Propositional Satisfiability

The DPLL procedure is the basis of some of the most successful propositional satisfiability solvers to date. Although originally devised as a proofprocedure for first-order logic, it has been used almost exclusively for propositional logic so far because of its highly inefficient treatment of quantifiers, based on instantiation into ground formulas. The recent FDPLL calculus by Baumgartner was the first successful attempt to lift the procedure to the first-order level without resorting to ground instantiations. FDPLL lifts to the first-order case the core of the DPLL procedure, the splitting rule, but ignores other aspects of the procedure that, although not necessary for completeness, are crucial for its effectiveness in practice. In this paper, we present a new calculus loosely based on FDPLL that lifts these aspects as well. In addition to being a more faithful lifting of the DPLL procedure, the new calculus contains a more systematic treatment of universal literals, one of FDPLL's optimizations, and so has the potential of leading to much faster implementations.

We first introduce Abstract DPLL, a rule-based formulation of the Davis--Putnam--Logemann--Loveland (DPLL) procedure for propositional satisfiability. This abstract framework allows one to cleanly express practical DPLL algorithms and to formally reason about them in a simple way. Its properties, such as soundness, completeness or termination, immediately carry over to the modern DPLL implementations with features such as backjumping or clause

This paper introduces GRASP (Generic seaRch Algorithm for the Sati$ability Problem), a new search algorithm for Propositional Satisjability (SAT). GRASP incorporates several search-pruning techniques, some of which are spec$c to SAT whereas others find equivalent in other f i e h ofArt$cial Intelligence. GRASP is premised on the inevitability of conflicts during search and its most distinguishing feature is the augmentation of basic backtracking search with a powe@l conjlict analysis procedure. Analyzing conflicts to determine their causes enables GRASP to backtrack non-chronologically to earlier kveh in the search tree, potentially pruning large portions of the search space. In addition, by "recording" the causes of conflicts, GRASP can recognize andpreempt the occurrence of similar conflicts later on in the search. Finally, straightforward bookkeeping of the causality chains leading up to conjicts allows GRASP to identi3 assignments that are necessary for a solution to be found. fiperimental results obtained5om a large number of benchmarks indicate that application of the proposed conflict analysis techniques to SAT algorithms can be extremely effective for a large number of representative classes of SAT instances.

Symbolic Model Checking has proven to be a powerful technique for the verification of reactive systems. BDDs [2] have traditionally been used as a symbolic representation of the system. In this paper we show how boolean decision procedures, like Stålmarck's Method or the Davis & Putnam Procedure [7], can replace BDDs. This new technique avoids the space blow up of BDDs, generates counterexamples much faster, and sometimes speeds up the verification. In addition, it produces counterexamples of minimal length. We introduce a bounded model checking procedure for LTL which reduces model checking to propositional satisfiability. We show that bounded LTL model checking can be done without a tableau construction. We have implemented a model checker BMC, based on bounded model checking, and preliminary results are presented.

Propositional satisfiability (SAT) problem is fundamental to the theory of NP-completeness. Indeed, using the concept of “polynomial-time reducibility” all NP-complete problems can be polynomially reduced to SAT. Thus, any new technique for satisfiability problems will lead to general approaches for thousands of hard combinatorial problems. In this paper, we introduce the incremental propositional satisfiability problem that consists of maintaining the satisfiability of a propositional formula anytime a conjunction of new clauses is added. More precisely, the goal here is to check whether a solution to a SAT problem continues to be a solution anytime a new set of clauses is added and if not, whether the solution can be modified efficiently to satisfy the old formula and the new clauses. We will study the applicability of systematic and approximation methods for solving incremental SAT problems. The systematic method is based on the branch and bound technique while the approximation methods rely on stochastic local search and genetic algorithms.

Marriage in Honey Bees Optimization (MBO) is a new swarm intelligence technique inspired by the marriage process of honey bees. It has been shown to be very effective in solving the propositional satisfiability problem known as 3-SAT (each clause has exactly three literals). The objective of this paper is to test a conventional annealing approach as a basis for determining the pool of drones (fathers). This modified MBO algorithm is tested using a group of randomly generated hard 3-SAT problems to compare its behavior and efficiency against previous implementations. We find that the proposed annealing function significantly improved the performance of the committee machine. The improvement is found to be quite dramatic.

Stochastic local search methods such as GSAT, WalkSAT and their variants have been used successfully at solving propositional satisfiability problems (SAT). The key operation in these local search algorithms is the speed of variable flipping. We present a parallel FPGA designs for CSAT capable of one flip per clock cycle which is achieved by exploiting maximal parallelism and "multi-try" pipelining. Experimental results show that a speedup of two orders of magnitude over software implementations can be achieved.

Schaefer proved in 1978 that the Boolean constraint satisfaction problem for a given constraint language is either in P or is NP-complete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P ≠ NP). We show that if one considers AC0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct).

This paper presents a reduction from the problem of solving parity games to the satisfiability problem in propositional logic (SAT). The reduction is done in two stages, first into difference logic, i.e. SAT combined with the theory of integer differences, an instance of the SAT modulo theories (SMT) framework. In the second stage the integer variables and constraints of the difference logic encoding are replaced with a set of Boolean variables and constraints on them, giving rise to a pure SAT encoding of the problem. The reduction uses Jurdziński's characterisation of winning strategies via progress measures. The reduction is motivated by the success of SAT solvers in symbolic verification, bounded model checking in particular. The paper reports on prototype implementations of the reductions and presents some experimental results.

One promising family of search strategies to alleviate runtime and storage requirements of ILP systems is that of stochastic local search methods, which have been successfully applied to hard propositional tasks such as satisfiability. Stochastic local search algorithms for propositional satisfiability benefit from the ability to quickly test whether a truth assignment satisfies a formula. Because of that many possible solutions can be tested and scored in a short time. In contrast, testing whether a clause covers an example in ILP takes much longer, so that far fewer possible solutions can be tested in the same time. Therefore in this paper we investigate stochastic local search in ILP using a relational propositionalized problem instead of directly use the first-order clauses space of solutions.

The paper compares two popular strategies for solving propositional satisfiability, backtracking search and resolution, and analyzes the complexity of a directional resolution algorithm (DR) as a function of the "width" (w * ) of the problem's graph. Our empirical evaluation confirms theoretical prediction, showing that on low-w * problems DR is very efficient, greatly outperforming the backtracking-based Davis-Putnam-Logemann-Loveland procedure (DP). We also emphasize the knowledge-compilation properties of DR and extend it to a tree-clustering algorithm that facilitates query answering. Finally, we propose two hybrid algorithms that combine the advantages of both DR and DP. These algorithms use control parameters that bound the complexity of resolution and allow time/space trade-offs that can be adjusted to the problem structure and to the user's computational resources. Empirical studies demonstrate the advantages of such hybrid schemes.

Many problems of practical relevance (e. g., equivalence checking and model check-ing of RTL designs, verification of timed and hybrid systems, temporal planning) are not easily expressible in propositional logic. Rather, they can be conveniently expressed in extensions of propositional logics, with boolean combinations of propo-sitional variables and mathematical constraints. For instance, the timing constraints typical of timed automata can

AbstractÐThis paper introduces GRASP (Generic seaRch Algorithm for the Satisfiability Problem), a new search algorithm for Propositional Satisfiability (SAT). GRASP incorporates several search-pruning techniques that proved to be quite powerful on a wide variety of SAT problems. Some of these techniques are specific to SAT, whereas others are similar in spirit to approaches in other fields of Artificial Intelligence. GRASP is premised on the inevitability of conflicts during the search and its most distinguishing feature is the augmentation of basic backtracking search with a powerful conflict analysis procedure. Analyzing conflicts to determine their causes enables GRASP to backtrack nonchronologically to earlier levels in the search tree, potentially pruning large portions of the search space. In addition, by ªrecordingº the causes of conflicts, GRASP can recognize and preempt the occurrence of similar conflicts later on in the search. Finally, straightforward bookkeeping of the causality chains leading up to conflicts allows GRASP to identify assignments that are necessary for a solution to be found. Experimental results obtained from a large number of benchmarks indicate that application of the proposed conflict analysis techniques to SAT algorithms can be extremely effective for a large number of representative classes of SAT instances.

In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden numbers as an example, we show that these problems can be represented by parameterized propositional theories in such a way that decisions concerning their satisfiability determine the numbers (function) in question. We show that by using general-purpose complete and local-search techniques for testing propositional satisfiability, this approach becomes effective -competitive with specialized approaches. By following it, we were able to obtain several new results pertaining to the problem of computing van der Waerden numbers. We also note that due to their properties, especially their structural simplicity and computational hardness, propositional theories that arise in this research can be of use in development, testing and benchmarking of SAT solvers. * This is an expanded and updated version of the conference paper .

Many problems of practical relevance (e. g., equivalence checking and model check-ing of RTL designs, verification of timed and hybrid systems, temporal planning) are not easily expressible in propositional logic. Rather, they can be conveniently expressed in extensions of propositional logics, with boolean combinations of propo-sitional variables and mathematical constraints. For instance, the timing constraints typical of timed automata can

Monotone literal fixing (MLF) and learning are well-known lookahead and lookback mechanisms in propositional satisfiability (SAT). When considering Quantified Boolean Formulas (QBFs), their separate implementation leads to significant speed-ups in state-of-the-art DPLL-based solvers. This paper is dedicated to the efficient implementation of MLF in a QBF solver with learning. The interaction between MLF and learning is far from being obvious, and it poses some nontrivial questions about both the detection and the propagation of monotone literals. Complications arise from the presence of learned constraints, and are related to the question about whether learned constraints have to be considered or not during the detection and/or propagation of monotone literals. In the paper we answer to this question both from a theoretical and from a practical point of view. We discuss the advantages and the disadvantages of various solutions, and show that our solution of choice, implemented in our solver QuBE, produces significant speed-ups in most cases. Finally, we show that MLF can be fundamental also for solving some SAT instances, taken from the 2002 SAT solvers competition.

Tackling real-world problems often requires to take various types of constraints into account. Such constraint types range from simple numerical comparators to complex resources. This article describes how planning techniques can be integrated with general constraintsolving frameworks, like SAT, IP and CP. In many cases, the complete planning problem can be cast in these frameworks.

Algorithms based on following local gradient information are surprisingly effective for certain classes of constraint satisfaction problems. Unfortunately, previous local search algorithms are notoriously incomplete: They are not guaranteed to find a feasible solution if one exists and they cannot be used to determine unsatisfiability. We present an algorithmic framework for complete local search and discuss in detail an instantiation for the propositional satisfiability problem (SAT). The fundamental idea is to use constraint learning in combination with a novel objective function that converges during search to a surface without local minima. Although the algorithm has worst-case exponential space complexity, we present empirical resulls on challenging SAT competition benchmarks that suggest that our implementation can perform as well as state-of-the-art solvers based on more mature techniques. Our framework suggests a range of possible algorithms lying between tree-based search a...

The MAXimum propositional SATisfiability problem (MAXSAT) is a well known NP-hard optimization problem with many theoretical and practical applications in artificial intelligence and mathematical logic. Heuristic local search algorithms are widely recognized as the most effective approaches used to solve them. However, their performance depends both on their complexity and their tuning parameters which are controlled experimentally and remain a difficult task. Extremal Optimization (EO) is one of the simplest heuristic methods with only one free parameter, which has proved competitive with the more elaborate general-purpose method on graph partitioning and coloring. It is inspired by the dynamics of physical systems with emergent complexity and their ability to self-organize to reach an optimal adaptation state. In this paper, we propose an extremal optimization procedure for MAXSAT and consider its effectiveness by computational experiments on a benchmark of random instances. Comparative tests showed that this procedure improves significantly previous results obtained on the same benchmark with other modern local search methods like WSAT, simulated annealing and Tabu Search (TS).

The MAXimum propositional SATisfiability problem (MAXSAT) is a well known NP-hard optimization problem with many theoretical and practical applications in artificial intelligence and mathematical logic. Heuristic local search algorithms are widely recognized as the most effective approaches used to solve them. However, their performance depends both on their complexity and their tuning parameters which are controlled experimentally and remain a

We study the backbone and the backdoors of propositional satisfiability problems. We make a number of theoretical, al-gorithmic and experimental contributions. From a theoret-ical perspective, we prove that backbones are hard even to approximate. From an algorithmic perspective, we present a number of different procedures for computing backdoors. From an empirical perspective, we study the correlation be-tween being in the backbone and in a backdoor. Experiments show that there tends to be very little overlap between back-bones and backdoors. We also study problem hardness for the Davis Putnam procedure. Problem hardness appears to be correlated with the size of strong backdoors, and weakly correlated with the size of the backbone, but does not appear to be correlated to the size of weak backdoors nor their num-ber. Finally, to isolate the effect of backdoors, we look at problems with no backbone.

This paper follows on previous papers which presented and evaluated various decision procedures for modal logics. It con rms previous experimental results in showing that SAT based decision procedures, i.e., the procedures built on top of decision procedures for propositional satis ability, are more efcient than tableau based decision procedures. It also con rms previous evidence of an easy-hard-easy pattern in the satis ability curve for modal K. Finally, it provides further experimental results, suggesting that SAT based decision procedures are also more e cient than the decision procedures based on Ohlbach's translation method. Our results contradict some of the results presented in previous papers. SPARC10, 32MBRam workstation on SunOS 4.1.4. Spass and KsatC are compiled by gcc 2.7.2, option -O2.

This paper presents the Job Shop Scheduling Problem (JSSP) represented as the well known Satisfiabilty Problem (SAT). Even though the representation of JSSP in SAT is not a new issue, presented here is a new codification that needs fewer clauses in the SAT formula for JSSP instances than those used in previous works. The codification proposed, which has been named the Reduced Sat Formula (RSF), uses the value of the latest starting time of each operation in a JSSP instance to evaluate RSF. The latest starting time is obtained using a procedure that finds the critical path in a graph. This work presents experimental results and analytical arguments showing that the new representation improves efficiency in finding a starting solution for JSSP instances.

We study the runtime distributions of backtrack procedures for propositional satisfiability and constraint satisfaction. Such procedures often exhibit a large variability in performance. Our study reveals some intriguing properties of such distributions: They are often characterized by very long tails or "heavy tails". We will show that these distributions are best characterized by a general class of distributions that can have infinite moments (i.e., an infinite mean, variance, etc.). Such nonstandard distributions have recently been observed in areas as diverse as economics, statistical physics, and geophysics. They are closely related to fractal phenomena, whose study was introduced by Mandelbrot. We also show how random restarts can effectively eliminate heavy-tailed behavior. Furthermore, for harder problem instances, we observe long tails on the left-hand side of the distribution, which is indicative of a non-negligible fraction of relatively short, successful runs. A rapid restart strategy eliminates heavy-tailed behavior and takes advantage of short runs, significantly reducing expected solution time. We demonstrate speedups of up to two orders of magnitude on SAT and CSP encodings of hard problems in planning, scheduling, and circuit synthesis.

Stochastic local search (SLS) algorithms for the propositional satisfiability problem (SAT) have been successfully applied to solve suitably encoded search problems from various domains. One drawback of these algorithms is that they are usually incomplete. We refine the notion of incompleteness for stochastic decision algorithms by introducing the notion of "probabilistic asymptotic completeness" (PAC) and prove for a number of well-known SLS algorithms whether or not they have this property. We also give evidence for the practical impact of the PAC property and show how to achieve the PAC property and significantly improved performance in practice for some of the most powerful SLS algorithms for SAT, using a simple and general technique called "random walk extension".

The implementation of effective reasoning tools for deciding the satisfiability of Quantified Boolean Formulas (QBFs) is an important issue in several research fields such as Formal Verification, Planning, and Reasoning about Knowledge. Several QBF solvers have been implemented in the last few years, most of them extending the well-known Davis, Putnam, Logemann, Loveland procedure (DPLL) for propositional satisfiability (SAT). At the same time, a substantial breed of QBF benchmarks emerged, both in the form of statistical models for the generation of random formulas, and in the form of real-world instances. In this paper we report about the -first ever -evaluation of QBF solvers that was run as a joint event to SAT'03 Conference on Theory and Applications of Satisfiability Testing. Owing to the relative youngness of QBF tools and applications, we decided to run the comparison on a non-competitive basis, using the same technology that powered SAT'02 and SAT'03 competitions of SAT solvers. Running the evaluation enabled us to collect all sorts of data regarding the relative strength of different solvers and methods, the quality of the benchmarks, and to understand some of the current challenges for researchers involved in the QBF arena.

Representing and reasoning about time dependent information is a key research issue in many areas of computer science and artificial intelligence. One of the best known and widely used formalisms for representing interval-based qualitative temporal information is Allen's interval algebra (IA). The fundamental reasoning task in IA is to find a scenario that is consistent with the given information. This problem is in general NP-complete.

We first introduce Abstract DPLL, a rule-based formulation of the Davis--Putnam--Logemann--Loveland (DPLL) procedure for propositional satisfiability. This abstract framework allows one to cleanly express practical DPLL algorithms and to formally reason about them in a simple way. Its properties, such as soundness, completeness or termination, immediately carry over to the modern DPLL implementations with features such as backjumping or clause learning.We then extend the framework to Satisfiability Modulo background Theories (SMT) and use it to model several variants of the so-called lazy approach for SMT. In particular, we use it to introduce a few variants of a new, efficient and modular approach for SMT based on a general DPLL(X) engine, whose parameter X can be instantiated with a specialized solver SolverT for a given theory T, thus producing a DPLL(T) system. We describe the high-level design of DPLL(X) and its cooperation with SolverT, discuss the role of theory propagation, ...

ABSTRACT The main objectives of Built-In Self Test (BIST) are the design of test pattern generator circuits which achieve the highest fault cov-erage, require the shortest sequence of test vectors and utilize the minimum circuit area. This paper targets the problem of generat-ing test ...

Recent experience suggests that branching algorithms are among the most attractive for solving propositional satisfiability problems. A key factor in their success is the rule they use to decide on which variable to branch next. We attempt to explain and improve the performance of branching rules with an empirical model-building approach. One model is based on the rationale given for the Jeroslow-Wang rule, variations of which have performed well in recent work, The model is refuted by carefully designed computational experiments. A second model explains the success of the Jeroslow-Wang rule, makes other predictions confirmed by experiment, and leads to the design of branching rules that are clearly superior to Jeroslow-Wang.

This paper presents CO3L, a compact, expressive and easy to use language for ontology representation. CO3L reflects the O3F model of ontology representation proposed elsewhere. The proposed language enables the representation of basic ontological entities, their relationships, and arbitrary axioms of the domain. Ontological entities include classes, properties, methods, facets and types. Relationships include n-ary associations and inheritance. Axioms may be used to capture complex constraints and relations between entities, to define relational and functional methods, and to represent the effects of the execution of action methods in the world. CO3L is based on the language of the first order logic, with explicit world states; the relational operator State/1, which is true of world state designators; the functional operator Do/2, which returns the world state designator resulting of the execution of an action in a given world state; and the modal operator Holds/2 which is true of propositions satisfied in given world states. Frame Name EntityDeclaration Abstract Kind of OntologicalProposition, AtomicProposition Description Declares the existence of ontological entities such as classes, properties, methods, types An entity declaration is also abstract. Only its sub-frames (e.g., ClassDeclaration, PropertyDeclaration, MethodDeclaration) can have concrete instances. Frame Name OntologyRelationship Abstract Kind of OntologicalProposition, AtomicProposition Description Declares the existence of a relationship between two or more ontological entities (e.g., the relation between a property and its type)

The goal of this paper is to describe and thoroughly test a decision procedure, called Ksat, checking satisfiability in the terminological logic ALC. Ksat is said to be SAT-based as it is defined in terms of a decision procedure for propositional satisfiability (SAT). The tests are performed comparing Ksat with, among other procedures, Kris, a state-of-the-art tableau-based implementation of a decision procedure for ALC. Ksat outperforms Kris of orders of magnitude. Furthermore, the empirical results highlight an intrinsic weakeness that tableau-based decision procedures have with respect to SAT-based decision procedures.

In the past few years, general-purpose propositional satisability (SAT) solvers have improved dramatically in performance and have been used to tackle many new problems. It has also been shown that certain simple fragments of rst-order logic can be decided e ciently by rst translating the problem into an equivalent SAT problem and then using a fast SAT solver. In this paper, we describe an alternative but similar approach to using SAT in conjunction with a more expressive fragment of rst-order logic. However, rather than translating the entire formula up front, the formula is incrementally translated during a search for the solution. As a result, only that portion of the translation that is actually relevant to the solution is obtained. We describe a number of obstacles that had to be overcome before developing an approach which was ultimately very e ective, and give results on veri cation benchmarks using CVC (Cooperating Validity Checker), which includes the Cha SAT solver. The results show a performance gain of several orders of magnitude over CVC without Cha and indicate that the method is more robust than the heuristics found in CVC's predecessor, SVC.

The implementation of effective reasoning tools for deciding the satisfiability of Quantified Boolean Formulas (QBFs) is an important research issue in Artificial Intelligence. Many decision procedures have been proposed in the last few years, most of them based on the Davis, Logemann, Loveland procedure (DLL) for propositional satisfiability (SAT). In this paper we show how it is possible to extend the conflict-directed backjumping schema for SAT to QBF: when applicable, it allows to jump over existentially quantified literals while backtracking. We introduce solution-directed backjumping, which allows the same for universally quantified literals. Then, we show how it is possible to incorporate both conflict-directed and solution-directed backjumping in a DLL-based decision procedure for QBF satisfiability. We also implement and test the procedure: The experimental analysis shows that, because of backjumping, significant speed-ups can be obtained. While there have been several proposals for backjumping in SAT, this is the first time -as far as we know-this idea has been proposed, implemented and experimented for QBFs.

The implementation of effective reasoning tools for deciding the satisfiability of Quantified Boolean Formulas (QBFs) is an important research issue in Artificial Intelligence. Many decision procedures have been proposed in the last few years, most of them based on the Davis, Logemann, Loveland procedure (DLL) for propositional satisfiability (SAT). In this paper we show how it is possible to extend the conflict-directed backjumping schema for SAT to QBF: when applicable, it allows to jump over existentially quantified literals while backtracking. We introduce solution-directed backjumping, which allows the same for universally quantified literals. Then, we show how it is possible to incorporate both conflict-directed and solution-directed backjumping in a DLL-based decision procedure for QBF satisfiability. We also implement and test the procedure: The experimental analysis shows that, because of backjumping, significant speed-ups can be obtained. While there have been several proposals for backjumping in SAT, this is the first time -as far as we know-this idea has been proposed, implemented and experimented for QBFs.

We address lower bounds on the time complexity of algorithms solving the propositional satisfiability problem. Namely, we consider two DPLL-type algorithms, enhanced with the unit clause and pure literal heuristics. Exponential lower bounds for solving satisfiability on provably satisfiable formulas are proven.

MAX-SAT is a well-known optimisation problem that can be seen as a generalisation of the propositional satisfiability problem. In this study, we in- vestigate how the performance of stochastic local search (SLS) algorithms — a large and prominent class of algorithms that includes, for example, Tabu Search, Evolutionary Algorithms and Simulated Annealing — depends on features of the underlying search

In this paper, we investigate whether hard combinatorial problems such as the Hamiltonian circuit problem HCP (an NP-complete problem from graph theory) can be practically solved by transformation to the propositional satis ability problem (SAT) and application of fast universal SAT-algorithms like GSAT to the transformed problem instances. By using the e cient transformation method proposed by Iwama and Miyazaki in 1994, one obtains a class of SAT-problems which are especially hard for SAT-algorithms based on local search such as GSAT. We identify structural properties of these problems which are responsible for GSAT's poor performance on this problem class. An empirical analysis indicates that several methods which have been designed to improve GSAT on structured problems are not e ective for SAT-transformed HCP-instances.

Stochastic local search algorithms based on the WalkSAT ar- chitecture are among the best known methods for solving hard and large instances of the propositional satisfiability problem (SAT). The performance and behaviour of these algorithms critically depends on the setting of the noise parameter, which controls the greediness of the search process. The optimal set- ting for the noise parameter