Complexity of Description Logics with Concrete Domains [PhD Thesis]

225 Pages·2002·1.465 MB·English

by  Carsten Lutz

The Complexity of Description Logics with
Concrete Domains
Von der Fakulta¨t fu¨r Mathematik, Informatik und
Naturwissenschaften der Rheinisch-Westf¨alischen Technischen
Hochschule Aachen zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Informatiker Carsten Lutz
aus Hamburg
Berichter: Universit¨atsprofessor Dr.-Ing. Franz Baader
Privatdozent Dr. rer.-nat. Frank Wolter
Tag der mu¨ndlichen Pru¨fung: 8. M¨arz 2002
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfu¨gbar.

To my parents
Acknowledgements
First of all, I would like to thank Franz Baader, my thesis supervisor, for his support
and advice. My work during the last three years was shaped by my teachers Franz
Baader, Ulrike Sattler, and Frank Wolter. Without any of them, this thesis would not
havebeenpossible. Moreover, IowetomycolleaguesfromAachen: SebastianBrandt,
Ralf Ku¨sters, Ralf Molitor, Ramin Sadre, Stephan Tobies, and Anni-Yasmin Turhan.
Working with you was a great pleasure. A special mention deserves Uli Sattler: I owe
her countless beers for supporting me in just as many ways. Thanks a lot!
On the private side, most importantly I would like to thank Anja for bearing with the
distance; Jakob for providing the best possible reason for starting to write my thesis;
my parents for always supporting me without ever pushing; my brother Patrick for
running the best “hotel” in Hamburg; and Daniel and Petra for never growing tired
of me being tired. There are many other people from Aachen, Hamburg, and the
scientific community who made the last three years worth living.
Contents
1 Introduction 1
2 Preliminaries 9
2.1 Description Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Introducing ALC . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Extensions of ALC . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 TBox and ABox Formalisms . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 TBoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 ABoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Description Logics with Concrete Domains. . . . . . . . . . . . . . . . 24
2.3.1 Introducing ALC(D) . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.2 Extensions of ALC(D) . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Examples of Concrete Domains . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Unary Concrete Domains and ALCf(D) . . . . . . . . . . . . . 31
2.4.2 Expressive Concrete Domains . . . . . . . . . . . . . . . . . . . 36
2.4.3 Temporal Concrete Domains . . . . . . . . . . . . . . . . . . . 36
3 Reasoning with ALCF(D) 41
3.1 Concept Satisfiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.2 The Completion Algorithm . . . . . . . . . . . . . . . . . . . . 44
3.1.3 Correctness and Complexity . . . . . . . . . . . . . . . . . . . . 49
3.2 ABox Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.1 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.2 Correctness and Complexity . . . . . . . . . . . . . . . . . . . . 62
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Acyclic TBoxes and Complexity 67
4.1 PSpace Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1.1 ALC with Acyclic TBoxes . . . . . . . . . . . . . . . . . . . . . 68
4.1.2 A Rule of Thumb . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.1.3 ALC-ABox Consistency . . . . . . . . . . . . . . . . . . . . . . 76
4.2 A Counterexample: ALCF . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 The Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
i
5 Extensions of ALC(D) 95
5.1 A NExpTime-complete Variant of the PCP . . . . . . . . . . . . . . . 96
5.2 A Concrete Domain for Encoding the PCP . . . . . . . . . . . . . . . 102
5.3 Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.1 ALC(D)-concept Satisfiability w.r.t. Acyclic TBoxes . . . . . . 110
5.3.2 ALC(cid:117)(D)-concept Satisfiability . . . . . . . . . . . . . . . . . . 113
5.3.3 ALC−(D)-concept Satisfiability . . . . . . . . . . . . . . . . . . 115
5.3.4 ALCP(D)-concept Satisfiability . . . . . . . . . . . . . . . . . . 120
5.3.5 ALCrp(D)-concept Satisfiability . . . . . . . . . . . . . . . . . . 123
5.4 The Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4.1 The Completion Algorithm . . . . . . . . . . . . . . . . . . . . 130
5.4.2 Termination, Soundness, and Completeness . . . . . . . . . . . 134
5.4.3 Adding Acyclic TBoxes . . . . . . . . . . . . . . . . . . . . . . 149
5.5 Comparison with ALCF . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.5.1 ALCF(cid:117)-concept Satisfiability . . . . . . . . . . . . . . . . . . . 154
5.5.2 Undecidability of ALCF− . . . . . . . . . . . . . . . . . . . . . 156
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6 Concrete Domains and General TBoxes 161
6.1 An Undecidability Result . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2 ALC(P) with General TBoxes . . . . . . . . . . . . . . . . . . . . . . . 163
6.2.1 Temporal Reasoning with ALC(P) . . . . . . . . . . . . . . . . 164
6.2.2 A Modelling Example . . . . . . . . . . . . . . . . . . . . . . . 165
6.2.3 Deciding Concept Satisfiability . . . . . . . . . . . . . . . . . . 168
6.2.4 Deciding ABox Consistency . . . . . . . . . . . . . . . . . . . . 184
6.3 Related Work and Discussion . . . . . . . . . . . . . . . . . . . . . . . 194
7 Summary and Outlook 197
Bibliography 201
Index 214
ii
Chapter 1
Introduction
Description Logics (DLs) are knowledge representation formalisms that allow to rep-
resent and reason about conceptual and terminological knowledge in a structured and
semantically well-understood manner. The basic entities for representing knowledge
using DLs are so-called concepts, which correspond to formulas with one free vari-
able in mathematical logic. Complex concepts are built from concept names (unary
predicates), role names (binary predicates), and concept constructors. For example,
using the basic propositionally closed Description Logic ALC, we can describe fathers
having at least one daughter using the concept
Male(cid:117)∀child.Human(cid:117)∃child.Female.
In this example, Male, Human, and Female are concept names while child is a role
name.
A major limitation of knowledge representation with Description Logics such as
ALC is that “concrete qualities” of real world entities cannot be adequately rep-
resented. For example, if we want to describe husbands that are younger than
their spouses, we can only do this by using an abstract, unary predicate such as
YoungerThanSpouse:
Male(cid:117)∃spouse.Female(cid:117)YoungerThanSpouse.
However, this approach is by no means satisfactory since it does not really capture
the semantics of what we were trying to say: we can at the same time demand that a
husbandis YoungerThanSpouse and50yearsold(usinganotherpredicate50YearsOld),
and that his spouse is 40YearsOld without obtaining a contradiction. Other concrete
qualitieswhoserepresentationleadstosimilarproblemsincludeweights,temperatures,
durations, and shape of real world entities.
To allow an adequate representation of concrete qualities, Description Logics can
be extended by so-called concrete domains. A concrete domain consists of a set such
as the natural numbers and a set of n-ary predicates such as the binary “<” with
the obvious (fixed) extension. The integration of concrete domains into Description
Logics is achieved by adding (i) a concrete domain-based concept constructor and
(ii) a new sort of role names that allows to associate values from the concrete domain
1
2 Chapter 1. Introduction
with abstract, logical objects. For example, the husbands that are younger than their
spouse can be described using the concept
Male(cid:117)∃spouse.Female(cid:117)∃age,spouseage.<,
where spouse is a functional role, age is one of the “concrete domain roles”, and
∃age,spouseage.<isanapplicationoftheconcretedomainconceptconstructor,which
must not be confused with the existential value restriction constructor used in the
subconcept ∃spouse.Female—both constructors start with the same symbol “∃”.
Some form of concrete domain can be found in many Description Logic formalisms
and in many implemented DL systems used in applications. However, although it is
generally considered very important to determine the decidability and computational
complexity of reasoning with Description Logics, the complexity of reasoning with
concrete domains has never been formally investigated. In this thesis, we close this
gap by determining the complexity of a many common Description Logics providing
for concrete domains.
The remainder of this introduction is concerned with a more detailed description
of the addressed research problems. Formal definitions and bibliographic references
are deferred until Chapter 2.
Description Logics
A Description Logic usually consists of a concept language and so-called TBox and
ABox formalisms. While the concept language is used for constructing complex con-
cepts and roles, TBoxes allow the representation of terminological knowledge and of
background knowledge from the application domain, and ABoxes store assertional
knowledge about the current state of affairs in a particular “world”. For example, the
concept language ALC mentioned above provides the concept constructors negation
(¬), conjunction ((cid:117)), disjunction ((cid:116)), universal value restriction of roles (∀), and ex-
istential value restriction of roles (∃). As we shall see later, there exist many other
concept and role constructors giving rise to more powerful concept languages.
LetusbrieflydescribetheuseofTBoxesandABoxesforknowledgerepresentation.
There exist various flavors of TBoxes with vast differences in expressivity. However,
even the weakest form of TBox, called acyclic TBox, can be used to represent termi-
nological knowledge about the application domain. For example, we can assign the
notion “younger husband” to the husbands describe above:
.
YoungerHusband = Male(cid:117)∃spouse.Female(cid:117)∃age,spouseage.<.
Intuitively, acyclic TBoxes can be thought of as (acyclic) macro definitions. Using a
more expressive type of TBox, so called general TBoxes, complex background knowl-
edge can be described. For example, we can express that all humans are either male
or female and no human is both male and female:
.
Human = Male(cid:116)Female
.
(cid:62) = ¬(Male(cid:117)Female).
3
Here, (cid:62) is a special concept that, intuitively, stands for “everything”.
In contrast to TBoxes, which represent knowledge of a general nature, ABoxes
store knowledge about particular situations. If, for example, we want to describe that
Johnis42yearsoldandmarriedtoMary, whois40yearsold, wecanusethefollowing
ABox:
John : Male John : ∃age.=
42
Mary : Female Mary : ∃age.=
40
(John,Mary) : spouse
Notethat,ifweaddthefactJohn : YoungerHusband,thentheresultingABoxdescribes
an “impossible” world since spouse is functional and John is not younger than his
(only) spouse Mary. Such inconsistencies can be detected by the Description Logic
reasoning services. More precisely, standard reasoning services offered by Description
Logics include the following:
• decide whether a concept C is satisfiable, i.e., whether it can have any instances
(concept satisfiability);
• decide whether a concept C is subsumed by a concept D, i.e., whether every
instance of C is necessarily also an instance of D (concept subsumption); and
• decide whether a given ABox is consistent, i.e., whether a world as described by
the ABox may exist (ABox consistency).
All these reasoning tasks can be considered with and without reference to TBoxes.
Several other reasoning services have been considered in the literature, but many of
them can be reduced to the basic ones listed above. Note that the top-most task in
the list corresponds to formula satisfiability in mathematical logic.
Apart from being a standard tool for knowledge representation and reasoning,
Description Logics are used in several other application areas such as reasoning about
entity relationship (ER) diagrams or reasoning about ontologies for the semantic web.
In all these application areas, DLs are expected to meet the following two demands:
1. Reasoning should be decidable. Moreover, since it is desirable to implement rea-
soning services in DL systems with an acceptable run-time behavior, reasoning
should be of low worst-case complexity.
2. The logic should be as expressive as possible to allow capturing all relevant
aspects of the application domain.
The trade-off between complexity and expressivity induced by these two demands is
one of the driving forces behind Description Logic research: the task is to develop
logics that are sufficiently expressive, yet for which reasoning is of an acceptable com-
plexity. During the last years, the reading of “acceptable complexity” has changed
dramatically. In contrast to the early years of DL research, where researchers were
striving to develop logics for which reasoning is tractable, nowadays there exist im-
plementations of Description Logics for which reasoning is ExpTime-complete, and
4 Chapter 1. Introduction
these implementations exhibit a reasonable run-time behavior on “real-world” prob-
lems. Nevertheless, the investigation of the worst-case complexity of reasoning with
DescriptionLogicsisstilloneofthemostimportantresearchtopicsinthefield. Know-
ingtheexactcomplexityclassofaproblemisnecessaryfordevisingoptimalalgorithms
and it provides useful information concerning the run-time behavior to be expected
from implemented DL reasoners.
Concrete Domains
Although several Description Logic systems also provided for some form of concrete
domain, the first formal treatment was given by Baader and Hanschke in [1991a]. The
authors propose to extend the basic Description Logic ALC with a concrete domain
D, thus obtaining the logic ALC(D). More precisely, ALC(D) extends ALC with
(i) functional roles called abstract features, (ii) the above mentioned “concrete domain
roles”, which are called concrete features and are also required to be functional, and
(iii) a concrete domain concept constructor. The difference between abstract and
concrete features is that the former are just functional binary predicates in a standard
first-order sense while the latter provide the link between abstract, logical objects and
objects fromthe concrete domain. The concrete domainconcept constructorprovided
byALC(D)hastheform∃u ,...,u .P,wheretheu areconcrete paths,i.e.,sequences
1 n i
f ···f g of finitely many abstract features followed by a single concrete feature, and
1 k
P is a predicate of arity n from the concrete domain D. For example, using ALC(D)
together with an appropriate concrete domain, we can describe people for whom the
wage of the boss of the father is smaller than the wage of the mother:
Human(cid:117)∃(father boss wage),(mother wage).<
Inthisexample, father, boss, andmother areabstractfeatureswhilewageisaconcrete
feature and concrete paths are written in parenthesis. It is important to note that the
concretedomainDcanbeviewedasaparametertothelogicALC(D),i.e.,Baaderand
Hanschke’s approach is not committed to any particular concrete domain. Instead,
their logic can be instantiated with any concrete domain suitable for representing
knowledge from the application domain at hand. In the literature, one can find a
broad spectrum of concrete domains that allow, e.g., to represent knowledge about
numbers (ages, weights, temperatures), temporal relationships, or spatial extensions.
In their 1991 paper, Baader and Hanschke prove that reasoning with ALC(D) is
decidable if the satisfiability of finite conjunctions of predicates from D is decidable
(and, additionally, D satisfies some minor technical conditions). However, although
concrete domains play an important role in DL research and ALC(D) can be regarded
as the basic Description Logic with concrete domains, this fundamental result has
never been further elaborated: first, the exact complexity of reasoning with ALC(D)
has never been determined; second, extensions of ALC(D) with standard concept and
role constructors not available in ALC or with TBox and ABox formalisms have only
rarely been defined. Complexity results for such logics were not available. The main