=Paper=
{{Paper
|id=Vol-48/paper-3
|storemode=property
|title=Contributions to the Axiomatic Foundations of Upper-Level Ontologies
|pdfUrl=https://ceur-ws.org/Vol-48/herre_degen.pdf
|volume=Vol-48
}}
==Contributions to the Axiomatic Foundations of Upper-Level Ontologies==
https://ceur-ws.org/Vol-48/herre_degen.pdf
Contributions to the Axiomatic Foundation
of Upper-Level Ontologies
Wolfgang Degen1 , Heinrich Herre2
IMMD, Universitat Erlangen, degen@informatik.uni-erlangen.de1
Institut fur Informatik, Universitat Leipzig, herre@informatik.uni-leipzig.de2
Abstract. In the present paper we make some contributions to the the-
ory of upper level ontologies. Every domain-speci c ontology must use as
a framework some upper-level ontology which describes the most general,
domain-independent categories of reality. It turns out that the top-level
ontology of the well-known standard modelling languages KIF (Knowl-
edge Interchange Format), F-logic (Frame-Logic), and CyCL, is based
on set-theoretical construction principles, so that the latter are essen-
tially limited by the extensionalism of set theory. In the current paper
we outline a new approach to upper-level ontologies which is based on
recent results in formal ontology. We formulate some criteria which any
upper-level ontology should satisfy although we shall not give a de nite
de nition. Every such ontology should include the three ontological cate-
gories of sets, universals, individuals and a system R of formal relations.
R contains, among others, membership, part-whole and inherence. The
individuals are divided in substantoids, moments and situoids. We re-
view a modelling language GOL (General Ontological Language) whose
top-level ontology satis es these criteria. Finally, we compare our top-
level ontology to others, in particular to KIF, and to the ontologies of
Russell-Norvig, J.Sowa, and to the standardization project SUO.
1 Introduction
The current paper is devoted to an analysis of upper level categories. Upper
level ontologies are concerned with theories of such highly general (domain-
independent) categories as: time, space, inherence, instantiation, identity, pro-
cesses, events, attributes, etc. The development of a widely accepted and well-
established upper level ontology is an important and ambitious task, important
because it determines a framework for the construction of domain-speci c and
generic ontologies, ambitious because it is concerned with highly debated philo-
sophical problems. An upper level ontology could be considered as an integral
part of a knowledge modelling and representation language. In our approach we
assume the point of view of realism, i.e. the position that the kind of things we
are speaking about have objective existence; hence, ontological research should
address reality itself. Thus, ontological research is not limited to building up
conceptual schemes. We believe that the real world itself plays an important
role to achieve a uni ed and consistent top-level ontology. We use the term cate-
gory to denote collections of entities. We classify the collection of all entities into
the categories sets, individuals, universals, and formal relations. In section 2 we
study the categories of sets, individuals, and universals. Section 3 is devoted to
formal relations. The categories of section 2 and section 3 present the highest
level of generality. In section 4 the category of individuals is further classi ed
and some formal relations are considered in more detail. In section 4.2 condi-
tions are formulated that - in our opinion- should be satis ed by every upper
level ontology, and in section 4.3 a formal modelling language GOL is reviewed
whose ontological basis satis es the criteria of an upper level ontology. GOL is
an ongoing research project at the University of Leipzig, [10]. In section 5 we
compare our ontology to other upper-level ontologies.
2 Individuals, Universals, and Sets
In our approach the entities of the real world can be partitioned in sets and
urelements. Urelements are entities which are not sets, and urelements are clas-
si ed into individuals and universals. Individuals are thought of as a realm of
concretely existing things in the world (not sets), within the con nes of space
and time. An individual is a single thing located in some single region of space-
time. A universal is an entity that can be instantiated by a number of di erent
individuals which are similar in some respect. Universals are patterns of features
which are not related to time and space. We assume that universals exist in the
individuals which instantiate them (thus they exist in re); thus, our attitude
is broadly Aristotelian in spirit. Individuals are correlated with universals via
the formal relation of instantiation. We write a :: u to denote that the individ-
ual a is an instance of the universal u. In the philosophical tradition there are
several conceptions of universals and individuals, respectively, [13].For example,
universals have been conceived as sets of individuals; and individuals have been
conceived as sets of universals. If one adopts both conceptions, then individuals
are sets of sets of individuals; and universals are sets of sets of universals. And
so on. We conclude that not both conceptions are admissible. Which of the two
conceptions has to be given up? We think that both have to be given up. A de-
tailed justi cation of this view is long and will be expounded in the full version
of the paper.
We assume that sets are neither in Space nor in Time. Hence, sets are not
individuals. A universal is determined by its intention which captures among
others a certain granularity and a certain view of its instances. We assume the
basic axiom that sets, individuals and universals are pairwise disjoint categories.
3 Formal Relations
According to the present standard doctrine, every n-ary relation is a set (or
class) of ordered n-tuples. This conception is applied indiscriminately to both
so-called empirical and mathematical relations. Thus, when John is kissing Mary,
the ordered pair hJohn, Maryi is an element of a certain set A of ordered pairs
that is denoted by "kissing". Also, that 2 < 5 is analysed by 2; 5i 2 B where B
6
is the set f x; yi : x < yg of ordered pairs. This uniform analysis harmonizes with
conceiving of both sentences as having the same syntactic form, namely R(a; b).
6
The usual semantics of rst-order logic, i. e. the semantics with respect to which
Godel's completeness theorem is proved, interprets any sentence R(a1 ; : : : ; an )
as stating that ha1 ; : : : ; an i 2 R.
Admittedly, the sentences "John is kissing Mary" and "2 < 5" have su-
cient super cial similarity to warrant the same syntactic and semantic analysis.
However, these sentences have radically di erent linguistic environments. For in-
stance, "John is kissing Mary" can be extended to "John is kissing Mary twice",
but "2 < 5" cannot be extended to "2 < 5 twice". Furthermore, the sentence
"John is kissing Mary" is implied by "John is kissing Mary twice". We have
therefore to use a logical form that will support this implication. While for 2
and 5 to be related as 2 < 5 no further entities are required to exist, there have
to exist entities beyond John and Mary if "John is kissing Mary (twice)" is to
be true, namely at least one (at least two) kiss(es). A plausible formulation is
the following:
1. 9x(x is a kiss ^ does(J; x) ^ su ers(M; y))
2. 9x; y(x is a kiss^x is a kiss^:x = y^does(J; x)^it does(J; y)^su ers(M; x)^
su ers(M; y))
Obviously 2. implies 1. in rst-order predicate logic. These considerations lead
to the distinction between formal and material relations. A relation is formal if
it holds as soon as its relata are given. Formal relations are called equivalently
immediate relations since they hold of their relata without mediating additional
individuals. The smaller-than relation < for numbers, the instantiation relation,
and membership are formal relations. Also the relations denoted by "does" and
"su ers" in (1) and (2) are formal. In contradistinction to this, the kissing rela-
tion does not hold as soon as its relata exist; a kiss (understood as an individual
event) must happen in addition to its relata. Therefore, we call the kissing rela-
tion a material relation. Formal relations may be classi ed with respect to the
main categories: sets, universals and indviduals. Membership is a formal relation
between sets and between urelements and sets, and instantiation is a formal re-
lation between individuals and universals. Universals and sets are connected by
the formal relation of extension. Examples of formal relations between individ-
uals are inherence, occupation, framing, containment which will be explained in
section 4.1.
4 Top Level Categories and Upper level Ontologies
In this section further categories of individuals are introduced, and some formal
relations are considered in more detail. We call all these categories as ontologi-
cally basic because they are the most fundamental.
4.1 Ontologically Basic Categories
The individuals are further subcategorized into moments, substantoid, chronoids,
topoids and situoids1 -terms which will be explained in more detail in what fol-
lows. This yields (at least) the following ontologically basic predicates (also called
basic types): Mom(x), Subst(x), Sit(x), Chron(x), Top(x). Chronoids can be un-
derstood as temporal durations, and topoids as spatial regions with a certain
mereotopological structure. A substance is that which bears individual proper-
ties or is connected to other substances by relational moments. Every substance
possesses matter. The ultimate (or prime) matter of a substance has no mo-
ments or qualities. This implies that an ultimate matter cannot appear since
every appearance has to take place via individual moments inhering in the ap-
pearing matter. We use the term substantoid to cover the whole range between
prime matter and those pieces of matter (called substances in the proper sense)
which possesses moments including forms, motions and qualities. An alignment
ha1 ; : : : ; an i is a sequence of substantiods a1 ; : : : ; an , the ai 's are called compo-
nents. An alignment is an individual in contradistinction to a list which is a
set.
The origin of the notion of moment lies in the theory of \individual accidents"
developed by Aristotle in his Metaphysics and Categories [1]. An accident is an
individualized property, event or process which is not a part of the essence of a
thing. We use the term \moment" in a more general sense and do not distinguish
between essential and inessential moments. Moments include individual qualities,
actions and passions, a blush, a handshake, thoughts and so on; moments thus
comprehend what are sometimes referred to as \events". The loss of a moment
in this more general sense may change the essence of a thing. Moments have in
common that they are all dependent on substantoids. Relational moments are
dependent on a plurality of substantoids.
A situoid is a part of the world that can be comprehended as a whole and
which takes into account the courses and histories of the ontological entities
occurring in it. Situations are special types of situoids: they are situoids at a
time, so that they present a snap-shot view of parts of the world. Situations can
be de ned as projections of situoids onto atomic (or very small) time intervals
or equivalently as situoids with an atomic (or very small) framing chronoid.
An elementary situation is composed of individual substances and the relational
moments which glue them together. Situations are constituted by using further
ontologically basic relations such as causality and intentionality. Analogously, we
distinguish elementary situoids from situoids. We assume that every situoid is
framed by a chronoid and by a topoid. For every situoid s there is a nite number
of universals which are associated with s. ass(s; u) has the meaning: s is a situoid
and u is a universal associated with s. There is a basic predicate Sitel(x) for
elementary situoid, and a predicate Sit(x) for situoids in general. Our approach
to situations di ers essentially from that of Barwise [2], [3]. Barwise did not
elaborate an ontology of relations; thus in particular, the relation of inherence
is missing from his theory.
1
This classi cation is, of course, tentative.
What are the ontologically basic relations needed to glue together the entities
mentioned above? Basic relations are membership denoted by 2, and part-of
relations, denoted by <, (re exive part-of). Other basic relations are denoted
by symbols starting with a colon. In the current paper we additionally consider
the basic relation of inherence, denoted by : , the relativized ternary part-of
relation, symbolized by :, the instantiation relation, denoted by ::, the framing
relation, designated by : <, the containment relation, denoted by :, and the
foundation relation, denoted by :?.
The phrase \inherence in a subject" can be understood as the translation
of the Latin expression in subjecto esse, in contradistinction to de subjecto dici,
which may be translated as \ predicated of a subject". The inherence relation :
- sometimes called ontic predication - glues moments to the substances, which
are their bearers, for example it glues your smile to your face, or the charge in
this conductor to the conductor itself. The part-of relations <; should have in
every case an individual as its second argument, i.e. if a < b; a b, then b is an
individual. The ternary part-whole relation :(x; y; z ) has the meaning: \z is a
universal and x qua instance of z is a part of y". Obviously, :(x; y; z ) implies
x y, but not conversely. These universal-dependent part-whole relations are
important in applications. The symbol :: denotes the instantiation relation, its
rst argument is an individual, and its second a universal. The instances of a
universal u are \individualizations" of the time- and space-independent pattern
of features captured by u. The containment relation : captures the constituents
of a situoid. x: y means \x is a constituent of the situoid y". Among the
constituents of a situoid s belong the occurring substantoids and the moments
inhering in them, but also the universals associated to s.
The binary relation : < glues chronoids or topoids to situations. We pre-
sume that every situation is framed by a chronoid and a topoid, and that every
chronoid or topoid frames a situation. The relation x: < y is to be read \the
chronoid (topoid) x frames the situation y". The de ned binary relation :?(x; y)
expresses the foundation relation between a substance and a moment. :?(x; y)
has the meaning that x is the foundation of y or x founds the moment y, as for
example your hair founds the moment of your being hairy, and the moment of be-
ing hairy inheres in your person. The de ned binary relation :occ(x; y) describes
a relation between substances and topoids having the meaning: the substantoid
x occupies the topoid y.
4.2 Upper Level-Ontologies
We formulate certain neccessary criteria which an upper level ontology - in our
opinion- should satisfy.
Upper-Level Ontology. Every upper level ontology includes at least the three
ontological categories: sets, individuals and universals and a system R of rela-
tions and predicates containing the basic relations and basic types described in
section 4.1.2
We consider the described system of basic categories as a constituent core
ontology because individuals are composed of substances and moments or are
contained in situoids. This system has to be extended by further basic relations
for treating space, time, and shapes. One possibility is to introduce the relation
x is boundary of y, and the coincidence relation [17].
The core ontology should be extended by unity conditions and comprehension
principles. Unity conditions are universals capturing kinds of wholes. An exam-
ple of a unity condition is the universal substantoid having a boundary. Compre-
hension principles allow to construct new universals from given universals. One
method to de ne comprehension principles is to introduce a language L whose
expressions can be used to formally specify universals. Then, certain expressions
can be understood as formal representations of universals. A L-comprehension
principle determines which of the L-expressions represent (specify) universals.
Using the core ontology (extended by some mereotopological notions) one
may introduce following categories: substrate , process, event, state, and contin-
uant. Here, substrate is a universal, and processes, events and states are parts of
situoids. A continuant can be understood as a substantoid with boundary.
4.3 Ontology of GOL
In this section we expound the partially axiomatized ontology underlying the
modelling language GOL (General Ontological Language). GOL is an ongoing
research project at the university of Leipzig [6] which is carried out in collab-
oration with ontologists at the University of Bu alo [15]. GOL includes the
categories which are considered in section 4.1. The modelling language GOL is
formalized in a rst-order language. We presume a basic ontological vocabulary,
denoted by Bas, containing the following groups of symbols:
Unary basic symbols.
Ur(x) (urelement) Set(x) (set) Ind(x) (individual)
Univ(x) (universal) Mom(x) (moment) Subst(x) (substantoid)
Sit(x) (situoid) Top(x) (topoid) Chron(x) (chronoid)
Symbols for binary and ternary basic relations.
2 (membership) :: (instantiation) : (inherence)
< (part-of) :(x; y; z ) (rel. part-of) :< (framing)
: ass(x; y) (y ass. to y) : (is contained in) : occ(x; y) (x occupies y)
ext(x; y) (is extension)
2
We emphasize that this (neccessary) criterion is not sucient for an ontology to be
upper level.
Furthermore, we use the following symbols: Sub (for the universal \substance"),
Time (for the universal \time"), Space (for the universal \space"). To the basic
vocabulary we may add further symbols used for domain speci c areas; the
domain speci c vocabulary is called an ontological signature, denoted by . An
ontological signature is determined by a set S of symbols used to denote sets (in
particular extensional relations), by a set U of symbols used to denote universals,
and by a set K of symbols used to denote individuals. An ontological signature
is summarized by a tuple = (S ,U; K).
The syntax of the language GOL( ) is de ned by the set of all expressions
containing the atomic formulas and closed with respect to the application of the
logical functors _; ^; !; :; $, and the quanti ers 8; 9. We use untyped variables
x; y; z; : : :; terms are variables or elements from U [ S [ K; r; s; si ; t denote terms.
Among the atomic formulas are expressions of the following form: r = s, t 2 s,
r : hs1 ; : : : ; sn i, v :: u, t: < s, t: s.
The language includes an axiomatization capturing the semantics of the on-
tologically basic relations. We do not present the axiomatization in full here, but
rather illustrate the main groups of axioms by selecting some typical examples.
We introduce three groups of axioms, whose union form the axioms Ax(GOL)
associated with the language GOL. Besides the logical axioms we have the fol-
lowing groups of axioms.
Axioms of Basic Ontology
(a) Sort and Existence Axioms
1. 9x(Set(x)),
2. 9x(Ur(x))
3. 8x(Set(x) _ Ur(x))
4. :9x(Set(x) ^ Ur(x))
5. 8x(Ur(x) $ Ind(x) _ Univ(x))
6. :9x(Ind(x) ^ Univ(x))
7. 8xy(x 2 y ! Set(y) ^ (Set(x) _ Ur(x)))
(b) Instantiation
D. ext(x; y) =df Univ(x) ^ Set(y) ^ 8u(u :: x ! u 2 y).
1. 8xy(x :: y ! Ind(x) ^ Univ(y))
2. 8x(Univ(x) ! 9y(Set(y) ^ 8u(u 2 y ! u :: x)))
(c) Axioms about sets
1. 8uv9x(Set(x) ^ x = fu; vg)
2. f Set j 2 ZF g, where Set is the relativization
of the formula to the basic symbol Set(x).3
3
In the formula Set all variables and quanti ers in are restricted to sets.
ZF is the system of Zermelo-Fraenkel. By 1. and 2. there exists arbitrary nite
sets over the urelements. Furthermore, we may construct nite lists. Note, that
lists and alignments are di erent sorts of entities.
(d) Axioms for Moments, Substances, and Inherence
1. 8x(Subst(x) ! 9y(Mom(y) ^ y : x))
2. 8x(Mom(x) ! 9yy ^ x : y))
3. 8xyz (Mom(x) ^ x : y ^ x : z ! y = z )
Axiom 3 is called the non-migration principle: the substances (considered as an
alignment) in which a moment inheres are uniquely determined.
(e) Axioms about Part-of
D1 ov(x; y) =df 9z (z x ^ z y), (overlap)
1. 8x(:x < x)
2. 8xyz (x < y ^ y < z ! x z )
3. 8xy(8z (z < x ! ov(z; y)) ! x y))
4. 8xyz (:(x; y; z ) ! Univ(z ) ^ x y)
5. 8xyzu(:(x; y; u) ^ :(y; z; u) ! :(x; z; u))
(f) Axioms governing Chronoids and Topoids.
Topoids are three-dimensional spatial regions, chronoids are temporal durations.
1. 8x(Sit(x) ! 9t1 t2 (Chron(t1 ) ^ Top(t2) ^ t1 : < x ^ t2 : < x))
2. 8x(Chron(x) ! 9x1 9s(Chron(x1 ) ^ Sit(s) ^ x x1 ^ x1 : < s))
3. 8x(Top(x) ! 9x1 9s(Top(x1 ) ^ Sit(s) ^ x x1 ^ x1 : < s))
4. 8xt(occ(x; t) ! Subst(x) ^ Top(t)) ^ 8x(Subst(x) ! 9t(occ(x; t)))
(g) Axioms about situations
D1. Cont (s) =df fm j m: sg
D2. s v t =df Cont(s) Cont(t).
1. 8x(Mom(x) ! 9s(Sit(s) ^ x: s)) ^ 8x(Subst(x) ! 9s(Sit(s) ^ x: s))
2. 8x(Sit(x) ! 9y(Subst(y) ^ y: x))
3. 8xy(Sit(x) ^ Sit(y) ! 9z (Sit(z ) ^ x v z ^ y v z ))
4. :9x(Sit(x) ^ 8y(Sit(y) ! y v x))
5. 8x(Sit(x) ! 9y(Univ(y) ^ ass(x; y))
Furthermore, we add the following axioms which are related to . Univ(u)
for every u 2 U, Set(R) for every R 2 S, Ind(c) for every c 2 K. A knowledge
base about a speci c domain w.r.t. the signature is determined by a set of
formulas from GOL( ) which are not basic axioms.
5 Comparison to other Upper-level Ontologies
5.1 Knowledge Interchange Format.
Knowledge Interchange Format (KIF) is a formal language for the interchange
of knowledge among computer programs, written by di erent programmers, at
di erent times, in di erent languages. KIF can be considered as a lower-level
knowledge modelling language.
The ontological basis of KIF can be extracted from [9]; we summarize the
main points. The most general ontological entity in KIF is an object. The no-
tion of an object, used in KIF, is quite broad: objects can be concrete (e.g. a
speci c carbon, Nietzsche, the moon) or abstract (the concepts of justice, the
number two); objects can be primitive or composite, and even ctional (e.g. a
unicorn). There is no ontological classi cation of the basic entitities. In KIF, a
fundamental distinction is drawn between individuals and sets. A set is a col-
lection of objects; an individual is any object that is not a set. KIF adopts a
version of the Neumann-Bernays-Godel set theory, GOL assumes ZF set theory.
The functions and relations in KIF are introduced as sets of nite lists; here
the term \set" corresponds to the term "class". Obviously, the relations and
functions in KIF correspond in GOL to the extensional relations. KIF does not
provide ontologically basic relations like our inherence, part-whole and the like.
Hence, the ontological basis of KIF is much weaker than that of GOL. GOL
can be considered as a proper extension of KIF; KIF can be understood as the
extensional part of GOL.
5.2 Upper Level Ontology of Russell and Norvig
The most general categories of this ontology are Abstract Objects and Events
[14]. Abstract objects are divided into Sets, Numbers, Representational Objects;
events are classi ed into Intervals , Places, Physical Objects, and Processes. This
ontology does not satisfy the criteria of section 4 because there is no clear dis-
tinction between sets, universals, and individuals. Also, there is no category of
formal relations. The class of universals (here called \categories") is a subclass
of sets. This pure extensional view of universals seems to be not correct. Fur-
thermore, numbers are considered as di erent from sets. But, it seems to be
plausible to introduce numbers as special sets, as in GOL. Obviously, classical
mathematics can be reduced to set theory. The instantiation relation is identi-
ed with membership. Besides this the ontology provides the part-of-relation.
The categories of events may be reconstructed within the category of situoids
in GOL. An event in the Russell-Norvig ontology is a \chunk" of a particular
universe with both temporal and spatial extent. An interval is an event that
includes as subevents all events occurring in a given time period. Such intervals
can be, in a sense, understood as situoids. But the di erence is, that situoids
are parts of the real world that can be comprehended as a whole. This implies
that to situoids are associated certain universals capturing the granularity and
the view of this part of the world.
5.3 Upper Level Ontology of Sowa
Sowa's ontology [19] does not satisfy the conditions of section 4.2. There is no
clear distinction between sets, universals and individuals. A main distinction is
made between classes and entities. Since these notions are interpreted in KIF
they can be understood within GOL as sets and urelements. There are the fol-
lowing two-place primitive relations: has, instance-of, sub-class of, temp-part,
spatial-part. The instance-of relation is interpreted by the membership-relation,
and the ontological status of the has-relation is unclear. Since these notions are
interpreted in KIF they are considered as set-theoretical notions. We want to
emphasize that formal relations of the real world are not relations in the sense of
set-theory. Sowa's ontology uses further two epistemic non-KIF operators: nec
and poss; but the ontological character of these modal operators is not clear.
In our opinion, modal operators should be eliminated from ontology. In Sowa's
ontology several classes are introduced, as for example relative, mediating, phys-
ical, abstract, continuant, ocurrent. These notions may be rede ned within GOL
giving them a clear ontological status.
5.4 Upper Level Ontology of LADSEB
In the papers [12] and [8] some principles for a top-level ontology are outlined.
The rudimentary top-level ontology, implicitly given in these papers, partially
satis es our criteria of 4.2. Formal relations are considered in [8] as relations
involving entities in all \material spheres"; these have a di erent meaning from
formal relations in our sense which we understand as being \immediate". Then
certain formal relations are discussed: intantiation and membership, parthood,
connection, location and extension, and dependence. The considered formal
properties include concreteness, abstractness, extensionality, unity, plurality, de-
pendence and independence. Intantiation, membership, and parthood are basic
relations in our sense; the inherence relation is missing. The extension relation
E (x; y) = x is the extension of y can be modelled by our relation occ(x,y) (the
entity x occupies the topoid y). The connections relation from [8] can be de ned
in GOL by using the coincidence relation and the notion of boundary.
5.5 SUO-Project
SUO is a project recently intiated by the Institute of Electrical and Electronic
Engineers to develop a \Standard Upper level Ontology" based on KIF [18].
This is designed to provide de nitions for between 1000 and 2000 general pur-
pose terms in such a way as to yield a common structure for low-level domain
ontologies of much larger size and more speci c scope. The ontologies were rst
divided into two classes, those de ning very high-level concepts and those de n-
ing lower-level notions. The highest level contains J. Sowa's upper-level ontology
and Russell-Norvig's upper-level ontology, and the second level contains a num-
ber of further concepts. From this and the above considerations of sections 5.2
and 5.3 follows that the SUO-ontology does not satisfy our criteria for an upper-
level ontology.
6 Conclusions
The development of an axiomatized and well-established upper-level ontology
is an important step towards a foundation for the science of Formal Ontology
in Information Systems. Every domain-speci c ontology must use as a framework
some upper-level ontology which describes the most general, domain-independent
categories of reality. For this purpose it is important to understand what an
upper-level category means, and we proposed some conditions that every upper-
level ontology should satisfy. The development of a well-founded upper-level
ontology is a dicult task that requires a cooperative e ort to make signi cant
progress.
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