Conflict-driven Clause Learning Solvers Research Articles

Design and implementation of modern CDCL ASP solvers

Answer Set Programming (ASP) is a declarative language oriented towards solving complex combinatorial problems. In fact, ASP has been successfully used to address problems in various academic and industrial domains. The success of ASP can be attributed to its concise syntax, intuitive semantics, and the availability of several efficient solvers based on the Conflict-Driven Clause Learning (CDCL) algorithm. This paper details the design and implementation of contemporary CDCL solvers, emphasizing both algorithmic descriptions and their effective and efficientimplementation.

Runtime vs. Extracted Proof Size: An Exponential Gap for CDCL on QBFs

Conflict-driven clause learning (CDCL) is the dominating algorithmic paradigm for SAT solving and hugely successful in practice. In its lifted version QCDCL, it is one of the main approaches for solving quantified Boolean formulas (QBF). In both SAT and QBF, proofs can be efficiently extracted from runs of (Q)CDCL solvers. While for CDCL, it is known that the proof size in the underlying proof system propositional resolution matches the CDCL runtime up to a polynomial factor, we show that in QBF there is an exponential gap between QCDCL runtime and the size of the extracted proofs in QBF resolution systems. We demonstrate that this is not just a gap between QCDCL runtime and the size of any QBF resolution proof, but even the extracted proofs are exponentially smaller for some instances. Hence searching for a small proof via QCDCL (even with non-deterministic decision policies) will provably incur an exponential overhead for some instances.

Preprocessing of Propagation Redundant Clauses

The propagation redundant (PR) proof system generalizes the resolution and resolution asymmetric tautology proof systems used by conflict-driven clause learning (CDCL) solvers. PR allows short proofs of unsatisfiability for some problems that are difficult for CDCL solvers. Previous attempts to automate PR clause learning used hand-crafted heuristics that work well on some highly-structured problems. For example, the solver SaDiCaL incorporates PR clause learning into the CDCL loop, but it cannot compete with modern CDCL solvers due to its fragile heuristics. We present PReLearn, a preprocessing technique that learns short PR clauses. Adding these clauses to a formula reduces the search space that the solver must explore. By performing PR clause learning as a preprocessing stage, PR clauses can be found efficiently without sacrificing the robustness of modern CDCL solvers. On a large portion of SAT competition benchmarks we found that preprocessing with PReLearn improves solver performance. In addition, there were several satisfiable and unsatisfiable formulas that could only be solved after preprocessing with PReLearn. PReLearn supports proof logging, giving a high level of confidence in the results. Lastly, we tested the robustness of PReLearn by applying other forms of preprocessing as well as by randomly permuting variable names in the formula before running PReLearn, and we found PReLearn performed similarly with and without the changes to the formula.

Too much information: Why CDCL solvers need to forget learned clauses.

Conflict-driven clause learning (CDCL) is a remarkably successful paradigm for solving the satisfiability problem of propositional logic. Instead of a simple depth-first backtracking approach, this kind of solver learns the reason behind occurring conflicts in the form of additional clauses. However, despite the enormous success of CDCL solvers, there is still only a limited understanding of what influences the performance of these solvers in what way. Considering different measures, this paper demonstrates, quite surprisingly, that clause learning (without being able to get rid of some clauses) can not only help the solver but can oftentimes deteriorate the solution process dramatically. By conducting extensive empirical analysis, we furthermore find that the runtime distributions of CDCL solvers are multimodal. This multimodality can be seen as a reason for the deterioration phenomenon described above. Simultaneously, it also gives an indication of why clause learning in combination with clause deletion is virtually the de facto standard of SAT solving, in spite of this phenomenon. As a final contribution, we show that Weibull mixture distributions can accurately describe the multimodal distributions. Thus, adding new clauses to a base instance has an inherent effect of making runtimes long-tailed. This insight provides an explanation as to why the technique of forgetting clauses is useful in CDCL solvers apart from the optimization of unit propagation speed.

Better Decision Heuristics in CDCL through Local Search and Target Phases

On practical applications, state-of-the-art SAT solvers dominantly use the conflict-driven clause learning (CDCL) paradigm. An alternative for satisfiable instances is local search solvers, which is more successful on random and hard combinatorial instances. Although there have been attempts to combine these methods in one framework, a tight integration which improves the state of the art on a broad set of application instances has been missing. We present a combination of techniques that achieves such an improvement. Our first contribution is to maximize in a local search fashion the assignment trail in CDCL, by sticking to and extending promising assignments via a technique called target phases. Second, we relax the CDCL framework by again extending promising branches to complete assignments while ignoring conflicts. These assignments are then used as starting point of local search which tries to find improved assignments with fewer unsatisfied clauses. Third, these improved assignments are imported back to the CDCL loop where they are used to determine the value assigned to decision variables. Finally, the conflict frequency of variables in local search can be exploited during variable selection in branching heuristics of CDCL. We implemented these techniques to improve three representative CDCL solvers (Glucose, MapleLcm DistChronoBT, and Kissat). Experiments on benchmarks from the main tracks of the last three SAT Competitions from 2019 to 2021 and an additional benchmark set from spectrum allocation show that the techniques bring significant improvements, particularly and not surprisingly, on satisfiable real-world application instances. We claim that these techniques were essential to the large increase in performance witnessed in the SAT Competition 2020 where Kissat and Relaxed LcmdCbDl NewTech were leading the field followed by CryptoMiniSAT-Ccnr, which also incorporated similar ideas.

Learn to relax: Integrating 0-1 integer linear programming with pseudo-Boolean conflict-driven search

Conflict-driven pseudo-Boolean solvers optimize 0-1 integer linear programs by extending the conflict-driven clause learning (CDCL) paradigm from SAT solving. Though pseudo-Boolean solvers have the potential to be exponentially more efficient than CDCL solvers in theory, in practice they can sometimes get hopelessly stuck even when the linear programming (LP) relaxation is infeasible over the reals. Inspired by mixed integer programming (MIP), we address this problem by interleaving incremental LP solving with cut generation within the conflict-driven pseudo-Boolean search. This hybrid approach, which for the first time combines MIP techniques with full-blown conflict analysis operating directly on linear inequalities using the cutting planes method, significantly improves performance on a wide range of benchmarks, approaching a “best-of-both-worlds” scenario between SAT-style conflict-driven search and MIP-style branch-and-cut.

Going deeper with optimal software products selection using many-objective optimization and satisfiability solvers

In search-based software engineering, one actively studied problem is the optimal software product selection from a feature model using multiple (usually more than three) optimization objectives simultaneously. This can be represented as a many-objective optimization problem. The primary goal of solving this problem is to search for diverse and high-quality valid products as rapidly as possible. Previous studies have shown that combining search-based techniques with satisfiability (SAT) solvers was promising for achieving this goal, but it remained open that how different solvers affect the performance of a search algorithm, and that whether the ways to randomize solutions in the solvers make a difference. Moreover, we may need further investigation on the necessity of mixing different types of SAT solving techniques. In this paper, we address the above open research questions by performing a series of empirical studies on 21 features models, most of which are reverse-engineered from industrial software product lines. We examine four conflict-driven clause learning solvers, two stochastic local search solvers, and two different ways to randomize solutions. Experimental results suggest that the performance can be indeed affected by different SAT solvers, and by the ways to randomize solutions in the solvers. This study serves as a practical guideline for choosing and tuning SAT solvers for the many-objective optimal software product selection problem.