{SCL} with Theory Constraints

We lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution inferences by a partial model assumption instead of an a priori fixed order as done for instance in hierarchic superposition. The model representation consists of ground background theory literals and ground foreground first-order literals. One major advantage of the model guided approach is that clauses generated by SCL(T) enjoy a non-redundancy property that makes expensive testing for tautologies and forward subsumption completely obsolete. SCL(T) is a semi-decision procedure for pure clause sets that are clause sets without first-order function symbols ranging into the background theory sorts. Moreover, SCL(T) can be turned into a decision procedure if the considered combination of a first-order logic modulo a background theory enjoys an abstract finite model property

SCL with Theory Constraints

We lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution inferences by a partial model assumption instead of an a priori fixed order as done for instance in hierarchic superposition. The model representation consists of ground background theory literals and ground foreground first-order literals. One major advantage of the model guided approach is that clauses generated by SCL(T) enjoy a non-redundancy property that makes expensive testing for tautologies and forward subsumption completely obsolete. SCL(T) is a semi-decision procedure for pure clause sets that are clause sets without first-order function symbols ranging into the background theory sorts. Moreover, SCL(T) can be turned into a decision procedure if the considered combination of a first-order logic modulo a background theory enjoys an abstract finite model property.Comment: 22 page

SCL with Theory Constraints

22 pagesWe lift the SCL calculus for first-order logic without equality to the SCL(T) calculus for first-order logic without equality modulo a background theory. In a nutshell, the SCL(T) calculus describes a new way to guide hierarchic resolution inferences by a partial model assumption instead of an a priori fixed order as done for instance in hierarchic superposition. The model representation consists of ground background theory literals and ground foreground first-order literals. One major advantage of the model guided approach is that clauses generated by SCL(T) enjoy a non-redundancy property that makes expensive testing for tautologies and forward subsumption completely obsolete. SCL(T) is a semi-decision procedure for pure clause sets that are clause sets without first-order function symbols ranging into the background theory sorts. Moreover, SCL(T) can be turned into a decision procedure if the considered combination of a first-order logic modulo a background theory enjoys an abstract finite model property

Proceedings of SAT Competition 2021 : Solver and Benchmark Descriptions

Non peer reviewe

Machine Learning for SAT Solvers

Boolean SAT solvers are indispensable tools in a variety of domains in computer science and engineering where efficient search is required. Not only does this relieve the burden on the users of implementing their own search algorithm, they also leverage the surprising effectiveness of modern SAT solvers. Thanks to many decades of cumulative effort, researchers have made persistent improvements to SAT technology to the point where nowadays the best solvers are routinely used to solve extremely large instances with millions of variables. Even though our current paradigm of SAT solvers runs in worst-case exponential time, it appears that the techniques and heuristics embedded in these solvers avert the worst-case exponential time in practice. The implementations of these various solver heuristics and techniques are vital to the solvers effectiveness in practice. The state-of-the-art heuristics and techniques gather data during the run of the solver to inform their choices like which variable to branch on next or when to invoke a restart. The goal of these choices is to minimize the solving time. The methods in which these heuristics and techniques process the data generally do not have theoretical underpinnings. Consequently, understanding why these heuristics and techniques perform so well in practice remains a challenge and systematically improving them is rather difficult. This goes to the heart of this thesis, that is to utilize machine learning to process the data as part of an optimization problem to minimize solving time. Research in machine learning exploded over the past decade due to its success in extracting useful information out of large volumes of data. Machine learning outclasses manual handcoding in a wide variety of complex tasks where data are plentiful. This is also the case in modern SAT solvers where propagations, conflict analysis, and clause learning produces plentiful of data to be analyzed, and exploiting this data to the fullest is naturally where machine learning comes in. Many machine learning techniques have a theoretical basis that makes them easy to analyze and understand why they perform well. The branching heuristic is the first target for injecting machine learning. First we studied extant branching heuristics to understand what makes a branching heuristics good empirically. The fundamental observation is that good branching heuristics cause lots of clause learning by triggering conflicts as quickly as possible. This suggests that variables that cause conflicts are a valuable source of data. Another important observation is that the state-of-the-art VSIDS branching heuristic internally implements an exponential moving average. This highlights the importance of accounting for the temporal nature of the data when deciding to branch. These observations led to our proposal of a series of machine learning-based branching heuristics with the common goal of selecting the branching variables to increase probability of inducing conflicts. These branching heuristics are shown empirically to either be on par or outcompete the current state-of-the art. The second area of interest for machine learning is the restart policy. Just like in the branching heuristic work, we first study restarts to observe why they are effective in practice. The important observation here is that restarts shrink the assignment stack as conjectured by other researchers. We show that this leads to better clause learning by lowering the LBD of learnt clauses. Machine learning is used to predict the LBD of the next clause, and a restart is triggered when the LBD is excessively high. This policy is shown to be on par with state-of-the-art. The success of incorporating machine learning into branching and restarts goes to show that machine learning has an important role in the future of heuristic and technique design for SAT solvers

Towards Next Generation Sequential and Parallel SAT Solvers

This thesis focuses on improving the SAT solving technology. The improvements focus on two major subjects: sequential SAT solving and parallel SAT solving. To better understand sequential SAT algorithms, the abstract reduction system Generic CDCL is introduced. With Generic CDCL, the soundness of solving techniques can be modeled. Next, the conflict driven clause learning algorithm is extended with the three techniques local look-ahead, local probing and all UIP learning that allow more global reasoning during search. These techniques improve the performance of the sequential SAT solver Riss. Then, the formula simplification techniques bounded variable addition, covered literal elimination and an advanced cardinality constraint extraction are introduced. By using these techniques, the reasoning of the overall SAT solving tool chain becomes stronger than plain resolution. When using these three techniques in the formula simplification tool Coprocessor before using Riss to solve a formula, the performance can be improved further. Due to the increasing number of cores in CPUs, the scalable parallel SAT solving approach iterative partitioning has been implemented in Pcasso for the multi-core architecture. Related work on parallel SAT solving has been studied to extract main ideas that can improve Pcasso. Besides parallel formula simplification with bounded variable elimination, the major extension is the extended clause sharing level based clause tagging, which builds the basis for conflict driven node killing. The latter allows to better identify unsatisfiable search space partitions. Another improvement is to combine scattering and look-ahead as a superior search space partitioning function. In combination with Coprocessor, the introduced extensions increase the performance of the parallel solver Pcasso. The implemented system turns out to be scalable for the multi-core architecture. Hence iterative partitioning is interesting for future parallel SAT solvers. The implemented solvers participated in international SAT competitions. In 2013 and 2014 Pcasso showed a good performance. Riss in combination with Copro- cessor won several first, second and third prices, including two Kurt-Gödel-Medals. Hence, the introduced algorithms improved modern SAT solving technology