SAT Formula Research Articles

Solving Satisfiability Modulo Counting for Symbolic and Statistical AI Integration with Provable Guarantees

Satisfiability Modulo Counting (SMC) encompasses problems that require both symbolic decision-making and statistical reasoning. Its general formulation captures many real-world problems at the intersection of symbolic and statistical AI. SMC searches for policy interventions to control probabilistic outcomes. Solving SMC is challenging because of its highly intractable nature (NP^PP-complete), incorporating statistical inference and symbolic reasoning. Previous research on SMC solving lacks provable guarantees and/or suffers from suboptimal empirical performance, especially when combinatorial constraints are present. We propose XOR-SMC, a polynomial algorithm with access to NP-oracles, to solve highly intractable SMC problems with constant approximation guarantees. XOR-SMC transforms the highly intractable SMC into satisfiability problems by replacing the model counting in SMC with SAT formulae subject to randomized XOR constraints. Experiments on solving important SMC problems in AI for social good demonstrate that XOR-SMC outperforms several baselines both in solution quality and running time.

Refining neural network predictions using background knowledge

Recent work has shown learning systems can use logical background knowledge to compensate for a lack of labeled training data. Many methods work by creating a loss function that encodes this knowledge. However, often the logic is discarded after training, even if it is still helpful at test time. Instead, we ensure neural network predictions satisfy the knowledge by refining the predictions with an extra computation step. We introduce differentiable refinement functions that find a corrected prediction close to the original prediction. We study how to effectively and efficiently compute these refinement functions. Using a new algorithm called iterative local refinement (ILR), we combine refinement functions to find refined predictions for logical formulas of any complexity. ILR finds refinements on complex SAT formulas in significantly fewer iterations and frequently finds solutions where gradient descent can not. Finally, ILR produces competitive results in the MNIST addition task.

Comparison of Two Convergence Criteria for the Variable-Assignment Lopsided Lovasz Local Lemma

The Lopsided Lovász Local Lemma (LLLL) is a probabilistic tool which is a cornerstone of the probabilistic method of combinatorics, which shows that it is possible to avoid a collection of "bad" events as long as their probabilities and interdependencies are sufficiently small. The strongest possible criterion that can be stated in these terms is due to Shearer (1985), although it is technically difficult to apply to constructions in combinatorics.
 
 The original formulation of the LLLL was non-constructive; a seminal algorithm of Moser & Tardos (2010) gave an efficient constructive algorithm for nearly all applications of it, including applications to $k$-SAT instances with a bounded number of occurrences per variables. Harris (2015) later gave an alternate criterion for this algorithm to converge, which appeared to give stronger bounds than the standard LLL. Unlike the LLL criterion or its variants, this criterion depends in a fundamental way on the decomposition of bad-events into variables.
 
 In this note, we show that the criterion given by Harris can be stronger in some cases even than Shearer's criterion. We construct $k$-SAT formulas with bounded variable occurrence, and show that the criterion of Harris is satisfied while the criterion of Shearer is violated. In fact, there is an exponentially growing gap between the bounds provable from any form of the LLLL and from the bound shown by Harris.

Characterizing the Temperature of SAT Formulas

The remarkable advances in SAT solving achieved in the last years have allowed to use this technology to solve many real-world applications, such as planning, formal verification and cryptography, among others. Interestingly, these industrial SAT problems are commonly believed to be easier than classical random SAT formulas, but estimating their actual hardness is still a very challenging question, which in some cases even requires to solve them. In this context, realistic pseudo-industrial random SAT generators have emerged with the aim of reproducing the main features of these application problems to better understand the success of those SAT solving techniques on them. In this work, we present a model to estimate the temperature of real-world SAT instances. This temperature represents the degree of distortion into the expected structure of the formula, from highly structured benchmarks (more similar to real-world SAT instances) to the complete absence of structure (observed in the classical random SAT model). Our solution is based on the popularity–similarity random model for SAT, which has been recently presented to reproduce two crucial features of application SAT benchmarks: scale-free and community structures. This model is able to control the hardness of the generated formula by introducing some randomizations in the expected structure. Using our regression model, we observe that the estimated temperature of the applications benchmarks used in the last SAT Competitions correlates to their hardness in most of the cases.

Proof of the satisfiability conjecture for large $k$

We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_{\rm sat}(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with high probability satisfiable for $\alpha\lt \alpha_{\rm sat}$, and unsatisfiable for $\alpha>\alpha_{\rm sat}$. We show that the threshold $\alpha_{\rm sat}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics. The proof develops a new analytic method for moment calculations on random graphs, mapping a high-dimensional optimization problem to a more tractable problem of analyzing tree recursions. We believe that our method may apply to a range of random CSPs in the $1$-RSB universality class.

Faster Algorithms for Weak Backdoors

A weak backdoor, or simply a backdoor, for a Boolean SAT formula F into a class of SAT formulae C is a partial truth assignment T such that F[T] is in C and satisfiability is preserved. The problem of finding a backdoor from class C1 into class C2, or WB(C1,C2), can be stated as follows: Given a formula F in C1, and a natural number k, determine whether there exists a backdoor for F into C2 assigning at most k variables. The class 0-Val contains all Boolean formulae with at least one negative literal in each clause. We design a new algorithm for WB(3CNF, 0-Val) by reducing it to a local search variant of 3-SAT. We show that our algorithm runs in time O*(2.562^k), improving on the previous state-of-the-art of O*(2.85^k). Here, the O* notation is a variant of the big-O notation that allows to omit polynomial factors in the input size. Next, we look at WB(3CNF, Null), where Null is the class consisting of the empty formula. This problem was known to have a trivial running time upper bound of O*(6^k) and can easily be solved in O*(3^k) time. We use a reduction to Conflict-Free-d-Hitting-Set to prove an upper bound of O*(2.2738^k), and also prove a lower bound of 2^o(k) assuming the Exponential Time Hypothesis. Finally, Horn is the class of formulae with at most one positive literal per clause. We improve the previous O*(4.54^k) running time for WB(3CNF, Horn) problem to O*(4.17^k), by exploiting the structure of the SAT instance to give a novel proof of the non-existence of the slowest cases after a slight restructuring of the branching priorities.

Advances and Challenges for Model and Data Engineering

Advances and Challenges for Model and Data Engineering

Popularity-similarity random SAT formulas

In the last decades, we have witnessed a remarkable success of algorithms solving the Boolean Satisfiability problem (SAT) on instances encoding application or real-world problems arising from a very diverse number of domains, such as hardware and software verification, planning or cryptography. These algorithms are the so known Conflict-Driven Clause Learning (CDCL) SAT solvers. Interestingly enough, the reasons for the success of these solvers on this diverse range of problems are not completely understood yet.A common issue when facing this open challenge is the heterogeneity of this set of benchmarks. Another problem is the limited number of existing instances. In this context, random models of SAT formulas capturing features shared by the majority of these application benchmarks become crucial, for both theoretical and practical purposes. On the one hand, it is undoubtedly necessary to have random models where theoretical properties, like hardness, can be studied. Therefore, realistic random SAT models may contribute to explain the success of these solvers on these industrial problems. On the other hand, the limited number of benchmarks and their hardness in practice makes the evaluation of new solving techniques a costly task. Therefore, these realistic random SAT generators can provide an unlimited number of pseudo-industrial random SAT instances with some desired properties.In this work, we present a random SAT instances generator based on the notion of locality. This notion is complementary to the popularity of variables, which is present in the scale-free structure, observable in actual application problems and achievable by previous generators. Our random SAT model combines both locality and popularity, and we show that they are two decisive dimensions of attractiveness among the variables of a formula, and how CDCL SAT solvers take advantage of them. Locality is closely related to the community structure, another important feature of application SAT benchmarks, which is indirectly achieved by this model. To the best of our knowledge, this is the first random SAT model that generates both scale-free structure and community structure at once.

Community-based 3-SAT formulas with a predefined solution

It is crucial to generate crafted SAT formulas with predefined solutions for the testing and development of SAT solvers since many SAT formulas from real-world applications have solutions. Although some generating algorithms have been proposed to generate SAT formulas with predefined solutions, community structures of SAT formulas are not considered. We propose a 3-SAT formula generating algorithm that not only guarantees the existence of a predefined solution, but also simultaneously considers community structures and clause distributions. The proposed 3-SAT formula generating algorithm controls the quality of community structures through controlling (1) the number of clauses whose variables have a common community, which we call intra-community clauses, and (2) the number of variables that only belong to one community, which we call intra-community variables. To study the combined effect of community structures and clause distributions on the hardness of SAT formulas, we measure solving runtimes of two solvers, gluHack (a leading CDCL solver) and CPSparrow (a leading SLS solver), on the generated SAT formulas under different groups of parameter settings. Through extensive experiments, we obtain some noteworthy observations on the SAT formulas generated by the proposed algorithm: (1) The community structure has little or no effects on the hardness of SAT formulas with regard to CPSparrow but a strong effect with regard to gluHack. (2) Only when the proportion of true literals in a SAT formula in terms of the predefined solution is 0.5, SAT formulas are hard-to-solve with regard to gluHack; when this proportion is below 0.5, SAT formulas are hard-to-solve with regard to CPSparrow. (3) When the ratio of the number of clauses to that of variables is around 4.25, the SAT formulas are hard-to-solve with regard to both gluHack and CPSparrow.

Community-based 3-SAT formulas with a predefined solution

Community-based 3-SAT formulas with a predefined solution

Satune: synthesizing efficient SAT encoders

Modern SAT solvers are extremely efficient at solving boolean satisfiability problems, enabling a wide spectrum of techniques for checking, verifying, and validating real-world programs. What remains challenging, though, is how to encode a domain problem (e.g., model checking) into a SAT formula because the same problem can have multiple distinct encodings, which can yield performance results that are orders-of-magnitude apart, regardless of the underlying solvers used. We develop Satune, a tool that can automatically synthesize SAT encoders for different problem domains. Satune employs a DSL that allows developers to express domain problems at a high level and a search algorithm that can effectively find efficient solutions. The search process is guided by observations made over example encodings and their performance for the domain and hence Satune can quickly synthesize a high-performance encoder by incorporating patterns from examples that yield good performance. A thorough evaluation with JMCR, SyPet, Dirk, Hexiom, Sudoku, and KillerSudoku demonstrates that Satune can easily synthesize high-performance encoders for different domains including model checking, synthesis, and games. These encoders generate constraint problems that are often several orders of magnitude faster to solve than the original encodings used by the tools.

Kolmogorov complexity based upper bounds for the unsatisfiability threshold of random k-SAT

We consider Boolean formulas with k literals per clause, or k-SAT formulas. According to the fixed-length random k-SAT model, we can construct a random k-SAT formula by selecting, uniforly at random, m = rn clauses out of the possible ones, for a fixed real number r > 0. As it has been experimentally observed that, as the ratio r of the number m of clauses to the number n of variables is increased from below the approximate value 4.2 to above, then the probability that a random k-SAT formula is satisfiable falls from, asymptotically, 1 to, asymptotically, 0, as n tends to infinity. It is conjectured that a constant value rk exists for all k, called the unsatisfiability threshold for random k-SAT formulas, such that arbitrarily close to it the probabiliy that a random 3-SAT formula is satisfiable falls, abruptly, from 1 to 0. In this paper we deploy techniques from Kolmogorov complexity in order to derive an upper bound to the unsagisfiabliity threshold value generalilzing a similar result from random 3-SAT.

An MDD-based SAT encoding for pseudo-Boolean constraints with at-most-one relations

Pseudo-Boolean (PB) constraints are ubiquitous in Constraint Satisfaction Problems. Encoding such constraints to SAT has proved to be an efficient solving approach. A commonly used technique for encoding PB constraints consists in representing the constraint as a Binary Decision Diagram (BDD), and then encoding this BDD to SAT. A key point in this technique is to obtain small BDD representations to generate small SAT formulas. In many problems, some subsets of the Boolean variables occurring in a PB constraint may also have other constraints imposed on them. In this work we introduce a way to take advantage of those constraints in order to obtain smaller SAT encodings. The main idea is that decision diagrams may be smaller if they avoid to represent truth assignments that are already forbidden by some other constraints. In particular, we present encodings for monotonic decreasing PB constraints, in conjunction with other constraints such as at-most-one, exactly-one and implication chains on subsets of their variables. We provide empirical evidence of the usefulness of this technique to reduce the size of the encodings as well as the solving time.

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.

Beyond NP: Quantifying over Answer Sets

AbstractAnswer Set Programming (ASP) is a logic programming paradigm featuring a purely declarative language with comparatively high modeling capabilities. Indeed, ASP can model problems in NP in a compact and elegant way. However, modeling problems beyond NP with ASP is known to be complicated, on the one hand, and limited to problems in $\[\Sigma _2^P\]$ on the other. Inspired by the way Quantified Boolean Formulas extend SAT formulas to model problems beyond NP, we propose an extension of ASP that introduces quantifiers over stable models of programs. We name the new language ASP with Quantifiers (ASP(Q)). In the paper we identify computational properties of ASP(Q); we highlight its modeling capabilities by reporting natural encodings of several complex problems with applications in artificial intelligence and number theory; and we compare ASP(Q) with related languages. Arguably, ASP(Q) allows one to model problems in the Polynomial Hierarchy in a direct way, providing an elegant expansion of ASP beyond the class NP.

Anti-SAT: Mitigating SAT Attack on Logic Locking

Logic locking is a technique that is proposed to protect outsourced IC designs from piracy and counterfeiting by untrusted foundries. A locked IC preserves the correct functionality only when a correct key is provided. Recently, the security of logic locking is threatened by a new attack called SAT attack, which can decipher the correct key of most logic locking techniques within a few hours even for a reasonably large key-size. This attack iteratively solves SAT formulas which progressively eliminate the incorrect keys till the circuit is unlocked. In this paper, we present a circuit block (referred to as Anti-SAT block) to enhance the security of existing logic locking techniques against the SAT attack. We show using a mathematical proof that the number of SAT attack iterations to reveal the correct key in a circuit comprising an Anti-SAT block is an exponential function of the key-size thereby making the SAT attack computationally infeasible. Besides, we address the vulnerability of the Anti-SAT block to various removal attacks and investigate obfuscation techniques to prevent these removal attacks. More importantly, we provide a proof showing that these obfuscation techniques for making Anti-SAT un-removable would not weaken the Anti-SAT block’s resistance to SAT attack. Through our experiments, we illustrate the effectiveness of our approach to securing modern chips fabricated in untrusted foundries.

Parallelizing SAT-based de-camouflaging attacks by circuit partitioning and conflict avoiding

As one of the most effective proactive countermeasures against reverse engineering, circuit camouflaging has emerged to be a hot research topic and it is becoming a mature technology with the development of various de-camouflaging attacks. Among them, the SAT-based method is the most powerful one to defeat circuit camouflaging. However, SAT-based attacks have scalability problem due to the complexity of the underlying SAT solvers, and straightforward approach to parallelize SAT-based attacks will fail. In this article, we propose a novel parallelization framework for SAT-based attacks, which consists of a two-level partition method (independent module partitioning and k-medoids clustering), together with a novel conflict avoidance strategy. Specifically, we first break down the camouflaged netlist into multiple independent modules that can be solved in parallel. However, any good circuit camouflaging approach should produce one or multiple large modules. To solve this problem, we further apply the k-medoids algorithm to partition the large modules into multiple “high cohesion” and “low coupling” clusters. Utilizing the relative independence of these clusters, we propose a two-stage attack method to avoid conflicts among clusters. Experimental results on OpenSparc T1 microprocessor controller demonstrate that our approach can on average reduce the scales of the SAT formulas by more than 50%, reduce the attack iteration number by 45%, and achieves on average 3.6× and maximum 10× speed up over the best-known SAT-based de-camouflaging tool.

The Number of Satisfying Assignments of Random Regulark-SAT Formulas

Let Φ be a randomk-SAT formula in which every variable occurs preciselydtimes positively anddtimes negatively. Assuming thatkis sufficiently large and thatdis slightly below the critical degree where the formula becomes unsatisfiable with high probability, we determine the limiting distribution of the number of satisfying assignments.

Selecting and covering colored points

Let U be a set of colored objects, where the color of an object is an integer between 1 and n. Let P={C1,…,Cn} be the partition of U into color classes, i.e., the set Ci consists of all objects in U colored i. We study two families of problems on color classes, namely, rainbow problems and covering problems where the covering objects have a certain property. The goal in a rainbow problem is to select an optimal rainbow, subject to the conditions of the problem, where a rainbow is a subset of U that contains at most one object from each color class. For example, in the problem known as rainbow minmax gap the color classes are pairs of points on the x-axis, and one needs to select exactly one point from each color class (i.e., a complete rainbow), such that the maximum distance between a pair of consecutive selected points is minimized. This problem was studied by Consuegra and Narasimhan, who left the question of its complexity unresolved. We prove that it is NP-hard. For our proof we obtain the following auxiliary result. A 3-SAT formula is an LSAT formula if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. We prove that the problem of deciding whether an LSAT formula is satisfiable or not is NP-complete. Using the LSAT result, we also prove that rainbow piercing and rainbow covering are NP-complete.In the covering problems that we consider, U is a set of points and only conflict-free objects are allowed, where an object is conflict free if it covers at most one point from each color class. The goal is to find a minimum-cardinality set of conflict-free objects that cover at least (or exactly) one point from each color class. We prove that problems in this family are NP-hard (or NP-complete) and present several constant-factor approximation algorithms.

TotSAT - Totally-Ordered Hierarchical Planning Through SAT

In this paper, we propose a novel SAT-based planning approach for hierarchical planning by introducing the SAT-based planner totSAT for the class of totally-ordered HTN planning problems. We use the same general approach as SAT planning for classical planning does: bound the problem, translate the problem into a formula, and if the formula is not satisfiable, increase the bound. In HTN planning, a suitable bound is the maximum depth of decomposition. We show how totally-ordered HTN planning problems can be translated into a SAT formula, given this bound. Furthermore, we have conducted an extensive empirical evaluation to compare our new planner against state-of-the-art HTN planners. It shows that our technique outperforms any of these systems.