot be represented efficiently at the bit level with BDDs have simple representations at the word level with BMDs. Furthermore, BMDs can represent Boolean functions with around the same complexity as BDDs. We propose a hierarchical approach to verifying arithmetic circuits, wherecomponentmodulesare first shownto implement their wordlevel specifications. The overall circuit functionality is then verified by composing the component functions and comparing the result to the wordlevel circuit specification. Multipliers with word sizes of up to 256 bits hav...
HighLevel Power Modeling, Estimation, and Optimization
 IEEE Trans. On Computer Aided Design
, 1998
"... Abstract—Silicon area, performance, and testability have been, so far, the major design constraints to be met during the development of digital verylargescaleintegration (VLSI) systems. In recent years, however, things have changed; increasingly, power has been given weight comparable to the othe ..."
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Cited by 89 (11 self)
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Abstract—Silicon area, performance, and testability have been, so far, the major design constraints to be met during the development of digital verylargescaleintegration (VLSI) systems. In recent years, however, things have changed; increasingly, power has been given weight comparable to the other design parameters. This is primarily due to the remarkable success of personal computing devices and wireless communication systems, which demand highspeed computations with low power consumption. In addition, there exists a strong pressure for manufacturers of highend products to keep power under control, due to the increased costs of packaging and cooling this type of devices. Last, the need of ensuring high circuit reliability has turned out to be more stringent. The availability of tools for the automatic design of lowpower VLSI systems has thus become necessary. More specifically, following a natural trend, the interests of the researchers have lately shifted to the investigation of power modeling, estimation, synthesis, and optimization techniques that account for power dissipation during the early stages of the design flow. This paper surveys representative contributions to this area that have appeared in the recent literature. Index Terms — Behavioral and logic synthesis, low power design, power management. I.
Symmetry Detection and Dynamic Variable Ordering of Decision Diagrams
, 1996
"... Knowing that some variables are symmetric in a function has numerous applications; in particular, it can help produce better variable orders for Binary Decision Diagrams (BDDs) and related data structures (e.g., Algebraic Decision Diagrams). It has been observed that there often exists an optimum ..."
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Cited by 55 (2 self)
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Knowing that some variables are symmetric in a function has numerous applications; in particular, it can help produce better variable orders for Binary Decision Diagrams (BDDs) and related data structures (e.g., Algebraic Decision Diagrams). It has been observed that there often exists an optimum order for a BDD wherein symmetric variables are contiguous. We propose a new algorithm for the detection of symmetries, based on dynamic reordering, and we study its interaction with the reordering algorithm itself. We show that combining sifting with an efficient symmetry check for contiguous variables results in the fastest symmetry detection algorithm reported to date and produces better variable orders for many BDDs. The overhead on the sifting algorithm is negligible. 1
The octahedron abstract domain
 In Static Analysis Symposium (2004
, 2004
"... NOTICE: This is the author’s version of a work that was accepted for publication in Science of Computer Programming. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this docu ..."
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Cited by 47 (1 self)
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NOTICE: This is the author’s version of a work that was accepted for publication in Science of Computer Programming. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. A definitive version was subsequently published in Science of Computer Programming, 64(2007):115139.
Ordered Binary Decision Diagrams and the DavisPutnam Procedure
 IN PROC. OF THE 1ST INTERNATIONAL CONFERENCE ON CONSTRAINTS IN COMPUTATIONAL LOGICS
, 1994
"... We compare two prominent decision procedures for propositional logic: Ordered Binary Decision Diagrams (obdds) and the DavisPutnam procedure. Experimental results indicate that the DavisPutnam procedure outperforms obdds in hard constraintsatisfaction problems, while obdds are clearly superior for ..."
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Cited by 44 (1 self)
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We compare two prominent decision procedures for propositional logic: Ordered Binary Decision Diagrams (obdds) and the DavisPutnam procedure. Experimental results indicate that the DavisPutnam procedure outperforms obdds in hard constraintsatisfaction problems, while obdds are clearly superior for Boolean functional equivalence problems from the circuit domain, and, in general, problems that require the schematization of a large number of solutions that share a common structure. The two methods illustrate the different and often complementary strengths of constraintoriented and searchoriented procedures.
Casefactor diagrams for structured probabilistic modeling
 In Proceedings of the Twentieth Conference on Uncertainty in Artificial Intelligence (UAI’04
, 2004
"... We introduce a probabilistic formalism subsuming Markov random fields of bounded tree width and probabilistic context free grammars. Our models are based on a representation of Boolean formulas that we call casefactor diagrams (CFDs). CFDs are similar to binary decision diagrams (BDDs) but are more ..."
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Cited by 42 (0 self)
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We introduce a probabilistic formalism subsuming Markov random fields of bounded tree width and probabilistic context free grammars. Our models are based on a representation of Boolean formulas that we call casefactor diagrams (CFDs). CFDs are similar to binary decision diagrams (BDDs) but are more concise than BDDs for circuits of bounded tree width and can concisely represent the set of parse trees over a given string under a given context free grammar (unlike BDDs). A probabilistic model consists of a CFD defining a feasible set of Boolean assignments and a weight (or cost) for each individual Boolean variable. We give an insideoutside algorithm for simultaneously computing the marginal of each Boolean variable, and a Viterbi algorithm for finding the minimum cost variable assignment. Both algorithms run in time proportional to the size of the CFD. 1 1