This issue of the International Journal of Intelligent Systems presents approaches to knowledge discovery based on rough set theory.[1][2][3][4][5][6][7][8] It is often the case that there are imperfections in raw input data needed for knowledge acquisition: uncertainty, vagueness, and incompleteness.Uncertainty arises in any measuring process where the observed value of a variable x tends to fluctuate from one measurement to the next.9 Sensors have varying accuracy.Sensor readings can fluctuate and can sometimes be inaccurate due to noisy environments or faulty sensor components.Hence, there is keen interest in having measures of uncertainty.In the context of data mining and knowledge discovery, there is interest in quantifying the certainty factor of a decision rule. 2 In rough set theory, every decision rule has two conditional probabilities associated with it: certainty and coverage factors.8 These two factors are closely related to two fundamental concepts of rough set theory, namely, lower approximation and upper approximation.It has been shown that the certainty and coverage factors satisfy Bayes' rule.8 In addition, a frequency-based estimate of the conditional probability that an object x belongs to a set X has also been introduced in rough set theory 4 (see also Ref. 3).Other rough set approaches to measurement in the presence of uncertainty have also been given ( see for example Refs. 3 and 6).Vagueness is yet another nettlesome problem in data mining and knowledge discovery.Two common sources of vagueness have been identified: error in physical measurements due to inaccurate measuring devices, as well as the mixture of noise and pure signals

This issue of the International Journal of Intelligent Systems presents perspectives on a rough set approach to reasoning about data.Rules derived from decision tables instantiate a reasoning process for particular data sets, and reflect our evaluations of a data set.In this special issue, a number of foundation articles on the discovery and significance of decision rules are given.Underlying the study of the rough set approach to information systems is an interest in the discovery of effective means of approximating concepts reflected in data sets.A number of articles in this issue also pave the way toward what might be described as rough computation.This form of computing utilizes a rough set approach to reasoning about data in guiding the actions of agents and in facilitating communication between distributed agents.Conditional probabilities can be used to advantage in explaining conditions for decisions in decision rules.In this issue of IJIS, Pawlak presents an approach to exchanging mutual conditions and decisions in rules.Fundamental concepts concerning rules derived from decision tables as well as an approach to drawing conclusions from data are presented by Pawlak.''Inversed'' decision rules provide an explanation for decisions relative to conditions.Stefanowski and Vanderpooten introduce a procedure called Explore for extracting from data all decision rules that satisfy requirements.Explore is compared with the Grzymala-Busse algorithm LEM2, which is a rough set based rule induction approach to generating classification rules.Grzymala-Busse and Stefanowski introduce three discretization methods performed during rule induction.Rules induced by the new methods are shown to be simpler and stronger.Szczuka represents hyperplane-based decision rules in neural networks.In this approach to decision rules, an attribute-value space is partitioned into subsets bounded by hyperplanes.Classification of objects proceeds according to the position of rules relative to hyperplanes.Skowron and Stepaniuk have shown how information granules can be defined by sets of decision rules.Granules defined by rules are examples of sequences of granules.Rule-based information granules provide a basis for reasoning in a distributed environment.Agents in such an environment Ž .can be designed so that concepts from a source server agent can be approximated by a receiving agent using a rough set approach in constructing information granules.In the paper by Yao, the focus is on information granulation and Ž .

We introduce basic notions related to granular computing, namely the information granule syntax and semantics as well as the inclusion and closeness (similarity) relations of granules. Different information sources (units, agents) are equipped with two kinds of operations on information granules: operations transforming tuples of information granules definable by a given agent into information granules definable by this agent and approximation operations for computing by agents approximations of information granules delivered by other agents. More complex granules are constructed by means of these operations and approximation operations from some input information granules. The construction of information granules is described by expressions called terms. We discuss a problem of synthesis of robust terms, i.e., descriptions of information granules, satisfying a given specification. This is an important problem for granular computing and its applications for spatial reasoning or knowledge discovery and data mining. © 2001 John Wiley & Sons, Inc.

This paper considers the construction of Petri nets to simulate the computation performed by decision systems. Algorithms are given to construct Petri nets which correspond to decision rules, information systems, and real-time decision systems. Rough as well as rough fuzzy Petri net extensions of colored and generalized fuzzy Petri nets are used to create highly parallel programs to simulate reasoning system computations. Constructed nets make it possible to evaluate the design of decision system tables, and to trace computations in rules derived from decision tables. Start places of nets are connected to Dill process receptors which await input from the environment. Time consumption during the propagation of outputs from sensors in a decision system is monitored with timers called approximate time windows, which measure durations between firings of decision transitions relative to time granules with names such as early, ontime, and late. Guards on decision transitions are propositional functions which permit a rule to fire for some sensor values and not for others. In addition, the design of guards makes allowance for multivalued logic, where conditional sensor readings are assessed in terms of their degree of membership in sensor measurement granules. In some cases, a rule can fire if the degree of truth of its guard (premise) is above some threshold. Through simulation, designers can arrive at reasonable estimates of the period of timers on decision transitions. The approach to simulating computations by decision systems presented in this paper results in fast, massively parallel programs implementable on a multiprocessor. © 1999 John Wiley & Sons, Inc.