Fuzzy information processing

Guilherme A. Barreto
Ricardo Coelho (Eds.)

Communications in Computer and Information Science

Fuzzy Information
Processing
37th Conference of the North American Fuzzy
Information Processing Society, NAFIPS 2018
Fortaleza, Brazil, July 4–6, 2018
Proceedings

123

831


Communications
in Computer and Information Science
Commenced Publication in 2007
Founding and Former Series Editors:
Alfredo Cuzzocrea, Xiaoyong Du, Orhun Kara, Ting Liu, Dominik Ślęzak,
and Xiaokang Yang

Editorial Board
Simone Diniz Junqueira Barbosa
Pontifical Catholic University of Rio de Janeiro (PUC-Rio),
Rio de Janeiro, Brazil
Phoebe Chen
La Trobe University, Melbourne, Australia
Joaquim Filipe

Polytechnic Institute of Setúbal, Setúbal, Portugal
Igor Kotenko
St. Petersburg Institute for Informatics and Automation of the Russian
Academy of Sciences, St. Petersburg, Russia
Krishna M. Sivalingam
Indian Institute of Technology Madras, Chennai, India
Takashi Washio
Osaka University, Osaka, Japan
Junsong Yuan
University at Buffalo, The State University of New York, Buffalo, USA
Lizhu Zhou
Tsinghua University, Beijing, China

831


More information about this series at />

Guilherme A. Barreto Ricardo Coelho (Eds.)
•

Fuzzy Information
Processing
37th Conference of the North American Fuzzy
Information Processing Society, NAFIPS 2018
Fortaleza, Brazil, July 4–6, 2018
Proceedings

123

Editors
Guilherme A. Barreto
Department of Teleinformatics Engineering
Federal University of Ceará
Fortaleza, Ceará
Brazil

Ricardo Coelho
Department of Statistics & Applied
Mathematics
Federal University of Ceará
Fortaleza, Ceará
Brazil

ISSN 1865-0929
ISSN 1865-0937 (electronic)
Communications in Computer and Information Science
ISBN 978-3-319-95311-3
ISBN 978-3-319-95312-0 (eBook)
/>Library of Congress Control Number: 2018947460
© Springer International Publishing AG, part of Springer Nature 2018
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Preface

In 1965, Lofti Asker Zadeh published the seminal paper “Fuzzy Sets” (Information and
Control, 8, 338–353), which describe the first ideas about a formal mathematical
modelling intended to bridge the gap between classic binary modelling and the subjective way that humans relate to day-to-day situations. Despite these ideas being
ambitious, this preliminary work inspired many researchers around the world and today
his ideas are found in almost all branches of science. According to the website Google
Scholar, this seminal paper has been cited in more than 100,000 scholarly works, and
many consumer products and software have been built based on its mathematical
concepts. Unfortunately, Professor Zadeh died in September 2017, and this book is a
modest tribute to the generous, gentle continuous and always friendly support that the
authors received over the years from Professor Lofti A. Zadeh.
It can be noted that the research field of fuzzy sets and systems has undergone
tremendous growth since 1965. This growth is in no small measure the result of the
emergence of some important scientific societies in North America (North American
Fuzzy Information Processing Society – NAFIPS – and IEEE Computational Intelligence Society – IEEE CIS), Europe (European Society for Fuzzy Logic and Technology – EUSFLAT), Asia (Japan Society for Fuzzy Theory and Intelligent Informatics
– JSFTII), and South America, especially in Brazil, (Brazilian Society of Automatics –
SBA – and Brazilian Society of Computational and Applied Mathematics – SBMAC).
There is also a transnational scientific society (International Fuzzy Systems Association

– IFSA). These societies promote scientific events in order to spread the state of the art,
its applications, and technological advances. A quick search in the Scopus database
gives an idea of the number of published articles about fuzzy sets and systems. By
dividing time from 1965 until today into four periods, we obtain the following:
(a) 4,754 published papers until 1990; (b) 27,773 published papers from 1991 to 2000;
(c) 93,012 from 2001 to 2010; and (d) 105,604 from 2011 to the current date (May
2018). This search was made by using the words “Fuzzy Sets” or “Fuzzy Systems” or
“Fuzzy Logic” as title, abstract, or keywords.
Among the societies mentioned, NAFIPS is the premier fuzzy society in North
America, which was founded in 1981. The purpose of NAFIPS is “the promotion of the
scientific study of, the development of an educational institution for the instruction in,
and the dissemination of educational materials in the public interest including, but not
limited to, theories and applications of fuzzy sets through publications, lectures, scientific meetings, or otherwise.” In this role, we understand the importance and
necessity of developing a strong intellectual base and encouraging new and innovative
applications. In addition, we recognize our leading role in promoting interactions and
technology transfer to other national and international organizations so as to bring the
benefits of this technology to North America and the world. The scientific event
organized by the NAFIPS has been contributing for more than 30 editions to the


VI

Preface

growth of the number of articles published in the fuzzy sets and systems field. The first
edition took place in the city of Logan, Utah, USA, in 1982, and it is held annually.
One of the objectives of NAFIPS is to expand the network of collaborators and
enthusiasts of fuzzy thinking beyond the borders of North American countries. The
37th North American Fuzzy Information Processing Society Annual Conference
(NAFIPS 2018) was held during July 4–6, 2018, in the beautiful city of Fortaleza,

capital of the state of Ceará, located on the sunny northeast coast of Brazil. This event
was held simultaneously with the 5th Brazilian Congress on Fuzzy Systems (CBSF
2018), bringing together researchers, engineers, and practitioners to share and present
the latest achievements and innovations in the area of fuzzy information processing, to
discuss thought-provoking developments and challenges, and to consider potential
future directions. Bearing this in mind, the NAFIPS 2018 meeting was the first edition
of the meeting to be organized outside the USA, Canada, and Mexico. NAFIPS 2018
had an international Program Committee including researchers from industry and
academia worldwide.
The organization of NAFIPS 2018 and CBSF 2018 was the result of a joint action
of the Brazilian Computational Intelligence Society (SBIC), the Brazilian Society of
Computational and Applied Mathematics (SBMAC), the Federal University of Ceará
(UFC), and the Brazilian funding agencies CAPES, process 88887.155510/2017-00,
and CNPq, project 407666/2017-6, in addition to the executive boards of NAFIPS and
CBSF.
This book is a collection of high-quality papers ranging over a large spectrum of
topics, including theory and applications of fuzzy numbers and sets, fuzzy logic, fuzzy
inference systems, fuzzy clustering, fuzzy pattern classification, neuro-fuzzy systems,
fuzzy control systems, fuzzy modeling, fuzzy mathematical morphology, fuzzy
dynamical systems, time series forecasting, and making decision under uncertainty.
We received 73 submissions from 11 countries, from which 54 papers were
accepted. The authors were from Brazil, Chile, Colombia, Czech Republic, India, Iran,
Mexico, Romania, Spain, Turkey, and the USA. Each submitted paper was reviewed
by at least three independent referees. The acceptance/rejection decision used the
following criteria: every paper with two positive reviews was accepted, and those with
two negative reviews were rejected. Borderline papers, those with one positive and one
negative review, were analyzed carefully by the conference chairs in order to evaluate
the reasons given for acceptance or rejection. Our final decision on these submissions
took into account mainly the potential of each paper to foster fruitful discussions and
the future development of the research on the theory and applications of fuzzy sets and

systems in Brazil and, for extension, in the whole of Latin America.
We are enormously grateful to all reviewers for their goodwill in cooperating for the
success of the aforementioned events. We very much appreciate their willingness for
hard work and prompt feedback, which certainly guaranteed the high quality of the
technical program.
We wish NAFIPS a long life. And we wish a long life for the Brazilian community,
who organizes CBSF, with which we share this mutual congress.
June 2018

Guilherme A. Barreto
Ricardo Coelho


Organization

General Co-chairs
Guilherme Barreto
Ricardo Coelho

Federal University of Ceará, Brazil
Federal University of Ceará, Brazil

Organizing Committee
Fernando Gomide
Guilherme Barreto
Laecio Carvalho de Barros
Patricia Melin
Ricardo Coelho
Weldon Lodwick
Centro Acadêmico de

Matemática Industrial

University of Campinas, Brazil
Federal University of Ceará, Brazil
University of Campinas, Brazil
Tijuana Institute of Technology, Mexico
Federal University of Ceará, Brazil
University of Colorado Denver, USA
CAMI

Web Masters
Felipe Albuquerque
Francisco Yuri Martins

Federal University of Ceará, Brazil
Federal University of Ceará, Brazil

NAFIPS Officers
Patricia Melin (President)
Martine Ceberio (President-Elect)
Christian Servin (Treasurer)
Valerie Cross (Secretary)

NAFIPS Board of Directors
Ildar Batyrshin
Ricardo Coelho
Martine De Cock
Scott Dick
Juan Carlos Figueroa Garcia
Weldon A. Lodwick

Marek Reformat
Shahnaz Shahbazova
Mark Wierman
Dongrui Wu


VIII

Organization

Program Committee
Giovanni Acampora
Plamen Angelov
Krassimir Atanassov
Adrian I. Ban
Guilherme Barreto
Laecio Carvalho de Barros
Ildar Batyrshin
Fernando Bobillo
Giovanni Bortolan
Tadeusz Burczynski
João Paulo Carvalho
Oscar Castillo
Martine Ceberio
Wojciech Cholewa
Ricardo Coelho
Lucian Coroianu
Valerie Cross
Bernard de Baets
Didier Dubois

Robert Fuller
Takeshi Furuhashi
Fernando Gomide
Wladyslaw Homenda
Sungshin Kim
Peter Klement
Vladik Kreinovich
Jonathan Lee
Weldon A. Lodwick
Francesco Marcelloni
Radko Mesiar
Vesa Niskanen
Fabrício Nogueira
Vilem Novak
Reinaldo Martinez Palhares
Irina Perfilieva
Henri Prade
Radu Emil Precup
Alireza Sadeghian
Yabin Shao
Andrzej Skowron
Umberto Straccia
Ricardo Tanscheit

University of Naples Federico II, Italy
Lancaster University, UK
Bulgarian Academy of Science, Bulgaria
University of Oradea, Romania
Universidade Federal do Ceará, Brazil
University of Campinas, Brazil

Instituto Politécnico Nacional, Mexico
University of Zaragoza, Spain
Institute of Neuroscience, IN-CNR, Italy
Institute of Fundamental Technological Research,
Poland
Instituto Superior Tecnico/INESC-ID, Portugal
Tijuana Institute of Technology, Mexico
University of Texas at El Paso, USA
Silesian University of Technology, Poland
Universidade Federal do Ceará, Brazil
University of Oradea, Romania
Miami University, USA
Ghent University, Belgium
Université Paul Sabatier, France
Óbuda University, Hungary
Nagoya University, Japan
University of Campinas, Brazil
Warsaw University of Technology, Poland
Pusan National University, South Korea
Johannes Kepler University, Austria
University of Texas at El Paso, USA
National Central University, Taiwan
University of Colorado Denver, USA
University of Pisa, Italy
Slovak University of Technology, Slovakia
University of Helsinki, Finland
Universidade Federal do Ceará, Brazil
University of Ostrava, Czech Republic
Federal University of Minas Gerais, Brazil
University of Ostrava, Czech Republic

Université Paul Sabatier, France
Politehnica University of Timisoara, Romania
Ryerson University, Canada
Northwest University for Nationalities, China
University of Warsaw, Poland
Consiglio Nazionale delle Ricerche, ISTI-CNR, Italy
Pontifícia Universidade Católica do Rio de Janeiro,
Brazil


Organization

Jose Luis Verdegay
Yiyu Yao
Hao Ying
Fusheng Yu
Slawomir Zadrozny
M. H. Fazel Zarandi
Guangquan Zhang
Hans J. Zimmermann

University of Granada, Spain
University of Regina, Canada
Wayne State University, USA
Beijing Normal University, China
Polish Academy of Sciences, Poland
Amirkabir University of Technology, Iran
University of Technology Sydney, Australia
RWTH Aachen University, Germany

Additional Reviewers
João Fernando Alcântara
Aluizio Araújo
Rodrigo Araújo
Romis Attux
Iury Bessa
Arthur Braga
Luiz Cordovil
Pedro Coutinho
Alexandre Evsukoff
Carmelo J.A. Bastos Filho
Heriberto Román Flores
João Paulo Pordeus Gomes

Jose Manuel Soto Hidalgo
Daniel Leite
Adi Lin
José Everardo Bessa Maia
Sebastia Massanet
Ajalmar Rêgo da Rocha Neto
Rudini Sampaio
Peter Sussner
George Thé
Marcos Eduardo Valle
Bin Wang
Zhen Zhang

IX

Contents

Formal Verification of a Fuzzy Rule-Based Classifier Using
the Prototype Verification System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Solomon Gebreyohannes, Ali Karimoddini, Abdollah Homaifar,
and Albert Esterline
Regularized Fuzzy Neural Network Based on Or Neuron
for Time Series Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Paulo Vitor de Campos Souza and Luiz Carlos Bambirra Torres
Design and Implementation of Fuzzy Expert System Based
on Evolutionary Algorithms for Diagnosing the Intensity
Rate of Hepatitis C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mehrnaz Behrooz and Mohammad Hossein Fazel Zarandi
Evolving Granular Fuzzy Min-Max Modeling. . . . . . . . . . . . . . . . . . . . . . .
Alisson Porto and Fernando Gomide
Combination of Spatial Clustering Methods Using Weighted Average
Voting for Spatial Epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Laisa Ribeiro de Sá, José Carlos da Silva Melo,
Jordana de Almeida Nogueira, and Ronei Marcos de Moraes
Fuzzy Logic Applied to eHealth Supported by a Multi-Agent System . . . . . .
Afonso B. L. Neto, João P. B. Andrade, Tibério C. J. Loureiro,
Gustavo A. L. de Campos, and Marcial P. Fernandez
Some Examples of Relations Between F-Transforms
and Powerset Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jiří Močkoř
Numerical Solutions for Bidimensional Initial Value Problem
with Interactive Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Vinícius F. Wasques, Estevão Esmi, Laécio C. Barros,
and Peter Sussner
Relevance of Classes in a Fuzzy Partition. A Study from a Group

of Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fabián Castiblanco, Camilo Franco, Javier Montero,
and J. Tinguaro Rodríguez
Interactive Fuzzy Process: An Epidemiological Model . . . . . . . . . . . . . . . . .
Francielle Santo Pedro, Laécio Carvalho de Barros,
and Estevão Esmi

1

13

24
37

49

61

72

84

96

108


XII

Contents

IoT Resources Ranking: Decision Making Under Uncertainty
Combining Machine Learning and Fuzzy Logic . . . . . . . . . . . . . . . . . . . . .
Renato Dilli, Amanda Argou, Renata Reiser,
and Adenauer Yamin
Least Squares Method with Interactive Fuzzy Coefficient:
Application on Longitudinal Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nilmara J. B. Pinto, Vinícius F. Wasques, Estevão Esmi,
and Laécio C. Barros
Using the Choquet Integral in the Pooling Layer in Deep
Learning Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Camila Alves Dias, Jéssica C. S. Bueno, Eduardo N. Borges,
Silvia S. C. Botelho, Graçaliz Pereira Dimuro, Giancarlo Lucca,
Javier Fernandéz, Humberto Bustince,
and Paulo Lilles Jorge Drews Junior
Consensus Image Feature Extraction with Ordered Directionally
Monotone Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cedric Marco-Detchart, Graçaliz Pereira Dimuro,
Mikel Sesma-Sara, Aitor Castillo-Lopez, Javier Fernandez,
and Humberto Bustince
Characterization of Lattice-Valued Restricted Equivalence Functions . . . . . . .
Eduardo Palmeira, Benjamín Bedregal, and Rogério R. Vargas
Aggregation with T-Norms and LexiT-Orderings and Their
Connections with the Leximin Principle . . . . . . . . . . . . . . . . . . . . . . . . . . .
Henrique Viana and João Alcântara
Fuzzy Formal Concept Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abner Brito, Laécio Barros, Estevão Laureano, Fábio Bertato,
and Marcelo Coniglio
Representing Intuistionistic Fuzzy Bi-implications Using
Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lucas Agostini, Samuel Feitosa, Anderson Avila, Renata Reiser,
André DuBois, and Maurício Pilla

119

132

144

155

167

179
192

206

Interval Version of Generalized Atanassov’s Intuitionistic Fuzzy Index . . . . .
Lidiane Costa, Mônica Matzenauer, Adenauer Yamin,
Renata Reiser, and Benjamín Bedregal

217

Fuzzy Ontologies: State of the Art Revisited . . . . . . . . . . . . . . . . . . . . . . .
Valerie Cross and Shangye Chen

230

Contents

Image Processing Algorithm to Detect Defects in Optical Fibers . . . . . . . . . .
Marcelo Mafalda, Daniel Welfer,
Marco Antônio De Souza Leite Cuadros,
and Daniel Fernando Tello Gamarra
A Comparative Study Among ANFIS, ANNs, and SONFIS
for Volatile Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jairo Andres Perdomo-Tovar, Eiber Arley Galindo-Arevalo,
and Juan Carlos Figueroa-García

XIII

243

253

Equilibrium Point of Representable Moore Continuous
n-Dimensional Interval Fuzzy Negations . . . . . . . . . . . . . . . . . . . . . . . . . .
Ivan Mezzomo, Benjamín Bedregal, and Thadeu Milfont

265

Color Mathematical Morphology Using a Fuzzy Color-Based
Supervised Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mateus Sangalli and Marcos Eduardo Valle

278

Fuzzy Kernel Associative Memories with Application in Classification . . . . .

Aline Cristina de Souza and Marcos Eduardo Valle

290

(T, N)-Implications and Some Functional Equations . . . . . . . . . . . . . . . . . .
Jocivania Pinheiro, Benjamin Bedregal, Regivan Santiago,
Helida Santos, and Graçaliz Pereira Dimuro

302

The Category of Semi-BCI Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jocivania Pinheiro, Rui Paiva, and Regivan Santiago

314

A Fuzzy Based Recommendation System for Stock Trading . . . . . . . . . . . . .
Érico Augusto Nunes Pinto, Leizer Schnitman,
and Ricardo A. Reis

324

Evolving Fuzzy Kalman Filter: A Black-Box Modeling Approach
Applied to Rocket Trajectory Forecasting. . . . . . . . . . . . . . . . . . . . . . . . . .
Danúbia Soares Pires and Ginalber Luiz de Oliveira Serra
Crisp Fuzzy Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jocivania Pinheiro, Benjamin Bedregal, Regivan Santiago,
and Helida Santos

336
348

Stock Market Price Forecasting Using a Kernel Participatory
Learning Fuzzy Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
R. Vieira, L. Maciel, R. Ballini, and Fernando Gomide

361

A Fuzzy Approach Towards Parking Space Occupancy Detection
Using Low-Quality Magnetic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Renato Lopes Moura and Peter Sussner

374


XIV

Contents

Adaptive Fuzzy Learning Vector Quantization (AFLVQ)
for Time Series Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Renan Fonteles Albuquerque, Paulo D. L. de Oliveira,
and Arthur P. de S. Braga
A Fuzzy C-means-based Approach for Selecting Reference Points
in Minimal Learning Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
José A. V. Florêncio, Madson L. D. Dias,
Ajalmar R. da Rocha Neto, and Amauri H. de Souza Júnior
Prey-Predator Model Under Fuzzy Uncertanties . . . . . . . . . . . . . . . . . . . . .
Chryslayne M. Pereira, Moiseis S. Cecconello,
and Rodney C. Bassanezi

385

398

408

Modeling and Simulation of Methane Dispersion
in the Dam of Santo Antonio – Rondônia/Brazil . . . . . . . . . . . . . . . . . . . . .
Geraldo L. Diniz and Evanizio M. Menezes Jr.

419

Solution to Convex Variational Problems with Fuzzy Initial
Condition Using Zadeh’s Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michael M. Diniz, Luciana T. Gomes, and Rodney C. Bassanezi

431

Necessary Optimality Conditions for Interval Optimization Problems
with Inequality Constraints Using Constrained Interval Arithmetic. . . . . . . . .
Gino G. Maqui-Huamán, Geraldo Silva, and Ulcilea Leal

439

Order Relations, Convexities, and Jensen’s Integral Inequalities
in Interval and Fuzzy Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tiago Mendonça da Costa, Yurilev Chalco-Cano,
Laécio Carvalho de Barros, and Geraldo Nunes Silva
Fuzzy Initial Value Problem: A Short Survey . . . . . . . . . . . . . . . . . . . . . . .
Marina Tuyako Mizukoshi

Zadeh’s Extension of the Solution of the Euler-Lagrange Equation
for a Quadratic Functional Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jônathas D. S. Oliveira, Luciana T. Gomes,
and Rodney C. Bassanezi
Estimating the Xenobiotics Mixtures Toxicity on Aquatic Organisms:
The Use of a-level of the Fuzzy Number . . . . . . . . . . . . . . . . . . . . . . . . . .
Magda S. Peixoto, Claudio M. Jonsson, Lourival C. Paraiba,
Laécio C. Barros, and Weldon A. Lodwick
An Approach for Solving Interval Optimization Problems . . . . . . . . . . . . . .
Fabiola Roxana Villanueva and Valeriano Antunes de Oliveira

450

464

477

489

500


Contents

A Brief Review of a Method for Bounds on Polynomial Ranges
over Simplexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ralph Baker Kearfott and Dun Liu
Moore: Interval Arithmetic in C++20. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
W. F. Mascarenhas
Towards Foundations of Fuzzy Utility: Taking Fuzziness

into Account Naturally Leads to Intuitionistic Fuzzy Degrees . . . . . . . . . . . .
Christian Servin and Vladik Kreinovich
Solving Transhipment Problems with Fuzzy Delivery Costs
and Fuzzy Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Juan Carlos Figueroa-García, Jhoan Sebastian Tenjo-García,
and Camilo Alejandro Bustos-Tellez
How to Gauge Repair Risk? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Francisco Zapata and Vladik Kreinovich

XV

508
519

530

538

551

Interval Type II Fuzzy Rough Set Rule Based Expert System
to Diagnose Chronic Kidney Disease . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mona Abdolkarimzadeh, M. H. Fazel Zarandi, and O. Castillo

559

Parameter Optimization for Membership Functions of Type-2 Fuzzy
Controllers for Autonomous Mobile Robots Using
the Firefly Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Marylu L. Lagunes, Oscar Castillo, Fevrier Valdez, Jose Soria,

and Patricia Melin

569

Differential Evolution Algorithm Using a Dynamic Crossover
Parameter with Fuzzy Logic Applied for the CEC 2015
Benchmark Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Patricia Ochoa, Oscar Castillo, and José Soria

580

Corporate Control with a Fuzzy Network: A Knowledge
Engineering Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gustavo Pérez Hoyos

592

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

601


Formal Verification of a Fuzzy Rule-Based
Classifier Using the Prototype
Verification System
Solomon Gebreyohannes, Ali Karimoddini(B) , Abdollah Homaifar,
and Albert Esterline
Department of Electrical and Computer Engineering,
North Carolina A&T State University, 1601 East Market Street,
Greensboro, NC 27411, USA

{shgebrey,akarimod,homaifar,esterlin}@ncat.edu

Abstract. This paper presents the formal specification and verification
of a Type-1 (T1) Fuzzy Logic Rule-Based Classifier (FLRBC) using the
Prototype Verification System (PVS). A rule-based system models a
system as a set of rules, which are either collected from subject matter experts or extracted from data. Unlike many machine learning techniques, rule-based systems provide an insight into the decision making
process. In this paper, we focus on a T1 FLRBC. We present the formal definition and verification of the T1 FLRBC procedure using PVS.
This helps mathematically verify that the design intent is maintained
in its implementation. A highly expressive language such as PVS, which
is based on a strongly-typed higher-order logic, allows one to formally
describe and mathematically prove that there is no contradiction or false
assumption in the procedure. We show this by (1) providing the formal
definition of the T1 FLRBC in PVS and then (2) formally proving or
deducing rudimentary properties of the T1 FLRBC from the formal specification.
Keywords: Formal verification
Prototype verification system

1

· Fuzzy rule-based classifier

Introduction

Unlike many machine learning techniques, which treat systems as “black boxes”
and model them using input-output relationships, Rule-Based Systems (RBSs)
model systems using rules that can provide an insight into their decision making processes. RBCs are effective tools for encoding a human expert’s knowledge
into an automated system [1]. They can be used for different applications such as
prediction and control applications. Recently, they have been used for a classification purpose. A fuzzy logic based RBS that is used for classification is called a
Fuzzy Logic Rule-Based Classifier (FLRBC). It translates the expert’s heuristic
knowledge into fuzzy “IF-THEN” statements (i.e., rules), which, along with an

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G. A. Barreto and R. Coelho (Eds.): NAFIPS 2018, CCIS 831, pp. 1–12, 2018.
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S. Gebreyohannes et al.

appropriate inference engine, are used for system classification [2,3]. FLRBCs
have been used to classify various kinds of items such as text, image [4], gesture
[5], and video [2,3] and have been applied to several areas such as battlefield
ground vehicles [6], sentiment analysis [7], and so on. FLRBCs are also combined with genetic algorithms [8], deep learning [9], decision trees, and other
techniques to optimize their performance.
Despite the great capabilities of FLRBCs and their wide range of applications, a major challenge is that how can we be sure (i.e., mathematically verify) that the design intent of an FLRBC is maintained in its implementation?
Regarding to design procedures, FLRBC is no exception to the general point
that natural language lacks the formality needed for requirement verification,
i.e., with natural language, one cannot show (and hence cannot ensure) consistency and completeness of the procedure. Hence, a formal way (from a mathematical perspective) of representing the FLRBC design procedure and verifying
its properties is necessary.
Formal verification can be defined as a “systematic process based on mathematical reasoning in order to verify that the design intent (specifications) is
maintained during implementation” [10]. There are different tools developed for
conducting formal verification such as Prototype Verification System (PVS) [11],
B [12], HOL [13], and Coq [14]. Using a higher-order theorem-proving system,
such as PVS, it is possible to reach a much higher level of confidence compared
to lighter formal methods. Formalizing specifications using PVS has been used
in a range of applications. In [15], a formal specification and verification of the
requirements for an airline reservation system using PVS is presented. As a more
sophisticated use of PVS in industrial applications, [16] uses PVS for verification
of two hardware examples, the pipelined microprocessor and n-bit ripple-carry
adder. Butler [17] also uses PVS for formal capturing of requirements of an
autopilot (related to an early Boeing-737 autopilot). In [18], PVS is used for

analysis of a space shuttle software requirements. Despite the progress made on
employing PVS for different verification applications, we are not aware of any
PVS formalization of the FLRBC.
This paper, therefore, proposes to use formal verification techniques to systemically and formally verify a fuzzy RBC whether it is consistent with required
specifications. We present FLRBC specifications and verify them using the PVS
framework. Our focus in this paper is on Type-1 (T1) FLRBC; however, the
approach can be extended to other fuzzy RBCs. We encode PVS theories for the
T1 FLRBC main components (viz., fuzzifier, inference, and defuzzifier), keeping
their generality. This means that the developed PVS theories are not dependent
on any particular application. Therefore, one can call and instantiate them to
be used for any other application that utilizes the FLRBC technique.
The rest of the paper is organized as follows. Section 2 discusses a fuzzy logic
rule-based classification. Section 3 presents a brief introduction to PVS. Section 4
presents the formal definition and verification of the T1 FLRBC. PVS theories
are developed and formal proofs are shown. Section 5 concludes this paper.


Formal Verification of a Fuzzy Rule-Based Classifier Using PVS

2

3

Fuzzy Logic Rule-Based Classification: An Informal
Description

In this section, we present fuzzy sets and systems preliminaries, and will discuss
fuzzy rule-based classification, mainly borrowed from [2,3,19].
2.1

Fuzzy Sets and Systems Preliminaries

This section introduces fuzzy sets and fuzzy logic systems. A fuzzy set is characterized by a membership function (MF), mapping the elements of a domain
space or universe of discourse to the interval [0, 1]. A type-1 fuzzy set can be
defined as follows:
Definition 1. A type-1 fuzzy set A is a set function on universe X into [0, 1],
i.e., μA : X → [0, 1].
A = {(x, μA (x))|x ∈ X, 0 ≤ μA (x) ≤ 1}

(1)

where the MF of A is denoted μA (x) and is called a type-1 MF.
A fuzzy system that operates on type-1 fuzzy sets (and crisp sets) is called a
type-1 fuzzy system.
Definition 2. A Type-1 fuzzy system contains four components - rules, fuzzifier,
inference engine, and defuzzifier - that are interconnected as shown in Fig. 1.
Once the rules have been established, the fuzzy system can be viewed as a mapping
from p inputs x = {x1 , . . . , xp } to an output y, and the mapping can be expressed
quantitatively as y = f (x).
Figure 1 shows a type-1 fuzzy logic system. Note that x is a specific value of x.
The components of this fuzzy system are described below.

Fig. 1. A type-1 fuzzy system [2, 3]

Rules are sets of IF-THEN statements that model the system and can have
two commonly different structures.


4

S. Gebreyohannes et al.

1. Zadeh’s lth rule, l = 1, . . . , M , has a form:
l
RZ
: IF x1 is F1l and . . . , xp is Fpl , THEN y is Gl

(2)

where Fil is the ith antecedent MF and Gl is the consequent MF of the lth
rule.
2. Takagi, Sugeno, and Kang (TSK) lth rule, l = 1, . . . , M , has a form:
RTl SK : IF x1 is F1l and . . . , xp is Fpl , THEN y is g l (x1 , . . . , xp )

(3)

where Fil is the ith antecedent MF and g l is the function for the lth rule.
The fuzzifier maps a crisp input x = {x1 , . . . , xp } into a fuzzy set in X. There
are two kinds of fuzzifiers: singleton and non-singleton. A singleton fuzzification
maps a specific value xi into μFil (xi ) ∈ [0, 1] while a non-singleton maps into a
type-1 fuzzy number.
In the fuzzy inference engine, fuzzy logic principles are used to map fuzzy
input sets in X1 ×. . .×Xp , that flow through an IF-THEN rule (or a set of rules),
into fuzzy output sets in Y . Each rule is interpreted as a fuzzy implication.
A defuzzifier maps a fuzzy output of the inference engine to a crisp
output y.
2.2

Type-1 Fuzzy Logic Rule-Based Classifier

This section discusses a singleton T1 FLRBC. The classifier consists of five
components, as shown in Fig. 2, by adding a comparator to a fuzzy system.
The T1 FLRBC is used for a binary classification, i.e., it classifies its inputs as
either Class 1 or Class 2.

Fig. 2. A type-1 fuzzy logic rule-based classifier

Recall that x is a set of p input features, i.e., x = {x1 , x2 , . . . , xp }, and y is
an output of the fuzzy system.
The rules of an FLRBC are a special case of Zadeh’s rule, in which the
consequent is a singleton or TSK rule with a constant function [3]. They are


Formal Verification of a Fuzzy Rule-Based Classifier Using PVS

5

characterized by MFs. For the lth Mamdani rule [20], l = 1, . . . , M, we have Rl :
Al → Gl , which can be represented by μAl →Gl (x, y), where Al = F1l × . . . × Fpl .
For the consequent, a crisp value +1 is used for Class 1 and −1 is used for
Class 2.
yl =

1
−1

Class 1
Class 2

(4)

Correspondingly, for the consequent sets, Gl , the MFs can be defined as
1
0

μGl (y) =

y = yl
otherwise

(5)

where y l could be either +1 for Class 1 or −1 for Class 2.
The fuzzy inference engine for this rule-base classifier is shown in Fig. 3.

Fig. 3. Fuzzy inference engine ([3], p. 108)

The membership function of each fired rule for singleton input can be calculated using a t-norm as:
μB l (y) =

p
Ti=1
μFil (xi ) = f l (x),
0,

y = yl
y = yl

(6)

where μFil (xi ), i = 1, . . . , p, represents a singleton fuzzification and T is a t-norm
operation.
Using height defuzzification, the output can be calculated as:
yRBC (x) =

M
l
l
l=1 f (x)y
,
M
l
i=1 f (x)

y l = ±1

(7)

A classification then can be performed as:
If yRBC (x) > 0,
If yRBC (x) ≤ 0,

3

classify x as Class 1
classify x as Class 2

(8)

Prototype Verification System

PVS [11] is a formal verification environment which provides a highly expressive
specification language based on a strongly-typed higher-order logic.


6

S. Gebreyohannes et al.

The type system of PVS supports the use of both interpreted and uninterpreted types. Uninterpreted types support abstraction (using type) with a minimum of assumptions on the type; e.g., var1: TYPE. Interpreted types, on the
other hand, detail the type. For example, var2: TYPE = nat declaration introduces the type name var2 (interpreted) as a natural number. The PVS prelude
[21] provides definitions of a comprehensive collection of basic interpreted types,
such as booleans, natural numbers, integers, reals, etc. The type system can be
easily extended by defining new types using well-known type operators, such as
functional, tuple and record combinators. It is also possible to define enumerated types and predicate subtypes as well as abstract data types, such as lists,
stacks, binary trees, etc. Details on the PVS language may be found in the PVS
Language Reference [22].
Since the PVS language is so expressive, the type checking process is not
decidable. For that reason, the type checker usually generates additional proof
obligations, which must be proved by the user in order to verify the type consistency of the specification. Such obligations are called Type-Correctness Conditions (TCC).
Formalizations in PVS are organized in theories, which include type, constant, variable, and formula definitions. Formulas can be constructed using
propositional operators as well as first-order and higher-order quantifications.
Every formula definition can be stated as an axiom or a theorem. Axioms are
assumed to be valid in the specification, but formal proofs must be provided
by the user for theorems. Specifications for many foundational and standard
theories are preloaded into PVS as prelude theories.

Fig. 4. Theorem proving in PVS

The theorem prover implemented in PVS is based on a formalism called

sequent calculus. A proof in such a calculus can be seen as a tree, where every
node is a sequent composed of two collections of formulas, called antecedents and
consequents, respectively.
antecedents
(9)
|
consequents
The intuition behind the notion of a sequent is that it represents the logical
consequence between the conjunction of its antecedents and the disjunction of
its consequents. In order to prove that a formula, for example, α → β, is valid,
the user must start with the sequent α β (the operator ‘ ’ denotes a sequent)


Formal Verification of a Fuzzy Rule-Based Classifier Using PVS

7

and try to reach trivially valid sequents by applying predefined proof rules to the
leaves of the proof tree. Although proofs in PVS are constructed interactively,
it does provide a considerable degree of automatization for a wide spectrum
of cases. Details about proofs can be found in the PVS prover guide [23]. The
steps in theorem proving using PVS as are shown in Fig. 4. There are also PVS
tutorials and applications developed in [15,16].

4

Formal Description and Verification of Fuzzy
Rule-Based Classification

This section presents the formal representation and verification of a singleton T1

FLRBC using PVS. Table 1 summarizes the semantic mapping of T1 FLRBC
components to PVS.
Table 1. Fuzzy RBC constructs to PVS mapping
Fuzzy RBC

PVS

Input feature

Uninterpreted Type

Output

Enumerated type

Input, antecedent

Finite sequence

µ, consequent

Subtypes

Membership function, rule Record type
Inference, defuzzification

RECURSIVE function

To formally describe a fuzzy RBC, we start by defining basic objects in the
following sections.

4.1

Basic TYPE Definition

The first step in the design of a T1 FLRBC system is selecting features that act
as antecedents. They are usually represented using their MFs. The antecedent
features are encoded as PVS (uninterpreted) TYPEs and their MFs as RECORD
TYPEs defined in the fbasic defs PVS theory. An MF consists of a range of
values and a function that maps a value (within the range) into [0, 1].
fbasic_defs: THEORY
BEGIN
Feature:
TYPE
m:
TYPE = nat
mu:
TYPE = {u:real| 0 <= u AND u <= 1}
MF:
TYPE = [# range: [real,real],
f: [{x:real|range‘1<=x


8

S. Gebreyohannes et al.

AND x<=range‘2}-> mu] #]
Feat_MF:

TYPE = [# feat: Feature,

mf:
MF #]

A rule (R) maps a set of antecedents (A = F1 × F2 . . . × Fp ) to a consequent
(G), i.e., R : A → G; it is usually described by the MF μR (X, y) = μA→G (X, y).
In an RBC, the consequent is a singleton where ‘Class 1’ is represented by +1
and ‘Class 2’ is represented by −1. Hence, a consequence is represented as a
subtype in PVS. Each rule has a specifier l <= M , where M is the total number
of rules. An input X has p features, x1 , x2 , x3 , . . . , xp . We represent the input
as a finite sequence of real values. A fuzzy rule is encoded using PVS RECORD
with accessors rule number, antecedents, and consequents.
Input:
Output:
Antecedents:
Consequent:
Rule:

Rules:
END fbasic_defs
4.2

TYPE
TYPE
TYPE
TYPE
TYPE

=
=
=

=
=

finseq[real]
{class1, class2}
finseq[Feat_MF]
{n: nat| n = -1 OR n = 1}
[# rulen: l,
antcs: Antecedents,
consq: Consequent #]
TYPE = finseq[Rule]

Describing Fuzzy RBC Using PVS

This section formally defines the T1 FLRBC using PVS. We put the definition
in a fls theory. We start by importing fbasic defs theory defined in Sect. 4.1
and declaring input and rule variables. Let X,r, and rs be input, a rule, and a
set of rules, respectively. There are p inputs and M rules.
fls: THEORY
BEGIN
IMPORTING fbasic_defs
p,M: VAR nat
X:
VAR Input
r:
VAR Rule
rs: VAR Rules
The number of antecedents in a rule r should be the same as the length of
the input vector x. This is encoded as a PVS AXIOM.
inpt_length: AXIOM FORALL (X,r): X‘length = r‘antcs‘length

Now, we can represent the inference engine. Based on Eq. 6, the inference engine
can be represented as a PVS RECURSIVE function as follows:


Formal Verification of a Fuzzy Rule-Based Classifier Using PVS

9

fl(X,r,p): RECURSIVE mu =
IF p=0 THEN 0
ELSE r‘antcs‘seq(p)‘mf‘f(X‘seq(p))*fl(X,r,p-1)
ENDIF
Measure p
The defuzzified output of the RBC can be obtained using Eq. 7. The numerator
of Eq. 7 is encoded using a PVS RECURSIVE function as:
sumfy(X,rs,M): RECURSIVE mu =
IF M=0 THEN 0
ELSE
fl(X,rs‘seq(M),X‘length)*rs‘seq(M)‘consq + sumfy(X,rs,M-1)
ENDIF
Measure M
Similarly, the denominator of Eq. 7 is encoded using a PVS as:
sumf(X,rs,M): RECURSIVE mu =
IF M=0 THEN 0
ELSE fl(X,rs‘seq(M),X‘length) + sumf(X,rs,M-1)
ENDIF
Measure M
Therefore, Eq. 7 can now be represented using PVS as:
yRBC(X,rs): real = sumfy(X,rs,rs‘length)/sumf(X,rs,rs‘length)
The final decision of the classifier is based on the sign of the defuzzified output

as shown in Eq. 8. This is encoded using a PVS IF-ELSE statement.
decision(X,rs): Output =
IF (yRBC(X,rs) > 0) THEN class 1
ELSE class 2
ENDIF
END fls
This completes the formal definition of the T1 FLRBC process.
4.3

Formal Verification

This section presents the formal verification - well-formedness (to verify our specification is correct, i.e., free of contradiction or false assumption) and requirement
verification (to verify a given requirement or property can be deduced from the
specification).


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S. Gebreyohannes et al.

Well-Formedness. PVS requires theorem proving in order to guarantee that
the specification is type correct [15]. Type checking of the fbasic defs theory
generates no TCCs, but the fls theory does generate TCCs. For example, one
TCC generated is
yRBC_TCC1: OBLIGATION
FORALL (X:Input, rs: Rules): sumf(X, rs, rs‘length) /= 0
The denominator of Eq. 7 needs to be non-zero to avoid division by zero. This
was not contained in our definition. Therefore, we add a PVS AXIOM for this.
nz_tnorm: AXIOM FORALL (X,rs): NOT (sumf(X,rs,rs‘length) = 0)
Any TCC should be discharged if we are to guarantee the correctness (no contradiction nor false assumptions) of the specification. We revise the theory for

no TCC. We successfully discharged some of the TCCs using the axioms. The
rest of the TCCs are proved automatically by a PVS standard strategy (tcc).
Once all the TCCs are discharged, the theories are well-formed; i.e., there is
no contradiction (nor false assumption) in the declaration. However, we do not
know yet whether it satisfies any given properties (or requirements). We show
this in Sect. 4.3.
Verification of Properties. Now, since the T1 FLRBC is formally represented,
one can mathematically verify properties of the T1 FLRBC by encoding them
as PVS THEOREMs and proving them interactively using appropriate prover
commands. In this section, we prove some rudimentary properties.
Lemma 1. The output of the T1 FLRBC should be either Class 1 or Class 2.
We define two PVS predicates: first we check independently that if a given output
is Class 1 or Class 2.
class1?(X,rs): bool = decision(X,rs) = class1
class2?(X,rs): bool = decision(X,rs) = class2
Then, the theorem will be the XOR of the two predicates.
bin_class: THEOREM FORALL(X,rs):xor(class1?(X,rs),class2?(X,rs))
This theorem is discharged automatically by a prover command grind. Q.E.D.
Let us check the dependence of the output of the classifier on the rules.
Lemma 2. (a) If the output of the classifier is Class 1, then there exists a rule
with a consequent of 1.
class1_det: THEOREM
decision(X,rs) = class1 IMPLIES
EXISTS r:member(rs,r) AND r‘consq = 1
(b) If the output of the classifier is Class 2, then there exists a rule with a
consequent of −1.