Recently published Research Reports (1995) of the Institute of Computer Science, W.U.T. ICS WUT Reports on Rough Set, 1995 can be ordered by: e-mail: bsk@ii.pw.edu.pl fax: (+48 22) 25 16 35 or by mail Bozena Skalska Institute of Computer Science Reports Warsaw Uniwersity of Technology ul. Nowowiejska 15/19 00 665 Warsaw Poland 1/95 Zdzislaw Pawlak [zpw@ii.pw.edu.pl] Rough Real Functions and Rough Controllers ABSTRACT: The aim of this paper is to give some basic ideas of rough functions, i.e. functions which can be understood as approximations of real functions, based on the rough set theory. 2/95 Krzysztof Slowinski [slowik@pozn1v.tup.edu.pl], Jan Fibak, Waldemar Jankowiak Verification of Conclusions from Rough Set Analysis of Highly Selective Vagotomy (HSV) During a Follow-up Program ABSTRACT: 213 duodenal ulcer patients were operated using highly selective vagotomy. These patients took part in a follow-up program divided into three time periods: 1976-1982 (77 patients), 1983-1986 (45 patients), 1987-1992 (91 patients). All of them were described by 11 pre-operating attributes concerning 4 anamnesis ones and 7 gastric secretion ones and classified from viewpoint of long term result of the operation into 4 classes. These data were collected in view of investigating dependencies existing among the pre-operating attributes and the long term result of operation. To carry out this investigation the rough set methodology have been used. Application of rough sets results from two periods of the follow-up program to decision making concerning patients in the third period, gave an improvement of operational results. 3/95 Mikhail Moshkov [skowron@mimuw.edu.pl] Unimprovable Upper Bounds on Complexity of Decision Trees over Information Systems ABSTRACT: For arbitrary information systems without constant attributes the complexity of deterministic and nondeterministic decision trees using attributes from this information system is considered. 4/95 Mikhail Moshkov [skowron@mimuw.edu.pl] Complexity of Deterministic and Nondeterministic Decision Trees. Global Approach ABSTRACT: Relations between the complexity of a task description and the minimal complexity of deterministic and nondeterministic decision trees solving this task are studied. The global approach to investigation of decision trees where arbitrary checks from check system can be used for constructing decision trees is considered. 5/95 Ibrahim Tentush [polk@mimuw.edu.pl] On Rough Mereological Cech Topologies ABSTRACT: We introduce a topological structure into a rough mereological space. The introduced Cech topological spaces as subspaces the quasi-topological spaces introduced by Clark in the calculus of individuals based on connection. The starting point is a rough inclusion introduced in [8] as a generalization of rough membership function [7]. 7/95 Bozena Kostek [bozenka@next.elka.pg.gda.pl] Statistical versus Artificial Intelligence Based Processing of Subjective Test Results ABSTRACT: A comparison of both statistical and non-statistical methods applied to the processing of subjective testing results has been presented. Some principles underlying the new proposed method were reviewed. Some exemplary data derived from listening tests were processed using parametric, non-parametric and rough sets-based methods. Conclusions concerning both statistical and non-statistical approaches to the processing of acoustic data will be discussed. 8/95 Andrzej Lenarcik, Zdzislaw Piasta [{ztp-al,mat-zp}@serv1.tu. kielce.pl] Minimizing the Number of Rules in Deterministic Rough Classifiers ABSTRACT: A method for generating an inductive decision algorithm is presented. The algorithm construction is based on an information about finite set of objects which is involved in a knowledge representation system. The case of knowledge representation systems with continuous condition attributes and one discrete decision attribute is considered. The main goal of the used approach is to minimize the number of decision rules in the algorithm under the assumption that the rules are deterministic. In the first stage of the algorithm generation the minimal global discretization of continuous condition attributes space is found. In the paper the optimal coding is performed using the rough set approach. The procedure involves replacing continuous condition attributes by binary attributes. The method is illustrated with the aid of three real-life examples. 9/95 Adam Obtulowicz [adamo@impan.impan.gov.pl] Some Remarks on Rough Real Functions SUMMARY: In the paper The Intermediate Value Theorem for roughly continuous functions is discussed and the coincidence of certain roughly continuous functions with homomorphisms of some Heyting algebra valued sets is presented. 10/95 Zdzislaw Pawlak [zpw@ii.pw.edu.pl] Rough Set Approach to Knowledge-Based Decision Support ABSTRACT: The rough set concept is a new mathematical approach to imprecision, vagueness and uncertainty. To some extend it overlaps with fuzzy set theory and evidence theory - nevertheless the rough set theory can be viewed in its own right, as an independent discipline. Many real-life applications of the theory have proved its practical usefulness. The paper presents the basic assumptions underlying the rough set philosophy, gives its fundamental concepts and discusses briefly some areas of applications, in particular in decision support. Finally further problems are shortly outlined. 11/95 Andrzej Skowron [skowron@mimuw.edu.pl], Son H. Nguyen [son@mimuw.edu.pl] Quantization of Real Value Attributes: Rough Set and Boolean Reasoning Approach ABSTRACT: The quantization of real value attributes is a crucial problem to be solved in synthesis of decision rules from data tables with real value attributes. We present a novel approach to this problem based on rough set approach and boolean reasoning. The main result states that the problem of optimal quantization of real value attributes is polynomially reducible to the problem of minimal reduct finding, so it is NP-hard. We construct efficient heuristics for finding suboptimal quantization of real value attributes. 12/95 Andrzej Skowron [skowron@mimuw.edu.pl] Synthesis of Adaptive Decision Systems from Experimental Data ABSTRACT: We discuss two basic questions related to the analysis and synthesis of behavior of cooperating intelligent agents in adaptive systems. The first one can be formulated as follows: What strategies are used by an intelligent agent allowing him to discover the decision rules from experimental data? We propose to build these strategies on the basis of rough set methods and boolean reasoning techniques. We present some applications of these methods for extracting decision rules from decision tables used to represent experimental data. The second one can be formulated as follows: What is a general framework for approximate reasoning about behavior of teams of intelligent agents? We will assume agents to be organized on the rough mereological principles to assembly (construct) complex objects satisfying in a satisfactory degree a given specification. We discuss how this approach can be used for building the foundations for approximate reasoning. Our approach is based on recently developed rough mereology being an extension of Lesniewski's mereology. 13/95 Marek Kretowski [mkret@ii.pb.bialystok.pl], Lech Polkowski, Andrzej Skowron [skowron@mimuw.edu.pl], Jaroslaw Stepaniuk [jstepan@ii.pb.bialystok.pl] Data Reduction Based on Rough Set Theory ABSTRACT: We present a data reduction technique allowing to reduce the number of examples and the number of attributes involved in the learning process from examples. The presented technique adopts the notions of an attribute reduct and a absorbent set of object set. 14/95 Jakub Wr¢blewski [jakubw@alfa.mimuw.edu.pl] Finding Minimal Reducts Using Genetic Algorithm SUMMARY: An application of genetic algorithm to short reducts finding is presented. Experimental results, including optimal values of algorithm parameters, are described. 15/95 Andrzej Skowron, Zbigniew Suraj [{skowron,suraj}@mimuw.edu.pl] Discovery of Concurrent Data Models from Experimental Data Tables: A Rough Set Approach ABSTRACT: The discovery of relations between data and of data models are the main objectives of the machine discovery. In the paper we describe a method for discovery of data models from experimental tables. The data models are represented by concurrent systems. Our method is based on a decomposition of systems specified by experimental data tables. The presented method has been applied for automatic data models discovery from experimental tables. As a model for concurrency Petri nets have been chosen. 16/95 Jakub Wr¢blewski [jakubw@alfa.mimuw.edu.pl] Finding Minimal Reducts Using Genetic Algorithm (extended version) ABSTRACT: An application of three types of genetic algorithm to short reducts finding is presented. The first mehtod: classical genetic algorithm with individuals represented by bit strings, appeared to be fast, but sometimes fails to find the global optimum. The second and third method bases on permutational coding and "greedy" algorithms. The results are much better, but the computation time increases. 17/95 Jan Komorowski [Jan.Komorowski@idt.unit.no], Lech T. Polkowski [polk@mimuw.edu.pl], Andrzej Skowron [skowron@mimuw.edu.pl] Towards a Rough Mereology-Based Logic for Approximate Solution Synthesis, Part 1 ABSTRACT: We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as schemes constructed in order to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely. 18/95 Rafal Deja [RD@labeto.keto.gliwi] Conflict Analysis Based on the Distance Function ABSTRACT: Some further issues on a conflict analysis regarding ideas of Z. Pawlak and A. Skowron are presented in this paper. To analyse conflicts effectively we need possibly simple and powerful formal model. As it was showed by Z. Pawlak in "On Conflicts" and then in "On Some Issues Connected with Conflict Analysis", we can use the rough set theory and decision table model for analysing the situation and possibly for finding the solution (consensus). In that model the relation of each agent to the specific issue is depicted in the form of the table with agents in rows and issues in the columns. The value assigned to each agent and to each attribute (issue) could be one from the triple {-1, 0, 1}, where -1 means, that the agent is against, 0 neutral and 1 - favourable toward the issue. 19/95 Robert M. Colomb, Jacek Sienkiewicz [jhs@ii.pw.edu.pl] Analysis of Redundancy in Expert Systems Case Data ABSTRACT: This paper reports a method of describing systematic redundancy in the attribute-space representation of cases input to classification-type expert systems. The redundancy is described in terms of partial functional dependencies among the attributes, estimated using methods derived from the theory of rough set. The method is validated by application to the well-known Garvan ESI data set. The description of redundancy can be used to test whether a data vector is a valid case before processing it by an expert system, and as a basis of a simulator generating random instances of the process as input to the expert system. Several other possible uses are noted. 20/95 Dorota Nejman [dnm@ii.pw.edu.pl Rough Sets in Handwritten Numerals Recognition ABSTRACT: In the paper we apply rough sets to decision making concerned with handwritten numerals recognition. On a basis of a set of numerals we built a rule based system which was tested on a different set of numerals. We examined several rule based systems and some ways of matching for unknown information vectors. It occurs that with the knowledge from the decision rules we can correctly classify a lot of confusing information vectors. 21/95 Zdzislaw Pawlak [zpw@ii.pw.edu.pl] On Some Issues Connected with Roughly Continuous Functions ABSTRACT: - 22/95 Barbara Marszal-Paszek, Piotr Paszek, Alicja Wakulicz-Deja, [paszek@usctoux1.cto.us.edu.pl] Applying Rough Sets to Diagnose in Children's Neurology, ABSTRACT: The work contains an example of applying the rough set theory to decide about the necessity of further tests and making final decisions connected with a diagnose, mode by a physician, about progressive encephalopathy in a child. Making the final decisions requires a series of invasive tests and that is why it is essential to carry out processes of appropriate preliminary classification. The process of the preliminary classification of patients with the use of the rough set theory is presented in the work. 23/95 Mikhail Moshkov [skowron@mimuw.edu.pl] Relationships Between Depth of Deterministic and Nondeterministic Programs Computing Functions of k-Valued Logic, ABSTRACT: Two types of programs computing functions of k-valued logic are considered: decision trees and acyclic programs over a complete finite basis. Relationships between the minimal depth of deterministic and the minimal depth of nondeterministic programs computing a function are studied. 28/95 Adam Obtulowicz [adamo@impan.impan.gov.pl] Differential Equations for Discrete Functions ABSTRACT: It is introduced and discussed a concept of differential equation for discrete functions. It is shown that this concept coincide with an idea of defining functions by using some induction schemes. 29/95 M. K. Chakraborty and Sanjukta Basu, Approximate Reasoning Methods in Vagueness: Graded and Rough Consequences ABSTRACT: - 30/95 Krzysztof Slowinski [slowik@pozn1v.tup.edu.pl] Diagnostic Peritoneal Lavage for Multiple Injuries Patients, Analysis of Experience Using Rough Set Approach ABSTRACT: - 32/95 Zdzislaw Pawlak [zpw@ii.pw.edu.pl[ Rough Sets Present State and Further Prospects ABSTRACT: The rough set theory is a new mathematical approach to vagueness and uncertainty. To some extend it overlaps with some other mathematical tools developed to deal with imperfect knowledge, in particular with fuzzy set theory and evidence theory - nevertheless the rough set theory can be viewed in its own rights, as an independent discipline. Many real-life applications of the theory have proved its usefulness. The paper characterizes the philosophy underlying the rough set theory, gives its rudiments and discusses briefly some areas of applications. At the end some further problems are briefly outline. 33/95 Andrzej Lenarcik and Zdzislaw Piasta [{ztp-al,mat-zp}@serv1.tu.kielce.pl ] Rough Classifiers with Mixtures of Discrete and Continuous Condition Attributes ABSTRACT: The purpose of this paper is to generalize the procedure of the rough classifier construction to the mixed case discrete and continuous attributes. The attribute values can be expressed now in binary, nominal or ordinal scale as well as in interval or ratio scale. The rough classifiers combine features of machine learning classification models with probabilistic methods of data structure description. A minimization of misclassification cost is used as the criterion of the rough classifier generation. The method is illustrated on two datasets from the credit assessment domain. 34/95 Andrzej Czyzewski and Andrzej Kaczmarek [zid@next.elka.pg.gda.pl] Speaker-Independent Recognition of Isolated Words Using Rough Sets ABSTRACT: The aim of the presented research is to elaborate and to test the speaker-independent system for the man-machine voice interfacing. The trajectory tracking method was implemented to feature vector extraction from speech signal. Statistical method was employed to the quantization of attribute values. The rough set method was used to derive decision rules for the recognition of speech patterns. Results and conclusions on the effectiveness of implemented algorithmic solutions were presented. 35/95 Bozena Kostek [bozenka@next.elka.pg.gda.pl] Computer Based Recognition of Musical Phrases Using the Rough Set Approach ABSTRACT: The aim of this work was to study problems related to the recognition of musical pieces using the rough set approach. A date base consisted of MIDI files based on Bach's fugues was prepared. Some modified forms of musical phrases were included. Methods of converting MIDI files into raw data at first, and then into parametrial vectors were presented. A rough-set based system was used for the recognition of musical phrases. Problems related to the automatic recognition of musical phrases were discussed. 36/95 Jaroslaw Stepaniuk and Marek Kretowski [jstepan@ii.pb.bialystok.pl, mkret@ii.pb.bialystok.pl] Decision System Based on Tolerance Rough Sets ABSTRACT: We present a decision system, which is based on tolerance relations and the rough set theory. It consists from a few almost separate subsystems: searching for tolerance thresholds, data reduction, tolerance decision rules generation. We investigate various similarity measures and tolerance thresholds to find out tolerance sets. We proposed technique whose aim is to reduce the number of examples and the number of attributes involved in the process of learning from examples. We also present algorithm for decision rules generation and classification of new instances. 38/95 Salvatore Greco, Benedetto Matarazzo and Roman Slowinski [slowinski@pozn1v.tup.edu.pl] Rough Set Approach to Multi-Attribute Choice and Ranking Problems ABSTRACT: We propose an original way of applying the rough set theory to the analysis of multi-attribute preference systems in the choice (Pþ) and ranking (Pþ) decision problematics. From the viewpoint of rough set theory, this approach implies to consider a pairwise comparison table, i.e. an information table whose objects are pairs of actions instead of single actions, and whose entries are binary relations instead of attribute values. From the viewpoint of multi-attribute decision methodology, this approach allows both representation of decision maker's (DM's) preferences in terms of "if ...then..." rules and their use for recommendation in Pþ and Pþ problematics, without assessing such preference parameters as importance weights and substitution rates. The rule representation of DM's preferences is alternative to traditionally used functional or relational models. The rough set methodology is explained in detail and illustrated by a didactic example. 40/95 Edward Bryniarski, Urszula Wybraniec-Skardowska [edlog@.uni.opole.pl, uws@.uni.opole.pl] Generalized Rough Sets in Contextual Spaces ABSTRACT: This paper originates from the conceptions of rough setspresented by Pawlak (1982, 1992). It also refers to Ziarko's conception (1993) and the conceptions of Blizard's multisets (1989a, 1989b), Zadeh's fuzzy sets (1965), and the authors'conceptions of generalized rough sets and an approximation space called the contextual space (1995). The authors' conception of contextual space is inspired by Ziarko's approach (1993) to rough sets. Rough sets introduced by Pawlak (1982) are particular cases of contextual rough sets defined in the contextual approximation space. This space is defined axiomatically by means of so called con\-text relations. Every contextual rough set determined by set X can be determined by the sum of the lower approximation of X and a subset of the boundary of X. One of the important notions of the conception is the notion of an element of a contextual rough set which allows for formulating and proving the counterpart of the axiom of extensionality for contextual rough sets. 41/95 Zdzislaw Pawlak [zpw@ii.pw.edu.pl] On Rough Derivatives, Rough Integrals and Rough Differential Equations ABSTRACT: In this paper we define rough (discrete) lower and upper representation of real functions and define and investigate some properties of these representations, such as rough continuity, rough derivatives, rough integral and rough differential equations - which can be viewed as discrete counterparts of real functions. An illustrative example of the introduced concepts is given. The presented approach can be used to synthesis and analysis of discrete dynamic system, in particular in control theory. It is also related to the qualitative reasoning methods.