A lower bound on CNF encodings of the at-most-one constraint

Constraint "at most one" is a basic cardinality constraint which requires that at most one of its $n$ boolean inputs is set to $1$. This constraint is widely used when translating a problem into a conjunctive normal form (CNF) and we investigate its CNF encodings suitable for this purpose. An encoding differs from a CNF representation of a function in that it can use auxiliary variables. We are especially interested in propagation complete encodings which have the property that unit propagation is strong enough to enforce consistency on input variables. We show a lower bound on the number of clauses in any propagation complete encoding of the "at most one" constraint. The lower bound almost matches the size of the best known encodings. We also study an important case of 2-CNF encodings where we show a slightly better lower bound. The lower bound holds also for a related "exactly one" constraint.Comment: 38 pages, version 3 is significantly reorganized in order to improve readabilit

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]

Ecotoxicology of Copper in Horticultural Soils: A Review

Nowadays, the world is facing the problem of environmental pollution because of the increase of man’s needs requires developmentin life activities, progress industrialization, transportation tools, enhancement of agriculture and exploitation of natural resources. Soil andwater resources are extremely exposed to pollution from different aspects. Agrochemicals in particular, have created severe problems, sincethey release thousands of chemicals to the environment. Several studies on the effect of environmental pollutants on agroecosystem have beencarried out. On the other hand, the importance of trace elements as environmental pollutants is well known and well documented in literature.Cu contamination to agricultural soils has been accelerated due to its wide and repeated use in agriculture and horticulture as fertilizers orfungicides to protect vines, citrus trees, and other fruit crops against fungus diseases. Applied Cu from different agrochemical sources to agroenvironment may be adsorbed and are transported to the groundwater table and pollute it besides polluting the soils. The use of Cu-basedfungicides in vineyard soils is widely documented worldwide. It has been found that many countries contain concentrations in excess of 100mg kg−1. Importance of study of transport of Cu arises due to the fact that Cu is absorbed in soils and also reaches the groundwater table,thus polluting both soil and ground water. It is often more important to be able to estimate the mobile fraction, the readily soluble fraction,the exchangeable fraction, or the plant available fraction of Cu content of a soil as a more direct indication of the likelihood of deleteriousor toxic effects on soils and groundwater. Therefore, the aim of present work was to highlight the behavior and ecotoxicological effects ofcopper on horticultural soils

Ecotoxicology of Copper in Horticultural Soils: A Review

Nowadays, the world is facing the problem of environmental pollution because of the increase of man’s needs requires development in life activities, progress industrialization, transportation tools, enhancement of agriculture and exploitation of natural resources. Soil and water resources are extremely exposed to pollution from different aspects. Agrochemicals in particular, have created severe problems, since they release thousands of chemicals to the environment. Several studies on the effect of environmental pollutants on agroecosystem have been carried out. On the other hand, the importance of trace elements as environmental pollutants is well known and well documented in literature. Cu contamination to agricultural soils has been accelerated due to its wide and repeated use in agriculture and horticulture as fertilizers or fungicides to protect vines, citrus trees, and other fruit crops against fungus diseases. Applied Cu from different agrochemical sources to agroenvironment may be adsorbed and are transported to the groundwater table and pollute it besides polluting the soils. The use of Cu-based fungicides in vineyard soils is widely documented worldwide. It has been found that many countries contain concentrations in excess of 100 mg kg−1. Importance of study of transport of Cu arises due to the fact that Cu is absorbed in soils and also reaches the groundwater table, thus polluting both soil and ground water. It is often more important to be able to estimate the mobile fraction, the readily soluble fraction, the exchangeable fraction, or the plant available fraction of Cu content of a soil as a more direct indication of the likelihood of deleterious or toxic effects on soils and groundwater. Therefore, the aim of present work was to highlight the behavior and ecotoxicological effects of copper on horticultural soils