341 research outputs found
On variables with few occurrences in conjunctive normal forms
We consider the question of the existence of variables with few occurrences
in boolean conjunctive normal forms (clause-sets). Let mvd(F) for a clause-set
F denote the minimal variable-degree, the minimum of the number of occurrences
of variables. Our main result is an upper bound mvd(F) <= nM(surp(F)) <=
surp(F) + 1 + log_2(surp(F)) for lean clause-sets F in dependency on the
surplus surp(F).
- Lean clause-sets, defined as having no non-trivial autarkies, generalise
minimally unsatisfiable clause-sets.
- For the surplus we have surp(F) <= delta(F) = c(F) - n(F), using the
deficiency delta(F) of clause-sets, the difference between the number of
clauses and the number of variables.
- nM(k) is the k-th "non-Mersenne" number, skipping in the sequence of
natural numbers all numbers of the form 2^n - 1.
We conjecture that this bound is nearly precise for minimally unsatisfiable
clause-sets.
As an application of the upper bound we obtain that (arbitrary!) clause-sets
F with mvd(F) > nM(surp(F)) must have a non-trivial autarky (so clauses can be
removed satisfiability-equivalently by an assignment satisfying some clauses
and not touching the other clauses). It is open whether such an autarky can be
found in polynomial time.
As a future application we discuss the classification of minimally
unsatisfiable clause-sets depending on the deficiency.Comment: 14 pages. Revision contains more explanations, and more information
regarding the sharpness of the boun
Multigrid solver for axisymmetrical 2D fluid equations
We have developed an efficient algorithm for steady axisymmetrical 2D fluid
equations. The algorithm employs multigrid method as well as standard implicit
discretization schemes for systems of partial differential equations. Linearity
of the multigrid method with respect to the number of grid points allowed us to
use grid, where we could achieve solutions in several minutes.
Time limitations due to nonlinearity of the system are partially avoided by
using multi level grids(the initial solution on grid was
extrapolated steady solution from grid which allowed using
"long" integration time steps). The fluid solver may be used as the basis for
hybrid codes for DC discharges.Comment: preliminary version; presented at 28 ICPIG, July 15-20, 2007, Prague,
Czech Republi
An Improved Exact Algorithm for the Exact Satisfiability Problem
The Exact Satisfiability problem, XSAT, is defined as the problem of finding
a satisfying assignment to a formula in CNF such that exactly one
literal in each clause is assigned to be "1" and the other literals in the same
clause are set to "0". Since it is an important variant of the satisfiability
problem, XSAT has also been studied heavily and has seen numerous improvements
to the development of its exact algorithms over the years.
The fastest known exact algorithm to solve XSAT runs in time,
where is the number of variables in the formula. In this paper, we propose
a faster exact algorithm that solves the problem in time. Like
many of the authors working on this problem, we give a DPLL algorithm to solve
it. The novelty of this paper lies on the design of the nonstandard measure, to
help us to tighten the analysis of the algorithm further
Extensive study of nuclear uncertainties and their impact on the r-process nucleosynthesis in neutron star mergers
Theoretically predicted yields of elements created by the rapid neutron
capture (r-) process carry potentially large uncertainties associated with
incomplete knowledge of nuclear properties as well as approximative
hydrodynamical modelling of the matter ejection processes. We present an
in-depth study of the nuclear uncertainties by systematically varying
theoretical nuclear input models that describe the experimentally unknown
neutron-rich nuclei. This includes two frameworks for calculating the radiative
neutron capture rates and six, four and four models for the nuclear masses,
-decay rates and fission properties, respectively. Our r-process nuclear
network calculations are based on detailed hydrodynamical simulations of
dynamically ejected material from NS-NS or NS-BH binary mergers plus the
secular ejecta from BH-torus systems. The impact of nuclear uncertainties on
the r-process abundance distribution and early radioactive heating rate is
found to be modest (within a factor for individual nuclei and
a factor 2 for the heating rate), however the impact on the late-time heating
rate is more significant and depends strongly on the contribution from fission.
We witness significantly larger sensitivity to the nuclear physics input if
only a single trajectory is used compared to considering ensembles of
200-300 trajectories, and the quantitative effects of the nuclear
uncertainties strongly depend on the adopted conditions for the individual
trajectory. We use the predicted Th/U ratio to estimate the cosmochronometric
age of six metal-poor stars to set a lower limit of the age of the Galaxy and
find the impact of the nuclear uncertainties to be up to 2 Gyr.Comment: 26 pages, 22 figures, submitted to MNRA
Exploiting Resolution-based Representations for MaxSAT Solving
Most recent MaxSAT algorithms rely on a succession of calls to a SAT solver
in order to find an optimal solution. In particular, several algorithms take
advantage of the ability of SAT solvers to identify unsatisfiable subformulas.
Usually, these MaxSAT algorithms perform better when small unsatisfiable
subformulas are found early. However, this is not the case in many problem
instances, since the whole formula is given to the SAT solver in each call. In
this paper, we propose to partition the MaxSAT formula using a resolution-based
graph representation. Partitions are then iteratively joined by using a
proximity measure extracted from the graph representation of the formula. The
algorithm ends when only one partition remains and the optimal solution is
found. Experimental results show that this new approach further enhances a
state of the art MaxSAT solver to optimally solve a larger set of industrial
problem instances
On SAT representations of XOR constraints
We study the representation of systems S of linear equations over the
two-element field (aka xor- or parity-constraints) via conjunctive normal forms
F (boolean clause-sets). First we consider the problem of finding an
"arc-consistent" representation ("AC"), meaning that unit-clause propagation
will fix all forced assignments for all possible instantiations of the
xor-variables. Our main negative result is that there is no polysize
AC-representation in general. On the positive side we show that finding such an
AC-representation is fixed-parameter tractable (fpt) in the number of
equations. Then we turn to a stronger criterion of representation, namely
propagation completeness ("PC") --- while AC only covers the variables of S,
now all the variables in F (the variables in S plus auxiliary variables) are
considered for PC. We show that the standard translation actually yields a PC
representation for one equation, but fails so for two equations (in fact
arbitrarily badly). We show that with a more intelligent translation we can
also easily compute a translation to PC for two equations. We conjecture that
computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing
proof of the transformation from AC-representations to monotone circuits,
improved wording and literature review; 3rd v. updated literature,
strengthened treatment of monotonisation, improved discussions; 4th v. update
of literature, discussions and formulations, more details and examples;
conference v. to appear LATA 201
Unified characterisations of resolution hardness measures
Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this paper we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. Our main contribution is a unified game-theoretic characterisation of these measures. As consequences we obtain new relations between the different hardness measures. In particular, we prove a generalised version of Atserias and Dalmau's result on the relation between resolution width and space from [5]
Generalising unit-refutation completeness and SLUR via nested input resolution
We introduce two hierarchies of clause-sets, SLUR_k and UC_k, based on the
classes SLUR (Single Lookahead Unit Refutation), introduced in 1995, and UC
(Unit refutation Complete), introduced in 1994.
The class SLUR, introduced in [Annexstein et al, 1995], is the class of
clause-sets for which unit-clause-propagation (denoted by r_1) detects
unsatisfiability, or where otherwise iterative assignment, avoiding obviously
false assignments by look-ahead, always yields a satisfying assignment. It is
natural to consider how to form a hierarchy based on SLUR. Such investigations
were started in [Cepek et al, 2012] and [Balyo et al, 2012]. We present what we
consider the "limit hierarchy" SLUR_k, based on generalising r_1 by r_k, that
is, using generalised unit-clause-propagation introduced in [Kullmann, 1999,
2004].
The class UC, studied in [Del Val, 1994], is the class of Unit refutation
Complete clause-sets, that is, those clause-sets for which unsatisfiability is
decidable by r_1 under any falsifying assignment. For unsatisfiable clause-sets
F, the minimum k such that r_k determines unsatisfiability of F is exactly the
"hardness" of F, as introduced in [Ku 99, 04]. For satisfiable F we use now an
extension mentioned in [Ansotegui et al, 2008]: The hardness is the minimum k
such that after application of any falsifying partial assignments, r_k
determines unsatisfiability. The class UC_k is given by the clause-sets which
have hardness <= k. We observe that UC_1 is exactly UC.
UC_k has a proof-theoretic character, due to the relations between hardness
and tree-resolution, while SLUR_k has an algorithmic character. The
correspondence between r_k and k-times nested input resolution (or tree
resolution using clause-space k+1) means that r_k has a dual nature: both
algorithmic and proof theoretic. This corresponds to a basic result of this
paper, namely SLUR_k = UC_k.Comment: 41 pages; second version improved formulations and added examples,
and more details regarding future directions, third version further examples,
improved and extended explanations, and more on SLUR, fourth version various
additional remarks and editorial improvements, fifth version more
explanations and references, typos corrected, improved wordin
Macrostate Data Clustering
We develop an effective nonhierarchical data clustering method using an
analogy to the dynamic coarse graining of a stochastic system. Analyzing the
eigensystem of an interitem transition matrix identifies fuzzy clusters
corresponding to the metastable macroscopic states (macrostates) of a diffusive
system. A "minimum uncertainty criterion" determines the linear transformation
from eigenvectors to cluster-defining window functions. Eigenspectrum gap and
cluster certainty conditions identify the proper number of clusters. The
physically motivated fuzzy representation and associated uncertainty analysis
distinguishes macrostate clustering from spectral partitioning methods.
Macrostate data clustering solves a variety of test cases that challenge other
methods.Comment: keywords: cluster analysis, clustering, pattern recognition, spectral
graph theory, dynamic eigenvectors, machine learning, macrostates,
classificatio
Degree-distribution Stability of Growing Networks
In this paper, we abstract a kind of stochastic processes from evolving
processes of growing networks, this process is called growing network Markov
chains. Thus the existence and the formulas of degree distribution are
transformed to the corresponding problems of growing network Markov chains.
First we investigate the growing network Markov chains, and obtain the
condition in which the steady degree distribution exists and get its exact
formulas. Then we apply it to various growing networks. With this method, we
get a rigorous, exact and unified solution of the steady degree distribution
for growing networks.Comment: 12 page
- …