Clark W. Barrett, R. Sebastiani, S. Seshia, C. Tinelli
ion YES NO YES Increase solution bound to cover satisfying solution
离子是否是增加溶液边界以覆盖满意的溶液
{"title":"Handbook of Satisfiability","authors":"Clark W. Barrett, R. Sebastiani, S. Seshia, C. Tinelli","doi":"10.3233/faia336","DOIUrl":"https://doi.org/10.3233/faia336","url":null,"abstract":"ion YES NO YES Increase solution bound to cover satisfying solution","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128645202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-02DOI: 10.3233/978-1-58603-929-5-185
Henry A. Kautz, Ashish Sabharwal, B. Selman
Research on incomplete algorithms for satisfiability testing lead to some of the first scalable SAT solvers in the early 1990’s. Unlike systematic solvers often based on an exhaustive branching and backtracking search, incomplete methods are generally based on stochastic local search. On problems from a variety of domains, such incomplete methods for SAT can significantly outperform DPLL-based methods. While the early greedy algorithms already showed promise, especially on random instances, the introduction of randomization and so-called uphill moves during the search significantly extended the reach of incomplete algorithms for SAT. This chapter discusses such algorithms, along with a few key techniques that helped boost their performance such as focusing on variables appearing in currently unsatisfied clauses, devising methods to efficiently pull the search out of local minima through clause re-weighting, and adaptive noise mechanisms. The chapter also briefly discusses a formal foundation for some of the techniques based on the discrete Lagrangian method.
{"title":"Incomplete Algorithms","authors":"Henry A. Kautz, Ashish Sabharwal, B. Selman","doi":"10.3233/978-1-58603-929-5-185","DOIUrl":"https://doi.org/10.3233/978-1-58603-929-5-185","url":null,"abstract":"Research on incomplete algorithms for satisfiability testing lead to some of the first scalable SAT solvers in the early 1990’s. Unlike systematic solvers often based on an exhaustive branching and backtracking search, incomplete methods are generally based on stochastic local search. On problems from a variety of domains, such incomplete methods for SAT can significantly outperform DPLL-based methods. While the early greedy algorithms already showed promise, especially on random instances, the introduction of randomization and so-called uphill moves during the search significantly extended the reach of incomplete algorithms for SAT. This chapter discusses such algorithms, along with a few key techniques that helped boost their performance such as focusing on variables appearing in currently unsatisfied clauses, devising methods to efficiently pull the search out of local minima through clause re-weighting, and adaptive noise mechanisms. The chapter also briefly discusses a formal foundation for some of the techniques based on the discrete Lagrangian method.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"464 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127684256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-02DOI: 10.3233/978-1-58603-929-5-131
Joao Marques-Silva, I. Lynce, S. Malik
One of the most important paradigm shifts in the use of SAT solvers for solving industrial problems has been the introduction of clause learning. Clause learning entails adding a new clause for each conflict during backtrack search. This new clause prevents the same conflict from occurring again during the search process. Moreover, sophisticated techniques such as the identification of unique implication points in a graph of implications, allow creating clauses that more precisely identify the assignments responsible for conflicts. Learned clauses often have a large number of literals. As a result, another paradigm shift has been the development of new data structures, namely lazy data structures, which are particularly effective at handling large clauses. These data structures are called lazy due to being in general unable to provide the actual status of a clause. Efficiency concerns and the use of lazy data structures motivated the introduction of dynamic heuristics that do not require knowing the precise status of clauses. This chapter describes the ingredients of conflict-driven clause learning SAT solvers, namely conflict analysis, lazy data structures, search restarts, conflict-driven heuristics and clause deletion strategies.
{"title":"Conflict-Driven Clause Learning SAT Solvers","authors":"Joao Marques-Silva, I. Lynce, S. Malik","doi":"10.3233/978-1-58603-929-5-131","DOIUrl":"https://doi.org/10.3233/978-1-58603-929-5-131","url":null,"abstract":"One of the most important paradigm shifts in the use of SAT solvers for solving industrial problems has been the introduction of clause learning. Clause learning entails adding a new clause for each conflict during backtrack search. This new clause prevents the same conflict from occurring again during the search process. Moreover, sophisticated techniques such as the identification of unique implication points in a graph of implications, allow creating clauses that more precisely identify the assignments responsible for conflicts. Learned clauses often have a large number of literals. As a result, another paradigm shift has been the development of new data structures, namely lazy data structures, which are particularly effective at handling large clauses. These data structures are called lazy due to being in general unable to provide the actual status of a clause. Efficiency concerns and the use of lazy data structures motivated the introduction of dynamic heuristics that do not require knowing the precise status of clauses. This chapter describes the ingredients of conflict-driven clause learning SAT solvers, namely conflict analysis, lazy data structures, search restarts, conflict-driven heuristics and clause deletion strategies.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116573435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the last twenty years a significant amount of effort has been devoted to the study of randomly generated satisfiability instances. While a number of generative models have been proposed, uniformly random k-CNF formulas are by now the dominant and most studied model. One reason for this is that such formulas enjoy a number of intriguing mathematical properties, including the following: for each k≥3, there is a critical value, rk, of the clauses-to-variables ratio, r, such that for rrk it is unsatisfiable with probability that tends to 1 as n→∞. Algorithmically, even at densities much below rk, no polynomial-time algorithm is known that can find any solution even with constant probability, while for all densities greater than rk, the length of every resolution proof of unsatisfiability is exponential (and, thus, so is the running time of every DPLL-type algorithm). By now, the study of random k-CNF formulas has also attracted attention in areas such as mathematics and statistical physics and is at the center of an area of intense research activity. At the same time, random k-SAT instances are a popular benchmark for testing and tuning satisfiability algorithms. Indeed, some of the better practical ideas in use today come from insights gained by studying the performance of algorithms on them. We review old and recent mathematical results about random k-CNF formulas, demonstrating that the connection between computational complexity and phase transitions is both deep and highly nuanced.
{"title":"Random Satisfiabiliy","authors":"D. Achlioptas","doi":"10.3233/FAIA200993","DOIUrl":"https://doi.org/10.3233/FAIA200993","url":null,"abstract":"In the last twenty years a significant amount of effort has been devoted to the study of randomly generated satisfiability instances. While a number of generative models have been proposed, uniformly random k-CNF formulas are by now the dominant and most studied model. One reason for this is that such formulas enjoy a number of intriguing mathematical properties, including the following: for each k≥3, there is a critical value, rk, of the clauses-to-variables ratio, r, such that for rrk it is unsatisfiable with probability that tends to 1 as n→∞. Algorithmically, even at densities much below rk, no polynomial-time algorithm is known that can find any solution even with constant probability, while for all densities greater than rk, the length of every resolution proof of unsatisfiability is exponential (and, thus, so is the running time of every DPLL-type algorithm). By now, the study of random k-CNF formulas has also attracted attention in areas such as mathematics and statistical physics and is at the center of an area of intense research activity. At the same time, random k-SAT instances are a popular benchmark for testing and tuning satisfiability algorithms. Indeed, some of the better practical ideas in use today come from insights gained by studying the performance of algorithms on them. We review old and recent mathematical results about random k-CNF formulas, demonstrating that the connection between computational complexity and phase transitions is both deep and highly nuanced.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"87 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131672691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Olaf Beyersdorff, Mikoláš Janota, Florian Lonsing, M. Seidl
Solvers for quantified Boolean formulas (QBF) have become powerful tools for tackling hard computational problems from various application domains, even beyond the scope of SAT. This chapter gives a description of the main algorithmic paradigms for QBF solving, including quantified conflict driven clause learning (QCDCL), expansion-based solving, dependency schemes, and QBF preprocessing. Particular emphasis is laid on the connections of these solving approaches to QBF proof systems: Q-Resolution and its variants in the case of QCDCL, expansion QBF resolution calculi for expansion-based solving, and QRAT for preprocessing. The chapter also surveys the relations between the various QBF proof systems and results on their proof complexity, thereby shedding light on the diverse performance characteristics of different solving approaches that are observed in practice.
{"title":"Quantified Boolean Formulas","authors":"Olaf Beyersdorff, Mikoláš Janota, Florian Lonsing, M. Seidl","doi":"10.3233/FAIA201015","DOIUrl":"https://doi.org/10.3233/FAIA201015","url":null,"abstract":"Solvers for quantified Boolean formulas (QBF) have become powerful tools for tackling hard computational problems from various application domains, even beyond the scope of SAT. This chapter gives a description of the main algorithmic paradigms for QBF solving, including quantified conflict driven clause learning (QCDCL), expansion-based solving, dependency schemes, and QBF preprocessing. Particular emphasis is laid on the connections of these solving approaches to QBF proof systems: Q-Resolution and its variants in the case of QCDCL, expansion QBF resolution calculi for expansion-based solving, and QRAT for preprocessing. The chapter also surveys the relations between the various QBF proof systems and results on their proof complexity, thereby shedding light on the diverse performance characteristics of different solving approaches that are observed in practice.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114955861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This chapter covers an application of propositional satisfiability to program analysis. We focus on the discovery of programming flaws in low-level programs, such as embedded software. The loops in the program are unwound together with a property to form a formula, which is then converted into CNF. The method supports low-level programming constructs such as bit-wise operators or pointer arithmetic.
{"title":"Software Verification","authors":"Daniel Kroening","doi":"10.3233/FAIA201004","DOIUrl":"https://doi.org/10.3233/FAIA201004","url":null,"abstract":"This chapter covers an application of propositional satisfiability to program analysis. We focus on the discovery of programming flaws in low-level programs, such as embedded software. The loops in the program are unwound together with a property to form a formula, which is then converted into CNF. The method supports low-level programming constructs such as bit-wise operators or pointer arithmetic.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130839297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-02DOI: 10.3233/978-1-58603-929-5-205
O. Kullmann
“Search trees”, “branching trees”, “backtracking trees” or “enumeration trees” are at the heart of many (complete) approaches towards hard combinatorial problems, constraint problems, and, of course, SAT problems. Given many choices for branching, the fundamental question is how to guide the choices so that the resulting trees are (relatively) small. Despite (or perhaps because) of its apparently more narrow scope, especially in the SAT area several approaches from theory and applications have found together, and the rudiments of a theory of branching heuristics emerged. In this chapter the first systematic treatment is given. So a general theory of heuristics guiding the construction of “branching trees” is developed, ranging from a general theoretical analysis to the analysis of the historical development of branching heuristics for SAT solvers, and also to heuristics beyond SAT solving.
{"title":"Fundaments of Branching Heuristics","authors":"O. Kullmann","doi":"10.3233/978-1-58603-929-5-205","DOIUrl":"https://doi.org/10.3233/978-1-58603-929-5-205","url":null,"abstract":"“Search trees”, “branching trees”, “backtracking trees” or “enumeration trees” are at the heart of many (complete) approaches towards hard combinatorial problems, constraint problems, and, of course, SAT problems. Given many choices for branching, the fundamental question is how to guide the choices so that the resulting trees are (relatively) small. Despite (or perhaps because) of its apparently more narrow scope, especially in the SAT area several approaches from theory and applications have found together, and the rudiments of a theory of branching heuristics emerged. In this chapter the first systematic treatment is given. So a general theory of heuristics guiding the construction of “branching trees” is developed, ranging from a general theoretical analysis to the analysis of the historical development of branching heuristics for SAT solvers, and also to heuristics beyond SAT solving.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125895343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maximum satisfiability (MaxSAT) is an optimization version of SAT that is solved by finding an optimal truth assignment instead of just a satisfying one. In MaxSAT the objective function to be optimized is specified by a set of weighted soft clauses: the objective value of a truth assignment is the sum of the weights of the soft clauses it satisfies. In addition, the MaxSAT problem can have hard clauses that the truth assignment must satisfy. Many optimization problems can be naturally encoded into MaxSAT and this, along with significant performance improvements in MaxSAT solvers, has led to MaxSAT being used in a number of different application areas. This chapter provides a detailed overview of the approaches to MaxSAT solving that have in recent years been most successful in solving real-world optimization problems. Further recent developments in MaxSAT research are also overviewed, including encodings, applications, preprocessing, incomplete solving, algorithm portfolios, partitioning-based solving, and parallel solving.
{"title":"Maximum Satisfiabiliy","authors":"F. Bacchus, M. Järvisalo, R. Martins","doi":"10.3233/FAIA201008","DOIUrl":"https://doi.org/10.3233/FAIA201008","url":null,"abstract":"Maximum satisfiability (MaxSAT) is an optimization version of SAT that is solved by finding an optimal truth assignment instead of just a satisfying one. In MaxSAT the objective function to be optimized is specified by a set of weighted soft clauses: the objective value of a truth assignment is the sum of the weights of the soft clauses it satisfies. In addition, the MaxSAT problem can have hard clauses that the truth assignment must satisfy. Many optimization problems can be naturally encoded into MaxSAT and this, along with significant performance improvements in MaxSAT solvers, has led to MaxSAT being used in a number of different application areas. This chapter provides a detailed overview of the approaches to MaxSAT solving that have in recent years been most successful in solving real-world optimization problems. Further recent developments in MaxSAT research are also overviewed, including encodings, applications, preprocessing, incomplete solving, algorithm portfolios, partitioning-based solving, and parallel solving.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124784657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-02DOI: 10.3233/978-1-58603-929-5-271
C. Gomes, Ashish Sabharwal
It has become well know over time that the performance of backtrack-style complete SAT solvers can vary dramatically depending on “little” details of the heuristics used, such as the way one selects the next variable to branch on and in what order the possible values are assigned to the variable. Extreme variations can result even from simple tie breaking mechanisms necessarily employed in all SAT solvers. The discovery of this extreme runtime variation has been both a stumbling block and an opportunity. This chapter focuses on providing an understanding of this intriguing phenomenon, particularly in terms of the so-called heavy tailed nature of the runtime distributions of systematic SAT solvers. It describes a simple formal model based on expensive mistakes to explain runtime distributions seen in practice, and discusses randomization and restart strategies that can be used to effectively overcome the negative impact of heavy tailed behavior. Finally, the chapter discusses the notion of backdoor variables, which explain the unexpectedly short runs one also often sees in practice.
{"title":"Exploiting Runtime Variation in Complete Solvers","authors":"C. Gomes, Ashish Sabharwal","doi":"10.3233/978-1-58603-929-5-271","DOIUrl":"https://doi.org/10.3233/978-1-58603-929-5-271","url":null,"abstract":"It has become well know over time that the performance of backtrack-style complete SAT solvers can vary dramatically depending on “little” details of the heuristics used, such as the way one selects the next variable to branch on and in what order the possible values are assigned to the variable. Extreme variations can result even from simple tie breaking mechanisms necessarily employed in all SAT solvers. The discovery of this extreme runtime variation has been both a stumbling block and an opportunity. This chapter focuses on providing an understanding of this intriguing phenomenon, particularly in terms of the so-called heavy tailed nature of the runtime distributions of systematic SAT solvers. It describes a simple formal model based on expensive mistakes to explain runtime distributions seen in practice, and discusses randomization and restart strategies that can be used to effectively overcome the negative impact of heavy tailed behavior. Finally, the chapter discusses the notion of backdoor variables, which explain the unexpectedly short runs one also often sees in practice.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130464067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-02DOI: 10.3233/978-1-58603-929-5-155
Marijn J. H. Heule, H. Maaren
The chapter on look-a-head architecture based solvers provides a state of the art description of how heuristics, data structures and learning in this context have evolved over the past year. Contributions on the results of various scientific teams working with this architecture are described and unified. It also provides insight on its weakness when applied to a certain type of problems which are not appropriate to solve using this architecture. It aims to describe the complementary role of this architecture and that of conflict driven solving mechanisms.
{"title":"Look-Ahead Based SAT Solvers","authors":"Marijn J. H. Heule, H. Maaren","doi":"10.3233/978-1-58603-929-5-155","DOIUrl":"https://doi.org/10.3233/978-1-58603-929-5-155","url":null,"abstract":"The chapter on look-a-head architecture based solvers provides a state of the art description of how heuristics, data structures and learning in this context have evolved over the past year. Contributions on the results of various scientific teams working with this architecture are described and unified. It also provides insight on its weakness when applied to a certain type of problems which are not appropriate to solve using this architecture. It aims to describe the complementary role of this architecture and that of conflict driven solving mechanisms.","PeriodicalId":250589,"journal":{"name":"Handbook of Satisfiability","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122722496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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