Piotr Wasilewski

Foundations of near sets

This paper introduces an approach to the foundations of information science considered in the context of near sets. Perceptual information systems (or, more concisely, perceptual systems) provide stepping stones leading to nearness relations, near sets and a framework for classifying perceptual objects. This work has been motivated by an interest in finding a solution to the problem of how one goes about discovering affinities between perceptual granules such as images. Near set theory provides a formal basis for observation, comparison and classification of perceptual granules. This is made clear in this article by considering various nearness relations that define coverings of sets of perceptual objects that are near each other. In the near set approach, every perceptual granule is a set of objects that have their origin in the physical world. Objects that have, in some degree, affinities are considered perceptually near each other, i.e., objects with similar descriptions. This article includes a comparison of near sets with other approaches to approximate knowledge representation and a sample application in image analysis. The main contribution of this article is the introduction of a formal foundation for near sets and a demonstration that the family of near sets is a Gratzer slash lattice.

Tolerance spaces: Origins, theoretical aspects and applications

This article considers the origins, theoretical aspects and applications of tolerance spaces. In terms of the origin of tolerance spaces, this article calls attention to the seminal work by J.H. Poincare (1854-1912) and E.C. Zeeman (1925-) on establishing the foundations for tolerance spaces. During the period from 1895 to 1912, Poincare introduced sets of sensations and sequences of almost the same sensations as a means of characterizing the physical spectrum. The perception of physical objects that are almost the same leads to a tolerance space view of visual perception as well as other forms of perception such as touch and sound. Roughly 60years later (in 1962), Zeeman formally introduced the notion of a tolerance space as a useful means of investigating a geometry of visual perception. In addition to the general theory of tolerance spaces, this article also carries forward earlier work on perceptual tolerance relations and considers the resemblance (nearness) between tolerance spaces. From an information systems point of view, it can be observed that tolerance spaces have proved to be fruitful in a number of research areas. Evidence of the utility of tolerance spaces in information systems can be seen in the introduction of tolerance rough sets, tolerance approximation spaces, and tolerance near sets. The contribution of this article is an overview of tolerance spaces considered in the context of visual perception and a presentation of a formal basis for the study of perceptual tolerance spaces.

Granular Sets --- Foundations and Case Study of Tolerance Spaces

A novel approach to extend the notions of definability and rough set approximations in information systems with non-equivalence relations is proposed. The upper approximation is defined as set-theoretic complement of negative region of a given concept; therefore, it does not need to be definable. Fundamental properties of new approximation operators are compared with the previous ones reported in literature. The proposed idea is illustrated within tolerance approximation spaces. In particular, granulation based on maximal preclasses is considered.

Interactive information systems: Toward perception based computing

We discuss the role of interactions in the modeling of perception processes. Interactive information systems, introduced in this paper, play the important role in this modeling. Moreover, the proposed approach opens a new research direction in rough set theory. In this approach, partial information about the environment used for the approximation of concepts is changing dynamically in a network of interacting information systems contrary to static information systems typically used in rough set theory so far. In particular, we illustrate the use of such information systems for representation of actions or plans, their (changing in time) pre and post conditions. These information systems create a starting point for perception modeling, i.e., modeling of the process of understanding of sensory measurements. We also propose interactive grammars as a tool for modeling interactive computations in perception based computing.

Introduction to Perception Based Computing

We discuss basic notions of Perception Based Computing (PBC). Perception is characterized by sensory measurements and ability to apply them to reason about satisfiability of complex vague concepts used, e.g., as guards for actions or invariants to be preserved by agents. Such reasoning is often referred as adaptive judgment. Vague concepts can be approximated on the basis of sensory attributes rather than defined exactly. Approximations usually need to be induced by using hierarchical modeling. Computations require interactions between granules of different complexity, such as elementary sensory granules, granules representing components of agent states, or complex granules representing classifiers that approximate concepts. We base our approach to interactive computations on generalized information systems and rough sets. We show that such systems can be used for modeling advanced forms of interactions in hierarchical modeling. Unfortunately, discovery of structures for hierarchical modeling is still a challenge. On the other hand, it is often possible to acquire or approximate them from domain knowledge. Given appropriate hierarchical structures, it becomes feasible to perform adaptive judgment, starting from sensory measurements and ending with conclusions about satisfiability degrees of vague target guards. Thus, our main claim is that PBC should enable users (experts, researchers, students) to submit domain knowledge, by means of a dialog. It should be also possible to submit hypotheses about domain knowledge to be checked semi-automatically. PBC should be designed more like laboratories helping users in their research rather than fully automatic data mining or knowledge discovery toolkit. In particular, further progress in understanding visual perception – as a special area of PBC – will be possible, if it becomes more open for cooperation with experts from neuroscience, psychology or cognitive science. In general, we believe that PBC will soon become necessity in many research areas.

Perceptual tolerance intersection

This paper introduces a perceptual tolerance intersection of sets as an example of near set operations. Such operations are motivated by the need to consider similarities between digital images viewed as disjoint sets of points. The proposed approach is in keeping with work by E.C. Zeeman on tolerance spaces and visual perception and J.H. Poincare on sets of similar sensations used to define representative (aka tolerance) spaces such as visual, tactile and motile spaces. Perceptual tolerance intersection of sets is a direct consequence of recent work on near sets and a solution to the problem of how one goes about discovering affinities between digital images. The main contribution of this article is a description-based approach to assessing the resemblances between digital images.

Information systems in modeling interactive computations on granules

In this paper we discuss the importance of information systems in modeling interactive computations performed on (complex) granules and propose a formal approach to interactive computations based on information systems. The basic concepts of information systems and rough sets are interpreted in the framework of interactive computations. We also show that information systems can be used for modeling more advanced forms of interactions such as hierarchical ones. The role of hierarchical interactions is emphasized in modeling interactive computations. Some illustrative examples of interactions used in the hierarchical multimodal classification method as well as in the ACT-R 6.0 system are reported.

Concept lattices vs. approximation spaces

The aim of this paper is to compare concept lattices and approximation spaces. For this purpose general approximation spaces are introduced. It is shown that formal contexts and information systems on one hand and general approximation spaces on the other could be mutually represented e.g. for every information system exists a general approximation space such that both structures determines the same indiscernibility relation. A close relationship between Pawlaks approximation spaces and general approximation spaces also holds: for each approximation space exists a general approximation space such that both spaces determine the same definable sets. It is shown on the basis of these relationships that an extent of the every formal concept is a definable set in some Pawlaks approximation space. The problem when concept lattices are isomorphic to algebras of definable sets in approximation spaces is also investigated.

Granular approach in knowledge discovery: real time blockage management in fire service

This article is focused on the recognition and prediction of blockages in the fire stations using granular computing approach. Blockage refers to the situation when all fire units are out and a new incident occurs. The core of the method is an estimation of the expected return times for the fire brigades based on the granularisation of source data. This estimation, along with some other considerations allows for evaluation of the probability of the blockage.

Rough Sets and Sorites Paradox

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 Zdzis law 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.