Adaptive Application of SAT Solving Techniques

AbstractNew heuristics and strategies have enabled major advancements in SAT solving in recent years. However, experimentation has shown that there is no winning solution that works in all cases. A degradation of orders of magnitude can be observed if the wrong heuristic is chosen. The problem is that it is impossible to know, in advance, which heuristics are best for a given problem. Consequently, many ideas - those that turn out to be useful for a small subset of the cases, but significantly increase run times on most others - are discarded.We propose the notion of Adaptive Solving as a possible solution to this problem. In our framework, the SAT solver monitors the effectiveness of the search on-the-fly using a Performance Metric. The metric gives a score according to its assessment of the search progress. Based on this score, one or more heuristics are turned on or off. The goal is to use a specific heuristic or strategy when it is advantageous, and turn it off when it is not, before it does too much damage. We suggest several possible metrics, and compare their effectiveness. Our adaptive solver achieves significant speedups on a large set of examples. We also show that applying different heuristics on different parts of the search space can improve run times even beyond what can be achieved by the best heuristic on its own

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

Proceedings of SAT Race 2019 : Solver and Benchmark Descriptions

Non peer reviewe

AlphaSMT: A Reinforcement Learning Guided SMT Solver

Satisfiability Modulo Theories (SMT) solvers are programs that decide whether a first-order logic formula is satisfiable. Over the last two decades, these solvers have become central to many methods and tools in fields as diverse as software engineering, verification, security, and Artificial Intelligence. Most modern SMT solvers provide user-controllable strategies, i.e., users can define a strategy that customizes a solving algorithm for their own problems by combining individual tactics as building blocks. A tactic is a well-defined and implemented reasoning step provided by the SMT solver, which either simplifies, trans- forms, or solves the given input SMT formula. The flexibility of customizing a strategy to a specialized type of formula is important since no existing strategy is considered optimal for all instances. However, finding a good customized strategy is challenging even for experts. In this thesis we present a novel class of reinforcement-learning (RL) guided methods, implemented in the Z3 SMT solver and that we refer to as AlphaSMT, which adaptively constructs the expected best strategy for any given input SMT formula. Briefly, the AlphaSMT RL framework combines deep Monte-Carlo Tree Search (MCTS) and logical reasoning in a unique way in order to enable the RL agent to learn the best combination of tactics for a given class of formulas. In more detail, a deep neural network serves as both the value function and the policy, evaluating state-action pairs and making the decision of which tactic to choose at each step. The neural network is trained toward the optimal policy by learning from self-exploring sampling solving processes. MCTS is used as a lookahead planning step for each decision made in the sampling processes. We evaluated the performance of AlphaSMT on benchmark sets from three SMT logics, namely, quantifier-free non-linear real arithmetic (QF NRA), quantifier-free non-linear integer arithmetic (QF NIA), and quantifier-free bit-vector (QF BV). In all these logics, AlphaSMT outperforms its base solver, Z3, by solving up to 80.5% more instances in a testing set. Evaluation results also show that a reasonably longer tactic timeout helps solve more instances and a pre-solver contributes significantly to the speedup

Efficient Automated Planning with New Formulations

Problem solving usually strongly relies on how the problem is formulated. This fact also applies to automated planning, a key field in artificial intelligence research. Classical planning used to be dominated by STRIPS formulation, a simple model based on propositional logic. In the recently introduced SAS+ formulation, the multi-valued variables naturally depict certain invariants that are missed in STRIPS, make SAS+ have many favorable features. Because of its rich structural information SAS+ begins to attract lots of research interest. Existing works, however, are mostly limited to one single thing: to improve heuristic functions. This is in sharp contrast with the abundance of planning models and techniques in the field. On the other hand, although heuristic is a key part for search, its effectiveness is limited. Recent investigations have shown that even if we have almost perfect heuristics, the number of states to visit is still exponential. Therefore, there is a barrier between the nice features of SAS+ and its applications in planning algorithms. In this dissertation, we have recasted two major planning paradigms: state space search and planning as Satisfiability: SAT), with three major contributions. First, we have utilized SAS+ for a new hierarchical state space search model by taking advantage of the decomposable structure within SAS+. This algorithm can greatly reduce the time complexity for planning. Second, planning as Satisfiability is a major planning approach, but it is traditionally based on STRIPS. We have developed a new SAS+ based SAT encoding scheme: SASE) for planning. The state space modeled by SASE shows a decomposable structure with certain components independent to others, showing promising structure that STRIPS based encoding does not have. Third, the expressiveness of planning is important for real world scenarios, thus we have also extended the planning as SAT to temporally expressive planning and planning with action costs, two advanced features beyond classical planning. The resulting planner is competitive to state-of-the-art planners, in terms of both quality and performance. Overall, our work strongly suggests a shifting trend of planning from STRIPS to SAS+, and shows the power of formulating planning problems as Satisfiability. Given the important roles of both classical planning and temporal planning, our work will inspire new developments in other advanced planning problem domains

CDCL(Crypto) and Machine Learning based SAT Solvers for Cryptanalysis

Over the last two decades, we have seen a dramatic improvement in the efficiency of conflict-driven clause-learning Boolean satisfiability (CDCL SAT) solvers over industrial problems from a variety of applications such as verification, testing, security, and AI. The availability of such powerful general-purpose search tools as the SAT solver has led many researchers to propose SAT-based methods for cryptanalysis, including techniques for finding collisions in hash functions and breaking symmetric encryption schemes. A feature of all of the previously proposed SAT-based cryptanalysis work is that they are \textit{blackbox}, in the sense that the cryptanalysis problem is encoded as a SAT instance and then a CDCL SAT solver is invoked to solve said instance. A weakness of this approach is that the encoding thus generated may be too large for any modern solver to solve it efficiently. Perhaps a more important weakness of this approach is that the solver is in no way specialized or tuned to solve the given instance. Finally, very little work has been done to leverage parallelism in the context of SAT-based cryptanalysis. To address these issues, we developed a set of methods that improve on the state-of-the-art SAT-based cryptanalysis along three fronts. First, we describe an approach called \cdcl (inspired by the CDCL($T$) paradigm) to tailor the internal subroutines of the CDCL SAT solver with domain-specific knowledge about cryptographic primitives. Specifically, we extend the propagation and conflict analysis subroutines of CDCL solvers with specialized codes that have knowledge about the cryptographic primitive being analyzed by the solver. We demonstrate the power of this framework in two cryptanalysis tasks of algebraic fault attack and differential cryptanalysis of SHA-1 and SHA-256 cryptographic hash functions. Second, we propose a machine-learning based parallel SAT solver that performs well on cryptographic problems relative to many state-of-the-art parallel SAT solvers. Finally, we use a formulation of SAT into Bayesian moment matching to address heuristic initialization problem in SAT solvers

Learning dynamic algorithm portfolios

Algorithm selection can be performed using a model of runtime distribution, learned during a preliminary training phase. There is a trade-off between the performance of model-based algorithm selection, and the cost of learning the model. In this paper, we treat this trade-off in the context of bandit problems. We propose a fully dynamic and online algorithm selection technique, with no separate training phase: all candidate algorithms are run in parallel, while a model incrementally learns their runtime distributions. A redundant set of time allocators uses the partially trained model to propose machine time shares for the algorithms. A bandit problem solver mixes the model-based shares with a uniform share, gradually increasing the impact of the best time allocators as the model improves. We present experiments with a set of SAT solvers on a mixed SAT-UNSAT benchmark; and with a set of solvers for the Auction Winner Determination proble