1,609 research outputs found
The Fractal Dimension of SAT Formulas
Modern SAT solvers have experienced a remarkable progress on solving
industrial instances. Most of the techniques have been developed after an
intensive experimental testing process. Recently, there have been some attempts
to analyze the structure of these formulas in terms of complex networks, with
the long-term aim of explaining the success of these SAT solving techniques,
and possibly improving them.
We study the fractal dimension of SAT formulas, and show that most industrial
families of formulas are self-similar, with a small fractal dimension. We also
show that this dimension is not affected by the addition of learnt clauses. We
explore how the dimension of a formula, together with other graph properties
can be used to characterize SAT instances. Finally, we give empirical evidence
that these graph properties can be used in state-of-the-art portfolios.Comment: 20 pages, 11 Postscript figure
The Fractal Dimension of SAT Formulas
Modern SAT solvers have experienced a remarkable progress on solving industrial instances. Most of the techniques have been developed after an intensive experimental process. It is believed that these techniques exploit the underlying structure of industrial instances. However, there is not a precise definition of the notion of structure.
Recently, there have been some attempts to analyze this structure in terms of complex networks, with the long-term aim of explaining the success of SAT solving techniques, and possibly improving them.
We study the fractal dimension of SAT instances with the aim of complementing the model that describes the structure of industrial instances. We show that many industrial families of formulas are self-similar, with a small fractal dimension. We also show how this dimension is affected by the addition of learnt clauses during the execution of SAT solvers.Peer Reviewe
Clustering of matter in waves and currents
The growth rate of small-scale density inhomogeneities (the entropy
production rate) is given by the sum of the Lyapunov exponents in a random
flow. We derive an analytic formula for the rate in a flow of weakly
interacting waves and show that in most cases it is zero up to the fourth order
in the wave amplitude. We then derive an analytic formula for the rate in a
flow of potential waves and solenoidal currents. Estimates of the rate and the
fractal dimension of the density distribution show that the interplay between
waves and currents is a realistic mechanism for providing patchiness of
pollutant distribution on the ocean surface.Comment: 4 pages, 1 figur
Community Structure in Industrial SAT Instances
Modern SAT solvers have experienced a remarkable progress on solving
industrial instances. Most of the techniques have been developed after an
intensive experimental process. It is believed that these techniques exploit
the underlying structure of industrial instances. However, there are few works
trying to exactly characterize the main features of this structure.
The research community on complex networks has developed techniques of
analysis and algorithms to study real-world graphs that can be used by the SAT
community. Recently, there have been some attempts to analyze the structure of
industrial SAT instances in terms of complex networks, with the aim of
explaining the success of SAT solving techniques, and possibly improving them.
In this paper, inspired by the results on complex networks, we study the
community structure, or modularity, of industrial SAT instances. In a graph
with clear community structure, or high modularity, we can find a partition of
its nodes into communities such that most edges connect variables of the same
community. In our analysis, we represent SAT instances as graphs, and we show
that most application benchmarks are characterized by a high modularity. On the
contrary, random SAT instances are closer to the classical Erd\"os-R\'enyi
random graph model, where no structure can be observed. We also analyze how
this structure evolves by the effects of the execution of a CDCL SAT solver. In
particular, we use the community structure to detect that new clauses learned
by the solver during the search contribute to destroy the original structure of
the formula. This is, learned clauses tend to contain variables of distinct
communities
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