Scale-Free Random SAT Instances

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called \emph{scale-free random SAT instances}. It is based on the use of a non-uniform probability distribution $P(i)\sim i^{-\beta}$ to select variable $i$, where $\beta$ is a parameter of the model. This results into formulas where the number of occurrences $k$ of variables follows a power-law distribution $P(k)\sim k^{-\delta}$ where $\delta = 1 + 1/\beta$. This property has been observed in most real-world SAT instances. For $\beta=0$, our model extends classical random SAT instances. We prove the existence of a SAT-UNSAT phase transition phenomenon for scale-free random 2-SAT instances with $\beta<1/2$ when the clause/variable ratio is $m/n=\frac{1-2\beta}{(1-\beta)^2}$. We also prove that scale-free random k-SAT instances are unsatisfiable with high probability when the number of clauses exceeds $\omega(n^{(1-\beta)k})$. %This implies that the SAT/UNSAT phase transition phenomena vanishes when $\beta>1-1/k$, and formulas are unsatisfiable due to a small core of clauses. The proof of this result suggests that, when $\beta>1-1/k$, the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes

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

How to generate statistics and influence policy using participatory methods in research : reflections on work in Malawi, 1999-2002

In Malawi in 1999–2002 we conducted research studies using participatory methods to generate population estimates; estimates of the proportion of people in a population with certain characteristics (e.g. the very food insecure); and estimates of the proportion of people in a population that should be targeted by an intervention. We claim that the statistics generated in these studies are at least as reliable as statistics obtained from traditional methods such as surveys. A key requirement is to produce results from a representative sample, which can be generalised in order to reach conclusions for the population of interest. This implies working in a larger number of sites than is common for most studies that use participatory methods. However, we argue that the Malawi experience shows that it is possible to organise such studies at a reasonable cost. Other key requirements are for the study design to incorporate statistical principles; and for PRA tools to be adapted to meet the demands of standardisation and comparability of data produced across sites. We argue that this can be done without undermining participatory approaches, behaviours and methods. If research studies using participatory methods follow this approach, the data generated will be suitable for standard statistical analysis. The statistics produced by such studies should be capable of informing policy at national level. However, there are some concerns about empowerment and ethical issues, which present challenges for the future

Community structure in industrial SAT instances

Modern SAT solvers have experienced a remarkable progress on solving industrial instances. It is believed that most of these successful techniques exploit the underlying structure of industrial instances. 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, 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. Representing SAT instances as graphs, we show that most application benchmarks are characterized by a high modularity. On the contrary, random SAT instances are closer to the classical Erdös-Rényi 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, and observe that new clauses learned by the solver during the search contribute to destroy the original structure of the formula. Motivated by this observation, we finally present an application that exploits the community structure to detect relevant learned clauses, and we show that detecting these clauses results in an improvement on the performance of the SAT solver. Empirically, we observe that this improves the performance of several SAT solvers on industrial SAT formulas, especially on satisfiable instances.Peer ReviewedPostprint (published version

Reducing SAT to Max2SAT

In the literature we find reductions from 3SAT to Max2SAT. These reductions are based on the usage of a gadget, i.e., a combinatorial structure that allows translating constraints of one problem to constraints of another. Unfortunately, the generation of these gadgets lacks an intuitive or efficient method. In this paper, we provide an efficient and constructive method for Reducing SAT to Max2SAT and show empirical results of how MaxSAT solvers are more efficient than SAT solvers solving the translation of hard formulas for Resolution.Supported by projects PROOFS (PID2019-109137GB-C21) and EU-H2020-RIP LOGISTAR (No. 769142)

Signatures of triplet superconductivity in nu=2-chiral Andreev states

We study the behavior of the conductance and the current-noise in three-terminal configurations of edge modes of a quantum Hall system in the nu=2 filling factor with normal and s-wave superconducting contacts. We discuss the impact of spin-orbit coupling in the quantum Hall system and the possibility of effectively inducing triplet pairing in the egde states. We show that the presence of these correlations imprints very clear signatures in both the non-linear conductance and noise in these type of devices.Comment: 5 pages, 3 figures, supplemental materia

Sugar and overall macronutrient profile in the Brazilian family diet (2002-2003)

Avaliou-se a influência exercida pela aquisição de calorias de açúcar sobre a participação calórica dos demais macronutrientes da dieta. As estimativas deste estudo são baseadas em dados da Pesquisa de Orçamentos Familiares realizada no Brasil pelo Instituto Brasileiro de Geografia e Estatística entre julho de 2002 e junho de 2003. Modelos de regressão linear múltiplos foram utilizados para estudar a influência das calorias de açúcar sobre a participação calórica de cada um dos macronutrientes na aquisição domiciliar de alimentos com o controle do valor calórico total da aquisição de alimentos e variáveis sócio-demográficas. Cada caloria adquirida de açúcar eleva em 0,3 caloria a participação de gorduras na aquisição domiciliar de alimentos e diminui em 0,07 a participação de proteínas. Cada caloria de açúcar procedente de alimentos processados aumenta em 1,6 caloria a participação de gorduras e em 0,4 caloria de ácidos graxos saturados e diminui em 0,8 caloria a participação de outros carboidratos que não o açúcar. Os resultados encontrados trazem novas evidências sobre o papel prejudicial do açúcar à saúde humana.This study evaluated the impact of calories from sugar on the calorie share of other macronutrients in the Brazilian family diet. The study's estimates are based on data from the Household Budget Survey conducted by the Brazilian Institute of Geography and Statistics from July 2002 to June 2003. Multiple linear regression models were used to study the impact of sugar calories on each of the macronutrients in household food acquisition, controlling for total energy content in food acquisition and socio-demographic variables. Each calorie from sugar purchases increases the share of calories from fat by 0.3 calories and decreases the share of calories from protein by 0.7 calories. Each calorie of sugar from processed foods increases the share from fat by 1.6 calories and the share from saturated fatty acids by 0.4 calories and decreases the share from non-sugar carbohydrates by 0.8 calories. The findings shed further light on the harmful role of sugar for human health