3,321 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 power spectrum of solar convection flows from high-resolution observations and 3D simulations
We compare Fourier spectra of photospheric velocity fields from very high
resolution IMaX observations to those from recent 3D numerical
magnetoconvection models. We carry out a proper comparison by synthesizing
spectral lines from the numerical models and then applying to them the adequate
residual instrumental degradation that affects the observational data. Also,
the validity of the usual observational proxies is tested by obtaining
synthetic observations from the numerical boxes and comparing the velocity
proxies to the actual velocity values from the numerical grid.
For the observations, data from the SUNRISE/IMaX instrument with about 120 km
spatial resolution are used, thus allowing the calculation of observational
Fourier spectra well into the subgranular range. For the simulations, we use
four series of runs obtained with the STAGGER code and synthesize the IMaX
spectral line (FeI 5250.2 A) from them. Proxies for the velocity field are
obtained via Dopplergrams (vertical component) and local correlation tracking
(horizontal component).
A very good match between observational and simulated Fourier power spectra
is obtained for the vertical velocity data for scales between 200 km and 6 Mm.
Instead, a clear vertical shift is obtained when the synthetic observations are
not degraded. The match for the horizontal velocity data is much less
impressive because of the inaccuracies of the LCT procedure. Concerning the
internal comparison of the direct velocity values of the numerical boxes with
those from the synthetic observations, a high correlation (0.96) is obtained
for the vertical component when using the velocity values on the
log() = -1 surface in the box. The corresponding Fourier spectra are
near each other. A lower maximum correlation (0.5) is reached (at =
1) for the horizontal velocities as a result of the coarseness of the LCT
procedure.Comment: 12 pages, 9 figures, accepted in A&
Disjoint NP-pairs from propositional proof systems
For a proof system P we introduce the complexity class DNPP(P) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or complete for DNPP(P). Moreover we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for DNPP(P) and the reductions between the canonical pairs exist
The Deduction Theorem for Strong Propositional Proof Systems
This paper focuses on the deduction theorem for propositional logic. We define and investigate different deduction properties and show that the presence of these deduction properties for strong proof systems is powerful enough to characterize the existence of optimal and even polynomially bounded proof systems. We also exhibit a similar, but apparently weaker condition that implies the existence of complete disjoint NPUnknown control sequence '\mathsf' -pairs. In particular, this yields a sufficient condition for the completeness of the canonical pair of Frege systems and provides a general framework for the search for complete NPUnknown control sequence '\mathsf' -pairs
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