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 $256\times 256$ 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 $256\times 256$ grid was extrapolated steady solution from $128\times 128$ 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 $\varphi$ 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 $O(1.1730^n)$ time, where $n$ is the number of variables in the formula. In this paper, we propose a faster exact algorithm that solves the problem in $O(1.1674^n)$ 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, $\beta$-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 $\sim 20$ for individual $A>90$ 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 $\sim$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