Attribute Exploration of Properties of Functions on Sets
An approach for studying relations between properties of functions on sets is proposed. The approach is based on Attribute Exploration. 16 properties of functions are considered, among them monotonicity, idempotency, path independence, exchange properties, convexity, etc. Example functions are partially computer generated on the powersets of sets with 2, 3 and 4 elements. Attribute Exploration is run on contexts where objects are functions and attributes are 16 function properties. Minimal implication bases are presented. The list of proved implications is presented and discussed.
In this paper we consider k-meet-semidistributive lattices and we are interested in the computation of the set-colored poset associated to an implicational base. The parameter k is of interest since for any finite lattice there exists an integer k for which is k-meet-semidistributive. When
they are known as meet-semidistributive lattices.
We first give a polynomial time algorithm to compute an implicational base of a k-meet-semidistributive lattice from its associated colored poset. In other words, for a fixed k, finding a minimal implicational base of a k-meet-semidistributive lattice L from a context (FCA literature) of L can be done not just in output-polynomial time (which is open in the general case) but in polynomial time in the size of the input. This result generalizes that in . Second, we derive an algorithm to compute a set-colored poset from an implicational base which is based on the enumeration of minimal transversals of a hypergraph and turns out to be in polynomial time for k-meet-semidistributive lattices , . Finally, we show that checking whether a given implicational base describes a k-meet-semidistributive lattice can be done in polynomial time.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.