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1

On estimation of priority vectors derived from inconsistent pairwise comparison matrices

Kazibudzki Paweł Tadeusz

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 4

52--59

EN

The most critical and purely heuristic assumption about priority vector estimation on the basis of pairwise comparisons is that which states a positive relationship between the consistency of decision makers’ judgments and the quality of estimates of their priorities. As this issue constitutes the area of interest of the Multi-Criteria Decision Making theory in relation to AHP, it’s examined in this paper via Monte Carlo simulations from the perspective of a new measure of PCM consistency i.e. Index of Square Logarithm Deviations. It needs to be emphasized that such problems of applied mathematics have been already studied via computer simulations as the only way of this phenomenon examination.

2

Inverted Fuzzy Implications in Approximate Reasoning

Suraj Z., Lasek A., Lasek P.

Fundamenta Informaticae

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2016

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Vol. 143, nr 1/2

151--171

EN

In 1973 Lotfi Zadeh introduced the theory of fuzzy logic [17]. Fuzzy logic was an extension of Boolean logic so that it allowed using not only Boolean values to express reality. One kind of basic logical operations in fuzzy logic are so-called fuzzy implications. From over eight decades a number of different fuzzy implications have been described [3] - [16]. In the family of all fuzzy implications the partial order induced from [0,1] interval exists. Pairs of incomparable fuzzy implications can generate new fuzzy implications by usingmin(inf) andmax(sup) operations. As a result the structure of lattice is created ([1], page 186). This leads to the following question: how to choose the correct functions among basic fuzzy implications and other generated as described above. In our paper, we propose a new method for choosing implications. Our method allows to compare two fuzzy implications. If the truth value of the antecedent and the truth value of the implication are given, by means of inverse fuzzy implications we can easily optimize the truth value of the implication consequent. In other words, we can choose the fuzzy implication, which has the greatest or the smallest truth value of the implication consequent or which has greater or smaller truth value than another implication. Primary results regarding this problem are included in the paper [14].

3

Automated Generation of Logical Constraints on Approximation Spaces Using Quantifier Elimination

Doherty P., Szałas A.

Fundamenta Informaticae

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2013

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Vol. 127, nr 1-4

135--149

EN

This paper focuses on approximate reasoning based on the use of approximation spaces. Approximation spaces and the approximated relations induced by them are a generalization of the rough set-based approximations of Pawlak. Approximation spaces are used to define neighborhoods around individuals and rough inclusion functions. These in turn are used to define approximate sets and relations. In any of the approaches, one would like to embed such relations in an appropriate logical theory which can be used as a reasoning engine for specific applications with specific constraints. We propose a framework which permits a formal study of the relationship between properties of approximations and properties of approximation spaces. Using ideas from correspondence theory, we develop an analogous framework for approximation spaces. We also show that this framework can be strongly supported by automated techniques for quantifier elimination.

4

A New Class of Fuzzy Petri Nets for Knowledge Representation and Reasoning

Suraj Z.

Fundamenta Informaticae

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2013

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Vol. 128, nr 1-2

193--207

EN

This paper presents a new class of Petri nets called generalised fuzzy Petri nets. The new class extends the existing fuzzy Petri nets by introducing three operators in the form of triangular norms, which are supposed to function as substitute for the min, max and * (algebraic product) operators. To demonstrate the power and the usefulness of this model, an application of the generalised fuzzy Petri nets in the domain of train traffic control is provided. The new model is more flexible than the classical one as in the former class the user has the chance to define the input/output operators. The proposed approach can be used for knowledge representation and reasoning in decision support systems.

5

Rough Truth Degrees of Formulas and Approximate Reasoning in Rough Logic

She Y., He X., Wang G.

Fundamenta Informaticae

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2011

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Vol. 107, nr 1

67-83

EN

A propositional logic PRL for rough sets was proposed in [1]. In this paper, we initially introduce the concepts of rough (upper, lower) truth degrees on the set of formulas in PRL. Then, by grading the rough equality relations, we propose the concepts of rough (upper, lower) similarity degree. Finally, three different pseudo-metrics on the set of rough formulas are obtained, and thus an approximate reasoning mechanism is established

6

Rough Truth Degrees of Formulas and Approximate Reasoning in Rough Logic

She Y., He X., Wang G.

Fundamenta Informaticae

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2011

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Vol. 111, nr 2

223-239

EN

A propositional logic PRL for rough sets was proposed in [1]. In this paper, we initially introduce the concepts of rough (upper, lower) truth degrees on the set of formulas in PRL. Then, by grading the rough equality relations, we propose the concepts of rough (upper, lower) similarity degree. Finally, three different pseudo-metrics on the set of rough formulas are obtained, and thus an approximate reasoning mechanism is established.

7

Struktura bazy wiedzy w bibliotece FUZZLIB

Kudłacik P.

Studia Informatica

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2010

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Vol. 31, nr 2A

469-478

PL

Biblioteka FUZZLIB to zbiór narzędzi pozwalających tworzyć i zarządzać rożnymi systemami rozmytymi za pomocą prostego w użyciu interfejsu. Szczególnie ułatwiono konfigurację i zarządzanie regułowej bazy wiedzy opartej na strukturach dynamicznych. Niniejszy artykuł opisuje zastosowane rozwiązania oraz interfejs programistyczny.

EN

FUZZLIB library provide a set of tools that let to create and manage diverse fuzzy systems with an easy to use interface. Configuration and management of a rule base, based on dynamic structures, is especially simplified. Article describes developed solutions and program interface.

8

Modeling and Reasoning with Paraconsistent Rough Sets

Vitória A., Małuszyński J., Szałas A.

Fundamenta Informaticae

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2009

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Vol. 97, nr 4

405-438

EN

We present a language for defining paraconsistent rough sets and reasoning about them. Our framework relates and brings together two major fields: rough sets [23] and paraconsistent logic programming [9]. To model inconsistent and incomplete information we use a four-valued logic. The language discussed in this paper is based on ideas of our previous work [21, 32, 22] developing a four-valued framework for rough sets. In this approach membership function, set containment and set operations are four-valued, where logical values are t (true), f (false), i (inconsistent) and u (unknown). We investigate properties of paraconsistent rough sets as well as develop a paraconsistent rule language, providing basic computational machinery for our approach.

9

Analysis of Approximate Petri Nets by Means of Occurrence Graphs

Suraj Z., Fryc B.

Fundamenta Informaticae

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2007

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Vol. 79, nr 3-4

541-551

EN

Approximate Petri nets (AP-nets) can be used for the knowledge representation and approximate reasoning. The AP-net model is defined on the basis of the rough set approach, fuzzy Petri nets and coloured Petri nets. One of the main advantages of AP-net model is a possibility to present the reachability set of a given AP-net by means of an occurrence graph. Such graphs can serve, among others, for analyzing and evaluating an approximate reasoning realized by using AP-net model. The main contribution of the paper is to present the algorithms for construction and analysis of occurrence graphs for the AP-nets, especially in the context of searching for the best decision and finding the shortest distance in order to compute such decision. This approach can be applied to the design and analysis of the formal models for expert systems, control systems, communication systems, etc.

10

A Correspondence Framework between Three-Valued Logics and Similarity-Based Approximate Reasoning

Doherty P., Szałas A.

Fundamenta Informaticae

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2007

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Vol. 75, nr 1-4

179-193

EN

This paper focuses on approximate reasoning based on the use of similarity spaces. Similarity spaces and the approximated relations induced by them are a generalization of the rough set-based approximations of Pawlak [17, 18]. Similarity spaces are used to define neighborhoods around individuals and these in turn are used to define approximate sets and relations. In any of the approaches, one would like to embed such relations in an appropriate logic which can be used as a reasoning engine for specific applications with specific constraints. We propose a framework which permits a formal study of the relationship between approximate relations, similarity spaces and three-valued logics. Using ideas from correspondence theory for modal logics and constraints on an accessibility relation, we develop an analogous framework for three-valued logics and constraints on similarity relations. In this manner, we can provide a tool which helps in determining the proper three-valued logical reasoning engine to use for different classes of approximate relations generated via specific types of similarity spaces. Additionally, by choosing a three-valued logic first, the framework determines what constraints would be required on a similarity relation and the approximate relations induced by it. Such information would guide the generation of approximate relations for specific applications.

11

Timed Approximate Petri Nets

Suraj Z., Fryc B.

Fundamenta Informaticae

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2006

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Vol. 71, nr 1

83-99

EN

Time is one of the most important considerations in designing practical systems. The notion of time plays a vital role in performance evaluation of real-time systems. A new class of timed approximate Petri nets (TAP-nets) is proposed in the paper. This net model combines high-level Petri nets with time and uncertain information. The approach presented in the paper for modelling of uncertainty, imprecision and vagueness is based on rough set theory and fuzzy Petri nets. The TAP-nets can be used for modelling and evaluating of approximate reasoning used to build expert systems, control systems, communication systems, etc. The main advantage of modelling practical systems using the TAP-nets is that the resulting models are simple, intuitive and allow the system analyst to evaluate the performance of such system models.

12

Calculi of Approximation Spaces

Skowron A., Stepaniuk J., Peters J., Swinarski R.

Fundamenta Informaticae

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2006

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Vol. 72, nr 1-3

363-378

EN

This paper considers the problem of how to establish calculi of approximation spaces. Approximation spaces considered in the context of rough sets were introduced by Zdzisaw Pawlak more than two decades ago. In general, a calculus of approximation spaces is a system for combining, describing, measuring, reasoning about, and performing operations on approximation spaces. An approach to achieving a calculus of approximation spaces that provides a basis for approximating reasoning in distributed systems of cooperating agents is considered in this paper. Examples of basic concepts are given throughout this paper to illustrate how approximation spaces can be beneficially used in many settings, in particular for complex concept approximation. The contribution of this paper is the presentation of a framework for calculi of approximation spaces useful for approximate reasoning by cooperating agents.

13

A Petri Net System - an Overview

Suraj Z., Fryc B., Matusiewicz Z., Pancerz K.

Fundamenta Informaticae

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2006

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Vol. 71, nr 1

101-120

EN

Petri nets are one of well established tools in both theoretical analysis and practical modelling of concurrent systems as well as approximate reasoning. However, practical usage of Petri nets is limited by the lack of computer tools which would allow to handle large and complex nets in a comfortable way. Three things are essential for modelling and analyzing by means of Petri nets - good editor, simulator and powerful analysis engine. Moreover, a program should have a graphical user interface providing an opportunity to work directly with the graphical representations of Petri nets and should be able to read and write data in formats of other popular simulators of Petri Nets. This paper presents a set of integrated graphical Petri net tools called Petri Net system (PN-system, in short). PN-system is a following version of PN-tools. This system can be used for constructing, editing and analyzing of different classes of Petri nets. PN-system is enhanced on fuzzy and adaptive fuzzy Petri nets' modules which allow to perform fuzzy reasoning automatically. It has got a graphical user interface. Moreover, PN-system can cooperate with the ROSECON system which is an original software tool for discovering concurrent models from data tables. PN-system is run on IBM PC platform under MS Windows operating system.

14

Fast reasoning in a rule-based system with uncertainty

Jankowska B. M.

Foundations of Computing and Decision Sciences

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2006

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Vol. 31, No. 3-4

243-262

EN

The knowledge processed in empirical domains is more or less uncertain. In order to support people who deal with them, expert systems with uncertainty are used. The expert systems that serve for planning or simulation purposes are often implemented as rule-based systems. To express the uncertainty of facts and rules, different mathematical methods are used: from probability factors and modal logics to the Zadeh's fuzzy logic. The last method is the most general, and it helps to conclude very reliable hypotheses. In the simulation systems both the conclusions' reliability and the time necessary for reasonings are of great importance. In this paper we point at the rule of convergence as a method of reasoning which allows to speed up reasonings performed in rule-based systems with uncertainty. We discuss its advantages, limitations and possible applications.

15

Towards a Framework for Approximate Ontologies

Doherty P., Grabowski M., Łukaszewicz W., Szałas A.

Fundamenta Informaticae

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2003

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Vol. 57, nr 2-4

147--165

EN

Currently, there is a great deal of interest in developing tools for the generation and use of ontologies on the WWW. These knowledge structures are considered essential to the success of the semantic web, the next phase in the evolution of the WWW. Much recent work with ontologies assumes that the concepts used as building blocks are crisp as opposed to approximate. It is a premise of this paper that approximate concepts and ontologies will become increasingly more important as the semantic web becomes a reality. We propose a framework for specifying, generating and using approximate ontologies. More specifically, (1) a formal framework for defining approximate concepts, ontologies and operations on approximate concepts and ontologies is presented. The framework is based on intuitions from rough set theory; (2) algorithms for automatically generating approximate ontologies from traditional crisp ontologies or from large data sets together with additional knowledge are presented. The knowledge will generally be related to similarity measurements between individual objects in the data sets, or constraints of a logical nature which rule out particular constellations of concepts and dependencies in generated ontologies. The techniques for generating approximate ontologies are parameterizable. The paper provides specific instantiations and examples.

16

Modelowanie wnioskowania przybliżonego za pomocą rozmytych sieci Petriego

Olejko Barbara, Suraj Zbigniew

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2002

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z. 26 [199]

191-204

PL

Wnioskowanie przybliżone wykorzystywane jest m. in. przy projektowaniu systemów ekspertowych. Jednym z narzędzi do modelowania takiego wnioskowania są rozmyte sieci Petriego ([l], [2]). Celem referatu jest zaprezentowanie projektu i wstępnej implementacji systemu FPN dla rozmytych sieci Petriego. System ten będzie częścią składową systemu do projektowania i analizy modeli systemów współbieżnych tworzoneg o w Katedrze Podstaw Informatyki, Wyższej Szkoły Informatyki i Zarządzania w Rzeszowie. System FPN będzie wspomagał użytkownika w tworzeniu i symulacji rozmytych sieci Petriego. System będzie umożliwiał również weryfikację i optymalizację bazy wiedzy, niezbędnej przy projektowaniu systemu ekspertowego.

17

Automatyczne odkrywanie modeli współbieżnych z danych eksperymentalnych

Pancerz Krzysztof, Suraj Zbigniew

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

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2002

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z. 26 [199]

205-226

PL

Odkrywanie wiedzy z dużych baz danych jest jednym z istotnych i aktualnych problemów badawczych. W referacie przedstawione zostaną pewne metody automatycznego odkrywania modeli współbieżnych zdanych eksperymentalnych ([4],[5-6]). Metody te oparte są na teorii zbiorów przybliżonych [l] i teorii sieci Petriego [2]. Celem referatu jest prezentacja algorytmów realizujących odkrywanie z danych eksperymentalnych modeli współbieżnych reprezentowanych w formie sieci Petriego. Praktycznym wynikiem tych prac badawczych będzie moduł wspomagający odkrywanie modeli współbieżnych z danych. Moduł będzie częścią składową tworzonego w Katedrze Podstaw Informatyki WSIiZ w Rzeszowie systemu komputerowego do wspomagania projektowania i analizy modeli współbieżnych, działającego na komputerach klasy PC pod kontrolą systemu operacyjnego Windows. Wskazane zostaną także kierunki dalszych badań.

18

Mechanizmy wnioskowania przybliżonego w bazach danych

Małsiak B.

Studia Informatica

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2002

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Vol. 23, nr 4

153-172

PL

W artykule przedstawiono zastosowanie teorii zbiorów rozmytych oraz metod wnioskowania przybliżonego (na przykładzie modelu Mamdaniego) w procesie zadawania rozmytych (nieprecyzyjnych) pytań do bazy danych i generacji odpowiedzi na tak sformułowane pytania.

EN

This article presents fuzzy sets theory and approximate reasoning and their applications in database. Database can include precise or imprecise data, and queries can be precise or imprecise too.

19

Imposing Restrictions on Density Functions Utilised in Computing With Words

Gemeinder M.

International Journal of Applied Mathematics and Computer Science

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2002

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Vol. 12, no 3

383-390

EN

Applying the generalised extension principle within the area of Computing with Words typically leads to complex maximisation problems. If distributed quantities--such as, e.g., size distributions within human populations--are considered, density functions representing these distributions become involved. Very often the optimising density functions do not resemble those found in nature; for instance, an optimising density function could consist of two single Dirac pulses positioned near the opposite bounds of the interval limiting the possible values of the quantity considered. Therefore, in this article, density functions with certain shapes which enable us to overcome this lack of resemblance are considered. Furthermore, some considerations on solving the resulting maximisation problems are reported.

20

A Neuro-Fuzzy System Based on Logical Interpretation of If-then Rules

Łęski J., Henzel N.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 4

703-722

EN

Several important fuzzy implications and their properties are described on the basis of an axiomatic approach to the definition of the fuzzy implications. Then the idea of approximate reasoning using the generalized modus ponens and fuzzy implications is considered. The elimination of the non-informative part of the final fuzzy set before defuzzification plays the key role in this paper. After reviewing well-known fuzzy systems, a new artificial neural network based on logical interpretation of if-then rules (ANBLIR) is introduced. Moreover, this system automatically generates rules from numerical data. Applications of ANBLIR to pattern recognition on numerical examples using benchmark databases are indicated.