An analytic criterion for CSAT

Covering problem is a problem of extraction of a minimum cost subset from a given set that satisfies certain constraints expressed as a Boolean formula in conjunctive normal form. This problem is NP-hard, heuristic methods are thus of interest. The authors present two heuristic methods to finding a nearly minimal solution and compare them to each other. The authors derive the asymptotic complexity of the presented methods and report some computational results obtained for a number of randomly generated covering problems. >

This paper presents a dynamic substitution technique of Boolean functions. It first recovers a set of Boolean functions from Boolean formula in conjunctive normal form (CNF). Then these functions are used to reduce the size of the learnt clauses by substituting the input arguments by the output ones. Preliminary experiments show the feasibility of our approach on some classes of SAT instances taken from the recent SAT Race and competitions.

Let FeC1∧⋅⋅⋅∧Cm be a Boolean formula in conjunctive normal form over a set V of n propositional variables, s.t. each clause Ci contains at most three literals l over V. Solving the problem exact 3-satisfiability (X3SAT) for F means to decide whether there is a truth assignment setting exactly one literal in each clause of F to true (1). As is well known X3SAT is NP-complete [6]. By exploiting a perfect matching reduction we prove that X3SAT is deterministically decidable in time O(20.18674n). Thereby we improve a result in [2,3] stating X3SAT∈O(20.2072n) and a bound of O(20.200002n) for the corresponding enumeration problem nX3SAT stated in a preprint [1]. After that by a more involved deterministic case analysis we are able to show that X3SAT∈O(20.16254n).

In this paper, we discuss the computational complexity of Read-once resolution (ROR) with respect to Horn formulas. Recall that a Horn formula is a boolean formula in conjunctive normal form (CNF), such that each clause has at most one positive literal. Horn formulas find applications in a number of domains such as program verification and logic programming. It is well-known that deduction in ProLog is based on unification, which in turn is based on resolution and instantiation. Resolution is a sound and complete procedure to check whether a boolean formula in CNF is satisfiable. Although inefficient in general, resolution has been used widely in theorem provers, on account of its simplicity and ease of implementation. This paper focuses on two variants of resolution, viz., Read-once resolution and Unit Read-once resolution (UROR). Both these variants are sound, but incomplete. In this paper, the goal is to check for the existence of proofs (refutations) of Horn formulas under these variants. We also discuss the computational complexity of determining optimal length proofs where appropriate.

Boolean satisfiability (SAT) and its extensions are becoming a core technology for the analysis of systems. The SAT-based approach divides into three steps: encoding, preprocessing, and search. It is often argued that by encoding arbitrary Boolean formulas in conjunctive normal form (CNF), structural properties of the original problem are not reflected in the CNF. This should result in the fact that CNF-level preprocessing and SAT solver techniques have an inherent disadvantagecompared to related techniques applicable on the level of more structural SAT instance representations such as Boolean circuits. In this work we study the effect of a CNF-level simplification technique called blocked clause elimination (BCE). We show that BCE is surprisingly effective both in theory and in practice on CNFs resulting from a standard CNF encoding for circuits: without explicit knowledge of the underlying circuit structure, it achieves the same level of simplification as a combination of circuit-level simplifications and previously suggested polarity-based CNF encodings. Experimentally, we show that by applying BCE in preprocessing, further formula reduction and faster solving can be achieved, giving promise for applying BCE to speed up solvers.

The SAT-Problem fot boolean formulas in conjunctive normal form often arises in the area of artificial intelligence, its solution is known as mechanical theorem proving. Most mechanical theorem provers perform this by using a variant of the resolution principle. The time and space complexity of resolution strongly depends on the class of formulas. Horn formulas, where each clause may contain at most one positive literal, represent the most important class where resolution proofs with polynomial length exist.

We have verified that the van der Waerden number W(2, 6) is 1132, that is, 1132 is the smallest integer n = W(2, 6) such that whenever the set of integers {1, 2, . , n} is 2-colored, there exists a monochromatic arithmetic progression of length 6. This was accomplished by applying special preprocessing techniques that drastically reduced the required search space. The exhaustive search showing that W(2, 6) = 1132 was carried out by formulating the problem as a satisfiability (SAT) question for a Boolean formula in conjunctive normal form (CNF), and then using a SAT solver specifically designed for the problem. The parallel backtracking computation was run over multiple Beowulf clusters, and in the last phase, field programmable gate arrays (FPGAs) were used to speed up the search. The fact that W(2, 6) > 1131 was shown previously by the first author.

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.

Functional unification (FU) grammar is a general linguistic formalism based on the merging of feature-sets. An informal outline is given of how the definition of derivation within FU grammar can be used to represent the satisfiability of an arbitrary logical formula in conjunctive normal form. This suggests that the generation of a structure from an arbitrary arbitrary FU grammar is NP-hard, which is an undesirably high level of computational complexity.

SAT-based ATPG turned out to be a robust alternative to classical structural ATPG algorithms such as FAN. The number of unclassified faults can be significantly reduced using a SAT-based ATPG approach. In contrast to structural ATPG, SAT solvers work on a Boolean formula in conjunctive normal form (CNF). This results in some disadvantages for SAT solvers when applied to ATPG, e.g. CNF transformation time and loss of structural knowledge. As a result, SAT-based ATPG algorithms are very robust for hard-to-test faults, but suffer from the overhead for easy-to-test faults. We propose the SAT technique dynamic clause activation (DCA) in order to reduce the run time gap between structural and SAT-based ATPG algorithms and, at the same time, retain the high level of robustness. Using DCA, the SAT solver works on a partial formula of a logic circuit which is dynamically extended during the search process using structural knowledge. Furthermore, efficient dynamic learning techniques can be easily integrated within the proposed technique. The approach is evaluated on large industrial circuits.

We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics.In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness.Several open problems remain, some of which we mention in the concluding section.

In this paper, we present a Bayesian Moment Matching (BMM) in-processing technique for Conflict-Driven Clause-Learning (CDCL) SAT solvers. BMM is a probabilistic algorithm which takes as input a Boolean formula in conjunctive normal form and a prior on a possible satisfying assignment, and outputs a posterior for a new assignment most likely to maximize the number of satisfied clauses. We invoke this BMM method, as an in-processing technique, with the goal of updating the polarity and branching activity scores. The key insight underpinning our method is that Bayesian reasoning is a powerful way to guide the CDCL search procedure away from fruitless parts of the search space of a satisfiable Boolean formula, and towards those regions that are likely to contain satisfying assignments.

Building a meaningful model of biological regulatory network is usually done by specifying the components (e.g. the genes) and their interactions, by guessing the values of parameters, by comparing the predicted behaviors to the observed ones, and by modifying in a trial-error process both architecture and parameters in order to reach an optimal fitness. We propose here a different approach to construct and analyze biological models avoiding the trial-error part, where structure and dynamics are represented as formal constraints. We apply the method to Hopfield-like networks, a formalism often used in both neural and regulatory networks modeling. The aim is to characterize automatically the set of all models consistent with all the available knowledge (about structure and behavior). The available knowledge is formalized into formal constraints. The latter are compiled into Boolean formula in conjunctive normal form and then submitted to a Boolean satisfiability solver. This approach allows to formulate a wide range of queries, expressed in a high level language, and possibly integrating formalized intuitions. In order to explore its potential, we use it to find cycles for 3-nodes networks and to determine the flower morphogenesis regulatory network of Arabidopsis thaliana. Applications of this technique are numerous and concern the building of models from data as well as the design of biological networks possessing specified behaviors.

In this paper we describe and analyse an improved algorithm for solving the 3-Satisfiability problem. If F is a boolean formula in conjunctive normal form with n variables and r clauses, then we will show that this algorithm solves the Satisfiability problem for formulas with at most three literals per clause in time less than O(1,579n).

We presentAbstract CNF2dDNNF, a calculus describing an approach for compiling a formula in conjunctive normal form (CNF) into deterministic negation normal form (d-DNNF). It combines component-based reasoning with a model enumeration approach based on conflict-driven clause learning (CDCL) with chronological backtracking. Its properties, such as soundness and termination, carry over to implementations which can be modeled by it. We provide a correctness proof and a detailed example. The main conceptual differences to currently available tools targeting d-DNNF compilation are discussed and future research directions presented. The aim of this work is to lay the theoretical foundation for a novel method for d-DNNF compilation. To the best of our knowledge, our approach is the first knowledge compilation method using CDCL with chronological backtracking.