MaxSAT Resolution and Subcube Sums

We study the MaxRes rule in the context of certifying unsatisfiability. We show that it can be exponentially more powerful than tree-like resolution, and when augmented with weakening (the system MaxResW), p-simulates tree-like resolution. In devising a lower bound technique specific to MaxRes (and not merely inheriting lower bounds from Res), we define a new proof system called the SubCubeSums proof system. This system, which p-simulates MaxResW, can be viewed as a special case of the semialgebraic Sherali-Adams proof system. In expressivity, it is the integral restriction of conical juntas studied in the contexts of communication complexity and extension complexity. We show that it is not simulated by Res. Using a proof technique qualitatively different from the lower bounds that MaxResW inherits from Res, we show that Tseitin contradictions on expander graphs are hard to refute in SubCubeSums. We also establish a lower bound technique via lifting: for formulas requiring large degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums

Polynomial Calculus for MaxSAT

MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem. Weighted Polynomial Calculus is a natural generalization of MaxSAT-Resolution and weighted Resolution that manipulates polynomials with coefficients in a finite field and either weights in ? or ?. We show the soundness and completeness of these systems via an algorithmic procedure. Weighted Polynomial Calculus, with weights in ? and coefficients in ??, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in ?, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable

Proof complexity for the maximum satisfiability problem and its use in SAT refutations

MaxSAT, the optimization version of the well-known SAT problem, has attracted a lot of research interest in the past decade. Motivated by the many important applications and inspired by the success of modern SAT solvers, researchers have developed many MaxSAT solvers. Since most research is algorithmic, its significance is mostly evaluated empirically. In this paper, we want to address MaxSAT from the more formal point of view of proof complexity. With that aim, we start providing basic definitions and proving some basic results. Then we analyse the effect of adding split and virtual, two original inference rules, to MaxSAT resolution. We show that each addition makes the resulting proof system stronger, even when virtual is restricted to empty clauses (0-virtual). We also analyse the power of our proof systems in the particular case of SAT refutations. We show that our strongest system, ResSV, is equivalent to circular and dual rail with split. We also analyse empirically some known gadget-based reformulations. Our results seem to indicate that the advantage of these three seemingly different systems over general resolution comes mainly from their ability of augmenting the original formula with hypothetical inconsistencies, as captured in a very simple way by the virtual rule.Under project RTI2018-094403-B-C33 funded by MCIN/AEI/ 10.13039/501100011033 and FEDER ”Una manera de hacer Europa"Peer ReviewedPostprint (author's final draft

On Continuous Local BDD-Based Search for Hybrid SAT Solving

We explore the potential of continuous local search (CLS) in SAT solving by proposing a novel approach for finding a solution of a hybrid system of Boolean constraints. The algorithm is based on CLS combined with belief propagation on binary decision diagrams (BDDs). Our framework accepts all Boolean constraints that admit compact BDDs, including symmetric Boolean constraints and small-coefficient pseudo-Boolean constraints as interesting families. We propose a novel algorithm for efficiently computing the gradient needed by CLS. We study the capabilities and limitations of our versatile CLS solver, GradSAT, by applying it on many benchmark instances. The experimental results indicate that GradSAT can be a useful addition to the portfolio of existing SAT and MaxSAT solvers for solving Boolean satisfiability and optimization problems.Comment: AAAI 2

An Analysis of Core-Guided Maximum Satisfiability Solvers Using Linear Programming

Many current complete MaxSAT algorithms fall into two categories: core-guided or implicit hitting set. The two kinds of algorithms seem to have complementary strengths in practice, so that each kind of solver is better able to handle different families of instances. This suggests that a hybrid might match and outperform either, but the techniques used seem incompatible. In this paper, we focus on PMRES and OLL, two core-guided algorithms based on max resolution and soft cardinality constraints, respectively. We show that these algorithms implicitly discover cores of the original formula, as has been previously shown for PM1. Moreover, we show that in some cases, including unweighted instances, they compute the optimum hitting set of these cores at each iteration. We also give compact integer linear programs for each which encode this hitting set problem. Importantly, their continuous relaxation has an optimum that matches the bound computed by the respective algorithms. This goes some way towards resolving the incompatibility of implicit hitting set and core-guided algorithms, since solvers based on the implicit hitting set algorithm typically solve the problem by encoding it as a linear program

Enforcement in Abstract Argumentation via Boolean Optimization

Computational aspects of argumentation are a central research topic of modern artificial intelligence. A core formal model for argumentation, where the inner structure of arguments is abstracted away, was provided by Dung in the form of abstract argumentation frameworks (AFs). AFs are syntactically directed graphs with the nodes representing arguments and edges representing attacks between them. Given the AF, sets of jointly acceptable arguments or extensions are defined via different semantics. The computational complexity and algorithmic solutions to so-called static problems, such as the enumeration of extensions, is a well-studied topic. Since argumentation is a dynamic process, understanding the dynamic aspects of AFs is also important. However, computational aspects of dynamic problems have not been studied thoroughly. This work concentrates on different forms of enforcement, which is a core dynamic problem in the area of abstract argumentation. In this case, given an AF, one wants to modify it by adding and removing attacks in a way that a given set of arguments becomes an extension (extension enforcement) or that given arguments are credulously or skeptically accepted (status enforcement). In this thesis, the enforcement problem is viewed as a constrained optimization task where the change to the attack structure is minimized. The computational complexity of the extension and status enforcement problems is analyzed, showing that they are in the general case NP-hard optimization problems. Motivated by this, algorithms are presented based on the Boolean optimization paradigm of maximum satisfiability (MaxSAT) for the NP-complete variants, and counterexample-guided abstraction refinement (CEGAR) procedures, where an interplay between MaxSAT and Boolean satisfiability (SAT) solvers is utilized, for problems beyond NP. The algorithms are implemented in the open source software system Pakota, which is empirically evaluated on randomly generated enforcement instances

Pseudo-Boolean Constraint Encodings for Conjunctive Normal Form and their Applications

In contrast to a single clause a pseudo-Boolean (PB) constraint is much more expressive and hence it is easier to define problems with the help of PB constraints. But while PB constraints provide us with a high-level problem description, it has been shown that solving PB constraints can be done faster with the help of a SAT solver. To apply such a solver to a PB constraint we have to encode it with clauses into conjunctive normal form (CNF). While we can find a basic encoding into CNF which is equivalent to a given PB constraint, the solving time of a SAT solver significantly depends on different properties of an encoding, e.g. the number of clauses or if generalized arc consistency (GAC) is maintained during the search for a solution. There are various PB encodings that try to optimize or balance these properties. This thesis is about such encodings. For a better understanding of the research field an overview about the state-of-the art encodings is given. The focus of the overview is a simple but complete description of each encoding, such that any reader could use, implement and extent them in his own work. In addition two novel encodings are presented: The Sequential Weight Counter (SWC) encoding and the Binary Merger Encoding. While the SWC encoding provides a very simple structure – it is listed in four lines – empirical evaluation showed its practical usefulness in various applications. The Binary Merger encoding reduces the number of clauses a PB encoding needs while having the important GAC property. To the best of our knowledge currently no other encoding has a lower upper bound for the number of clauses produced by a PB encoding with this property. This is an important improvement of the state-of-the art, since both GAC and a low number of clauses are vital for an improved solving time of the SAT solver. The thesis also contributes to the development of new applications for PB constraint encodings. The programming library PBLib provides researchers with an open source implementation of almost all PB encodings – including the encodings for the special cases at-most-one and cardinality constraints. The PBLib is also the foundation of the presented weighted MaxSAT solver optimax, the PBO solver pbsolver and the WBO, PBO and weighted MaxSAT solver npSolver