Alexander Nadel

A clause-based heuristic for SAT solvers

We propose a new decision heuristic for DPLL-based propositional SAT solvers. Its essence is that both the initial and the conflict clauses are arranged in a list and the next decision variable is chosen from the top-most unsatisfied clause. Various methods of initially organizing the list and moving the clauses within it are studied. Our approach is an extension of one used in Berkmin, and adopted by other modern solvers, according to which only conflict clauses are organized in a list, and a literal-scoring-based secondary heuristic is used when there are no more unsatisfied conflict clauses. Our approach, implemented in the 2004 version of zChaff solver and in a generic Chaff-based SAT solver, results in a significant performance boost on hard industrial benchmarks.

Efficient MUS extraction with resolution

We report advances in state-of-the-art algorithms for the problem of Minimal Unsatisfiable Subformula (MUS) extraction. First, we demonstrate how to apply techniques used in the past to speed up resolution-based Group MUS extraction to plain MUS extraction. Second, we show that model rotation, presented in the context of assumption-based MUS extraction, can also be used with resolution-based MUS extraction. Third, we introduce an improvement to rotation, called eager rotation. Finally, we propose a new technique for speeding-up resolution-based MUS extraction, called path strengthening. We integrated the above techniques into the publicly available resolution-based MUS extractor HaifaMUC, which, as a result, now outperforms leading MUS extractors.

Efficient SAT solving under assumptions

In incremental SAT solving, assumptions are propositions that hold solely for one specific invocation of the solver. Effective propagation of assumptions is vital for ensuring SAT solving efficiency in a variety of applications. We propose algorithms to handle assumptions. In our approach, assumptions are modeled as unit clauses, in contrast to the current state-of-the-art approach that models assumptions as first decision variables. We show that a notable advantage of our approach is that it can make preprocessing algorithms much more effective. However, our initial scheme renders assumption-dependent (or temporary) conflict clauses unusable in subsequent invocations. To resolve the resulting problem of reduced learning power, we introduce an algorithm that transforms such temporary clauses into assumption-independent pervasive clauses. In addition, we show that our approach can be enhanced further when a limited form of look-ahead information is available. We demonstrate that our approach results in a considerable performance boost of the SAT solver on instances generated by a prominent industrial application in hardware validation.

Simultaneous SAT-Based model checking of safety properties

We present several algorithms for simultaneous SAT (propositional satisfiability) based model checking of safety properties. More precisely, we focus on Bounded Model Checking and Temporal Induction methods for simultaneously verifying multiple safety properties on the same model. The most efficient among our proposed algorithms for model checking are based on a simultaneous propositional satisfiability procedure (SSAT for short), which we design for solving related propositional objectives simultaneously, by sharing the learned clauses and the search. The SSAT algorithm is fully incremental in the sense that all clauses learned while solving one objective can be reused for the remaining objectives. Furthermore, our SSAT algorithm ensures that the SSAT solver will never re-visit the same sub-space during the search, even if there are several satisfiability objectives, hence one traversal of the search space is enough. Finally, in SSAT all SAT objectives are watched simultaneously, thus we can solve several other SAT objectives when the search is oriented to solve a particular SAT objective first. Experimental results on Intel designs demonstrate that our new algorithms can be orders of magnitude faster than the previously known techniques in this domain.

Ultimately Incremental SAT

Incremental SAT solving under assumptions, introduced in Minisat, is in wide use. However, Minisat’s algorithm for incremental SAT solving under assumptions has two main drawbacks which hinder performance considerably. First, it is not compliant with the highly effective and commonly used preprocessor SatELite. Second, all the assumptions are left in the formula, rather than being represented as unit clauses, propagated, and eliminated. Two previous attempts to overcome these problems solve either the first or the second of them, but not both. This paper remedies this situation by proposing a comprehensive solution for incremental SAT solving under assumptions, where SatELite is applied and all the assumptions are propagated. Our algorithm outperforms existing approaches over publicly available instances generated by a prominent industrial application in hardware validation.

Generating diverse solutions in SAT

This paper considers the DiversekSet problem in SAT, that is, the problem of efficiently generating a number of diverse solutions (satisfying assignments) given a propositional formula. We provide an extensive analysis of existing algorithms for this problem in a newly developed framework and propose new algorithms. While existing algorithms adapt modern SAT solvers to solve DiversekSet by changing their polarity selection heuristic, our new algorithms adapt the variable ordering strategy as well. Our experimental results demonstrate that the proposed algorithms improve the diversification quality of the solutions on large industrial instances of DiversekSet arising in SAT-based semiformal verification of hardware.

Towards a better understanding of the functionality of a conflict-driven SAT solver

We show that modern conflict-driven SAT solvers implicitly build and prune a decision tree whose nodes are associated with flipped variables. Practical usefulness of conflict-driven learning schemes, like 1UIP or AllUIP, depends on their ability to guide the solver towards refutations associated with compact decision trees. We propose an enhancement of 1UIP that is empirically helpful for real-world industrial benchmarks.

Assignment stack shrinking

Assignment stack shrinking is a technique that is intended to speed up the performance of modern complete SAT solvers. Shrinking was shown to be efficient in SAT’04 competition winners Jerusat and Chaff. However, existing studies lack the details of the shrinking algorithm. In addition, shrinking’s performance was not tested in conjunction with the most modern techniques. This paper provides a detailed description of the shrinking algorithm and proposes two new heursitics for it. We show that using shrinking is critical for solving well-known industrial benchmark families with the latest versions of Minisat and Eureka.

Bit-Vector Optimization

A variety of applications of Satisfiability Modulo Theories SMT require finding a satisfying assignment which optimizes some user-given function. Optimization in the context of SMT is referred to as Optimization Modulo Theories OMT. Current OMT research is mostly dedicated to optimization in arithmetic domains. This paper is about Optimization modulo Bit-Vectors OBV. We introduce two OBV algorithms which can easily be implemented in an eager bit-vector solver. We show that an industrial problem of fixing cell placement during the physical design stage of the CAD process can be reduced to optimization modulo either Bit-Vectors BV or Linear Integer Arithmetic LIA. We demonstrate that our resulting OBV tool can solve industrial instances which are out of reach of existing BV and LIA OMT solvers.

Implicative simultaneous satisfiability and applications

This paper proposes an efficient algorithm for the systematic learning of implications. This is done as part of a new search and restart strategy in the SAT solver. We evaluate the new algorithm within a number of applications, including BMC and induction with invariant strengthening for equivalence checking. We provide extensive experimental evidence attesting to a speedup of one and often two orders of magnitude with our algorithm, on a representative set of industrial and publicly available test suites, as compared to a basic version of invariant strengthening. Moreover, we show that the new invariant strengthening algorithm alone performs better than induction and interpolation, and that the absolutely best result is achieved when it is combined with interpolation. In addition, we experimentally demonstrate the superiority of an application of our new algorithm to BMC.