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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 149 (3 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 state-of-the-art 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.
SATzilla: Portfolio-based Algorithm Selection for SAT
"... It has been widely observed that there is no single “dominant ” SAT solver; instead, different solvers perform best on different instances. Rather than following the traditional approach of choosing the best solver for a given class of instances, we advocate making this decision online on a per-inst ..."
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Cited by 145 (22 self)
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It has been widely observed that there is no single “dominant ” SAT solver; instead, different solvers perform best on different instances. Rather than following the traditional approach of choosing the best solver for a given class of instances, we advocate making this decision online on a per-instance basis. Building on previous work, we describe SATzilla, an automated approach for constructing per-instance algorithm portfolios for SAT that use so-called empirical hardness models to choose among their constituent solvers. This approach takes as input a distribution of problem instances and a set of component solvers, and constructs a portfolio optimizing a given objective function (such as mean runtime, percent of instances solved, or score in a competition). The excellent performance of our SATzilla portfolios has been independently verified in the 2007 SAT Competition, where our SATzilla-07 solvers won three gold, one silver and one bronze medal. In this article, we go well beyond SATzilla-07 by making the portfolio construction scalable and completely automated, and improving it by integrating local search solvers as candidate solvers, by predicting performance score instead of runtime, and by using hierarchical hardness models that take into account different types of SAT instances. We demonstrate the effectiveness of these new techniques in extensive experimental results on data sets including instances from the most recent SAT competition. 1.
DPLL(T): Fast Decision Procedures
, 2004
"... The logic of equality with uninterpreted functions (EUF) and its extensions have been widely applied to processor verification, by means of a large variety of progressively more sophisticated (lazy or eager) translations into propositional SAT. Here we propose a new approach, namely a general DP ..."
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Cited by 141 (14 self)
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The logic of equality with uninterpreted functions (EUF) and its extensions have been widely applied to processor verification, by means of a large variety of progressively more sophisticated (lazy or eager) translations into propositional SAT. Here we propose a new approach, namely a general DPLL(X) engine, whose parameter X can be instantiated with a specialized solver Solver T for a given theory T , thus producing a system DPLL(T ). We describe this DPLL(T ) scheme, the interface between DPLL(X) and Solver T , the architecture of DPLL(X), and our solver for EUF, which includes incremental and backtrackable congruence closure algorithms for dealing with the built-in equality and the integer successor and predecessor symbols. Experiments with a first implementation indicate that our technique already outperforms the previous methods on most benchmarks, and scales up very well.
Scalable error detection using Boolean satisfiability
- In Proc. 32Ç È POPL. ACM
, 2005
"... We describe a software error-detection tool that exploits recent advances in boolean satisfiability (SAT) solvers. Our analysis is path sensitive, precise down to the bit level, and models pointers and heap data. Our approach is also highly scalable, which we achieve using two techniques. First, for ..."
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Cited by 118 (10 self)
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We describe a software error-detection tool that exploits recent advances in boolean satisfiability (SAT) solvers. Our analysis is path sensitive, precise down to the bit level, and models pointers and heap data. Our approach is also highly scalable, which we achieve using two techniques. First, for each program function, several optimizations compress the size of the boolean formulas that model the control- and data-flow and the heap locations accessed by a function. Second, summaries in the spirit of type signatures are computed for each function, allowing inter-procedural analysis without a dramatic increase in the size of the boolean constraints to be solved. We demonstrate the effectiveness of our approach by constructing a lock interface inference and checking tool. In an interprocedural analysis of more than 23,000 lock related functions in the latest Linux kernel, the checker generated 300 warnings, of which 179 were unique locking errors, a false positive rate of only 40%.
A Fast Pseudo-Boolean Constraint Solver
, 2003
"... Linear Pseudo-Boolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints. They can be used to compactly describe many discrete EDA problems with constraints on linea ..."
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Cited by 115 (1 self)
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Linear Pseudo-Boolean (LPB) constraints denote inequalities between arithmetic sums of weighted Boolean functions and provide a significant extension of the modeling power of purely propositional constraints. They can be used to compactly describe many discrete EDA problems with constraints on linearly combined, parameterized weights, yet also offer efficient search strategies for proving or disproving whether a satisfying solution exists. Furthermore, corresponding decision procedures can easily be extended for minimizing or maximizing an LPB objective function, thus providing a core optimization method for many problems in logic and physical synthesis. In this paper we review how recent advances in satisfiability (SAT) search can be extended for pseudo-Boolean constraints and describe a new LPB solver that is based on generalized constraint propagation and conflict-based learning. We present a comparison with other, state-of-the-art LPB solvers which demonstrates the overall efficiency of our method.
Generic ILP versus Specialized 0-1 ILP: An Update
- IN INTERNATIONAL CONFERENCE ON COMPUTER-AIDED DESIGN
, 2002
"... Optimized solvers for the Boolean Satisfiability (SAT) problem have many applications in areas such as hardware and software verification, FPGA routing, planning, etc. Further uses are complicated by the need to express "counting constraints" in conjunctive normal form (CNF). Expressing su ..."
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Cited by 97 (23 self)
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Optimized solvers for the Boolean Satisfiability (SAT) problem have many applications in areas such as hardware and software verification, FPGA routing, planning, etc. Further uses are complicated by the need to express "counting constraints" in conjunctive normal form (CNF). Expressing such constraints by pure CNF leads to more complex SAT instances. Alternatively, those constraints can be handled by Integer Linear Programming (ILP), but generic ILP solvers may ignore the Boolean nature of 0-1 variables. Therefore specialized 0-1 ILP solvers extend SAT solvers to handle these so-called "pseudo-Boolean" constraints. This work
Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure,”
- Journal on Satisfiability, Boolean Modeling, and Computation,
, 2007
"... Abstract In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving transcendental functions, we provide a tight integration of recent SAT solving techniques with interval-based arithmetic constraint solving. Our approach deviates subst ..."
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Cited by 90 (12 self)
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Abstract In order to facilitate automated reasoning about large Boolean combinations of nonlinear arithmetic constraints involving transcendental functions, we provide a tight integration of recent SAT solving techniques with interval-based arithmetic constraint solving. Our approach deviates substantially from lazy theorem proving approaches in that it directly controls arithmetic constraint propagation from the SAT solver rather than delegating arithmetic decisions to a subordinate solver. Through this tight integration, all the algorithmic enhancements that were instrumental to the enormous performance gains recently achieved in propositional SAT solving carry over smoothly to the rich domain of non-linear arithmetic constraints. As a consequence, our approach is able to handle large constraint systems with extremely complex Boolean structure, involving Boolean combinations of multiple thousand arithmetic constraints over some thousands of variables.
Predicting Learnt Clauses Quality in Modern SAT Solvers
, 2009
"... Beside impressive progresses made by SAT solvers over the last ten years, only few works tried to understand why Conflict Directed Clause Learning algorithms (CDCL) are so strong and efficient on most industrial applications. We report in this work a key observation of CDCL solvers behavior on this ..."
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Cited by 87 (13 self)
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Beside impressive progresses made by SAT solvers over the last ten years, only few works tried to understand why Conflict Directed Clause Learning algorithms (CDCL) are so strong and efficient on most industrial applications. We report in this work a key observation of CDCL solvers behavior on this family of benchmarks and explain it by an unsuspected side effect of their particular Clause Learning scheme. This new paradigm allows us to solve an important, still open, question: How to designing a fast, static, accurate, and predictive measure of new learnt clauses pertinence. Our paper is followed by empirical evidences that show how our new learning scheme improves state-of-the art results by an order of magnitude on both SAT and UNSAT industrial problems.
PBS: A backtrack search pseudo Boolean solver
- In Symposium on the theory and applications of satisfiability testing (SAT
, 2002
"... in areas such as hardware and software verification, FPGA routing, planning in AI, etc. Further uses are complicated by the need to express “counting constraints ” in conjunctive normal form (CNF). Expressing such constraints by pure CNF leads to more complex SAT instances. Alternatively, those cons ..."
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Cited by 86 (1 self)
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in areas such as hardware and software verification, FPGA routing, planning in AI, etc. Further uses are complicated by the need to express “counting constraints ” in conjunctive normal form (CNF). Expressing such constraints by pure CNF leads to more complex SAT instances. Alternatively, those constraints can be handled by Integer Linear Programming (ILP), but off-the-shelf ILP solvers tend to ignore the Boolean nature of 0-1 variables. This work attempts to generalize recent highly successful SAT techniques to new applications. First, we extend the basic Davis-Putnam framework to handle counting constraints and apply it to solve routing problems. Our implementation outperforms previously reported solvers for the satisfiability with “pseudo-Boolean ” constraints and shows significant speed-up over best SAT solvers when such constraints are translated into CNF,. Additionally, we solve instances of the Max-ONEs optimization problem which seeks to maximize the number of “true ” values over all satisfying assignments. This, and the related Min-ONEs problem are important due to reductions from Max-Clique and Min Vertex Cover. Our experimental results for various benchmarks are superior to all approaches reported earlier. 1
Automated Abstraction Refinement for Model Checking Large State Spaces Using SAT Based Conflict Analysis
- IN PROCEEDINGS OF FMCAD
, 2002
"... We introduce a SAT based auto338m abstraction refinement framework for model checking systems with several thomGG4 state variables in the com o influenceo f the specificatio8 The abstractmo del iscoK060mEN8 by designating a large numbero f state variables as invisible. In co trast to previoN wo rk ..."
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Cited by 84 (12 self)
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We introduce a SAT based auto338m abstraction refinement framework for model checking systems with several thomGG4 state variables in the com o influenceo f the specificatio8 The abstractmo del iscoK060mEN8 by designating a large numbero f state variables as invisible. In co trast to previoN wo rk where invisible variables were treated as free inputs we describe a co06NGmEG7430m mo0 advantageo3 approF h in which the abstract transitio relatio isappro ximated by pre-89889L6728 invisible variables during imageco8087FmEG0 The abstract co4 terexamplesorexamp fro mo del-checking the abstract mo del are symbo lically simulatedo the coG0K8K system using a state-oGNK7Kmo SAT checker. Ifno co43FK3 co4 terexample isfo640 a subseto f the invisible variables is reintro duced into the systemand thepro cess is repeated. The main co tributio o f this paper are two new algo37FmE fo identifying the relevant variablesto be reintro duced. Thesealgo78NNm mogo7 the SAT checking phase inom4F to analyze the impacto individual variables. Ourmetho d is co48NFF fo safetypro erties (AG p) in the sense that -- perfoN06G0 permitting -- a pro erty is either verifiedo dispro ved by aco4GKKm co4 terexample. Experimental results are givento demoGGmE40 the power of our method on real-world designs.
