We present an approach based on the Interactive Granular Computing (IGrC) model as the basis for developing foundations of Complex Intelligent Systems, i.e., Intelligent Systems dealing with complex phenomena (IS's).The generalization of GrC to IGrC was proposed to support the design of IS's treated in IGrC as examples of complex granules (c-granules) with control.To make such systems successful, it is necessary to enable such systems to have continuous interaction with the physical world.The control of c-granules aims to properly implement the physical semantics of specified transformations of c-granules in the physical world.This implementation is based on the discovery of relevant configurations of physical objects, which provides the basis for perceiving relevant data about these objects and their interactions through the control of c-granules.Additionally, to create high-quality models that serve as the basis for the behavior of IS's, these configurations must be adaptively adjusted by control to allow for the perception of relevant data used to induce those models.Unlike information granules from GrC, the correct implementation of c-granule transformations cannot be restricted to the abstract space.An important property of the IS's discussed here is that they cannot be separated from interactions with the physical world.Hence, they cannot be confined to an abstract space.In particular, the relevance of IGrC in searching for rough computational building blocks for cognition is discussed.These computational building blocks are modeled by complex granules (c-granules) and their networks.It is also proposed to use IGrC as the basis for developing IS's grounded on cognitive computing.

The article concerns the well-known RIONA algorithm.We focus on the explainability property of this algorithm.The theoretical results, formulated and proved in the paper, show the relationships of the RIONA classifiers to both instanceand rule-based classifiers.In particular, we show the equivalence (relative to the classification) of the RIONA algorithm with the rule-based algorithm generating all consistent and maximally general rules from the neighbourhood of the test case.

The theory of rough sets was founded by Zdzisław Pawlak as a framework for data and knowledge exploration.His seminal paper titled "Rough Sets" was published in 1982, in International Journal of Computer and Information Sciences.One of the key aspects that lets us use rough sets in practical scenarios is the notion of information system, which comes from even earlier Professor Pawlak's works.Information systems are the means for data and knowledge representation.They constitute the input to rough set mechanisms aimed at computing approximations of concepts and deriving compacted, interpretable decision models.In particular, the fundamental notion of the indiscernibility relation is defined on the basis of a given information system.Accordingly, we discuss to what extent information systems can serve as the basis for intelligent systems.We claim that in many cases it is not enough to treat a data set -represented as an information system -as a purely abstract object with no linkage to the data origins.Oppositely, we should give ourselves a technical possibility to construct information systems dynamically, taking into account interaction with physical environments where the data comes from.With this respect, we refer to the notions of interactive granular computing and we generally consider together the paradigms of rough sets, information systems, and information granulation.

A new method of decision tree construction from temporal data is proposed in the paper. This method uses the so-called temporal cuts for binary partition of data in tree nodes. The novelty of the proposed approach is that the quality of cuts is calculated not on the basis of the discernibility of o bjects (related to time points), but on the basis of the discernibility of time windows labeled with different decision classes. The paper includes results of experiments performed on our data sets and collections from machine learning repositories. In order to evaluate the presented method, we compared its performance with the classification results of a local discretization decision tree, and other methods well known from literature. Our new method outperforms these known methods.

Reaction system is a model of interactive computations which was motivated by the functioning of the living cell. It is an idealized mathematical model, also because it abstracts from the complex nature of the physical systems where only partial, incomplete information is available (e.g., about the ir states). The framework of rough sets was developed to deal with such incomplete information. In this paper we establish a connection between reaction systems and rough sets. This is done in a somewhat broader perspective of the relationship between “pure” mathematical models and “realistic models” that take into account the limitation of perceiving physical reality.

In the original publication, the Acknowledgments was published incorrectly. The correct Acknowledgments is provided in this Correction.

We discuss the rough set approach to approximation of vague concepts. There are already published several papers on rough sets and vague concepts staring from the seminal papers by Zdzisław Pawlak. However, only a few of them are discussing the relationships of rough sets with the sorites paradox. This paper contains a continuation of discussion on this issue.

Introduction of rough sets by Professor Zdzisław Pawlak has completed 35 years. The theory has already attracted the attention of many researchers and practitioners, who have contributed essentially to its development, from all over the world. The methods, developed based on rough set theory alone or in combination with other approaches, found applications in many areas. In this article, we outline some selected past and present research directions of rough sets. In particular, we emphasize the importance of searching strategies for relevant approximation spaces as the basic tools in achieving computational building blocks (granules or patterns) required for approximation of complex vague concepts. We also discuss new challenges related to problem solving by intelligent systems (IS) or complex adaptive systems (CAS). The concern is to control problems using interactive granular computing, an extension of the rough set approach, for effective realization of computations realized in IS or CAS. These challenges are important for the development of natural computing too.

We present an extension of logical structures, called interactive logical structures, for reasoning about interactive computations performed by Intelligent Systems or Complex Adaptive Systems. Reasoning based on such structures is called adaptive judgment and it goes beyond deduction, induction, an d abduction. An extension of logical structures, based on complex granules, couples the abstract world and the physical world of an agent’s environment, and transmits the features of interactions of physical objects realized in the physical world to the abstract world. This allows us to consider the problems of perception and action.

This article is a continuation of previous work, in which a new method of decision tree construction was presented.That method is based on the use of so-called verifying cuts, which can provide knowledge obtained from the attributes frequently eliminated when greedy methods of the choice of singleton best cuts are applied.Till now only one strategy of choosing verifying cuts was examined.It exploits a measure based on a number of pairs of objects discerned by a chosen cut.In this paper, we examine two additional measures used for determining the best verifying cuts.They are based on Gini's Index and Entropy.The paper includes the results of experiments that have been performed on data obtained from biomedical database and machine learning repositories.

This article introduces a new method of a decision tree construction. Such construction is performed using additional cuts applied for a verification of the cuts’ quality in tree nodes during the classification of objects. The presented approach allows us to exploit the additional knowledge represe nted in the attributes which could be eliminated using greedy methods. The paper includes the results of experiments performed on data sets from a biomedical database and machine learning repositories. In order to evaluate the presented method, we compared its performance with the classification results of a local discretization decision tree, well known from literature. Our new method outperforms the existing method, which is also confirmed by statistical tests.

In several papers we have discussed a computing model, called the Interactive Granular Computing (IGrC), for interactive computations on complex granules. In this paper, we compare two models of computing, namely the Turing model and the IGrC model.

Understanding the nature of interactions is regarded as one of the biggest challenges in projects related to complex adaptive systems. We discuss foundations for interactive computations in interactive intelligent systems (IIS), developed in the Wistech program and used for modeling complex systems. We emphasize the key role of risk management in problem solving by IIS. The considerations are based on experience gained in real-life projects concerning, e.g., medical diagnosis and therapy support, control of an unmanned helicopter, fraud detection algorithmic trading or fire commander decision support.

Looking back to Prof. Zadeh’s paradigm of Computing with Words (CWW) [28, 29, 30], one can notice that the initial attempt of such an endeavour was to set up a basic vocabulary of linguistic words, and fix their semantics based on fuzzy sets. Then a grammar was proposed to generate compound linguis tic expressions based on the primitive ones, and simultaneously based on the semantic interpretations of those basic linguistic expressions a general scheme for the semantics of the rest of linguistic expressions were proposed. Sentences involving linguistic quantifiers and vague predicates constitute a fragment of natural language. In this paper, we choose this fragment of the natural language, and explore the semantics from the perspective of rough sets [13, 14, 16, 17, 18, 21]. We fix a set of basic crisp quantifiers, mainly of proportional kind. A set of vague quantifiers are proposed to lie in a close vicinity of those crisp quantifiers in the sense that a particular vague quantifier can be visualized as a blurred, may be called rough, image of a set of crisp quantifiers. Semantics of the rest of the vague quantifiers can be obtained based on the subjective perception of the interrelations among the (vague) quantifiers.

We discuss an approach for dealing with uncertainty in complex systems. The approach is based on interactive computations over complex objects called here complex granules (c-granules, for short). Any c-granule consists of a physical part and a mental part linked in a special way. We begin from the rough set approach and next we move toward interactive computations on c-granules. From our considerations it follows that the fundamental issues of intelligent systems based on interactive computations are related to risk management in such systems. Our approach is a step toward realization of the Wisdom Technology (WisTech) program. The approach was developed over years of work on different real-life projects.

Information granules (infogranules, for short) are widely discussed in the literature. In particular, let us mention here the rough granular computing approach based on the rough set approach and its combination with other approaches to soft computin

This issue contains seven papers presented during the "Third Symposium on Cellular Automata-Journes Automates Cellulaires" (JAC 2012), held in La Marana, Corsica (France) in the period September 19th-21th

The problem considered in this paper is how to describe and compare visual objects. The solution to this problem stems from a consideration of nearness relations in two different forms of Efremovič proximity spaces. In this paper, the visual objects

This special issue of Fundamenta Informaticae contains a selection of papers initially presented at the 6th International Conference on Rough Sets and Knowledge Technology (RSKT'11) held during October 8-11, 2011 in Banff, Canada.RSKT is an international scientific conference series that has been held every year since 2006.The conferences serve as a major forum that brings researchers and industry practitioners together to discuss and deliberate on fundamental issues of knowledge processing and management and knowledge-intensive practical solutions in the current knowledge age.Experts from around the world meet to present state-of-the-art scientific results, to nurture academic and industrial interaction, and to promote collaborative research in rough sets and knowledge technology.We initially had twelve papers invited.After rigorous review, eight papers were selected to be included in this issue.They are substantially extended versions of respective conference papers.Each paper was review by three domain experts and went through at least two rounds of revisions.

We consider several issues related to reasoning about changes in systems interacting with the environment by sensors. In particular, we discuss challenging problems of reasoning about changes in hierarchical modeling and approximation of transition f

This special issue of Fundamenta Informaticae (FI) contains a selection of papers presented initially at the 5th International Conference on Rough Sets and Knowledge Technology (RSKT

We discuss the problem of measuring the quality of decision support (classification) system that involves granularity based on rough set concepts. We put forward the proposal for such quality measure in the case when the underlying granular system is

We discuss some generalizations of the approximation space definition introduced in 1994 [24, 25]. These generalizations are motivated by real-life applications. Rough set based strategies for extension of such generalized approximation spaces from samples of objects onto their extensions are discussed. This enables us to present the uniform foundations for inducing approximations of different kinds of granules such as concepts, classifications, or functions. In particular, we emphasize the fundamental role of approximation spaces for inducing diverse kinds of classifiers used in machine learning or data mining.

The minimal inhibitory rules for information systems can be used for construction of classifiers. We show that almost all information systems from a certain large class of information systems have relatively short minimal inhibitory rules. However, the number of such rules is not polynomial in the number of attributes and the number of objects. This class consists of all k-valued information systems, k ⩾ 2, with the number of objects polynomial in the number of attributes. Hence, for efficient construction of classifiers some filtration techniques in rule generation are necessary. Another way is to work with lazy classification algorithms based on inhibitory rules.

This paper outlines an approach to hierarchical modelling of complex patterns that is based on operations of sums with constraints on information systems. It is shown that such operations can be treated as a universal tool in hierarchical modelling of complex patterns.

This article introduces an approach to measures of information granules based on rough set theory. Informally, an information granule is a representation of a multiset (or bag) of real-world objects that are somehow indistinguishable, or similar, or which cause the same functionality. Examples of measures of information granules based on the rough set theory are inclusion, closeness, size, and enclosure. All of these measures are based on rough inclusion. This paper is limited to a consideration of measures of inclusion based on a straightforward extension of classical rough membership functions and closeness based on measurement of separation of equivalence classes in a partition of the universe containing information granules. Measurement of sensor-based information granules has been motivated by recent studies of sensor signals. A sensor signal is a non-empty, finite set of sample sensor signal values temporally ordered. Classification of sensor signals requires measurements of sample signal values over subintervals of time. This article introduces a rough set approach to measuring information granule inclusion and closeness.

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

We discuss information granule calculi as a basis of granular computing. They are defined by constructs like information granules, basic relations of inclusion and closeness between information granules as well as operations on them. The exact interpretation between granule languages of different information sources (agents) often does not exist. Hence (rough) inclusion and closeness of granules are considered instead of their equality. Examples of all the basic constructs of information granule calculi are presented. The construction of more complex information granules is described by expressions called terms. We discuss the synthesis problem of robust terms, i.e., descriptions of information granules, satisfying a given specification in a satisfactory degree. We also present a method for synthesis of information granules represented by robust terms (approximate schemes of reasoning) by means of decomposition of specifications for such granules. The discussed problems of granular computing are of special importance for many applications, in particular related to spatial reasoning as well as to knowledge discovery and data mining.

Rough Mereology is a paradigm allowing for a synthesis of main ideas of two potent paradigms for reasoning under uncertainty: Fuzzy Set Theory and Rough Set Theory. Approximate reasoning is based in this paradigm on the predicate of being a part to a degree. We present applications of Rough Mereology to the important theoretical idea put forth by Lotfi Zadeh (1996, 1997), i.e., Granularity of Knowledge: We define granules of knowledge by means of the operator of mereological class and we extend the idea of a granule over complex objects like decision rules as well as decision algorithms. We apply these notions and methods in the distributed environment discussing complex problems of knowledge and granule fusion. We express the mechanism of complex granule formation by means of a formal grammar called Synthesis Grammar defined over granules of knowledge, granules of classifying rules, or over granules of classifying algorithms. We finally propose hybrid rough‐neural schemes bridging rough and neural computations.

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.

The paper deals with an automatic concurrent control design method derived from the specification of a discrete event control system represented in the form of a decision table. The main stages of our approach are: the control specification by decision tables, generation of rules from the specification of the system behavior, and converting rules set into a concurrent program represented in the form of a Petri net. Our approach is based on rough set theory [17].

Rough Mereology has been proposed as a paradigm for approximate reasoning in complex information systems. Its primitive notion is that of a predicate of rough inclusion which gives for any two entities of discourse the degree in which one of them is a part of the other. Rough Mereology may be regarded as an extension of Rough Set Theory as it proposes to argue in terms of similarity relations induced from a rough inclusion instead of reasoning in terms of more strict indiscernibility relations. Rough Mereology is also a generalization of Mereology i.e. a theory of reasoning based on the notion of a part. Classical languages of mathematics are of two-fold kind: the language of set theory (naive or formal) expressing classes of objects as sets consisting of ”elements”, ”points” etc. suitable for objects perceived as built of ”atoms” and applied to structures perceived as discrete and the language of part relations suitable for e.g. continuous objects like solids, regions, etc. where two objects are related to each other by saying that one of them is a part of the other. Mereological theories for reasoning about complex structures are at the heart of Qualitative Spatial Reasoning. In this paper, we study basic aspects of Rough Mereology in Information Systems. Mereology makes the distinction between entities perceived as individuals (singletons), to which the part predicate may be applied, and entities perceived as distributive classes (sets, lists, general names etc.) of entities. This distinction is made formal and precise within Ontology i.e. Theory of Being based on the primitive notion of the copula is which is also a basic ingredient of theories for Spatial Reasoning. The practical aim of Ontology is to elaborate a system of concepts (notions, names, sets of entities) about which the reasoning is carried out. Therefore, we begin our study with an analysis of a simple rough set-based Ontology (the template ontology) in Information Systems and in this setting we present our approach to Mereology in Information Systems. In this framework we introduce Rough Mereology and we present some ways for defining rough inclusions. We demonstrate applications of Rough Mereology to approximate reasoning taking as the case subject Qualitative Spatial Reasoning. We address here some of its mereo-topological as well as mereo-geometrical aspects.

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.

We investigate here the possibility of approximating a language starting from a partial knowledge of its strings. This is understood as the possibility to read only a bounded part of a string: a prefix or a subword. In this way, indiscernibility rela

In this paper we present some strategies for synthesis of decision algorithms studied by us. These strategies are used by systems of communicating agents and lead from the original (input) data table to a decision algorithm. The agents are working wi

In a parallel communicating grammar system, several grammars work together, synchronously, on their own sentential forms, and communicate on request. No restriction is imposed usually about the communicated strings. We consider here two types of rest

We propose a method called analytical morphology for data filtering. The method was created on the basis of some ideas of rough set theory and mathematical morphology. Mathematical morphology makes an essential use of geometric structure of objects w

We generalize the notion of an approximation space introduced in [8]. In tolerance approximation spaces we define the lower and upper set approximations. We investigate some attribute reduction problems for tolerance approximation spaces determined b

We present some methods, based on the rough set and Boolean reasoning approaches, for extracting laws from decision tables. First we discuss several procedures for decision rules synthesis from decision tables. Next we show how to apply some near‐to‐functional relations between data to data filtration. Two methods of searching for new classifiers (features) are described: searching for new classifiers in a given set of logical formulas, and searching for some functions approximating near‐to‐functional relations.