## The Quest for Efficient Boolean Satisfiability Solvers (2002)

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### BibTeX

@INPROCEEDINGS{Zhang02thequest,

author = {Lintao Zhang and Sharad Malik},

title = {The Quest for Efficient Boolean Satisfiability Solvers},

booktitle = {},

year = {2002},

pages = {17--36},

publisher = {Springer}

}

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### Abstract

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.

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Citation Context ...everal SAT algorithms that have seen practical success. These algorithms are based on various principles such as resolution [5], search [6], local search and random walk [7], Binary Decision Diagrams =-=[8]-=-, Stälmarck’s algorithm [9], and others. Gu et al. [10] provide an excellent review of many of the algorithms developed thus far. Some of these algorithms are complete, while others are stochastic met... |

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Citation Context ... SAT solvers. This research has resulted in the development of several SAT algorithms that have seen practical success. These algorithms are based on various principles such as resolution [5], search =-=[6]-=-, local search and random walk [7], Binary Decision Diagrams [8], Stälmarck’s algorithm [9], and others. Gu et al. [10] provide an excellent review of many of the algorithms developed thus far. Some o... |

359 | GRASP a new search algorithm for satisfiability
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Citation Context ...orithm will choose an unassigned variable from the formula and branch on it for both phases. The solution process begins with calling the function DPLL() with an empty set of variable assignments. In =-=[25]-=-, the authors generalized many of the actual implementations of various solvers based on DPLL and rewrote it in an iterative manner as shown in Fig. 2. The algorithm described in Fig. 2 is an improvem... |

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Citation Context ...o clauses that generate conflicts. Non-chronological backtracking, sometimes referred to as conflict-directed backjumping, was proposed first in the Constraint Satisfaction Problem (CSP) domain (e.g. =-=[38]-=-). This, together with conflict-directed learning, were first incorporated into a SAT solver by Silva and Sakallah in GRASP [25], and by Bayardo and Schrag in rel_sat [18]. These techniques are essent... |

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Citation Context ...roblems that are exactly the same except for the variable order may require totally different times to solve by a certain SAT solver (e.g. one can be solved in seconds while the other takes days). In =-=[46]-=-, the authors proposed to use random restart to cope with this phenomenon. Random restart randomly throws away the already searched space and starts from scratch. This technique is applied in DC + - V... |

288 | Efficient Conflict Driven Learning in a Boolean Satisfiability Solver
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Citation Context ... SATO [19] and Chaff [20] have incorporated similar technique in the solving process. Previously, learning and non-chronological backtracking have been discussed by analyzing implication graphs (e.g. =-=[17, 39]-=-). Here we will formulate learning as an alternate but equivalent resolution process and discuss different schemes in this framework. Researchers have adapted the conflict analysis engine to some dedu... |

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Citation Context ...d satisfiable instances, stochastic methods may find solutions very quickly. Currently, two of the more successful approaches to the stochastic method are random walk based algorithms such as walksat =-=[7]-=- and Discrete Lagrangian-Based global search methods such as DLM [52]. For more about other SAT solving techniques, we refer the readers to a survey[10]. 5. Conclusions and Future Works In this paper,... |

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Citation Context ...4], 2cl [15] and CSAT [16] among others. They are still appearing occasionally in the literature for performance comparison reasons. In the mid 1990’s, Silva and Sakallah [17], and Bayardo and Schrag =-=[18]-=- proposed to augment the original DPLL algorithm with non-chronological backtracking and conflict-driven learning. These techniques greatly improved the efficiency of the DPLL algorithm for structured... |

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Citation Context ...s for about forty years. In the last ten years we have seen significant growth and success in SAT solver research based on the DPLL framework. Earlier SAT solvers based on DPLL include Tableau (NTAB) =-=[13]-=-, POSIT [14], 2cl [15] and CSAT [16] among others. They are still appearing occasionally in the literature for performance comparison reasons. In the mid 1990’s, Silva and Sakallah [17], and Bayardo a... |

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Citation Context ...ally presented in a Product of Sum form, usually called a Conjunctive Normal Form (CNF). It is not a limitation to require the instance to be presented in CNF. There exist polynomial algorithms (e.g. =-=[24]-=-) to transform any propositional formula into a CNF formula that has the same satisfiability as the original one. In the discussions that follow, we will assume that the problem is presented in CNF. A... |

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Citation Context ...forty years. In the last ten years we have seen significant growth and success in SAT solver research based on the DPLL framework. Earlier SAT solvers based on DPLL include Tableau (NTAB) [13], POSIT =-=[14]-=-, 2cl [15] and CSAT [16] among others. They are still appearing occasionally in the literature for performance comparison reasons. In the mid 1990’s, Silva and Sakallah [17], and Bayardo and Schrag [1... |

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Citation Context ... solving process in order to make the solver robust. Some researchers [48] also propose to randomize backtracking. All in all, it seems that randomization is quite important because of the heavy tail =-=[49]-=- nature of SAT solving process. 4. Other techniques used in SAT solvers In this section, we briefly discuss some of the other techniques used to solve SAT problems besides the basic DPLL search. The o... |

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Citation Context ...s. This has naturally led to the exploration of decision diagram style set representations. In [42] and [43], the authors have experimented with using Zero-suppressed Binary Decision Diagrams (ZBDDs) =-=[44]-=- to represent the clause database. A ZBDD representation of the clause database can detect not only tail subsumption but also head subsumption. Both authors report significant compression of the claus... |

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Citation Context .... These algorithms are based on various principles such as resolution [5], search [6], local search and random walk [7], Binary Decision Diagrams [8], Stälmarck’s algorithm [9], and others. Gu et al. =-=[10]-=- provide an excellent review of many of the algorithms developed thus far. Some of these algorithms are complete, while others are stochastic methods. For a given SAT instance, complete SAT solvers ca... |

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Citation Context ...of finding algorithms that are efficient enough for solving many interesting SAT instances. These instances arise from many diverse areas - many practical problems in AI planning [2], circuit testing =-=[3]-=-, software verification [4] can be formulated as SAT instances. This has motivated the research in practically efficient SAT solvers. This research has resulted in the development of several SAT algor... |

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Citation Context ...luations have also been done (e.g. [26, 27]). Early branching heuristics such as Bohm’s Heuristic (reported in [28]), Maximum Occurrences on Minimum sized clauses (MOM) (e.g. [14]), and Jeroslow-Wang =-=[29]-=- can be regarded as greedy algorithms that try to make the next branch generate the largest number of implications or satisfy most clauses. All these heuristics use some functions to estimate the effe... |

111 |
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Citation Context ...ease the robustness of certain SAT solvers. Researchers have been extending the randomization idea of random restart to other aspects of the SAT solving process as well. For example, portfolio design =-=[47]-=- aims at using different solving strategies during one solving process in order to make the solver robust. Some researchers [48] also propose to randomize backtracking. All in all, it seems that rando... |

83 |
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Citation Context ...maintain a transitive closure of all the 2 literal clauses. However, the overhead of maintaining this information seems to far outweigh any benefit gained from them on the average. Recursive Learning =-=[36]-=- is another reasoning technique originally proposed in the context of learning with a logic circuit representation of a formula. Subsequent research [37] has proposed to incorporate this technique in ... |

79 | Integrating equivalency reasoning into Davis–Putnam procedure
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Citation Context ...orating the pure literal rule will generally slow down the solving process for most of the benchmarks encountered. Another explored deduction mechanism is equivalence reasoning. In particular, eqsatz =-=[32]-=- incorporated equivalence reasoning into the satz [22] solver and found that it is effective on some particular classes of benchmarks. In that work, the equivalence reasoning is accomplished by a patt... |

78 | Branching rules for satisfiability
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Citation Context ...iciency of the solver. Over the years many different branching heuristics have been proposed by different researchers. Not surprisingly, comparative experimental evaluations have also been done (e.g. =-=[26, 27]-=-). Early branching heuristics such as Bohm’s Heuristic (reported in [28]), Maximum Occurrences on Minimum sized clauses (MOM) (e.g. [14]), and Jeroslow-Wang [29] can be regarded as greedy algorithms t... |

78 | Report on a sat competition
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Citation Context ...ve been proposed by different researchers. Not surprisingly, comparative experimental evaluations have also been done (e.g. [26, 27]). Early branching heuristics such as Bohm’s Heuristic (reported in =-=[28]-=-), Maximum Occurrences on Minimum sized clauses (MOM) (e.g. [14]), and Jeroslow-Wang [29] can be regarded as greedy algorithms that try to make the next branch generate the largest number of implicati... |

63 | A backbone-search heuristic for efficient solving of hard 3-sat formulae - Dubois, Dequen - 2001 |

59 | A discrete Lagrangian-based global-search method for solving satisfiability problems
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Citation Context ...quickly. Currently, two of the more successful approaches to the stochastic method are random walk based algorithms such as walksat [7] and Discrete Lagrangian-Based global search methods such as DLM =-=[52]-=-. For more about other SAT solving techniques, we refer the readers to a survey[10]. 5. Conclusions and Future Works In this paper, we briefly discussed some of the techniques employed in modern Boole... |

55 | Combining strengths of circuit-based and CNFbased algorithms for a high-performance SAT solver
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Citation Context ...AT solvers. Application specific knowledge can help a lot in the solving process as demonstrated in [56]. For a particular application, custom implementation of a SAT solver may also be helpful (e.g. =-=[57]-=-). All in all, we believe there are still many research topics to be explored. As more and more applications utilize SAT solvers as deduction and reasoning engine, we believe many new algorithms will ... |

53 | Implementing the Davis–Putnam method
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Citation Context ...linked lists in terms of cache misses that translates to substantial speed-up in the solving process. Researchers have proposed schemes other than sparse matrix representation for storing clauses. In =-=[41]-=-, the authors of the solver SATO proposed the use of a data structure called trie to store clauses. A trie is a ternary tree. Each internal node in the trie structure is a variable index, and its thre... |

49 | Conflict analysis in search algorithms for propositional satisfiability
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Citation Context ...Tableau (NTAB) [13], POSIT [14], 2cl [15] and CSAT [16] among others. They are still appearing occasionally in the literature for performance comparison reasons. In the mid 1990’s, Silva and Sakallah =-=[17]-=-, and Bayardo and Schrag [18] proposed to augment the original DPLL algorithm with non-chronological backtracking and conflict-driven learning. These techniques greatly improved the efficiency of the ... |

42 | The impact of branching heuristics in propositional satisfiability algorithms
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Citation Context ...iciency of the solver. Over the years many different branching heuristics have been proposed by different researchers. Not surprisingly, comparative experimental evaluations have also been done (e.g. =-=[26, 27]-=-). Early branching heuristics such as Bohm’s Heuristic (reported in [28]), Maximum Occurrences on Minimum sized clauses (MOM) (e.g. [14]), and Jeroslow-Wang [29] can be regarded as greedy algorithms t... |

41 | The propositional formula checker heerhugo
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Citation Context ...g SAT. Stalmärck’s algorithm use breath-first search in contrast to the depth-first search employed by DPLL. There are commercial implementations of SAT solvers based on this algorithm [50]. HeerHugo =-=[51]-=- is a publicly available solver that claims to be using an algorithm similar to the Stalmärk’s algorithm. Another approach is to use stochastic algorithms. Stochastic algorithms cannot prove a SAT ins... |

23 |
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Citation Context ...atisfiable even though they may be able to find assolution for certain kinds of satisfiable instances quickly. Stochastic methods have applications in domains such as AI planning [2] and FPGA routing =-=[11]-=-, where instances are likely to be satisfiable and proving unsatisfiability is not required. However, for many other domains (especially verification problems e.g. [4, 12]), the primary task is to pro... |

22 | Efficient data structures for backtrack search SAT solvers
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Citation Context ...s being watched. Therefore, it is significantly faster than both counter-based and head/tail mechanisms for BCP. In Fig. 3, we show a comparison of 2-literal watching and head/tail list mechanism. In =-=[31]-=-, the authors examined the mechanisms mentioned above and introduced some new deduction data structures and mechanisms. In particular, the experiments suggest that the mechanism called Head/Tail list ... |

14 | Complete unrestricted backtracking algorithms for satisfiability
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Citation Context ...pects of the SAT solving process as well. For example, portfolio design [47] aims at using different solving strategies during one solving process in order to make the solver robust. Some researchers =-=[48]-=- also propose to randomize backtracking. All in all, it seems that randomization is quite important because of the heavy tail [49] nature of SAT solving process. 4. Other techniques used in SAT solver... |

14 |
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Citation Context ...ions use SAT solvers as blackboxes and no interaction is possible between the applications and the SAT solvers. Application specific knowledge can help a lot in the solving process as demonstrated in =-=[56]-=-. For a particular application, custom implementation of a SAT solver may also be helpful (e.g. [57]). All in all, we believe there are still many research topics to be explored. As more and more appl... |

11 |
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Citation Context ...s. In the last ten years we have seen significant growth and success in SAT solver research based on the DPLL framework. Earlier SAT solvers based on DPLL include Tableau (NTAB) [13], POSIT [14], 2cl =-=[15]-=- and CSAT [16] among others. They are still appearing occasionally in the literature for performance comparison reasons. In the mid 1990’s, Silva and Sakallah [17], and Bayardo and Schrag [18] propose... |

8 | The role of a skeptic agent in testing and benchmarking of SAT algorithms
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Citation Context ...gorithms is more like an art than a science because it is easy to tune a solver to a given set of benchmarks, but the parameters may not work for the same benchmarks with some simple permutation (e.g.=-=[55]-=-). On the application side, currently most of the applications use SAT solvers as blackboxes and no interaction is possible between the applications and the SAT solvers. Application specific knowledge... |

7 |
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Citation Context ...istent with each other. We can also require that all the 2 literal clauses be consistent with each other and so on. Researchers have been exploring this idea in the deduction process in works such as =-=[34, 35]-=-. In particular, these approaches maintain a transitive closure of all the 2 literal clauses. However, the overhead of maintaining this information seems to far outweigh any benefit gained from them o... |

4 |
The complexity of theorem-proving procedures," The 3rd Annual
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Citation Context ...signment such that the formula evaluates to true is called the Boolean Satisfiability Problem, commonly abbreviated as SAT. SAT has seen much theoretical interest as the canonical NP-complete problem =-=[1]-=-. Given its NP-Completeness, it is very unlikely that there exists any polynomial algorithm for SAT. However, NP-Completeness does not exclude the possibility of finding algorithms that are efficient ... |

4 |
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Citation Context ...n learning, the learned clauses are generated by resolution. Resolution is a process to generate a clause from two clauses analogous to the process of consensus in the logic optimization domain (e.g. =-=[40]-=-). Resolution is given by (x + y ) (y’ +z)≡ (x + y ) (y’ +z)(x+z) The term (x + z) is called the resolvent of clause (x + y) and (y’ + z). Because of this, we have (x + y ) (y’ +z)→ (x+z) Similar to t... |

3 |
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Citation Context ...have seen practical success. These algorithms are based on various principles such as resolution [5], search [6], local search and random walk [7], Binary Decision Diagrams [8], Stälmarck’s algorithm =-=[9]-=-, and others. Gu et al. [10] provide an excellent review of many of the algorithms developed thus far. Some of these algorithms are complete, while others are stochastic methods. For a given SAT insta... |

3 |
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Citation Context ... ten years we have seen significant growth and success in SAT solver research based on the DPLL framework. Earlier SAT solvers based on DPLL include Tableau (NTAB) [13], POSIT [14], 2cl [15] and CSAT =-=[16]-=- among others. They are still appearing occasionally in the literature for performance comparison reasons. In the mid 1990’s, Silva and Sakallah [17], and Bayardo and Schrag [18] proposed to augment t... |

3 | J.P.: The puzzling role of simplification in propositional satisfiability
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Citation Context ...re too expensive to be applied at every node of the search tree. The preprocessor can be applied on-line (within the solver) or off-line (it produces an equivalent instance to be fed to a solver). In =-=[45]-=-, the authors give an overview of some of the existing preprocessing techniques and find that the result of applying simplification techniques before the regular search is actually mixed. The time req... |

2 |
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Citation Context ...are efficient enough for solving many interesting SAT instances. These instances arise from many diverse areas - many practical problems in AI planning [2], circuit testing [3], software verification =-=[4]-=- can be formulated as SAT instances. This has motivated the research in practically efficient SAT solvers. This research has resulted in the development of several SAT algorithms that have seen practi... |

2 |
Improving Satisfiability Algorithms by Using Recursive Learning," presented at
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Citation Context ...d from them on the average. Recursive Learning [36] is another reasoning technique originally proposed in the context of learning with a logic circuit representation of a formula. Subsequent research =-=[37]-=- has proposed to incorporate this technique in SAT solvers and found that it works quite well for some benchmarks generated from combinational circuit equivalence checking problems. 3.3 Conflict Analy... |

1 |
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Citation Context ...lude the possibility of finding algorithms that are efficient enough for solving many interesting SAT instances. These instances arise from many diverse areas - many practical problems in AI planning =-=[2]-=-, circuit testing [3], software verification [4] can be formulated as SAT instances. This has motivated the research in practically efficient SAT solvers. This research has resulted in the development... |

1 |
Symbolic Model Checking without BDDs," presented at Tools and Algorithms for the Analysis and Construction of Systems (TACAS'99
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Citation Context ...I planning [2] and FPGA routing [11], where instances are likely to be satisfiable and proving unsatisfiability is not required. However, for many other domains (especially verification problems e.g. =-=[4, 12]-=-), the primary task is to prove unsatisfiability of the instances. For these, complete SAT solvers are a requirement. In recent years search-based algorithms based on the well-known DavisLogemann-Love... |

1 | SATO: An efficient propositional prover," presented at - Zhang - 1997 |

1 |
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Citation Context ...g. [4, 11, 12]), which pushed these solvers to their limits and provided strong motivation for finding even more efficient algorithms. This led to a new generation of solvers such as SATO [19], Chaff =-=[20]-=-, and BerkMin [21] which pay a lot of attention to optimizing various aspects of the DPLL algorithm. The results are some very efficient SAT solvers that can often solve SAT instances generated from i... |

1 |
BerkMin: a Fast and Robust SAT-Solver," presented at Design Automation
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Citation Context ...at have too many literals. There are many heuristics to measure the usefulness of a learned clause. For example, rel_sat [18] proposes to use relevance to measure a clause’s usefulness, while BerkMin =-=[21]-=- use the number of conflicts that involve this clause in the history to measure a clause’s usefulness. These measures seem to work reasonably well. 3.4 Data Structure for Storing Clause Database Curre... |

1 |
and Anbulagan, "Heuristics based on unit propagation for satisfiability problems," presented at
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Citation Context ... some very efficient SAT solvers that can often solve SAT instances generated from industrial applications with tens of thousands or even millions of variables. On another front, solvers such as satz =-=[22]-=- and cnfs [23] keep pushing the ability to tackle hard random 3-SAT instances. These solvers, though very efficient on random instances, are typically not competitive on structured instances generated... |

1 |
An efficient algorithm for unit-propagation," presented at
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Citation Context ... dictates the data structure and organization of the solver. A simple and intuitive implementation for BCP is to keep counters for each clause. This scheme is attributed to Crawford and Auton [13] by =-=[30]-=-. Similar schemes are subsequently employed in many solvers such as GRASP [25], rel_sat [18], satz [22] etc. For example, in GRASP [25], each clause keeps two counters, one for the number of value 1 l... |