SAT Instances Research Articles

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.

On black-box optimization in divide-and-conquer SAT solving

ABSTRACT Solving hard instances of the Boolean satisfiability problem (SAT) in practice is an interestingly nontrivial area. The heuristic nature of SAT solvers makes it impossible to know in advance how long it will take to solve any particular SAT instance. One way of coping with this disadvantage is the Divide-and-Conquer approach when an original SAT instance is decomposed into a set of simpler subproblems. However, the way it is decomposed plays a crucial role in the resulting effectiveness of solving. In the present study, we reduce the problem of choosing a proper decomposition to a stochastic pseudo-Boolean black-box optimization problem. Several optimization algorithms of different types were used to analyse a number of hard SAT-based optimization problems, related to SAT-based cryptanalysis of state-of-the-art stream ciphers. A meticulous computational study showed that some of the considered optimization algorithms perform much better than the others in the context of the problems from the considered class. It turned out that the obtained results also pose some cryptographic interest.

Community Structure in Industrial SAT Instances

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is believed that most of these successful techniques exploit the underlying structure of industrial instances. Recently, there have been some attempts to analyze the structure of industrial SAT instances in terms of complex networks, with the aim of explaining the success of SAT solving techniques, and possibly improving them.
 In this paper, we study the community structure, or modularity, of industrial SAT instances. In a graph with clear community structure, or high modularity, we can find a partition of its nodes into communities such that most edges connect variables of the same community. Representing SAT instances as graphs, we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erdös-Rényi random graph model, where no structure can be observed. We also analyze how this structure evolves by the effects of the execution of a CDCL SAT solver, and observe that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. Motivated by this observation, we finally present an application that exploits the community structure to detect relevant learned clauses, and we show that detecting these clauses results in an improvement on the performance of the SAT solver. Empirically, we observe that this improves the performance of several SAT solvers on industrial SAT formulas, especially on satisfiable instances.

Critical branching processes in digital memcomputing machines

Memcomputing is a novel computing paradigm that employs time non-locality (memory) to solve combinatorial optimization problems. It can be realized in practice by means of non-linear dynamical systems whose point attractors represent the solutions of the original problem. It has been previously shown that during the solution search digital memcomputing machines go through a transient phase of avalanches (instantons) that promote dynamical long-range order. By employing mean-field arguments we predict that the distribution of the avalanche sizes follows a Borel distribution typical of critical branching processes with exponent . We corroborate this analysis by solving various random 3-SAT instances of the Boolean satisfiability problem. The numerical results indicate a power-law distribution with exponent , in very good agreement with the mean-field analysis. This indicates that memcomputing machines self-tune to a critical state in which avalanches are characterized by a branching process, and that this state persists across the majority of their evolution.

On conversions from CNF to ANF

In this paper we discuss conversion methods from the conjunctive normal form (CNF) to the algebraic normal form (ANF) of a Boolean function. Whereas the reverse conversion has been studied before, the CNF to ANF conversion has been achieved predominantly via a standard method which tends to produce many polynomials of high degree. Based on a block-building mechanism, we design a new blockwise algorithm for the CNF to ANF conversion which is geared towards producing fewer and lower degree polynomials. In particular, we look for as many linear polynomials as possible in the converted system and check that our algorithm finds them. As an application, we suggest a new algebraic extension of the resolution calculus which is tailored to the output of the standard CNF to ANF conversion algorithm. Experiments show that the ANF produced by our algorithm outperforms the standard conversion in “real life” examples originating from cryptographic attacks, and that our algebraic resolution algorithm solves some SAT instances which are notoriously hard for SAT solvers.

Pseudo-Boolean Black-Box Optimization Methods in the Context of Divide-and-Conquer Approach to Solving Hard SAT Instances

Pseudo-Boolean Black-Box Optimization Methods in the Context of Divide-and-Conquer Approach to Solving Hard SAT Instances

Revisiting the Learned Clauses Database Reduction Strategies

In this paper, we revisit an important issue of CDCL-based SAT solvers, namely the learned clauses database management policies. Our motivation takes its source from a simple observation on the remarkable performances of both random and size-bounded reduction strategies. We first derive a simple reduction strategy, called Size-Bounded Randomized strategy (in short SBR), that combines maintaining short clauses (of size bounded by k), while deleting randomly clauses of size greater than k. The resulting strategy outperform the state-of-the-art on SAT instances taken from the SAT competitions 2013 and 2018, and remains competitive on a broad range of SAT instances of the SAT Competition 2014. Reinforced by the interest of keeping short clauses, we propose several new dynamic variants, and we discuss their performance. We also propose different ways for adjusting dynamically the size-bounded parameter of the strategy.

Proceedings of the 15th International Workshop on the ACL2 Theorem Prover and Its Applications

We present a tool that primarily supports the ability to check bounded properties starting from a sequence of states in a run. The target design is compiled into an AIGNET which is then selectively and iteratively translated into an incremental SAT instance in which clauses are added for new terms and simplified by the assignment of existing literals. Additional applications of the tool can be derived by the user providing alternative attachments of constrained functions which guide the iterations and SAT checks performed. Some Verilog RTL examples are included for reference.

Knowledge Compilation Methods Based on the Clausal Relevance and Extension Rule

We introduce the concepts of Relevancematrix (RM) and Relevance-set (RS). And we construct the association between RM and the Knowledge compilation (KC) methods based on Extension rule (ER). Based on the basic parameters of RS and the relationship between RM and the KC methods based on ER, we design two efficient heuristics, called M2S (maximum sum of elements in RS and sum of literals in RS) and MNE (minimum number of maximum terms not extended by RS). Both of above heuristics intend to find the minimum set of maximum terms which cannot be extended by RS. Furthermore, we apply M2S and MNE on KCER. M2S_KCER (KCER with M2S) and MNE_KCER (KCER with MNE) are designed and implemented based on M2S and MNE, respectively. Experimentally, for the SAT instances with random lengths of clauses, M2S_KCER and MNE_KCER can improve the efficiency and quality of KCER sharply, and they are two best KC algorithms of EPCCL (each pair contains complementary literal) theory in all KC algorithms based on KCER.

Sensor Synthesis for POMDPs with Reachability Objectives

Partially observable Markov decision processes (POMDPs) are widely used in probabilistic planning problems in which an agent interacts with an environment using noisy and imprecise sensors. We study a setting in which the sensors are only partially defined and the goal is to synthesize “weakest” additional sensors, such that in the resulting POMDP, there is a small-memory policy for the agent that almost-surely (with probability 1) satisfies a reachability objective. We show that the problem is NP-complete, and present a symbolic algorithm by encoding the problem into SAT instances. We illustrate trade-offs between the amount of memory of the policy and the number of additional sensors on a simple example. We have implemented our approach and consider three classical POMDP examples from the literature, and show that in all the examples the number of sensors can be significantly decreased (as compared to the existing solutions in the literature) without increasing the complexity of the policies.

Local reduction

We reduce non-deterministic time T≥2n to a 3SAT instance ϕ of quasilinear size |ϕ|=T⋅logO(1)⁡T such that there is an explicit NC0 circuit C that encodes ϕ in the following way: on input a (log⁡|ϕ|)-bit index i, C outputs the ith clause of ϕ. The previous best result was C in NC1. Even in the simpler setting of polynomial size (|ϕ|=poly(T)), the previous best result was C in AC0.More generally, for any time T≥n and parameter r≤n we obtain |ϕ|=max⁡(T,2n/r)⋅(nlog⁡T)O(1), and each output bit of C is a decision tree of depth O(log⁡r).As an application, we tighten Williams' connection between satisfiability algorithms and circuit lower bounds (STOC 2010; SIAM J. Comput. 2013).

D2-SYNCHRONIZATION IN NONDETERMINISTIC AUTOMATA

We approach the problem of computing a D2-synchronizing word of minimum length for a given nondeterministic automaton via its encoding as an instance of SAT and invoking a SAT solver. In addition, we report some of the experimental results obtained when we had tested our method on randomly generated automata and certain benchmarks.

Bee swarm optimization for solving the MAXSAT problem using prior knowledge

This paper explores rule decomposition for solving the MAXSAT problem. Four approaches are proposed to steer a bee swarm optimization metaheuristic. Two decomposition methods are proposed: direct and indirect. The first one applies the Kmeans algorithm, while the second one transforms a MAXSAT instance into a transactional database before performing decomposition using the Apriori algorithm. Several experiments conducted on DIMACS benchmark instances, and some other hard and large SAT instances have been carried out. Results show clear improvement compared to the state-of-the-art MAXSAT algorithms in terms of the quality of the obtained solutions. They show that the proposed approaches are stable when dealing with hard instances such as Parity8 from DIMACS. Results also demonstrate the superiority of the proposed approaches for medium and large instances. The proposed approaches could be applied to other optimization problems such as the weighted MAXSAT problem, the MAXCSP and coloring problems. They may also be adapted for other metaheuristics and decomposition methods.

Sufficient conditions for convergence of the survey propagation algorithm

Message propagation algorithms are very effective at finding satisfying assignments for SAT instances, and hard regions of a random SAT have become narrower. However, message propagation algorithms do not always converge for some random SAT instances. Unfortunately, a rigorous theoretical proof of this phenomenon is still lacking. The survey propagation (SP) algorithm is very effective at solving SAT instances, and a theoretical analysis of SP is very important in designing other message passing algorithms. Through this study, we derived the sufficient conditions for convergence of SP with extending message [0,~1] to message $(-\infty,\infty)$. Finally, the experiment results show that the conditions for the convergence of SP are very effective in random 3-SAT instances.

A randomized diversification strategy for solving satisfiability problem with long clauses

Satisfiability problem (SAT) is a central problem in artificial intelligence due to its computational complexity and usefulness in industrial applications. Stochastic local search (SLS) algorithms are powerful to solve hard instances of satisfiability problems, among which CScoreSAT is proposed for solving SAT instances with long clauses by using greedy mode and diversification mode. In this paper, we present a randomized variable selection strategy to improve efficiency of the diversification mode, and thus propose a new SLS algorithm. We perform a number of experiments to evaluate the new algorithm comparing with the recently proposed algorithms, and show that our algorithm is comparative with others for solving random instances near the phase transition threshold.

Algorithms for Deciding Counting Quantifiers over Unary Predicates

We study algorithms for fragments of first order logic ex- tended with counting quantifiers, which are known to be highly complex in general. We propose a fragment over unary predicates that is NP-complete and for which there is a nor- mal form where Counting Quantification sentences have a single Unary predicate, thus call it the CQU fragment. We provide an algebraic formulation of the CQU satisfiability problem in terms of Integer Linear Programming based on which two algorithms are proposed, a direct reduction to SAT instances and an Integer Linear Programming version extended with a column generation mechanism. The latter is shown to lead to a viable implementation and experiments shows this algorithm presents a phase transition behavior.

A parallel SAT solving algorithm based on improved handling of conflict clauses

In this study, a new parallel algorithm for solving Boolean satisfiability problem (SAT) is suggested. This algorithm is based on the Conflict Driven Clause Learning (CDCL) algorithm. Given an original SAT instance, CDCL is launched on it. Some types of conflict clauses, which are produced by CDCL, are collected. Then the collected clauses are employed to construct a family of different SAT instances. In particular, each SAT instance from a family is produced by adding some subset of the collected conflict clauses to an initial set of clauses (which belong to the original instance). In order to solve the original SAT instance, it is sufficient to solve at least one SAT instance from the constructed family. The proposed algorithm was implemented in the form of a parallel SAT solver. It was compared with two high ranked state-of-the-art parallel solvers. As a test set, we used SAT instances which encode cryptanalysis of three keystream generators: summation generator, threshold generator, and Gifford generator. According to computational experiments, our solver outperforms those two solvers.

A new hybridized algorithm based on Population-Based Simulated Annealing with an experimental study of phase transition in 3-SAT

This paper is about experiments for satisfiability problem using a new algorithm (GR-MM-PBSA) that improves the algorithm Population-Based Simulated Annealing (PBSA). GR-MM-PBSA runs in a parallel way Simulated Annealing (SA) and Threshold Annealing (TA) algorithms with a Golden Ratio space search strategy and Markovian Model to select initial and final temperature. In this paper we execute differents hybridized Simulated Annealing (or Threshold Accepting) algorithms and compares the efficiency of these, using a metric based on transition phase effect. Simulated Annealing Algorithms (SAA) theoretically can reach the optimum if the control parameters and cooling scheme are chosen correctly. All algorithms are compared with a metric based on transition phase obtained for 3-SAT instances. This paper shows the results of SAA hybridizations are more efficient than the original algorithm, without increasing their computational complexity. We also show the experimental data about runs with 820 3-SAT instances with ratio clauses-variables between 2.0 to 6.0.

Beyond the structure of SAT formulas

Nowadays, many real-world problems are encoded into SAT instances and efficiently solved by modern SAT solvers. These solvers, usually known as Conflict-Driven Clause Learning (CDCL) SAT solvers, include a variety of sophisticated techniques, such as clause learning, lazy data structures, conflict-based adaptive branching heuristics, or random restarts, among others. However, the reasons of their efficiency in solving real-world, or industrial, SAT instances are still unknown. The common wisdom in the SAT community is that these technique exploit some hidden structure of real-world problems. In this thesis, we characterize some important features of the underlying structure of industrial SAT instances. Namely, they are the community structure and the self-similar structure. We observe that most industrial SAT formulas, viewed as graphs, have these two properties. This means that (i) in a graph with a clear community structure, i.e. having high modularity, we can find a partition of its nodes into communities such that most edges connect nodes of the same community; and (ii) in a graph with a self-similar pattern, i.e. being fractal, its shape is kept after re-scalings, i.e., grouping sets of nodes into a single node. We also analyze how these structures are affected by the effects of CDCL techniques during the search. Using the previous structural studies, we propose three applications. First, we face the problem of generating pseudo-industrial random SAT instances using the notion of modularity. Our model generates instances similar to (classical) random SAT formulas when the modularity is low, but when this value is high, our model is also adequate to model realistic pseudo-industrial problems. Second, we propose a method based on the community structure of the instance to detect relevant learnt clauses. Our technique augments the original instance with this set of relevant clauses, and this results into an overall improvement of the efficiency of several state-of-the-art CDCL SAT solvers. Finally, we analyze the classification of industrial SAT instances into families using the previously analyzed structure features, and we compare them to other classifiers commonly used in portfolio SAT approaches. In summary, this dissertation extends the understandings of the structure of SAT instances, with the aim of better explaining the success of CDCL techniques and possibly improve them, and propose a number of applications based on this analysis of the underlying structure of SAT formulas.

Hard satisfiable 3-SAT instances via autocorrelation

We establish a reduction of a combinatorial problem defined via autocorrelation to an instance of Boolean satisfiability. As a consequence, we obtain a family of hard satisfiable 3-SAT instances. The combinatorial problem that we reduce is the D-optimal matrices problem. We generated a family of 3-SAT instances from the D-optimal matrices problem with the motivation to solve interesting cases using the power of SAT solvers. We give a detailed construction of the generated instances that were submitted to SAT competition 2014. Our reduction techniques is fairly straightforward and can be adapted to various other problems that are defined via autocorrelation.