DPLL Algorithm Research Articles

A Verified Implementation of the DPLL Algorithm in Dafny

We present a DPLL SAT solver, which we call TrueSAT, developed in the verification-enabled programming language Dafny. We have fully verified the functional correctness of our solver by constructing machine-checked proofs of its soundness, completeness, and termination. We present a benchmark of the execution time of TrueSAT and we show that it is competitive against an equivalent DPLL solver implemented in C++, although it is still slower than state-of-the-art CDCL solvers. Our solver serves as a significant case study of a machine-verified software system. The benchmark also shows that auto-active verification is a promising approach to increasing trust in SAT solvers, because it combines execution speed with a high degree of trustworthiness.

The third AI summer: AAAI Robert S. Engelmore Memorial Lecture

The third AI summer: AAAI Robert S. Engelmore Memorial Lecture

An asynchronous P system with a DPLL algorithm for solving SAT

An asynchronous P system with a DPLL algorithm for solving SAT

Answering Skyline Queries Over Incomplete Data With Crowdsourcing

Due to the pervasiveness of incomplete data, incomplete data queries are vital in a large number of real-life scenarios. Current models and approaches for incomplete data queries mainly rely on the machine power. In this paper, we study the problem of <i>skyline queries over incomplete data with crowdsourcing</i> . We propose a novel query framework, termed as <inline-formula><tex-math notation="LaTeX">${\sf BayesCrowd}$</tex-math></inline-formula> , which takes into account the data correlation using the Bayesian network. We leverage the typical <i>c-table</i> model on incomplete data to represent objects. Considering budget and latency constraints, we present a suite of effective task selection strategies. Moreover, we introduce a <i>marginal utility</i> function to measure the benefit of crowdsourcing one task. In particular, the probability computation of each object being an answer object is at least as hard as #SAT problem. To this end, we propose an <i>adaptive</i> DPLL (i.e., Davis-Putnam-Logemann- Loveland) algorithm to speed up the computation. Extensive experiments using both real and synthetic data sets confirm the superiority of <inline-formula><tex-math notation="LaTeX">${\sf BayesCrowd}$</tex-math></inline-formula> to the state-of-the-art method, in terms of execution time, monetary cost, and latency minimization.

A Hybrid Approach to Solve SAT and UNSAT Problems

This work involves the construction of a hybrid approach to solve SAT and UNSAT problems. The solution combines techniques found in Stalmarck and DPLL algorithms for solving propositional satisfiability problems. We extend the Stalmarck rules in order to reduce computational cost during the deduction phase. We applied our solution to SAT and UNSAT instances. From the results it was found that this approach showed good results for UNSAT industrial instances. In some cases, our approach obtained gains of 45% to 60% in terms of efficient when compared to existing approaches. We compared our solution against the best known SAT solvers such as ZChaff and RSat. The techniques and methods involved are described in the article.

Grid based clustering for satisfiability solving

The originality of this work resides into the exploitation of data mining techniques for problem solving. Two major phases define this work. The first one is to determine the clustering technique that best suits each SAT instance based on the distribution of the later. The clustering technique is then applied to reduce the complexity of each instance by creating sub-instances that can be solved independently in the second phase. The latter consists into a resolution step where the DPLL or BSO algorithms are executed depending on the number of variables to be assigned in each cluster. This two-phase resolution strategy provides more efficient problem solving. The Boolean Satisfiability problem (SAT) is considered in this study because of its importance for the Artificial Intelligence (AI) community and the impact of its solving on other complex problems. Three different distributions of the problem were observed. The first distribution defines a space where the variables are dispersed forming regions of considerable density interspersed with regions of lower density or empty regions. On the other hand, the other two distributions do not show any significant shape, as the variables are randomly scattered, with one of these two dispersions having the particularity that almost all its variables are of high occurrence. To each of the three distributions, a clustering technique is associated. Density-based clustering techniques are the most appropriate type of clustering for the first distribution. Meanwhile, grid-based clustering and frequent patterns mining seem to be the most suitable clustering techniques for the second and third distributions. Investigations are undertaken on these latter issues and contributions are presented in this paper. Experiments were conducted on public benchmarks and the results showed the importance of the pre-processing step of data mining to solve the SAT problem.

Verifying the DPLL Algorithm in Dafny

Modern high-performance SAT solvers quickly solve large satisfiability instances that occur in practice. If the instance is satisfiable, then the SAT solver can provide a witness which can be checked independently in the form of a satisfying truth assignment. However, if the instance is unsatisfiable, the certificates could be exponentially large or the SAT solver might not be able to output certificates. The implementation of the SAT solver should then be trusted not to contain bugs. However, the data structures and algorithms implemented by a typical high-performance SAT solver are complex enough to allow for subtle programming errors. To counter this issue, we build a verified SAT solver using the Dafny system. We discuss its implementation in the present article.

Resolution over linear equations modulo two

We consider an extension of the resolution proof system that operates with disjunctions of linear equalities over F2; we denote this system by Res(⊕). It is well known that tree-like resolution is equivalent in behavior to DPLL algorithms for the Boolean satisfiability problem. Every DPLL algorithm splits the input problem into two by assigning two possible values to a variable; then it simplifies the two resulting formulas and makes two recursive calls. Tree-like Res(⊕)-proofs correspond to an extension of the DPLL paradigm, in which we can split by an arbitrary linear combination of variables modulo two. These algorithms quickly solve formulas that explicitly encode linear systems modulo two which were used for proving exponential lower bounds for conventional DPLL algorithms.We prove exponential lower bounds on the size of tree-like Res(⊕)-proofs. We also show that resolution and tree-like Res(⊕) do not simulate each other.We prove a space vs size tradeoff for Res(⊕)-proofs. We prove that Res(⊕) is implicationally complete and also that Res(⊕) is polynomially equivalent to its semantic version.We consider the proof system Res(⊕;⩽k) that is a restricted version of Res(⊕) operating with disjunctions of linear equalities such that at most k equalities depend on more than one variable. We simulate Res(⊕;⩽k) in the OBDD-based proof system with blowup 2k and in Polynomial Calculus Resolution with blowup 2nH(2k/n)poly(n), where n is the number of variables and H(p) is the binary entropy; the latter result implies exponential lower bounds on the size of Res(⊕;⩽δn)-proofs for some constant δ>0. Raz and Tzameret introduced the system R(lin) which operates with disjunctions of linear equalities with integer coefficients. We show that Res(⊕) can be p-simulated in R(lin).

An Exhaustive DPLL Algorithm for Model Counting

State-of-the-art model counters are based on exhaustive DPLL algorithms, and have been successfully used in probabilistic reasoning, one of the key problems in AI. In this article, we present a new exhaustive DPLL algorithm with a formal semantics, a proof of correctness, and a modular design. The modular design is based on the separation of the core model counting algorithm from SAT solving techniques. We also show that the trace of our algorithm belongs to the language of Sentential Decision Diagrams (SDDs), which is a subset of Decision-DNNFs, the trace of existing state-of-the-art model counters. Still, our experimental analysis shows comparable results against state-of-the-art model counters. Furthermore, we obtain the first top-down SDD compiler, and show orders-of-magnitude improvements in SDD construction time against the existing bottom-up SDD compiler.

Quantum-Walk Speedup of Backtracking Algorithms

We describe a general method to obtain quantum speedups of classical algorithms which are based on the technique of backtracking, a standard approach for solving constraint satisfaction problems (CSPs). Backtracking algorithms explore a tree whose vertices are partial solutions to a CSP in an attempt to find a complete solution. Assume there is a classical backtracking algorithm which finds a solution to a CSP on n variables, or outputs that none exists, and whose corresponding tree contains T vertices, each vertex corresponding to a test of a partial solution. Then we show that there is a bounded-error quantum algorithm which completes the same task using O(sqrt(T) n^(3/2) log n) tests. In particular, this quantum algorithm can be used to speed up the DPLL algorithm, which is the basis of many of the most efficient SAT solvers used in practice. The quantum algorithm is based on the use of a quantum walk algorithm of Belovs to search in the backtracking tree. We also discuss how, for certain distributions on the inputs, the algorithm can lead to an exponential reduction in expected runtime.

A Hybrid Quantum Genetic Algorithm and Local Search based DPLL for Max 3-SAT Problems

In this paper, we present a new framework for combining complete and incomplete methods in order to deal with the Max Sat problem. The objective is to find the best assignment for a set of Bo olean variables, which gives the maximum of verified clauses in a Boolean formula. Unfortunately, this problem has been shown to be NP-hard (non-deterministic polynomial-time hard) if the number of variables per clause is greater than 3. The proposed approach is based on quantum inspired genetic principles and the well known exact algorithm DPLL. The underlying idea is to harness the optimization capabilities of quantum genetic algorithm to achieve good quality solutions for Max SAT problem. A new local search based on DPLL algorithm has been embodied in the quantum genetic algorithm leading to an efficient hybrid framework which achieves better b alance between diversification and intensification search. The obtained results are very encouraging and show the feasibility and effectiveness of the proposed hybrid approach.

Parameterized Complexity of DPLL Search Procedures

We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game that models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k -clique requires n Ω(k) steps for a nontrivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked by Beyersdorff et al. [2012] of understanding the Resolution complexity of this family of formulas.

The complexity of inverting explicit Goldreich’s function by DPLL algorithms

Goldreich’s function has n binary inputs and n binary outputs. Every output depends on d inputs and is computed from them by a fixed predicate of arity d. Every Goldreich’s function is determined by its dependency graph G and a predicate P. In 2000, O. Goldreich formulated the conjecture that if G is an expander and P is a random predicate of arity d, then the corresponding function is one-way. In this paper, we give a simple proof of an exponential lower bound on the inversion of Goldreich’s function by myopix DPLL algorithms. The dependency graph G in our construction may be based on an arbitrary expander; in particular, one can use an explicit expander; while all previously known results are based on random dependency graphs. The predicate P can be linear or slightly nonlinear. Our construction can be used in the proof of lower bounds for drunken DPLL algorithms as well. Bibliography: 18 titles.

A Comparison of Local Reduction and Sat-Solver Based Algebraic Cryptanalysis of Jh And Keccak

ABSTRACT Local reduction methods can be used to assess the resistance of cryptosystems against algebraic attacks. The assessment is based on the separation of the attack into polynomial-time reduction algorithm, and exponential time guessing and backtracking. This approach is similar to that employed by the DPLL algorithm that is used as a core of various modern SAT-solvers. In the article we show the application of this method to evaluate the strength of (reduced versions of) two chosen SHA-3 candidates: JH, and Keccak, respectively. We compare the complexity estimates with the behavior of the full search algorithm. We also compare the results based on the local reduction with the attack based on the use of SAT-solvers PrecoSAT, and CryptoMiniSAT, respectively.

Extracting a DPLL Algorithm

We formalize a completeness proof for the DPLL proof system and extract a DPLL SAT solver from it. When applied to a propositional formula in conjunctive normal form the program produces either a satisfying assignment or a DPLL derivation which shows that it is unsatisfiable. We use non-computational quantifiers to remove redundant computational content from the extracted program and improve its performance. The formalization is carried out in the Minlog system.

Finding kernels or solving SAT

We begin by offering a new, direct proof of the equivalence between the problem of the existence of kernels in digraphs, KER, and satisfiability of propositional theories, SAT, giving linear reductions in both directions. Having introduced some linear reductions of the input graph, we present new algorithms for KER, with variations utilizing solvers of boolean equations. In the worst case, the algorithms try all assignments to either a feedback vertex set, F, or a set of nodes E touching only all even cycles. Hence KER is fixed parameter tractable not only in the size of F, as observed earlier, but also in the size of E. A slight modification of these algorithms leads to a branch and bound algorithm for KER which is virtually identical to the DPLL algorithm for SAT. This suggests deeper analogies between the two problems and the probable scenario of KER research facing the challenges known from the work on SAT. The algorithm gives also the upper bound O ⁎ ( 1.427 | G | ) on the time complexity of general KER and O ⁎ ( 1.286 | G | ) of KER for oriented graphs, where | G | is the number of vertices.

Some Properties of Pigeon-Hole Formulas

由鸽巢原理定义的鸽巢公式PH_nⁿ⁺¹ 是著名的消解难例之一,研究该公式的结构和性质有助于其他难例的构造.证明了PH_nⁿ⁺¹ 是一个极小不可满足公式,根据其极小不可满足性,给出了最大可满足真值指派的两种标准形式,Haken 关于PH_nⁿ⁺¹ 的难解证明用到了其中一种标准形式.公式PH_nⁿ⁺¹ 具有良好的子结构同构性质,如果DPLL 算法中允许使用同构规则,则存在PH_nⁿ⁺¹ 的反驳证明,其复杂性可以降至O(n³).

Formula Caching in DPLL

We consider extensions of the DPLL approach to satisfiability testing that add a version ofmemoization, in which formulas that the algorithm has previously shown to be unsatisfiable are remembered for later use. Suchformula cachingalgorithms have been suggested for satisfiability and stochastic satisfiability by several authors. We formalize these methods by developing extensions of the fruitful connection that has previously been developed between DPLL algorithms for satisfiability and tree-like resolution proofs of unsatisfiability. We analyze a number of variants of these formula caching methods and characterize their strength in terms of proof systems. These proof systems are new and simple, and have a rich structure. We compare them to several studied proof systems: tree-like resolution, regular resolution, general resolution, Res(k), and Frege systems and present both simulation and separations. One of our most interesting results is the introduction of a natural and implementable form of DPLL with caching, FCWreason. This system is surprisingly powerful: we prove that it can polynomially simulate regular resolution, and furthermore, it can produce short proofs of some formulas that require exponential-size resolution proofs.

ManySAT: a Parallel SAT Solver

In this paper, ManySAT a new portfolio-based parallel SAT solver is thoroughly described. The design of ManySAT benefits from the main weaknesses of modern SAT solvers: their sensitivity to parameter tuning and their lack of robustness. ManySAT uses a portfolio of complementary sequential algorithms obtained through careful variations of the standard DPLL algorithm. Additionally, each sequential algorithm shares clauses to improve the overall performance of the whole system. This contrasts with most of the parallel SAT solvers generally designed using the divide-and-conquer paradigm. Experiments on many industrial SAT instances, and the first rank obtained by ManySAT in the parallel track of the 2008 SAT-Race clearly show the potential of our design philosophy.

New updating criteria for conflict-based branching heuristics in DPLL algorithms for satisfiability

The paper is concerned with the computational evaluation and comparison of a new family of conflict-based branching heuristics for evolved DPLL Satisfiability solvers. Such a family of heuristics is based on the use of new scores updating criteria developed in order to overcome some of the typical unpleasant behaviors of DPLL search techniques. In particular, a score is associated with each literal. Whenever a conflict occurs, some scores are incremented with different values, depending on the character of the conflict. The branching variable is then selected by using the maximum among those scores. Several variants of this have been introduced into a state-of-the-art implementation of a DPLL SAT solver, obtaining several versions of the solver having quite different behavior. Experiments on many benchmark series, both satisfiable and unsatisfiable, demonstrate advantages of the proposed heuristics.