77 research outputs found
Learning for Dynamic subsumption
In this paper a new dynamic subsumption technique for Boolean CNF formulae is
proposed. It exploits simple and sufficient conditions to detect during
conflict analysis, clauses from the original formula that can be reduced by
subsumption. During the learnt clause derivation, and at each step of the
resolution process, we simply check for backward subsumption between the
current resolvent and clauses from the original formula and encoded in the
implication graph. Our approach give rise to a strong and dynamic
simplification technique that exploits learning to eliminate literals from the
original clauses. Experimental results show that the integration of our dynamic
subsumption approach within the state-of-the-art SAT solvers Minisat and Rsat
achieves interesting improvements particularly on crafted instances
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Increasing The Effectiveness of Deduction in Propositional SAT Solvers
The satisfiability (SAT) of a propositional formula is the decision problem to determine whether there is a satisfying assignment that can make the formula true or not. In the past few years, many successful SAT solvers based on the David-Putnam-Logemann-Loveland (DPLL) procedure [DP60, DLL62, MS99, MMZ+01, ES03] for formulae in conjunctive normal form (CNF) have been developed. Since the deduction procedure of DPLL is sound but not complete, its effects depend on which formula is selected to represent the input function. CNF transformations are among the most effective techniques to improve quality of the input formula by either simplifying clauses [ES03, EB05,|SE05, ZKKSV06, HS07, HS09] or learning new ones [MS99]. Specifically, effective CNF transformations can help SAT solvers to be sped up by allowing them to do more deductions and less enumerations.
In my dissertation, I characterize existing transformations in terms of their impact on the deductive power of the formula and their effects on the proof conciseness, that is, the sizes of the implication graphs. I also present two new techniques that try to increase deductive power. The first is a check performed during the computation of resolvents. The second is a new preprocessing algorithm based on distillation that combines simplification and increase of deductive power. Most current SAT solvers apply resolution at various stages to derive new clauses or simplify existing ones. The former happens during conflict analysis, while the latter is usually done during preprocessing. I show how subsumption of the operands by the resolvent can be inexpensively detected during resolution; I then discuss how this detection is used to improve three stages of the SAT solver: variable elimination, clause distillation, and conflict analysis. The on-the-fly subsumption check is easily integrated in a SAT solver. In particular, it is compatible with strong conflict analysis and the generation of unsatisfiability proofs. Experiments show the effectiveness of the new techniques.
SAT solvers also benefit from clauses learned by the DPLL procedure, even though they are by definition redundant. In addition to those derived from conflicts, the clauses learned by dominator analysis during the deduction procedure tend to produce smaller implication graphs and sometimes increase the deductive power of the input CNF formula. I extend dominator analysis with an efficient self-subsumption check. I also show how the information collected by dominator analysis can be used to detect redundancies in the satisfied clauses and, more importantly, how it can be used to produce supplemental conflict clauses. I characterize these transformations in terms of deductive power and proof conciseness. Experiments show that the main advantage of dominator analysis and its extensions lies in improving proof conciseness
A novel EGs-based framework for systematic propositional-formula simplification
Funding: Bowles is partially supported by Austrian FWF Meitner Fellowship M-3338 N.This paper presents a novel simplification calculus for propositional logic derived from Peirce’s Existential Graphs’ rules of inference and implication graphs. Our rules can be applied to arbitrary propositional logic formulae (not only in CNF), are equivalence-preserving, guarantee a monotonically decreasing number of clauses and literals, and maximise the preservation of structural problem information. Our techniques can also be seen as higher-level SAT preprocessing, and we show how one of our rules (TWSR) generalises and streamlines most of the known equivalence-preserving SAT preprocessing methods. We further show how this rule can be extended with a novel n-ary implication graph to capture all known equivalence-preserving preprocessing procedures. Finally, we discuss the complexity and implementation of our framework as a solver-agnostic algorithm to simplify Boolean satisfiability problems and arbitrary propositional formula.Postprin
Improving Model Finding for Integrated Quantitative-qualitative Spatial Reasoning With First-order Logic Ontologies
Many spatial standards are developed to harmonize the semantics and specifications of GIS data and for sophisticated reasoning. All these standards include some types of simple and complex geometric features, and some of them incorporate simple mereotopological relations. But the relations as used in these standards, only allow the extraction of qualitative information from geometric data and lack formal semantics that link geometric representations with mereotopological or other qualitative relations. This impedes integrated reasoning over qualitative data obtained from geometric sources and “native” topological information – for example as provided from textual sources where precise locations or spatial extents are unknown or unknowable. To address this issue, the first contribution in this dissertation is a first-order logical ontology that treats geometric features (e.g. polylines, polygons) and relations between them as specializations of more general types of features (e.g. any kind of 2D or 1D features) and mereotopological relations between them. Key to this endeavor is the use of a multidimensional theory of space wherein, unlike traditional logical theories of mereotopology (like RCC), spatial entities of different dimensions can co-exist and be related. However terminating or tractable reasoning with such an expressive ontology and potentially large amounts of data is a challenging AI problem. Model finding tools used to verify FOL ontologies with data usually employ a SAT solver to determine the satisfiability of the propositional instantiations (SAT problems) of the ontology. These solvers often experience scalability issues with increasing number of objects and size and complexity of the ontology, limiting its use to ontologies with small signatures and building small models with less than 20 objects. To investigate how an ontology influences the size of its SAT translation and consequently the model finder’s performance, we develop a formalization of FOL ontologies with data. We theoretically identify parameters of an ontology that significantly contribute to the dramatic growth in size of the SAT problem. The search space of the SAT problem is exponential in the signature of the ontology (the number of predicates in the axiomatization and any additional predicates from skolemization) and the number of distinct objects in the model. Axiomatizations that contain many definitions lead to large number of SAT propositional clauses. This is from the conversion of biconditionals to clausal form. We therefore postulate that optional definitions are ideal sentences that can be eliminated from an ontology to boost model finder’s performance. We then formalize optional definition elimination (ODE) as an FOL ontology preprocessing step and test the simplification on a set of spatial benchmark problems to generate smaller SAT problems (with fewer clauses and variables) without changing the satisfiability and semantic meaning of the problem. We experimentally demonstrate that the reduction in SAT problem size also leads to improved model finding with state-of-the-art model finders, with speedups of 10-99%. Altogether, this dissertation improves spatial reasoning capabilities using FOL ontologies – in terms of a formal framework for integrated qualitative-geometric reasoning, and specific ontology preprocessing steps that can be built into automated reasoners to achieve better speedups in model finding times, and scalability with moderately-sized datasets
Towards Next Generation Sequential and Parallel SAT Solvers
This thesis focuses on improving the SAT solving technology. The improvements focus on two major subjects: sequential SAT solving and parallel SAT solving.
To better understand sequential SAT algorithms, the abstract reduction system Generic CDCL is introduced. With Generic CDCL, the soundness of solving techniques can be modeled. Next, the conflict driven clause learning algorithm is extended with the three techniques local look-ahead, local probing and all UIP learning that allow more global reasoning during search. These techniques improve the performance of the sequential SAT solver Riss. Then, the formula simplification techniques bounded variable addition, covered literal elimination and an advanced cardinality constraint extraction are introduced. By using these techniques, the reasoning of the overall SAT solving tool chain becomes stronger than plain resolution. When using these three techniques in the formula simplification tool Coprocessor before using Riss to solve a formula, the performance can be improved further.
Due to the increasing number of cores in CPUs, the scalable parallel SAT solving approach iterative partitioning has been implemented in Pcasso for the multi-core architecture. Related work on parallel SAT solving has been studied to extract main ideas that can improve Pcasso. Besides parallel formula simplification with bounded variable elimination, the major extension is the extended clause sharing level based clause tagging, which builds the basis for conflict driven node killing. The latter allows to better identify unsatisfiable search space partitions. Another improvement is to combine scattering and look-ahead as a superior search space partitioning function. In combination with Coprocessor, the introduced extensions increase the performance of the parallel solver Pcasso. The implemented system turns out to be scalable for the multi-core architecture. Hence iterative partitioning is interesting for future parallel SAT solvers.
The implemented solvers participated in international SAT competitions. In 2013 and 2014 Pcasso showed a good performance. Riss in combination with Copro- cessor won several first, second and third prices, including two Kurt-Gödel-Medals. Hence, the introduced algorithms improved modern SAT solving technology
Subsumption dirigée par l'analyse de conflits
International audienceNon disponibl
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