Writing Declarative Specifications for Clauses

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.

Optimized solvers for the Boolean satisfiability (SAT) problem have many applications in areas such as hardware and software verification, FPGA routing, planning, and so forth. Further uses are complicated by the need to express "counting constraints" in conjunctive normal form (CNF). Expressing such constraints by pure CNF leads to more complex SAT instances. Alternatively, those constraints can be handled by integer linear programming (ILP), but generic ILP solvers may ignore the Boolean nature of 0-1 variables. Therefore, specialized 0-1 ILP solvers extend SAT solvers to handle these so-called "pseudo-Boolean" (PB) constraints. This work provides an update on the on-going competition between generic ILP techniques and specialized 0-1 ILP techniques. To make a fair comparison, we generalize recent ideas for fast SAT-solving to more general 0-1 ILP problems that may include counting constraints and optimization. This generalization is embodied in our PB constraint solver and optimizer PBS, which is compared with state-of-the-art CNF and generic ILP solvers. Another aspect of our comparison is the evaluation on 0-1 ILP benchmarks that originate in electronic design automation (EDA) but that cannot be directly solved by an SAT solver. Specifically, we solve instances of the max-SAT and max-ONEs optimization problems, which seek to maximize the number of satisfied clauses and the "true" values over all satisfying assignments, respectively. Those problems have straightforward applications to SAT-based routing and are additionally important due to reductions from max-cut, max-clique, and min vertex cover. Our experimental results show that specialized 0-1 techniques implemented in PBS tend to outperform generic ILP techniques on Boolean optimization problems, as well as on general EDA SAT problems.

Chapter 2 Satisfiability Solvers

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.

Boolean Satisfiability (SAT), the first problem proven to be NP-complete, is intractable on digital computers based on the von Neumann architecture. An efficient SAT solver can benefit many applications such as artificial intelligence, circuit design, and functional verification. Recently, a SAT solver approach based on a deterministic, continuous-time dynamical system (CTDS) was introduced [1]. This approach shows polynomial analog time-complexity on even the hardest k-SAT (k ≥ 3) problem instances, but at an energy cost dependent on exponentially growing auxiliary variables. This paper reports a novel analog hardware SAT solver, AC-SAT, implementing the CTDS via incorporating novel, analog circuit design ideas. AC-SAT is intended to be used as a co-processor and is programmable for handling different problem specifications. Furthermore, with its modular design, AC-SAT can be readily extended to solve larger size problems. SPICE simulation results show that AC-SAT can indeed solve the SAT problems, and it has speedup factors of ~10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> on even the hardest 3-SAT problems, when compared with a state-of-the-art SAT solver on digital computers.

Boolean satisfiability (SAT) solvers are at the core of efficient approaches for solving a vast multitude of practical problems. Moreover, albeit targeting an NP-complete problem, SAT solvers are increasingly used for tackling problems beyond NP. Despite the success of SAT in practice, modeling with SAT and more importantly implementing SAT-based problem solving solutions is often a difficult and error-prone task. This paper proposes the PySAT toolkit, which enables fast Python-based prototyping using SAT oracles and SAT-related technology. PySAT provides a simple API for working with a few state-of-the-art SAT oracles and also integrates a number of cardinality constraint encodings, all aiming at simplifying the prototyping process. Experimental results presented in the paper show that PySAT-based implementations can be as efficient as those written in a low-level language.

Multiway decision graph (MDG) is a canonical representation of a subset of many-sorted first-order logic. It generalises the logic of equality with abstract types and uninterpreted function symbols. The area of satisfiability (SAT) has been the subject of intensive research in recent years, with significant theoretical and practical contributions. In this paper, we propose a new design verification tool integrating MDG and SAT, to check the safety of a design by invariant checking. Using MDG to encode the set of states provides a powerful mean of abstraction. We use a SAT solver to search for paths of reachable states violating the property under certain encoding constraints. In addition, we introduce an automated conversion-verification methodology to convert a directed formula (DF) into a conjunctive normal form (CNF) formula that can be fed to a SAT solver. The formal verification of this conversion is conducted within the HOL theorem prover. Finally, we present experimental results and a case study to show the correctness and the efficiency of our proposed methodology.

Query-answering (QA) problems have attracted wider interest recently, owing partly to emerging applications involving integration between rules and ontologies in the Semantic Web's layered architecture. Success of using satisfiability (SAT) solvers as low-level solvers for dealing with several kinds of problems motivates us to apply SAT solvers to a class of QA problems. To find all elements of an answer set, a variant of a usual SAT solver, called an all-solution satisfiability (All-SAT) solver, is used for determining all models of a given input set of propositional clauses. Using an All-SAT solver, we propose a procedure for solving QA problems. The procedure generates input for an All-SAT solver from a high-level description of a given QA problem by problem transformation in a clause space that includes set-bounded variables. Based on the equivalent transformation principle, the correctness of the proposed procedure is shown.

Recently, Boolean Satisfiability (SAT) solving has been proposed to tackle the increasing complexity in high-level system design. Working well for system specifications with a limited amount of routing options, they tend to fail for densely connected computing platforms. This paper proposes an automated system design approach employing Answer Set Programming (ASP). ASP provides a stringent semantics, allowing for an efficient representation of routing options. An automotive case-study illustrates that the proposed ASP-based system design approach is competitive for sparsely connected computing platforms, while it outperforms SAT-based approaches for dense Networks-on-Chip by an order of magnitude.

In recent years backtrack search algorithms for propositional satisfiability (SAT) have been the subject of dramatic improvements. These improvements allowed SAT solvers to successfully solve instances with thousands or tens of thousands of variables. However, many new challenging problem instances are still too hard for current SAT solvers. As a result, further improvements to SAT technology are expected to have key consequences in solving hard real-world instances. This paper introduces a new idea: choosing the backtrack variable using a heuristic approach with the goal of diversifying the regions of the space that are explored during the search. The proposed heuristics are inspired by the heuristics proposed in recent years for the decision branching step of SAT solvers, namely, VSIDS and its improvements. Completeness conditions are established, which guarantee completeness for the new algorithm, as well as for any other incomplete backtracking algorithm. Experimental results on hundreds of instances derived from real-world problems show that the new technique is able to speed SAT solvers, while aborting fewer instances. These results clearly motivate the integration of heuristic backtracking in SAT solvers.

Boolean Satisfiability (SAT) solvers are popular engines used throughout the verification world. Bounded sequential problems such as bounded model checking and bounded sequential equivalence checking rely on fast and robust SAT solvers. In this work, we introduce a technique that improves the performance of the underlying SAT solver for bounded sequential problems by taking advantage of a design's don't care states. We develop cost effective methods of filtering, replicating and applying the don't care states to the original problem thus reducing the search space. Experiments demonstrate the effectiveness of the proposed method on ISCAS'89 benchmarks.

Maximum satisfiability (MaxSAT), employing propositional logic as the declarative language of choice, has turned into a viable approach to solving NP-hard optimization problems arising from artificial intelligence and other real-world settings. A key contributing factor to the success of MaxSAT is the rise of increasingly effective exact solvers that are based on iterative calls to a Boolean satisfiability (SAT) solver. The three types of SAT-based MaxSAT solving approaches, each with its distinguishing features, implemented in current state-of-the-art MaxSAT solvers are the core-guided, the implicit hitting set (IHS), and the objective-bounding approaches. The objective-bounding approach is based on directly searching over the objective function range by iteratively querying a SAT solver if the MaxSAT instance at hand has a solution under different bounds on the objective. In contrast, both core-guided and IHS are so-called unsatisfiability-based approaches that employ a SAT solver as an unsatisfiable core extractor to determine sources of inconsistencies, but critically differ in how the found unsatisfiable cores are made use of towards finding a provably optimal solution. Furthermore, a variety of different algorithmic variants of the core-guided approach in particular have been proposed and implemented in solvers. It is well-acknowledged that each of the three approaches has its advantages and disadvantages, which is also witnessed by instance and problem-domain specific runtime performance differences (and at times similarities) of MaxSAT solvers implementing variants of the approaches. However, the questions of to what extent the approaches are fundamentally different and how the benefits of the individual methods could be combined in a single algorithmic approach are currently not fully understood. In this work, we approach these questions by developing UniMaxSAT, a general unifying algorithmic framework. Based on the recent notion of abstract cores, UniMaxSAT captures in general core-guided, IHS and objective-bounding computations. The framework offers a unified way of establishing quite generally the correctness of the current approaches. We illustrate this by formally showing that UniMaxSAT can simulate the computations of various algorithmic instantiations of the three types of MaxSAT solving approaches. Furthermore, UniMaxSAT can be instantiated in novel ways giving rise to new algorithmic variants of the approaches. We illustrate this aspect by developing a prototype implementation of an algorithmic variant for MaxSAT based on the framework.

The problem of finding the minimum three-dimensional Kochen–Specker (KS) vector system, an important problem in quantum foundations, has remained open for over 55 years. We present a new method to address this problem based on a combination of a Boolean satisfiability (SAT) solver and a computer algebra system (CAS). Our approach improved the lower bound on the size of a KS system from 22 to 24. More importantly, we provide the first computer-verifiable proof certificate of a lower bound to the KS problem with a proof size of 41.6 TiB for order 23. The efficiency is due to the powerful combination of SAT solvers and CAS-based orderly generation.

Automatic construction of parallel portfolios via algorithm configuration

Multiway Decision Graph (MDG) is a canonical representation of a subset of many-sorted first-order logic. It generalizes the logic of equality with abstract types and uninterpreted function symbols. The area of Satisfiability (SAT) ha s been the subject of intensive research in recent years, with significant theoretical and practical contributions. From a practical perspective, a large number of very effective SAT solvers have recently been proposed, most of which based on improvements made to the original Davis-Putnam algorithm. Local search algorithms have allowed solving extremely large satisfiable instances of SAT. The combination between various verification methodologies will enhance the capabilities of each and overcome their limitations. In this paper, we introduce a model checking methodology for MDG based models using MDG tool and SAT solver. We use SAT solver searching for feasible paths of reachable states satisfying the property under certain encoding constraints. Finally, we provide a case study showing the correctness and the efficiency of our approach.