Solving hard industrial combinatorial problems with SAT

The Boolean Satisfiability Problem (SAT) is well known in computational complexity, representing the first decision problem to be proved NP-complete. SAT is also often the subject of work in proof complexity. Besides its theoretical interest, SAT finds a wide range of practical applications. Moreover, SAT solvers have been the subject of remarkable efficiency improvements since the mid-1990s, motivating their widespread use in many practical applications including Bounded and Unbounded Model Checking. The success of SAT solvers has also motivated the development of algorithms for natural extensions of SAT, including Quantified Boolean Formulas (QBF), Pseudo-Boolean constraints (PB), Maximum Satisfiability (MaxSAT) and Satisfiability Modulo Theories (SMT) which also see application in the model-checking context. This chapter first covers the organization of modern conflict-driven clause learning (CDCL) SAT solvers, which are used in the vast majority of practical applications of SAT. It then reviews the techniques shown to be effective in modern SAT solvers.

Software product lines are able to describe multiple products sharing a common base of features and are commonly described as feature models. For complex software product lines, automatic analyses are required to ensure validity and to improve the interactive configuration process. Modern SAT solvers are vital components for the validation process of feature models. The increasing variability of software product lines implies the need to use more expressive solvers like SMT solvers. To assist the development of feature modeling tools, we compare SAT and SMT solvers for the automated analysis of feature models. During this thesis, we create an abstract data type to formally define analyses for feature model defects and their explanations. The result shows that SAT solvers are more efficient at detecting the defects, while SMT solvers can find explanations for them multiple times faster. Feature models can be further expanded by attaching attributes to features. Such attributes may contain a numerical value. Additionally, one attribute can be defined for multiple features. In this thesis, we aim to support the interactive configuration process, by providing the range of the sum of values for an attribute. Such ranges depend on the remaining choices in a configuration of the product line. We provide an exact computation using SMT and an approximation using a heuristic. The evaluation results show that an SMT solver is not suitable for supporting interactive configuration. However, the approximated ranges of the provided heuristic were very close to the exact ones.

This dissertation is to illustrate the current state of the art of DNA computing achievements, especially of new approaches to solve theoretical 3-SAT problems and the hitting-set problem. Beginning with Adleman’s breakthrough which is an molecular algorithm for the solution of a NP-complete, combinatorial problem, the directed Hamiltonian path problem (HPP). Today, many researchers all over the world concentrate on proposing new methods to solve engineering or application problems with a DNA computing approach. Satisfiability problem is given a Boolean formula, and decide if a satisfying truth assignment exists. ( ) ( ) … ( ) ( ) is an example of Boolean formula. k-SAT problem means that each clause has exactly k literals. Not-All-Equal (NAE) 3-SAT problem and One-In-Three (1IN3) 3-SAT problem are both NP-complete problems. In this dissertation, we present molecular solutions to find all true assignments (3-SAT problem) and furthermore find Not-All-Equal (NAE) solutions and One-In-Three (1IN3) solutions in DNA-based Supercomputing. Hitting-set problem assume that there exists a collection C of subsets of a finite set S, and a positive integer K  |S|, and we need to know if there is a subset  S with | |  K such that contains at least one element of each subset in C. In other words, is the subset that intersects every subset in C and is called the hitting-set. In this dissertation, a DNA-based algorithm is proposed to solve the small hitting-set problem. A small hitting-set is a hitting-set with the smallest K value, i.e., the hitting-set with the smallest number of elements. Furthermore, an algorithm is introduced to find the number of ones from 2n combinations and minimum numbers of ones represents the small hitting-set since K is expected to be as small as possible. The complexity of all the presented DNA-based algorithms is also discussed. We describe time complexity and volume complexity of three Algorithms, numbers of test tube used and the longest library strand in solution space of all three Algorithms. Finally, the simulated experiment is applied to verify correctness of the proposed DNA-based algorithm for solving the One-In-Three (1IN3) 3-SAT problem, and simulation of Not-All-Equal (NAE) 3-SAT problem is similar. Also, another simulated experiment is applied to our proposed DNA-based algorithm 6-2, in order to solve the well-known hitting-set problem. This research has been motivated by the benefit and the application of DNA computing and gives new methods to solve two 3-SAT problems and the hitting-set problem which are NP-complete.

SAT solving has gained tremendous interest. On the practical side there have been considerable performance improvements, due to new highly efficient algorithms, new heuristics, and optimized data structures. There are new applications and reformulations of important classical problems, mainly in the context of formal methods, where SAT solving is also applied successfully in an industrial setting. These applications range from equivalence checking, configuration, over model checking to test case generation. SAT is becoming one of the most important core technology in all these areas. Many applications actually use Satisfiability Modulo Theory (SMT), which can be seen as an extension of SAT solving. SMT has it roots in automated theorem proving. But it heavily relies on SAT technology. We discuss some key technologies in practical SAT solving, e.g. how to write a fast solver, some aspects in lifting propositional SAT technology to richer domains, how competitions can help to improve the state-of-the-art and finally touch on applications in model checking, hardware and software verification.

Solvers for SAT Modulo Theories (SMT) can nowadays handle large industrial (e.g., formal hardware and software verification) problems over theories such as the integers, arrays, or equality. Here we show that SMT approaches can also efficiently solve problems that, at first sight, do not have a typical SMT flavor. In particular, here we deal with SAT and SMT problems where models M are sought such that a given cost function f(M) is minimized. For this purpose, we introduce a variant of SMT where the theory T becomes progressively stronger, and prove it correct using the Abstract DPLL Modulo Theories framework. We discuss two different examples of applications of this SMT variant: weighted Max-SAT and weighted Max-SMT. We show how, with relatively little effort, one can obtain a competitive system that, in the case of weighted Max-SMT in the theory of Difference Logic, can even handle well-known hard radio frequency assignment problems without any tailored heuristics. These results seem to indicate that Max-SAT/SMT techniques can already be used for realistic applications.

This paper describes a method for combining “off-the-shelf” SAT and constraint solvers for building an efficient Satisfiability Modulo Theories (SMT) solver for a wide range of theories. Our method follows the abstraction/refinement approach to simplify the implementation of custom SMT solvers. The expected performance penalty by not using an interweaved combination of SAT and theory solvers is reduced by generalising a Boolean solution of an SMT problem first via assigning don’t care to as many variables as possible. We then use the generalised solution to determine a thereby smaller constraint set to be handed over to the constraint solver for a background theory. We show that for many benchmarks and real-world problems, this optimisation results in considerably smaller and less complex constraint problems. The presented approach is particularly useful for assembling a practically viable SMT solver quickly, when neither a suitable SMT solver nor a corresponding incremental theory solver is available. We have implemented our approach in the ABsolver framework and applied the resulting solver successfully to an industrial case-study: the verification problems arising in verifying an electronic car steering control system impose non-linear arithmetic constraints, which do not fall into the domain of any other available solver.

In this paper we present a new progressive cooperating simplifier for deciding the satisfiability of a quantifier-free formula in the first-order theory of integers involving combinations of sublogics, referred to as Satisfiability Modulo Theories (SMT). Our approach, given an SMT problem, replaces each non-propositional theory atom with a Boolean indicator variable yielding a purely propositional formula to be decided by a SAT solver. Starting with the most abstract representation (the Boolean formula), the solver gradually integrates more complex theory solvers into the working decision procedure. Additionally, we propose a method to simplify “expensive” atoms into suitable conjunctions of “cheaper” theory atoms when conflicts occur. This process considerably increases the efficiency of the overall procedure by reducing the number of calls to the slower theory solvers. This is made possible by adopting our novel inter-logic implication framework, as proposed in this paper. We have implemented these methods in our Ario SMT solver by combining three different theory solvers within a DPLL-style SAT solver: a Unit-Two-Variable-Per-Inequality (UTVPI) solver, an integer linear programming (ILP) solver, and a solver for systems of equalities with uninterpreted functions. The efficiencies of our proposed algorithms are demonstrated and exhaustively investigated on a wide range of benchmarks in hardware and software verification domain. Empirical results are also presented showing the advantages/limitations of our methods over other modern techniques for solving these SMT problems.

The dramatic improvements in Boolean satisfiability (SAT) solving since the turn of the millennium have made it possible to leverage state-of-the-art conflict-driven clause learning (CDCL) solvers for many combinatorial problems in academia and industry, and the use of proof logging has played a crucial role in increasing the confidence that the results these solvers produce are correct. However, the fact that SAT proof logging is performed in conjunctive normal form (CNF) clausal format means that it has not been possible to extend guarantees of correctness to the use of SAT solvers for more expressive combinatorial paradigms, where the first step is an unverified translation of the input to CNF. In this work, we show how cutting-planes-based reasoning can provide proof logging for solvers that translate pseudo-Boolean (a.k.a. 0-1 integer linear) decision problems to CNF and then run CDCL. To support a wide range of encodings, we provide a uniform and easily extensible framework for proof logging of CNF translations. We are hopeful that this is just a first step towards providing a unified proof logging approach that will also extend to maximum satisfiability (MaxSAT) solving and pseudo-Boolean optimization in general.

Finding the Right Hybrid Algorithm - A Combinatorial Meta-Problem

We present our extension of ACL2 with Satisfiability Modulo Theories (SMT) solvers using ACL2's trusted clause processor mechanism. We are particularly interested in the verification of physical systems including Analog and Mixed-Signal (AMS) designs. ACL2 offers strong induction abilities for reasoning about sequences and SMT complements deduction methods like ACL2 with fast nonlinear arithmetic solving procedures. While SAT solvers have been integrated into ACL2 in previous work, SMT methods raise new issues because of their support for a broader range of domains including real numbers and uninterpreted functions. This paper presents Smtlink, our clause processor for integrating SMT solvers into ACL2. We describe key design and implementation issues and describe our experience with its use.

This paper addresses the problem of counting models in integer linear programming (ILP) using Boolean Satisfiability (SAT) techniques, and proposes two approaches to solve this problem. The first approach consists of encoding ILP instances into pseudo-Boolean (PB) instances. Moreover, the paper introduces a model counter for PB constraints, which can be used for counting models in PB as well as in ILP. A second alternative approach consists of encoding instances of ILP into instances of SAT. A two-step procedure is proposed, consisting of first mapping the ILP instance into PB constraints and then encoding the PB constraints into SAT. One key observation is that not all existing PB to SAT encodings can be used for counting models. The paper provides conditions for PB to SAT encodings that can be safely used for model counting, and proves that some of the existing encodings are safe for model counting while others are not. Finally, the paper provides experimental results, comparing the PB and SAT approaches, as well as existing alternative solutions.

Answer Set Programming (ASP) is a constraint-based programming paradigm proposed recently by AI researchers. In this paradigm one solves a problem by encoding constraints as logic program rules in such a way that the solutions to the original problem correspond to the so-called answer sets of the logic program. This new programming paradigm has found applications in problems ranging from combinatorial search problems, such as the graph coloring and Hamiltonian circuit problems, to product configuration and data encryption problems. In this dissertation, we make the following two contributions to the new programming paradigm ASP. First, we propose and implement a new ASP system using traditional SAT (propositional satisfiability) solvers. This is done by a translation from logic programs to sets of clauses in propositional logic. Our experimental results show that for many benchmark domains, our system performs better than existing specialized ASP systems. Secondly, we study the phase transition phenomenon in randomly generated logic programs, and identify a region in a class of randomly generated logic programs where the logic programs are hard for all existing ASP systems.

Many combinatorial problems in various fields can be translated to Maximum Satisfiability (MaxSAT) problems. Although the general problem is $\mathcal {N}\mathcal {P}$ -hard, more and more practical problems may be solved due to the significant effort which has been devoted to the development of efficient solvers. The art of constraints encoding is as important as the art of devising algorithms for MaxSAT. In this paper, we present several encoding methods of pseudo-Boolean constraints into Boolean satisfiability problems in Conjunctive Normal Form (CNF) formula, which are based on the idea of modular arithmetic and only generate auxiliary variables for each unique combination of weights. These techniques are efficient in encoding and solving MaxSAT problems. In particular, our solvers won the partial MaxSAT industrial category from 2010 through 2012 and ranked second in the 2017 main weighted track of the MaxSAT evaluation. We prove the correctness and the pseudo-polynomial space complexity of our encodings and also give a heuristics of the base selection for modular arithmetic. Our experimental results show that our encoding compactly encodes the constraints, and the obtained clauses are efficiently handled by a state-of-the-art SAT solver.

SMT solving is often a major source of cost in a broad range of techniques such as symbolic program analysis. Thus, speeding up SMT solving is still an urgent requirement. A dominant approach, which is known as eager SMT solving, is to reduce a first-order formula to a pure Boolean formula, which is handed to an expensive SAT solver to determine the satisfiability. We observe that the SAT solver can utilize the knowledge in the first-order formula to boost its solving efficiency. Unfortunately, despite much progress, it is still not clear how to make use of the knowledge in an eager SMT solver. This paper addresses the problem by introducing a new and fast method, which utilizes the interval and data-dependence information learned from the first-order formulas. We have implemented the approach as a tool called Trident and evaluated it on three symbolic analyzers (Angr, Qsym, and Pinpoint). The experimental results, based on seven million SMT solving instances generated for thirty real-world software systems, show that Trident significantly reduces the total solving time from 2.9X to 7.9X over three state-of-the-art SMT solvers (Z3, CVC4, and Boolector), without sacrificing the number of solved instances. We also demonstrate that Trident achieves end-to-end speedups for three program analysis clients by 1.9X, 1.6X, and 2.4X, respectively.

We describe an attempt to unify search-based and compilation-based approaches to multi-agent path finding (MAPF) through satisfiability modulo theories (SMT). The task in MAPF is to navigate agents in an undirected graph to given goal vertices so that they do not collide. We rephrase Conflict-Based Search (CBS), one of the state-of-the-art algorithms for optimal MAPF solving, in the terms of SMT. This idea combines SAT-based solving known from MDD-SAT, a SAT-based optimal MAPF solver, at the low level with conflict elimination of CBS at the high level. Where the standard CBS branches the search after a conflict occurs, we refine the propositional model with a disjunctive constraint instead. Our novel algorithm called SMT-CBS hence does not branch at the high-level but incrementally extends the propositional model that is consulted with the SAT solver at each iteration. We experimentally compare SMT-CBS with CBS and MDD-SAT.