Theory and Applications of Relational Structures as Knowledge Instruments: COST Action 274, TARSKI. Revised Papers

279 Pages·2003·2.42 MB·English

by  Rudolf Berghammer

Lecture Notes in Computer Science 2929
EditedbyG.Goos,J.Hartmanis,andJ.vanLeeuwen
3
Berlin
Heidelberg
NewYork
HongKong
London
Milan
Paris
Tokyo
Harrie de Swart Ewa Orłowska
Gunther Schmidt Marc Roubens (Eds.)
Theory and Applications
of Relational Structures
as Knowledge Instruments
COST Action 274, TARSKI
Revised Papers
1 3
SeriesEditors
GerhardGoos,KarlsruheUniversity,Germany
JurisHartmanis,CornellUniversity,NY,USA
JanvanLeeuwen,UtrechtUniversity,TheNetherlands
VolumeEditors
HarriedeSwart
TilburgUniversity,FacultyofPhilosophy
P.O.Box90153,5000LETilburg,TheNetherlands
E-mail:[e-mail]
EwaOrłowska
NationalInstituteofTelecommunications
ul.Szachowa1,04-894Warsaw,Poland
E-mail:[e-mail]
GuntherSchmidt
Universita¨tderBundeswehrMu¨nchen
Fakulta¨tfu¨rInformatik,Institutfu¨rSoftwaretechnologie
85577Neubiberg,Germany
E-mail:[e-mail]
MarcRoubens
UniversityofLie`ge,DepartmentofMathematics
SartTilmanBuilding,14GrandeTraverse,B37,4000Lie`ge1,Belgium
E-mail:[e-mail]
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Preface
Relational structures abound in the daily environment: relational databases,
data mining, scaling procedures, preference relations, etc. Reasoning about and
with relations has a long-standing European tradition. Today, there are strong
European research groups in the theoretical as well as the applied branches.
European research in the field may be divided into three broad areas:
1. Algebraic Logic: algebras of relations, relational semantics, and algebras and
logics derived from information systems.
2.ComputationalAspectsofAutomatedRelationalReasoning:decidabilityand
complexity of algorithms, network satisfaction.
3. Applications: Linguistics, Psychology, Economics, etc.
While there is a wealth of theoretical knowledge to be used, there has been
littleinteractionbetweenbasicandappliedresearchinthefield.Forthisreason,
a European Concerted Research Action has been implemented, designated as
COST Action 274: TARSKI (Theory and Applications of Relational Structures
as Knowledge Instruments).
The main objective of this book is to advance the understanding of relational
structuresandtheuseofrelationalmethodsinapplicableobjectdomains.There
are the following sub-objectives:
1.tostudythesemanticalandsyntacticalaspectsofrelationalstructuresarising
from ‘real world’ situations;
2. to investigate automated inference for relational systems, and, where possible
orfeasible,developdeductivesystemswhichcanbeimplementedintoindustrial
applications, such as diagnostic systems;
3. to develop non-invasive scaling methods for predicting relational data; and
4. to make software for dealing with relational systems commonly available.
Weareconfidentthatthepresentbookwillfurthertheunderstandingofinterdis-
ciplinaryissuesinvolvingrelationalreasoning.Thestudyandpossibleintegration
of different approaches to the same problem, which may have arisen at different
locations, will be of practical value to the developers of information systems.
The first five papers concern the mechanization of relational reasoning. This
group of mechanization papers starts with a comparative report on two already
existingsystemsbyRudolfBerghammer,GuntherSchmidt,andMichaelWinter.
The GUHA article by Petr Ha´jek, Martin Holenˇa, and Jan Rauch refers to the
well-developed system in Prague which derives information relations from infor-
mation systems and is therefore some sort of a program for relational data min-
ing. While there have been extensive studies in automated reasoning for propo-
sitional logics, Renate Schmidt and Ullrich Hustadt give a respective overview
for modal and description logic reasoning systems. An attempt to develop a for-
VI Preface
mal basis for theory extraction from relational data guided by some ontology is
undertaken by Gunther Schmidt. Pasquale Caianiello, Stefania Costantini, and
EugenioOmodeofocusondefinitionalextensionsappliedtorelationalformalisms
as a way of overcoming expressive limitations of logical formalisms.
The next three papers concern the field of relational scaling and preferences.
KimCao-VanandBernardDeBaetsdiscusshowaproperdefinitionofaranking
can be introduced into the framework of supervised learning. Agnieszka Rusi-
nowska gives an overview of axiomatic and strategic approaches to bargaining
problems. Harrie de Swart et al. give an overview of the four major categories
of voting procedures and their flaws.
The last four papers deal with the algebraic and logical foundations of real
world relations. Wojciech Buszkowski presents relational representability results
fortheclassesofalgebrasrelatedtotheLambeksyntacticcalculus.IvoDu¨ntsch
and Gu¨nther Gediga study modal-like approximation operators determined by
binaryrelationsandpresenttheirapplicationstopracticalproblemsthatrequire
a qualitative data analysis. Ivo Du¨ntsch, Ewa Orl(cid:2)owska and Anna Radzikowska
introduce and study a class of weak relation algebras based on not necessar-
ily distributive lattices. Ingrid Rewitzky developed a relational model of pro-
gramming languages whose commands may involve both angelic and demonic
non-determinism.
Referees
Ricardo Caferra Roger Maddux Dimiter Vakarelov
Jules Desharnais Ewa Orl(cid:2)owska Hui Wang
Saˇso Dˇzeroski Irina Perfilieva Michael Winter
Marcelo Frias Marc Roubens
Gu¨nther Gediga Gunther Schmidt
Wendy MacCaull Harrie de Swart
Acknowledgements
We owe much to the referees mentioned above and are most grateful to them.
Jozef Pijnenburg was instrumental in editing this book because of his highly
appreciated expertise in LATEX. The cooperation of many authors in this book
was supported by COST action 274, TARSKI, and is gratefully acknowledged.
Editors
Prof. Harrie de Swart, Chair, Tilburg University, The Netherlands
Prof. Ewa Orl(cid:2)owska, Institute for Telecommunications, Warsaw, Poland
Prof. Gunther Schmidt, Universita¨t der Bundeswehr, Mu¨nchen, Germany
Prof. Marc Roubens, Universit´e de Li`ege and Facult´e Polytechnique de Mons,
Belgium
Table of Contents
RelView and Rath – Two Systems for Dealing with Relations ......... 1
Rudolf Berghammer, Gunther Schmidt, and Michael Winter
The GUHA Method and Foundations of (Relational) Data Mining ....... 17
Petr H´ajek, Martin Holenˇa, and Jan Rauch
Mechanised Reasoning and Model Generation
for Extended Modal Logics.......................................... 38
Renate A. Schmidt and Ullrich Hustadt
Theory Extraction in Relational Data Analysis ........................ 68
Gunther Schmidt
An Environment for Specifying Properties of Dyadic Relations
and Reasoning about Them. I: Language Extension Mechanisms ......... 87
Pasquale Caianiello, Stefania Costantini, and Eugenio G. Omodeo
Consistent Representation of Rankings................................ 107
Kim Cao-Van and Bernard De Baets
Axiomatic and Strategic Approaches to Bargaining Problems ............ 124
Agnieszka Rusinowska
Categoric and Ordinal Voting: An Overview........................... 147
Harrie de Swart, Ad van Deemen, Eliora van der Hout, and Peter Kop
Relational Models of Lambek Logics.................................. 196
Wojciech Buszkowski
Approximation Operators in Qualitative Data Analysis ................ 214
Ivo Du¨ntsch and Gu¨nther Gediga
Lattice–Based Relation Algebras and Their Representability............. 231
Ivo Du¨ntsch, Ewa Orl(cid:1)owska, and Anna Maria Radzikowska
Binary Multirelations............................................... 256
Ingrid Rewitzky
Author Index ................................................. 273
RelView and Rath –
Two Systems for Dealing with Relations
Rudolf Berghammer1,(cid:1), Gunther Schmidt2,(cid:1), and Michael Winter3,(cid:1)
1 Institut fu¨r Informatik und Praktische Mathematik
Christian-Albrechts-Universit¨at zu Kiel
24098 Kiel, Germany
2 Fakult¨at fu¨r Informatik
Universit¨at der Bundeswehr Mu¨nchen
85577 Neubiberg, Germany
3 Computer Science Department
Brock University
St. Catharines, Ontario, Canada, L2S 3A1
Abstract. In this paper we present two systems for dealing with rela-
tions,theRelViewandtheRathsystem.Afterashortintroductionto
bothsystemsweexhibittheirusualdomainofapplicationbypresenting
some typical examples.
1 Introduction
In the area of logical reasoning, people began soon to look for subsets easier to
handlethan,forexample,fullpredicatelogic.Thisattemptresultednotleastin
relational reasoning. Already as early as 1915, Leopold Lo¨wenheim postulated
that one should resort to reasoning with relations in the “Gebietekalkul”, and
should “Schro¨derize” all of mathematics. This approach is certainly burdened
with a loss in expressiveness. Nevertheless, such a loss has been accepted in
the past by many scientists, as everything looks much simpler and it does not
deteriorate expressiveness too much.
Whenworkingwithrelationstoday,oneusuallyasksforadditionalcomputer
aid.Threesystemswithquitedifferentapproacheshavebeenproposedfromour
groups the last years. First, there may be just a specialized support in formula
manipulationasinRalf(see[7,8]),amendedevenbysomeautomatedfeatures.
A second approach is completely “on the model side” as with RelView. Here,
instead of working with binary predicates that may result in true or false, one
works with Boolean matrices. This is a paradigm shift allowing to incorporate
techniques known from linear algebra. In the RelView system this has been
elaborated in great detail to the extent that now something is available which
might be compared to a “numerics package” – this time however for relational
algebra. Thirdly, one may remain on the syntactic side, still avoiding to work
(cid:1) Co-operationforthispaperwassupportedbyEuropeanCOSTAction274“Theory
and Applications of Relational Structures as Knowledge Instruments” (TARSKI).
H.deSwartetal.(Eds.):TARSKI,LNCS2929,pp.1–16,2003.
(cid:1)c Springer-VerlagBerlinHeidelberg2003
2 Rudolf Berghammer, Gunther Schmidt, and Michael Winter
in a model. This means concentrating solely on the algebraic rules valid in the
relational fragment. This characterizes the Rath approach. Logical reasoning is
facilitatedsincetheRathsystemoffersprecisetypecontrol.Negation,e.g.,need
not be avoided, as due to the type restriction no unacceptably large result will
showup.Rathalsoworksifsomeoftherulesofrelationalgebraareabandoned
focusingonDedekindcategories,divisionallegories,etc.Allthecommonaspects
are handled simultaneously.
Considered in the context of the newly founded COST action 274: TARSKI,
all systems seem extremely well-suited to fostering mechanization. Given the
observationthatmanypeoplekeepinventingideastocopewithrelationalstruc-
tures arising around real-world phenomena, there is always the task to study
whether these ideas are really helpful – whether they really work. The systems
offer detailed computer help in different directions. Here, we exhibit in which
way they may be used. Since Ralf is currently not maintained, we concentrate
on RelView and Rath.
There is however a lot of work going on to further mechanize any form of
work with relations. On the one hand side, a successor of RALF is currently
under construction. On the other hand side, methods to decompose relations
with respect to different criteria have been developed recently [14].
2 Relation-Algebraic Preliminaries
In this section, we briefly introduce the basic concepts of relation algebra, some
special relations, and some relation-algebraic constructions. For more details
concerning the algebraic theory of relations, see e.g., [4,13].
Given non-empty sets X and Y, the set of all (set-theoretic or concrete)
relations with domain X and range Y is denoted by [X↔Y] and we write
R : X↔Y instead of R ∈ [X↔Y]. If X and Y are finite and of cardinality m
and n, respectively, then we may consider R as a Boolean matrix with m rows
and n columns. This matrix interpretation is well-suited for many purposes.
Therefore, in this paper we frequently will use matrix concepts and notations
also for relations. Especially, we will speak of rows and columns, and we will
denote membership by Rxy instead of (x,y)∈R.
We assume the reader to be familiar with the basic operations on relations,
viz. RT (transposition), R (negation), R ∪ S (union), R ∩ S (intersection), RS
(composition), R ⊆ S (inclusion, subrelation test), and the special relations O
(empty relation), L (universal relation), and I (identity relation). With the set-
theoreticoperations ,∪,∩,⊆andtheconstantsO,Lsuchrelations,respectively
Boolean, matrices form a complete Boolean lattice. Further well-known laws for
operations on relations are, for instance:
RTT =R Q(R∩S)⊆QR∩QS (RS)T =STRT
The theoretical framework for such laws to hold is that of a relation algebra.
First, such an algebraic structure is a category. I.e., there is a class of objects;
foreverypairA,BofobjectsthereisaclassRAB ofmorphisms,andforalltriples
RelView and Rath – Two Systems for Dealing with Relations 3
RAB, RBC, RAC there is a composition from RAB ×RBC to RAC such that
associativity holds and for all RAB there exists precisely one left identity from
RAA and one right identity from RBB. The morphisms are called (abstract)
relations and for their composition and the identity relations we use here the
same notation as for concrete relations. However, this category is extended by
a transposition operation mapping relations from RAB to RBA, where we use
again the notation of the concrete case. Furthermore, the following properties
are demanded to hold:
1. Every class RAB is a complete Boolean lattice with the usual operations
, ∪, ∩, the ordering ⊆, and the least (empty) relation O and greatest
(universal) relation L.
2. For all relations Q ∈ RAB, R ∈ RBC, and S ∈ RAC the following so-called
Schr¨oder equivalences hold:
QTS ⊆R ⇐⇒ QR⊆S ⇐⇒ SRT ⊆Q (1)
Often, in particular within the RelView system, the following so-called
Tarski rule is required as a further axiom; it is strongly connected to a gen-
eralization of the notion of simplicity known from universal algebra:
LRL=L ⇐⇒ R(cid:8)=O (2)
NotethatforR∈RBC intheequalityof(2)thereoccurthree–possibledifferent
– universal relations, viz. from RAB and RCD on the left-hand side and from
RAD on the right-hand, which all are denoted by the same symbol.
Let R be a (concrete or abstract) relation. Then R is called univalent (or
functional respectively a partial mapping) if RTR ⊆ I, and total if RL = L. As
usual, a mapping is a univalent and total relation. Relation R is called injective
ifRT isunivalentandsurjective ifRT istotal.Abijective relationisaninjective
and surjective relation.
Now, let R in addition be homogeneous, i.e., a relation for which the specific
product RR exists. (In the abstract case this is equivalent to R ∈ RAA and in
the concrete case this is equivalent to R:X↔X.) Then R is called reflexive if
I ⊆ R, transitive if RR ⊆ R, and antisymmetric if R∩RT ⊆ I. A partial order
is a reflexive, antisy(cid:1)mmetric, and transitive relation. The transitive closure of R
is defined as R+ = Ri, where R0 =I and Ri+1 =RRi for all i∈N. Using
i>0
R+,thereflexive-transitiveclosureR∗ ofRmaybedefinedthroughR∗ =I∪R+.
If R+ ⊆I, then R is said to be acyclic.
A relation v with v = vL is called a (row-) vector. In the case of a concrete
relation v : X↔Y this condition means that an element from X is either in
relation to none of the elements or to all elements of Y. Hence, v equals a
Cartesian product X(cid:3)×Y, where X(cid:3) is a subset of X. As for a concrete vector
the range is without relevance, we consider in the following frequently vectors
v : X↔1 with a singleton set 1 = {⊥} as range and write then vx instead of
vx⊥, i.e., suppress the second index. Such a vector v may be considered as a
Boolean matrix with exactly one column, i.e., as a Boolean column vector. It
describes the set X(cid:3) ={x∈X |vx}.