Springer advances in artificial intelligence SBIA 2004 (2005) ling OCR 7 0 2 6 lotb

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Lecture Notes in Artificial Intelligence

3171

Edited by J. G. Carbonell and J. Siekmann

Subseries of Lecture Notes in Computer Science

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Ana L.C. Bazzan Sofiane Labidi (Eds.)

Advances in
Artificial Intelligence –
SBIA 2004
17th Brazilian Symposium on Artificial Intelligence
São Luis, Maranhão, Brazil
September 29 – October 1, 2004
Proceedings

Springer
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eBook ISBN:
Print ISBN:

3-540-28645-4
3-540-23237-0

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Preface

SBIA, the Brazilian Symposium on Artificial Intelligence, is a biennial event
intended to be the main forum of the AI community in Brazil. The SBIA 2004
was the 17th issue of the series initiated in 1984. Since 1995 SBIA has been

accepting papers written and presented only in English, attracting researchers
from all over the world. At that time it also started to have an international
program committee, keynote invited speakers, and proceedings published in the
Lecture Notes in Artificial Intelligence (LNAI) series of Springer (SBIA 1995,
Vol. 991, SBIA 1996, Vol. 1159, SBIA 1998, Vol. 1515, SBIA 2000, Vol. 1952,
SBIA 2002, Vol. 2507).
SBIA 2004 was sponsored by the Brazilian Computer Society (SBC). It was
held from September 29 to October 1 in the city of São Luis, in the northeast
of Brazil, together with the Brazilian Symposium on Neural Networks (SBRN).
This followed a trend of joining the AI and ANN communities to make the joint
event a very exciting one. In particular, in 2004 these two events were also held
together with the IEEE International Workshop on Machine Learning and Signal
Processing (MMLP), formerly NNLP.
The organizational structure of SBIA 2004 was similar to other international
scientific conferences. The backbone of the conference was the technical program
which was complemented by invited talks, workshops, etc. on the main AI topics.
The call for papers attracted 209 submissions from 21 countries. Each paper
submitted to SBIA was reviewed by three referees. From this total, 54 papers
from 10 countries were accepted and are included in this volume. This made
SBIA a very competitive conference with an acceptance rate of 25.8%. The
evaluation of this large number of papers was a challenge in terms of reviewing
and maintaining the high quality of the preceding SBIA conferences. All these
goals would not have been achieved without the excellent work of the members
of the program committee – composed of 80 researchers from 18 countries – and
the auxiliary reviewers.
Thus, we would like to express our sincere gratitude to all those who helped
make SBIA 2004 happen. First of all we thank all the contributing authors;
special thanks go to the members of the program committee and reviewers for
their careful work in selecting the best papers. Thanks go also to the steering
committee for its guidance and support, to the local organization people, and to

the students who helped with the website design and maintenance, the papers
submission site, and with the preparation of this volume. Finally, we would like
to thank the Brazilian funding agencies and Springer for supporting this book.
Porto Alegre, September 2004

Ana L.C. Bazzan
(Chair of the Program Committee)
Sofiane Labidi
(General Chair)

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Organization

SBIA 2004 was held in conjunction with SBRN 2004 and with IEEE MMLP
2004. These events were co-organized by all co-chairs involved in them.

Chair
Sofiane Labidi (UFMA, Brazil)

Steering Committee
Ariadne Carvalho (UNICAMP, Brazil)
Geber Ramalho (UFPE, Brazil)
Guilherme Bitencourt (UFSC, Brazil)
Jaime Sichman (USP, Brazil)

Organizing Committee
Allan Kardec Barros (UFMA)
Alzio Arẳjo (UFPE)

Ana L.C. Bazzan (UFRGS)
Geber Ramalho (UFPE)
Osvaldo Ronald Saavedra (UFMA)
Sofiane Labidi (UFMA)

Supporting Scientific Society
SBC

Sociedade Brasileira de Computaỗóo

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Organization

VII

Program Committee
Luis Otavio Alvares
Analia Amandi

Univ. Federal do Rio Grande do Sul (Brazil)
Universidad Nacional del Centro de la Provincia
de Buenos Aires (Argentina)
John Atkinson
Universidad de Concepcin (Chile)
Pontifícia Universidade Católica, PR (Brazil)
Bráulio Coelho Avila
Flávia Barros
Universidade Federal de Pernambuco (Brazil)

Guilherme Bittencourt
Universidade Federal de Santa Catarina (Brazil)
Olivier Boissier
École Nationale Superieure des Mines
de Saint-Etienne (France)
University of Liverpool (UK)
Rafael H. Bordini
Dibio Leandro Borges
Pontifícia Universidade Católica, PR (Brazil)
University of Amsterdam (The Netherlands)
Bert Bredeweg
Jacques Calmet
Universität Karlsruhe (Germany)
Mario F. Montenegro Campos Universidade Federal de Minas Gerais (Brazil)
Universidade Federal do Ceará (Brazil)
Fernando Carvalho
Francisco Carvalho
Universidade Federal de Pernambuco (Brazil)
Institute of Psychology, CNR (Italy)
Cristiano Castelfranchi
Univ. Técnica Federico Santa María (Chile)
Carlos Castro
Université Montpellier II (France)
Stefano Cerri
Université Laval (Canada)
Ibrahim Chaib-draa
Universidade de Lisboa (Portugal)
Helder Coelho
Université Pierre et Marie Curie (France)
Vincent Corruble

Ernesto Costa
Universidade de Coimbra (Portugal)
Anna Helena Reali Costa
Universidade de São Paulo (Brazil)
Antônio C. da Rocha Costa Universidade Católica de Pelotas (Brazil)
Augusto C.P.L. da Costa
Universidade Federal da Bahia (Brazil)
Evandro de Barros Costa
Universidade Federal de Alagoas (Brazil)
Kerstin Dautenhahn
University of Hertfordshire (UK)
Keith Decker
University of Delaware (USA)
Marco Dorigo
Université Libre de Bruxelles (Belgium)
Michael Fisher
University of Liverpool (UK)
University of Bristol (UK)
Peter Flach
Ana Cristina Bicharra Garcia Universidade Federal Fluminense (Brazil)
Uma Garimella
AP State Council for Higher Education (India)
Lúcia Giraffa
Pontifícia Universidade Católica, RS (Brazil)
Claudia Goldman
University of Massachusetts, Amherst (USA)
Fernando Gomide
Universidade Estadual de Campinas (Brazil)
Gabriela Henning
Universidad Nacional del Litoral (Argentina)

Michael Huhns
University of South Carolina (USA)
Nitin Indurkhya
University of New South Wales (Australia)
Alípio Jorge
University of Porto (Portugal)
Celso Antơnio Alves Kaestner Pontifícia Universidade Católica, PR (Brazil)

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VIII

Organization

Franziska Klügl
Sofiane Labidi
Lluis Godo Lacasa
Marcelo Ladeira
Nada Lavrac
Christian Lemaitre
Victor Lesser
Vera Lúcia Strube de Lima
Jose Gabriel Pereira Lopes
Michael Luck
Ana Teresa Martins
Stan Matwin
Eduardo Miranda
Maria Carolina Monard
Valérie Monfort

Eugenio Costa Oliveira
Tarcisio Pequeno
Paolo Petta
Geber Ramalho
Solange Rezende
Carlos Ribeiro
Francesco Ricci
Sandra Sandri
Sandip Sen
Jaime Simão Sichman
Carles Sierra
Milind Tambe
Patricia Tedesco
Sergio Tessaris
Luis Torgo
Andre Valente
Wamberto Vasconcelos
Rosa Maria Vicari
Renata Vieira
Jacques Wainer
Renata Wasserman
Michael Wooldridge
Franco Zambonelli
Gerson Zaverucha

Universität Würzburg (Germany)
Universidade Federal do Maranhão (Brazil)
Artificial Intelligence Research Institute (Spain)
Universidade de Brasília (Brazil)
Josef Stefan Institute (Slovenia)

Lab. Nacional de Informatica Avanzada (Mexico)
University of Massachusetts, Amherst (USA)
Pontifícia Universidade Católica, RS (Brazil)
Universidade Nova de Lisboa (Portugal)
University of Southampton (UK)
Universidade Federal do Ceará (Brazil)
University of Ottawa (Canada)
University of Plymouth (UK)
Universidade de São Paulo at São Carlos (Brazil)
MDT Vision (France)
Universidade do Porto (Portugal)
Universidade Federal do Ceará (Brazil)
Austrian Research Institut for Artificial
Intelligence (Austria)
Universidade Federal de Pernambuco (Brazil)
Universidade de São Paulo at São Carlos (Brazil)
Instituto Tecnológico de Aeronáutica (Brazil)
Istituto Trentino di Cultura (Italy)
Artificial Intelligence Research Institute (Spain)
University of Tulsa (USA)
Universidade de São Paulo (Brazil)
Institut d’Investigació en Intel. Artificial (Spain)
University of Southern California (USA)
Universidade Federal de Pernambuco (Brazil)
Free University of Bozen-Bolzano (Italy)
University of Porto (Portugal)
Knowledge Systems Ventures (USA)
University of Aberdeen (UK)
Univ. Federal do Rio Grande do Sul (Brazil)
UNISINOS (Brazil)

Universidade Estadual de Campinas (Brazil)
Universidade de São Paulo (Brazil)
University of Liverpool (UK)
Università di Modena Reggio Emilia (Italy)
Universidade Federal do Rio de Janeiro (Brazil)

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Organization

IX

Sponsoring Organizations
By the publication of this volume, the SBIA 2004 conference received financial
support from the following institutions:
CNPq
CAPES
FAPEMA
FINEP

Conselho Nacional de Desenvolvimento Cientớfico e Tecnolúgico
Fundaỗóo Coordenaỗóo de Aperfeiỗoamento de Pessoal de Nớvel
Superior
Fundaỗóo de Amparo à Pesquisa do Estado do Maranhão
Financiadora de Estudos e Projetos

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X

Organization

Additional Reviewers
Mara Abel
Nik Nailah Bint Abdullah
Diana Adamatti
Stephane Airiau
João Fernando Alcântara
Teddy Alfaro
Luis Almeida
Marcelo Armentano
Dipyaman Banerjee
Dante Augusto Couto Barone
Gustavo Batista
Amit Bhaya
Reinaldo Bianchi
Francine Bica
Waldemar Bonventi
Flávio Bortolozzi
Mohamed Bouklit
Paolo Bouquet
Carlos Fisch de Brito
Tiberio Caetano
Eduardo Camponogara
Teddy Candale
Henrique Cardoso
Ariadne Carvalho
André Ponce de Leon F. de Carvalho

Ana Casali
Adelmo Cechin
Luciano Coutinho
Damjan Demsar
Clare Dixon
Fabrício Enembreck
Paulo Engel
Alexandre Evsukoff
Anderson Priebe Ferrugem
Marcelo Finger
Ricardo Freitas
Leticia Friske
Arjita Ghosh
Daniela Godoy
Alex Sandro Gomes
Silvio Gonnet
Marco Antonio Insaurriaga Gonzalez
Roderich Gross
Michel Habib

Juan Heguiabehere
Emilio Hernandez
Benjamin Hirsch
Jomi Hübner
Ullrich Hustadt
Alceu de Souza Britto Junior
Branko Kavsek
Alessandro Lameiras Koerich
Boris Konev
Fred Koriche

Luís Lamb
Michel Liquière
Peter Ljubic
Andrei Lopatenko
Gabriel Lopes
Emiliano Lorini
Teresa Ludermir
Alexei Manso Correa Machado
Charles Madeira
Pierre Maret
Graỗa Marietto
Lilia Martins
Claudio Meneses
Claudia Milaré
Márcia Cristina Moraes
Álvaro Moreira
Ranjit Nair
Marcio Netto
André Neves
Julio Cesar Nievola
Luis Nunes
Maria das Graỗas Volpe Nunes
Valguima Odakura
Carlos Oliveira
Flỏvio Oliveira
Fernando Osúrio
Flỏvio Pỏdua
Elias Pampalk
Marcelino Pequeno
Luciano Pimenta

Aloisio Carlos de Pina
Joel Plisson
Ronaldo Prati
Carlos Augusto Prolo
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Organization

Ricardo Prudêncio
Josep Puyol-Gruart
Sergio Queiroz
Violeta Quental
Leila Ribeiro
María Cristina Riff
Maria Rifqi
Ana Rocha
Linnyer Ruiz
Sabyasachi Saha
Luis Sarmento
Silvia Schiaffino
Hernan Schmidt
Antônio Selvatici
David Sheeren
Alexandre P. Alves da Silva
Flávio Soares Corrêa da Silva
Francisco Silva

XI

Klebson dos Santos Silva
Ricardo de Abreu Silva
Roberto da Silva
Valdinei Silva
Wagner da Silva
Alexandre Simões
Eduardo do Valle Simoes
Marcelo Borghetti Soares
Marcilio Carlos P. de Souto
Renata Souza
Andréa Tavares
Marcelo Andrade Teixeira
Clésio Luis Tozzi
Karl Tuyls
Adriano Veloso
Felipe Vieira Fernando Von Zuben
Alejandro Zunino

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Table of Contents

Logics, Planning, and Theoretical Methods
On Modalities for Vague Notions

Mario Benevides, Carla Delgado, Renata P. de Freitas,
Paulo A.S. Veloso, and Sheila R.M. Veloso

1

Towards Polynomial Approximations of Full Propositional Logic
Marcelo Finger

11

Using Relevance to Speed Up Inference. Some Empirical Results
Joselyto Riani and Renata Wassermann

21

A Non-explosive Treatment of Functional Dependencies
Using Rewriting Logic
Gabriel Aguilera, Pablo Cordero, Manuel Enciso, Angel Mora,
and Inmaculada Perez de Guzmán

31

Reasoning About Requirements Evolution
Using Clustered Belief Revision
Odinaldo Rodrigues, Artur d’Avila Garcez, and Alessandra Russo

41

Analysing AI Planning Problems in Linear Logic –
A Partial Deduction Approach

Peep Küngas

52

Planning with Abduction: A Logical Framework
to Explore Extensions to Classical Planning
Silvio do Lago Pereira and Leliane Nunes de Barros

62

High-Level Robot Programming:
An Abductive Approach Using Event Calculus
Silvio do Lago Pereira and Leliane Nunes de Barros

73

Search, Reasoning, and Uncertainty
Word Equation Systems: The Heuristic Approach
César Luis Alonso, Fátima Drubi, Judith Gómez-García,
and José Luis Monta
A Cooperative Framework
Based on Local Search and Constraint Programming
for Solving Discrete Global Optimisation
Carlos Castro, Michael Moossen, and María Cristina Riff

83

93

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XIV

Table of Contents

Machine Learned Heuristics to Improve Constraint Satisfaction
Marco Correia and Pedro Barahona

103

Towards a Natural Way of Reasoning
José Carlos Loureiro Ralha and Célia Ghedini Ralha

114

Is Plausible Reasoning a Sensible Alternative
for Inductive-Statistical Reasoning?
Ricardo S. Silvestre and Tarcísio H. C. Pequeno

124
134

Paraconsistent Sensitivity Analysis for Bayesian Significance Tests
Julio Michael Stern

Knowledge Representation and Ontologies
An Ontology for Quantities in Ecology
Virgínia Brilhante

144

Using Color to Help in the Interactive Concept Formation
Vasco Furtado and Alexandre Cavalcante

154

Propositional Reasoning for an Embodied Cognitive Model
Jerusa Marchi and Guilherme Bittencourt

164

A Unified Architecture to Develop Interactive Knowledge Based Systems
Vládia Pinheiro, Elizabeth Furtado, and Vasco Furtado

174

Natural Language Processing
Evaluation of Methods for Sentence and Lexical Alignment
of Brazilian Portuguese and English Parallel Texts
Helena de Medeiros Caseli, Aline Maria da Paz Silva,
and Maria das Graỗas Volpe Nunes
Applying a Lexical Similarity Measure
to Compare Portuguese Term Collections
Marcirio Silveira Chaves and Vera Lúcia Strube de Lima
Dialog with a Personal Assistant
Fabrício Enembreck and Jean-Paul Barthès
Applying Argumentative Zoning in an Automatic Critiquer
of Academic Writing
Valéria D. Feltrim, Jorge M. Pelizzoni, Simone Teufel,

Maria das Graỗas Volpe Nunes, and Sandra M. Aluớsio
DiZer: An Automatic Discourse Analyzer for Brazilian Portuguese
Thiago Alexandre Salgueiro Pardo, Maria das Graỗas Volpe Nunes,
and Lucia Helena Machado Rino

184

194
204

214

224

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A Comparison of Automatic Summarizers
of Texts in Brazilian Portuguese
Lucia Helena Machado Rino, Thiago Alexandre Salgueiro Pardo,
Carlos Nascimento Silla Jr., Celso Antônio Alves Kaestner,
and Michael Pombo

XV

235

Machine Learning, Knowledge Discovery,

and Data Mining
Heuristically Accelerated Q–Learning: A New Approach
to Speed Up Reinforcement Learning
Reinaldo A.C. Bianchi, Carlos H.C. Ribeiro, and Anna H.R. Costa
Using Concept Hierarchies in Knowledge Discovery
Marco Eugênio Madeira Di Beneditto and Leliane Nunes de Barros
A Clustering Method for Symbolic Interval-Type Data
Using Adaptive Chebyshev Distances
Francisco de A.T. de Carvalho, Renata M.C.R. de Souza,
and Fabio C.D. Silva

245
255

266

An Efficient Clustering Method for High-Dimensional Data Mining
Jae- Woo Chang and Yong-Ki Kim

276

Learning with Drift Detection
João Gama, Pedro Medas, Gladys Castillo, and Pedro Rodrigues

286

Learning with Class Skews and Small Disjuncts
Ronaldo C. Prati, Gustavo E.A.P.A. Batista,
and Maria Carolina Monard

296

Making Collaborative Group Recommendations
Based on Modal Symbolic Data
Sérgio R. de M. Queiroz and Francisco de A.T. de Carvalho

307

Search-Based Class Discretization
for Hidden Markov Model for Regression
Kate Revoredo and Gerson Zaverucha

317

SKDQL: A Structured Language
to Specify Knowledge Discovery Processes and Queries
Marcelino Pereira dos Santos Silva and Jacques Robin

326

Evolutionary Computation, Artificial Life,
and Hybrid Systems
Symbolic Communication in Artificial Creatures:
An Experiment in Artificial Life
Angelo Loula, Ricardo Gudwin, and João Queiroz

336
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XVI

Table of Contents

What Makes a Successful Society?
Experiments with Population Topologies in Particle Swarms
Rui Mendes and José Neves

346

Splinter: A Generic Framework
for Evolving Modular Finite State Machines
Ricardo Nastas Acras and Silvia Regina Vergilio

356

An Hybrid GA/SVM Approach for Multiclass Classification
with Directed Acyclic Graphs
Ana Carolina Lorena and André C. Ponce de Leon F. de Carvalho

366

Dynamic Allocation of Data-Objects in the Web,
Using Self-tuning Genetic Algorithms
Joaquín Pérez O., Rodolfo A. Pazos R., Graciela Mora O.,
Guadalupe Castilla V., José A. Martínez., Vanesa Landero N.,
Héctor Fraire H., and Juan J. González B.

376

Detecting Promising Areas by Evolutionary Clustering Search
Alexandre C.M. Oliveira and Luiz A.N. Lorena

385

A Fractal Fuzzy Approach to Clustering Tendency Analysis
Sarajane Marques Peres and Márcio Luiz de Andrade Netto

395

On Stopping Criteria for Genetic Algorithms
Martín Safe, Jessica Carballido, Ignacio Ponzoni, and Nélida Brignole

405

A Study of the Reasoning Methods Impact on Genetic Learning
and Optimization of Fuzzy Rules
Pablo Alberto de Castro and Heloisa A. Camargo

414

Using Rough Sets Theory and Minimum Description Length Principle
to Improve a
Fuzzy Revision Method for CBR Systems
Florentino Fdez-Riverola, Fernando Díaz, and Juan M. Corchado

424

Robotics and Computer Vision
Forgetting and Fatigue in Mobile Robot Navigation

Ls Correia and António Abreu

434

Texture Classification Using the Lempel-Ziv-Welch Algorithm
Leonardo Vidal Batista and Moab Mariz Meira

444

A Clustering-Based Possibilistic Method for Image Classification
Isabela Drummond and Sandra Sandri

454

An Experiment on Handshape Sign Recognition
Using Adaptive Technology: Preliminary Results
Hemerson Pistori and João José Neto

464
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XVII

Autonomous Agents and Multi-agent Systems
Recent Advances on Multi-agent Patrolling
Alessandro Almeida, Geber Ramalho, Hugo Santana, Patrícia Tedesco,
Talita Menezes, Vincent Corruble, and Yann Chevaleyre

On the Convergence to and Location
of Attractors of Uncertain, Dynamic Games
Eduardo Camponogara
Norm Consistency in Electronic Institutions
Marc Esteva, Wamberto Vasconcelos, Carles Sierra,
and Juan A. Rodríguez-Aguilar
Using the
for a Cooperative Framework
of MAS Reorganisation
Jomi Fred Hübner, Jaime Simão Sichman, and Olivier Boissier

474

484
494

506

A Paraconsistent Approach for Offer Evaluation in Negotiations
Fabiano M. Hasegawa, Bráulio C. Ávila,
and Marcos Augusto H. Shmeil

516

Sequential Bilateral Negotiation
Orlando Pinho Jr., Geber Ramalho, Gustavo de Paula,
and Patrícia Tedesco

526

Towards to Similarity Identification to Help in the Agents’ Negotiation
Andreia Malucelli and Eugénio Oliveira

536

Author Index

547

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On Modalities for Vague Notions
Mario Benevides1,2, Carla Delgado2, Renata P. de Freitas2,
Paulo A.S. Veloso2, and Sheila R.M. Veloso2
1
Instituto de Matemática
Programa de Engenharia de Sistemas e Computaỗóo, COPPE
Universidade Federal do Rio de Janeiro, Caixa Postal 68511, 21945-970
Rio de Janeiro, RJ, Brasil
2

{mario,delgado,naborges,veloso,sheila}@cos.ufrj.br

Abstract. We examine modal logical systems, with generalized operators, for the precise treatment of vague notions such as ‘often’, ‘a meaningful subset of a whole’, ‘most’, ‘generally’ etc. The intuition of ‘most’

as “all but for a ‘negligible’ set of exceptions” is made precise by means
of filters. We examine a modal logic, with a new modality for a local
version of ‘most’ and present a sound and complete axiom system. We
also discuss some variants of this modal logic.
Keywords: Modal logic, vague notions, most, filter, knowledge representation.

1

Introduction

We examine modal logical systems, with generalized operators, for the precise
treatment of assertions involving some versions of vague notions such as ‘often’,
‘a meaningful subset of a whole’, ‘most’, ‘generally’ etc. We wish to express these
vague notions and reason about them.
Vague notions, such as those mentioned above, occur often in ordinary language and in some branches of science, some examples being “most bodies expand when heated” and “typical birds fly”. Vague terms such as ‘likely’ and
‘prone’ are often used in more elaborate expressions involving ‘propensity’, e.g.
“A patient whose genetic background indicates a certain propensity is prone
to some ailments”. A precise treatment of these notions is required for reasoning about them. Generalized quantifiers have been employed to capture some
traditional mathematical notions [2] and defaults [10]. A logic with various generalized quantifiers has been suggested to treat quantified sentences in natural
language [1] and an extension of first-order logic with generalized quantifiers for
capturing a sense of ‘generally’ is presented in [5]. The idea of this approach
is formulating ‘most’ as ‘holding almost universally’. This seems quite natural,
once we interpret ‘most’ as “all, but for a ‘negligible’ set of exceptions”.
Modal logics are specification formalisms which are simpler to be handled
than first-order logic, due to the hiding of variables and quantifiers through the
modal operators (box and diamond). In this paper we present a modal counterpart of filter logic, internalizing the generalized quantifier through a new
A.L.C. Bazzan and S. Labidi (Eds.): SBIA 2004, LNAI 3171, pp. 1–10, 2004.
© Springer-Verlag Berlin Heidelberg 2004

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2

Mario Benevides et al.

modality whose behavior is intermediate between those of the classical modal
operators
and
Thus one will be able to express “a reply to a message
will be received almost always”:
“eventually a reply to a message
will be received almost always”:
“the system generally operates
correctly”:
etc.
An important class of problems involves the stable property detection. In
a more concrete setting consider the following situation. A stable property is
one which once it becomes true it remains true forever: deadlock, termination
and loss of a token are examples. In these problems, processes communicate
by sending and receiving messages. A process can record its own state and the
messages it sends and receives, nothing else.
Many problems in distributed systems can be solved by detecting global
states. An example of this kind of algorithm is the Chandy and Lamport Distributed Snapshots algorithm for determining global states of distributed systems
[6]. Each process records its own state and the two processes that a channel is
incident on cooperate in recording the channel state. One cannot ensure that
the states of all processes and channels will be recorded at the same instant,
because there is no global clock, however, we require that the recorded process
and channel states form a meaningful global state. The following text illustrates
this problem [6]: “The state detection algorithm plays the role of a group of

photographers observing a panoramic, dynamic scene, such as a sky filled with
migrating birds – a scene so vast that it cannot be captured by a single photograph. The photographers must take several snapshots and piece the snapshots
together to form a picture of the overall scene. The snapshots cannot all be taken
at precisely the same instant because of synchronization problems. Furthermore,
the photographers should not disturb the process that is being photographed;
(...) Yet, the composite picture should be meaningful. The problem before us
is to define ‘meaningful’ and then to determine how the photographs should be
taken.”
If we take the modality
to capture the notion of meaningful, then the
formula
means:
is true in a meaningful set of states”. Returning to the
example of Chandy and Lamport Algorithm, the formula:

would mean “if in a meaningful set of states, for each pair of processes and
the snapshot of process
local state has property
snapshot of process has
property
and the snapshot of the state of channel ij has property
then it is
always the case that global stable property holds forever”. So we can express
relationships among local process states, global system states and distributed
computation’s properties even if we cannot perfectly identify the global state at
each time; for the purpose of evaluating stable properties, a set of meaningful
states that can be figured out from the local snapshots collected should suffice.
Another interesting example comes from Game Theory. In Extensive Games
with Imperfect Information (well defined in [9]), a player may not be sure about
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On Modalities for Vague Notions

3

the complete past history that has already been played. But, based on a meaningful part of the history he/she has in mind, he/she may still be able to decide
which action to choose. The following formula can express this fact

The formula above means: “it is always the case that, if it is player’s turn and
properties
are true in a meaningful part of his/her history, then player
should choose action to perform”. This is in fact the way many absent-minded
players reason, especially in games with lots of turns like ‘War’, Chess, or even
a financial market game.
We present a sound and complete axiomatization for generalized modal logic
as a first step towards the development of a modal framework for generalized
logics where one can take advantage of the existing frameworks for modal logics
extending them to these logics.
The structure of this paper is as follows. We begin by motivating families,
such as filters, for capturing some intuitive ideas of ‘generally’. Next, we briefly
review a system for reasoning about generalized assertions in Sect. 3. In Sect.
4, we introduce our modal filter logic. In Sect. 5 we comment on how to adapt
our ideas to some variants of vague modal logics. Sect. 6 gives some concluding
remarks.

2

Assigning Precise Meaning to Generalized Notions

We now indicate how one can arrive at the idea of filters [4] for capturing some
intuitive ideas of ‘most’, ‘meaningful’, etc. Our approach relies on the familiar
intuition of ‘most’ as “all but for a ‘negligible’ set of exceptions” as well as on
some related notions. We discuss, trying to explain, some issues in the treatment
of ‘most’, and the same approach can be applied in treating ‘meaningful’, ‘often’,
etc.

2.1

Some Accounts for ‘Most’

Various interpretations seem to be associated with vague notions of ‘most’. The
intended meaning of “most objects have a given property” can be given either
directly, in terms of the set of objects having the property, or by means of the
set of exceptions, those failing to have it. In either case, a precise formulation
hinges on some ideas concerning these sets. We shall now examine some proposals
stemming from accounts for ‘most’.
Some accounts for ‘most’ try to explain it in terms of relative frequency
or size. For instance, one would understand “most Brazilians like football” as
the “the Brazilians that like football form a ‘likely’ portion”, with more than,
say, 75% of the population, or “the Brazilians that like football form a ‘large’
set”, in that their number is above, say, 120 million. These accounts of ‘most’
may be termed “metric”, as they try to reduce it to a measurable aspect, so
to speak. They seek to explicate “most people have property
as “the people
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having form a ‘likely’ (or ‘large’) set”, i.e. a set having ‘high’ relative frequency
(or cardinality), with ‘high’ understood as above a given threshold. The next
example shows a relaxed variant of these metric accounts.
Example 1. Imagine that one accepts the assertions “most naturals are larger
than fifteen” and “most naturals do not divide twelve” about the universe of
natural numbers. Then, one would probably accept also the assertions:
“Most naturals are larger than fifteen or even”
“Most naturals are larger than fifteen and do not divide twelve”
Acceptance of the first two assertions, as well as inferring
from them,
might be explained by metric accounts, but this does not seem to be the case
with assertion
A possible account for this situation is as follows. Both sets
F, of naturals below fifteen, and T, of divisors of twelve, are finite. So, their
union still form a finite set.
This example uses an account based on sizes of the extensions: it explains
“most naturals have property
as “the naturals failing to have form a ‘small’
set”, where ‘small’ is taken as finite. Similarly, one would interpret “most reals
are irrational” as “the rational reals form a ‘small’ set”, with ‘small’ now understood as (at most) denumerable. This account is still quantitative, but more
relaxed. It explicates “most objects have property
as “those failing to have
form a ‘small’ set”, in a given sense of ‘small’.
As more neutral names encompassing these notions, we prefer to use ‘sizable’,
instead of ‘large’ or ‘likely’, and ‘negligible’ for ‘unlikely’ or ‘small’. The previous
terms are vague, the more so with the new ones. This, however, may be advantageous. The reliance on a – somewhat arbitrary – threshold is less stringent and
they have a wider range of applications, stemming from the liberal interpretation
of ‘sizable’ as carrying considerable weight or importance. Notice that these notions of ‘sizable’ and ‘negligible’ are relative to the situation. (In “uninteresting

meetings are those attended only by junior staff”, the sets including only junior
staff members are understood as ‘negligible’.)

2.2

Families for ‘Most’

We now indicate how the preceding ideas can be conveyed by means of families,
thus leading to filters [4] for capturing some notions of ‘most’. One can understand “most birds fly” as “the non-flying birds form a ‘negligible’ set”. This
indicates that the intended meaning of “most objects have
may be rendered
as “the set of objects failing to have is negligible”, in the sense that it is in
a given family of negligible sets. The relative character of ‘negligible’ (and ‘sizable’) is embodied in the family of negligible sets, which may vary according
to the situation. Such families, however, can be expected to share some general
properties, if they are to be appropriate for capturing notions of ‘sizable’, such
as ‘large’ or ‘likely’. Some properties that such a family may, or may not, be
expected to have are illustrated in the next example.
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On Modalities for Vague Notions

5

Example 2. Imagine that one accepts the assertions:
“Most American males like beer”
“Most American males like sports” and
“Most American are Democrats or Republicans”
In this case, one is likely to accept also the two assertions:
“Most American males like beverages”

“Most American males like beer and sports”
Acceptance of
should be clear. As for
its acceptance may be explained by exceptions. (As the exceptional sets of non beer-lovers and of nonsports-lovers have negligibly few elements, it is reasonable to say that “negligibly
few American males fail to like beer or sports”, so “most American males like
beer and sports”.) In contrast, even though one accepts
neither one of the
assertions “most American males are Democrats” and “most American males
are Republicans” seems to be equally acceptable.
This example hinges on the following ideas: if
and B has ‘most’
elements, then W also has ‘most’ elements; if both and have ‘negligibly few’
elements, then
will also have ‘negligibly few’ elements; a union
may
have ‘most’ elements, without either D or R having ‘most’ elements.
We now postulate reasonable properties of a family
of negligible
sets (in the sense of carrying little weight or importance) of a universe V.
if
“subsets of negligible sets are negligible”.
“the empty set is negligible”.
“the universe V is not negligible”.
if
“unions of negligible sets are negligible”.
These postulates can be explained by means of a notion of ‘having about the
same importance’ [12]. Postulates
and (V ) concern the non-triviality of our
notion of ‘negligible’. Also,
is not necessarily satisfied by families that may

be appropriate for some weaker notions, such as ‘several’ or ‘many’. In virtue
of these postulates, the family
of negligible sets is non-empty and proper as
well as closed under subsets and union, thus forming an ideal. Dually, a family
of sizable sets – of those having ‘most’ elements – is a proper filter (but not
necessarily an ultrafilter [4]).
Conversely, each proper filter gives rise to a non-trivial notion of ‘most’. Thus,
the interpretation of “most objects have property
as “the set of objects failing
to have is negligible” amounts to “the set of objects having belongs to a
given proper filter”. The properties of the family
are intuitive and coincide
with those of ideals. As the notion of ‘most’ was taken as the dual of ‘negligible’,
it is natural to explain families of sizeable sets in terms of filters (dual of ideals).
So, generalized quantifiers, ranging over families of sets [1], appear natural to
capture these notions.

3

Filter Logic

Filter logic extends classical first-order logic by a generalised quantifier
whose
intended interpretation is ‘generally’. In this section we briefly review filter logic:
its syntax, semantics and axiomatics.

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Given a signature we let
be the usual first-order language (with equality
of signature and use
for the extension of
by the new operator
The formulas of
are built by the usual formation rules and a new
variable-binding formation rule for generalized formulas: for each variable if
is a formula in
then so is
Example 3. Consider a signature with a binary predicate L (on persons). Let
stand for
loves
Some assertions expressed by sentences of
are: “people generally love everybody” “somebody loves people in
general” –
and “people generally love each other” –
Let
be
is taller than
We can express “people generally are taller
than
by
and
is taller than people in general” by
The semantic interpretation for ‘generally’ is provided by enriching first-order
structures with families of subsets and extending the definition of satisfaction to

the quantifier
A filter structure
for a signature consists of a usual structure
for together with a filter over the universe A of
We extend the usual
definition of satisfaction of a formula in a structure under assignment to its
(free) variables, using the extension
as follows: for a formula
iff
is in
As usual, satisfaction of a formula hinges only on the realizations assigned to
its symbols. Thus, satisfaction for purely first-order formulas (without
does
not depend on the family of subsets. Other semantic notions, such as reduct,
model
and validity, are as usual [4, 7]. The behavior of is intermediate between those of the classical and
A deductive system for the logic of ‘generally’ is formalized by adding axiom
schemata, coding properties of filters, to a calculus for classical first-order logic.
To set up a deductive system for filter logic one takes a sound and complete
deductive calculus for classical first-order logic, with Modus Ponens (MP) as
the sole inference rule (as in [7]), and extend its set A of axiom schemata by
adding a set
of new axiom schemata (coding properties of filters), to form
This set
consists of all the generalizations of the following five
schemata (where
and are formulas of

for a new variable not occurring in
These schemata express properties of filters, the last one covering alphabetic variants. Other usual deductive notions, such as (maximal) consistent sets,

witnesses and conservative extension [4,7], can be easily adapted. So, filter derivability amounts to first-order derivability from the filter schemata:
iff
Hence, we have monotonicity of
and substitutivity of equivalents.
This deductive system is sound and complete for filter logic, which is a proper
conservative extension of classical first-order logic. It is not difficult to see that
we have a conservative extension of classical logic:
iff
for and
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