=Paper=
{{Paper
|id=Vol-2969/paper23-FOMI
|storemode=property
|title=Comparing Ontological Alternatives to Model Engineering Systems and Components
|pdfUrl=https://ceur-ws.org/Vol-2969/paper23-FOMI.pdf
|volume=Vol-2969
|authors=Francesco Compagno,Stefano Borgo,Claudio Masolo,Emilio M. Sanfilippo
|dblpUrl=https://dblp.org/rec/conf/jowo/CompagnoBMS21
}}
==Comparing Ontological Alternatives to Model Engineering Systems and Components==
https://ceur-ws.org/Vol-2969/paper23-FOMI.pdf
Comparing Ontological Alternatives to Model
Engineering Systems and Components
Francesco Compagno1,2 , Stefano Borgo1 , Claudio Masolo1 and Emilio M. Sanfilippo1
1 ISTC-CNR Laboratory for Applied Ontology, via alla cascata 56/C, 38123, Povo, Italy
2 Adige S.P.A, via per Barco, 11, Levico Terme, 38056, Italy
Abstract
In this paper, we study from an ontological viewpoint some modeling challenges related to the design,
realization, and maintenance of engineering systems such as what it means that an assembly complies to
a specification and how it persists through time when some of its components are missing or replaced.
These challenges intertwine with the missing component and the replacement problems, which are by
themselves fundamental from an engineering stance and are thus explicitly addressed in the paper. We
describe these topics and for each one we compare different modeling approaches dealing with them,
considering also the differences between the three- and four-dimensionalist views. In conclusion, we
summarize the comparison of the approaches by highlighting their advantages and shortcomings from an
engineering perspective.
Keywords
Engineering systems, Maintenance, Assembly modeling, Engineering design, Ontology, 3D, 4D
1. Introduction
This paper continues a line of research started in [1]. The goal is to systematically assess a number
of ontological models which focus on the same engineering scenario. This goal is achieved by
formalizing the given scenario according to different ontological viewpoints, and by technically
comparing the obtained formalizations. In this way, one acquires information on the formal
and conceptual consequences of adopting an ontological perspective as well as on the technical
aspects in which these perspectives differ. Beside highlighting advantages and disadvantages of
each ontological model, this analysis can help knowledge engineers and practitioners to choose
the (foundational, top-level) ontology most suited to their interest and purposes.
More specifically, our research line concentrates on the design, realization, and maintenance
of engineering systems, intended here as mereologically compound devices, i.e., assemblies of
inter-related components. The paper considers only general modeling approaches which are well-
known and widely used in ontology research. This allows us to make general assessments which
hold in general and shed light on commonalities across models that adopt the same ontological
perspective. Even though the analysis we present remains general for the given purposes, our
FOMI 2021: 11th International Workshop on Formal Ontologies meet Industry, held at JOWO 2021: Episode VII The
Bolzano Summer of Knowledge, September 11–18 , 2021, Bolzano, Italy
" francesco.compagno@loa.istc.cnr.it (F. Compagno); stefano.borgo@cnr.it (S. Borgo); claudio.masolo@cnr.it
(C. Masolo); emilio.sanfilippo@cnr.it (E. M. Sanfilippo)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
work is driven by real industrial scenarios that we analyze in collaboration with Adige S.P.A (of
the BLM Group1 ), a company specialized in the manufacturing of laser cutting machines.
Our analysis applies to systems which are characterized by a specification. Such specification
is the result of a design process and identifies a set of conditions that the system must satisfy.
A first challenge concerns the formal representation of these specifications and, related to it,
the clarification of what it means that a system must satisfy the conditions to be compliant with
the specification. A second challenge concerns the manner in which systems persist through
time. In industrial scenarios, there may be periods of time in which a system is not operative,
e.g., because it undergoes maintenance operations, which could have been scheduled earlier
or due to unexpected malfunctioning of some of its components. Maintenance can also end
up with the replacement of some components. Therefore, the modeler has to state whether
during these periods of maintenance the physical system still complies (perhaps in a weaker way)
with its specification, and to which kinds of components replacements it can survive. From an
ontological perspective, this aspect is intimately related to the spatio-temporal nature of systems
that one adopts. For instance, the standard ISO 15926 [2] embraces a four-dimensionalism (4D)
view. According to this view, systems persist through time by accumulating temporal parts. In a
three-dimensionalism (3D) view, one assumes that systems persist by being wholly present at
each time in which they exist. The 4D and 3D views are basic ontological distinctions that have
practical consequences on the models one obtain. We will analyze how these differences impact
the representation of systems’ compliance through time.
Starting from this general stand, we will pay particular attention to two more specific problems
in engineering modeling considered in the literature, e.g. [3, 4, 5]. The first problem, i.e., the
replacement problem, concerns components replacement. In experts’ terms, it is common to
claim that a system’s component can be replaced n-times, as if a component were something that
could be entirely substituted while keeping its identity. Understanding what components are in
this view becomes therefore crucial to make sense of experts’ talks. The second problem, called
missing component problem, concerns reference to systems or components when they are not
physically present or even realized. This can happen at design time, e.g., when experts talk about
the machine that they are designing claiming that, e.g., it has given dimensions (within certain
tolerances). A similar situation occurs during maintenance tasks, when components are removed
from their hosting systems, e.g., to be controlled, cleaned or repaired. During these periods, or
when a component has never been installed like in a system under construction, experts may
address the component as if it were present in its position; e.g., they may assemble the system
while referring to how the missing component should be connected to other systems’ parts. Again,
we wonder how to make sense of experts’ talks from an ontological perspective; e.g., whether it is
necessary to introduce a class for engineering items which exist without being physically present.
Of course, one can always take a revisionist approach, i.e., claiming that engineers’ expressions,
when taken ontologically, should be opportunely rephrased.Yet, before considering this option,
one should take engineers’ talk at face value since this is the information actually exchanged and
needed in engineering.
The paper is structured as follows. Section 2 introduces the notation and key notions that
we use in the formal discussion like feature, component, and specification. Section 3 presents
1 https://www.blmgroup.com/
alternative ways to formalize engineering notions and their usage in particular from the 3D vs.
4D ontological viewpoints. Section 4 draws some conclusions.
2. System’s Specifications
We introduce in this section the general view and notation that we will use in section 3 to discuss
systems from different ontological perspectives.
By engineering systems we mean engineering assemblies, i.e., physical objects based on an
explicit design and composed of several interrelated components. To exemplify the presentation,
we represent the structure of systems-types as (connected) graphs like the one depicted in Fig. 1.
Each node represents a systems’ component characterized in terms of (complex) features, labeled
Fi. A feature Fi is a Boolean combination of (non-relational) properties that may include types
(e.g., pump, hole) and/or properties describing shape, size, color, etc.2 For instance, we could
have that feature F1 characterizes physical objects with weigh 2kg ± 0.1kg, made of metal or
plastic but not wood, with a rectangular slot, etc. Usually, a feature does not identify a unique
object; this is necessarily true when the feature is associated with a replaceable component. Arcs
between nodes denote relations, labeled Ri, holding between the corresponding components.3
For simplicity, we consider only binary relations, however, the framework is general and applies
to n-ary relations as well. Thus, Fig. 1 shows a specification of systems with three components
(one for each node) characterized by features F1, F2 and F3, respectively, and satisfying relations
R1 and R2 as shown.
F1 → F2 → F3
R1 R2
Figure 1: A system-type with three components.
To avoid ambiguities, engineers tend to use tags to securely identify each component in a
system. These tags are useful for maintenance purposes, e.g., to exactly identify the component
upon which a technician needs to operate. Fig. 2 shows a graph similar to the one in Fig. 1
but now enriched with tags C1, C2, and C3. The tags are assumed to be unique identifiers of
components in the system. Note that in some graphs components’ features cannot explicitly play
an identification role. To see this, consider the diagram in Fig. 3. Here, the components C2 and
C4 are characterized by the same feature and are distinguished only because the relation R3 is
asymmetric.
The components Ci (like the features Fi ) could be recursively specified and further decomposed.
For simplicity, however, we will work with Ci components that do not have proper parts. It
follows that the components of a system do not have common parts, i.e., they are mutually disjoint.
2 Note that we use the term ‘feature’ in a more general sense in comparison to the engineering literature, where it
commonly refers to things such as holes, slots, bumps, etc. described from different viewpoints [6].
3 Things like wires, pipes, or sets of bolts and nuts can be explicitly taken into account by designers, hence they
can be treated as components linked to other components via specific relations. An example of such a relation is the
contact between a pin and its socket. Alternatively, designers can (partially) encapsulate them in the Ri-relations.
[C1] F1 → [C2] F2 → [C3] F3
R1 R2
Figure 2: The graph specification in Fig. 1 with tags C1, C2, C3.
R1 → [C2] F2 R2
→
[C1] F1
↓
R3 → [C3] F3
R1
→ R2
[C4] F2
Figure 3: A graph specification where the components C2 and C4 have the same feature F2.
Also, we will refer in the following sections to the example in Fig. 3, although the analysis applies
to arbitrary graphs of components/features and relations between them.
3. Modeling Options for Engineering
The purpose of this section is to make explicit the underlying assumptions that different ontologi-
cal approaches require to model the problems raised in the introduction.
3.1. 3D vs. 4D Approaches
The way the 3D and the 4D approaches understand and model systems is definitely different.
The difference is evident when we look at how systems and components relate to time. Let us
represent the existence of objects in time with the primitive EXt x standing for “at time t, the object
x is present”.4 We focus here only on objects that exist at some time, i.e., we assume formula (f1)
stating that for every object x there exists a time t at which it exists.5
f1 ∃t(EXt x)
According to 3D, objects persist in time by enduring, i.e., by being wholly present at every
time at which they exist. To account for change, 3D qualifies properties and relations on objects
with respect to time. For parthood we introduce x ≤t y (“at time t, the object x is part of the object
y”) characterized by (f2)-(f5), where the overlap relation ≬ is defined in (d1) and holds between
two objects at the times at which they have a common part. At time t, two objects coincide if, at
t, they have the same parts (d2).6 Parthood simpliciter can be defined as in (d3) which allows
the introduction of the mereological sum of a plurality of objects—we write x1 + . . . + xn for the
mereological sum of n objects—and the sum (called fusion) of all the objects satisfying some
property (in the language)—we write σ x(φ (x)) for the fusion of the objects satisfying property φ .
4 We write P x instead of Pxt to highlight the time-argument t.
t
5 Since it is clear from the language when a variable is quantified over time or over objects, we leave implicit these
quantification constraints.
6 Neither coincidence at t nor coincidence during the whole lives of objects imply their identity.
A mereological system is called closed when it assumes the existence (and uniqueness) of the
sum of any (finite) plurality of objects and of any fusion based on properties in the language [7].
d1 x ≬t y ≜ ∃z(z ≤t x ∧ z ≤t y)
d2 x ≡t y ≜ ∀z(z ≤t x ↔ z ≤t y)
d3 x ≤ y ≜ ∀t(EXt x → x ≤t y)
f2 x ≤t y → EXt x ∧ EXt y
f3 EXt x → x ≤t x
f4 x ≤t y ∧ y ≤t z → x ≤t z
f5 EXt x ∧ EXt y ∧ ¬x ≤t y → ∃z(z ≤t x ∧ ¬z ≬t y)
Regarding 4D within the engineering literature, we refer to West’s work [3],7 which follows
closely the standard ISO 15926 [2]. More generally, 4D is the thesis for which objects consist
of both spatial and temporal parts, i.e., they persist in time as they extend in space. This
hypothesis allows 4D models to take a classical (atemporal) extensional mereology axiomatized
as in (f6)-(f10) with overlap defined as in (d4). According to Sider [9], 4D can be characterized
by introducing the notion of temporal part (temporal slice) as in (d5)—here TPt xy stands for
“at time t, the object x is a temporal part of the object y”—and by assuming that at every time
at which an object exists it has a temporal part, as stated in (f11). From (f1) and (f6)-(f11) it is
possible to prove that the temporal part of an object at a given time is unique. To simplify the
notation, when x exists at t, we write x@t to indicate the unique object y such that TPt yx. In this
view, objects consist of the mereological sum of their temporal parts. For instance, if x exists only
at t and t̄, it has two different temporal parts at those times, i.e., respectively, x@t and x@t̄ , and x
itself is equal to their sum: x = x@t + x@t̄ .
4D allows for the definition of temporary parthood as in (d6), i.e., x@t ≤ y@t . This strategy
can be replicated for all temporally qualified predicates, i.e., Rt x1 . . . xn reduces to Rx1@t . . . xn@t .
Furthermore, the temporal coincidence x ≡t y reduces to identity of the corresponding temporal
parts, i.e., x@t = y@t .
Notice that, starting from (f1) and (f6)-(f11), all the axioms considered for the temporally
qualified parthood can be proven. The vice versa does not hold: 3D does not commit to temporal
parts. In this sense 4D makes a heavier ontological commitment with respect to 3D (see [10]).
d4 x ≬ y ≜ ∃z(z ≤ x ∧ z ≤ y)
d5 TPt xy ≜ EXt x ∧ EXt y ∧ ¬∃t̄(EXt̄ x ∧ t ̸= t̄) ∧ x ≤ y ∧ ∀z(z ≤ y ∧ EXt z → z ≬ x)
d6 x ≤t y ≜ ∃uv(TPt ux ∧ TPt vy ∧ u ≤ v)
f6 x ≤ y ∧ EXt x → EXt y
f7 x ≤ x
f8 x ≤ y ∧ y ≤ x → x = y
f9 x ≤ y ∧ y ≤ z → x ≤ z
7 The reader can refer to [8] for readings on 4D.
f10 ¬x ≤ y → ∃z(z ≤ x ∧ ¬(z ≬ y))
f11 EXt x → ∃y(TPt yx)
3.2. Specification and Compliance
We now consider how to logically represent the graph specifications introduced in Sect. 2 and the
compliance of individual systems with them.
A common approach consists in embracing 3D and seeing the specifications as (complex)
relations represented in first-order logic (FOL) via predicates. Let us assume that Fig. 3 depicts
the specification of a target system-type K. To state that four specific components, say x, y, z
and v, at a certain time t, satisfy the constraints in the specification we can use a single complex
relation S, see (f12).8 The component types can be characterized by introducing a specific
S-relational role; see, e.g., (f13) for C1. Following standard practice, one can introduce necessary
and sufficient conditions for being a K (at a given time) as in (f14) where Kt x is read as “at time t,
the physical object x is an instance of the type K”. Since we adopted a closed mereology, (f13)
and (f14) guarantee that an entity x is a C1-instance only if there exists a K-system which has x
among its components, i.e., a C1-component implies the existence of a K-system. In other words,
C1-components are existentially dependent on K-systems.
Introducing formula (f13) for C2-C4, too, one can trivially prove (f15), i.e., K-systems are the
mereological sums of the Ci-components considered in the specification. However the converse
of (f15) does not hold in general. For instance, assume that C1t a holds for St ab̄cd only, while
C1t b holds for St ābcd only (with a ̸= ā and b ̸= b̄), then there is no suitable 4-tuple satisfying St
to infer the converse of (f15).
f12 St xyzv ↔ F1t x ∧ F2t y ∧ F3t z ∧ F2t v ∧ R1t xy ∧ R1t xv ∧ R2t yz ∧ R2t vz ∧ R3t yv ∧
¬(x ≬t y) ∧ ¬(x ≬t z) ∧ ¬(x ≬t v) ∧ ¬(y ≬t z) ∧ ¬(y ≬t v) ∧ ¬(z ≬t v)
f13 C1t x ↔ ∃bcd(St xbcd)
f14 Kt x ↔ ∃abcd(St abcd ∧ x ≡t a + b + c + d)
f15 Kt x → ∃abcd(C1t a ∧ C2t b ∧ C3t c ∧ C4t d ∧ x ≡t a + b + c + d)
In a 4D setting, the previous formulas can be rewritten with atemporal predicates following
what done for parthood in (d6); e.g., (f14) can be rewritten as in (f16). Given this observation, we
will consider in the remaining of the paper only temporally qualified predicates and discuss 4D
only for some specific problems mainly concerning components (e.g., in Sect. 3.3).
f16 Kt x ↔ ∃abcd(Sa@t b@t c@t d@t ∧ x@t = (a+b+c+d)@t )
Another approach in conceptual modeling [11], even though less common than the ones just
presented, considers specifications as complex relations represented by individual constants. In
this view, the predicate St xyzu is replaced by xyzu ::t s where s represents the complex relation S
and :: is a sort of instantiation relation: here xyzu ::t s stands for “at time t, objects x, y, z, u satisfy
the relation S”. For simplicity, the different instantiation primitives corresponding to the arities of
8 We add the non-overalapping conditions for generality. Since by assumption our components do not have proper
parts, strictly speaking these constraints are not needed.
predicates are noted in the same way. We can then follow the previous discussion by observing
that the formulas (f12), (f13), (f14), and (f15) can be easily rewritten in this new framework; e.g.,
(f14) now becomes (f17). We will explicitly consider this option only in Sect.3.4 to manage the
problem of the missing component.
f17 x ::t k ↔ ∃abcd(abcd ::t s ∧ x ≡t a + b + c + d)
In this approach the main difficulty consists in characterizing the instantiation relation. How-
ever, the introduction of system-types in the domain of quantification allows taking into account
their intensional and intentional dimensions. Different types could have the same instances and
type differences may be grounded on contextual information; e.g., a design feature (or the entire
specification), since designed by a company, may have copyright, etc.
A further interesting point of this approach concerns the possibility to align the mereological
structure of a system-type with that of its instances. For instance, since K-systems have four
components, individuated by the Ci-tags, so the type K is composed by four components-types,
i.e., k = c1 + c2 + c3 + c4. Thus, K can be seen as a structural property composed by the
Ci-properties. The fact that the Ci-properties uniquely identify components rules out the cases
where the same property is part of the structural one several time as, for instance, H in H2 O (see
[12] for the debate on structural universals). The mereological alignment is partially guaranteed
by the rewriting of (f13)-(f15).
3.3. The Components of a Given System and the Replacement Problem
The notion of component in (f13) can be relativized to a system s as in (f18). Note that now (f19)
holds. The left to right arrow is trivial. To see that the opposite direction holds, assume C1t ax. By
(f18) there exist b, c, d such that St abcd and x ≡t a + b + c + d that, by (f14), imply Kt x.
f18 C1t xs ↔ ∃bcd(St xbcd ∧ s ≡t x + b + c + d)
f19 Kt x ↔ ∃abcd(C1t ax ∧ C2t bx ∧ C3t cx ∧ C4t dx)
In the context of a specific system, each tag Ci should have an identification role, i.e., the
relational constraints in the specification S should grant the fact that all the Cis are unambiguous
identifiers of the specific components composing the system, so that, e.g., (f20) would hold for
C1. For instance, in Fig. 3, C2-components can be distinguished from C4-components only when
the relation R3 is asymmetric.9 A specification S should indeed ensure (f21) from which (f20)
follows. Similarly for the other Cis.
f20 C1t xs ∧ C1t ys → x = y
f21 St xbcd ∧ St ybcd → x = y
Concerning the replacement problem, the replacement of the C1-component for system s can
be represented by formula C1t0 xs ∧ C1t1 ys ∧ C1t2 zs ∧ x ̸= y ̸= z, which shows a case where the role
of ‘being the C1-component of s’ is played, at different times, by three different objects.
9 Features themselves may not be enough because often systems have several components of the same type, e.g.,
the same kind of valve, electric engine or fastener can be present in several places.
According to 4D, there exists three temporal slices: x@t0 , y@t1 , and z@t2 . One can then consider
the mereological sum of all the C1-slices of s, see (f22), as the referent of the expression ‘the
Ci-component of s’. As claimed by West [3], systems’ components are existentially dependent
on the whole system and they are non-ordinary objects since they are submitted to discontinuous
changes, i.e., the installed component can be (instantaneously) replaced by a new one.
f22 C1xs ↔ x = σ y(∃tz(y = z@t ∧ y ≤ s ∧ C1y))
Vice versa, in a 3D-framework, to have, for a given system s, a single Ci-component and, at
the same time, allowing its replacement, one needs to add a type of individuals in the domain of
quantification. Call these new individuals stable components or simply s-components. Stable
components abstract from the actual objects installed in the system during its lifetime. Intuitively
they are essentially characterized in terms of the patterns of features and relations considered in
the specification (see (f18) for C1). This move is similar to the classical one where the statue is
distinguished from the clay, see [13]. To characterize s-components, (f20) is strengthened as in
(f23), i.e., the unicity is now guaranteed also diachronically. To state that at every time at which a
K-system exists it always has the same four s-components, we can rewrite (f19) as in (f24). Note
that in (f23) and (f24) the first argument of the Ci predicates is now an s-component. To account
for the replacement process, one could assume that the s-components may change constituents.
This requires to introduce a relation x ◁t y for “at time t, x constitutes y”. The identification
role of the Cis is preserved by requiring the unicity (at a given time) of the constituent of an
s-constitutent (f25). In this approach an s-component c that has been replaced twice will satisfy
formula C1t0 cs ∧ C1t1 cs ∧ C1t2 cs ∧ x ◁t0 c ∧ y ◁t1 c ∧ z ◁t2 c ∧ x ̸= y ̸= z. Note that according to 4D
constitution amounts to coincidence, i.e., x ◁t y reduces to x ≡t y which reduces to x@t = y@t .
f23 C1t xs ∧ C1t̄ ys → x = y
f24 ∃t(Kt x) → ∃abcd(∀t(Kt x ↔ C1t ax ∧ C2t bx ∧ C3t cx ∧ C4t dx))
f25 x ◁t c ∧ y ◁t c → x = y
3.4. Missing Components and Partial Compliance
According to the approaches discussed in the previous sections, both systems and components
can exist intermittently, i.e., their temporal extension may not be a temporal interval. Yet, when
the system exists, all its components must be physically present. Hence, to deal with the missing
component problem, i.e., the possibility that some components of a system are not physically
realized, one needs to conceive both notions of system and compliance in a more flexible way.
A way to realize this is to allow for partial compliance, i.e., systems are classified under a
type even though they do not fully match the corresponding specification. One can intend partial
compliance as a sort of specification-weakening; e.g., for K-systems, in (f13) and (f14), S can
be substituted with Š where St xyzv → Št xyzv but not vice versa,10 i.e., some constraints in the
specification S are relaxed or ruled out. K-systems and Ci components could then lack some
(optional) characteristics indicated in the original specification. Interestingly, one could consider
different conditions at different times. For instance, immediately after the completion of the
10 As done for S, we assume that components of Š do not have proper parts.
[C1] F1 [C2] F2 [C3] F1
R2
R1 → ↓ ← R3
[C4] F4
Figure 4: A graph specification with the “reference” component C4.
production process, a system could be required to fully comply with the specification, i.e., it
must satisfy S. However, through time, one can tolerate the loss of some characteristics of the
system, so that it is enough if it satisfies Š, especially when this loss does not compromise its
basic functionalities and is not attributable to production defects (e.g., cars can be scratched).
This approach allows representing optional characteristics of systems but it still requires the
presence of all the components. A second way to intend partial compliance is to consider optional
components (the two possibilities can clearly be combined). Suppose, for instance that K-systems
can lack C4-components, i.e., consider S̄ in (f26) and modify (f14) as in (f27) allowing K-systems
to lose or acquire components during their lifecycle.11 Unfortunately, by adopting (f26), i.e., by
tolerating that some components are missing, some Ci-tags could lose their effectiveness because
one can have troubles in re-identifying them through time: the second disjuncts in the definitions
(f28) and (f29) of, respectively, C2 and C4, are identical.
f26 S̄t xyz ↔ F1t x ∧ F2t y ∧ F3t z ∧ R1t xy ∧ R2t yz ∧ ¬(x ≬t y) ∧ ¬(x ≬t z) ∧ ¬(y ≬t z)
f27 Kt x ↔ (∃abcd(St abcd ∧ x ≡t a + b + c + d) ∨ ∃abc(S̄t abc ∧ x ≡t a + b + c))
f28 C2t xs ↔ (∃acd(St axcd ∧ s ≡t a + x + c + d) ∨ ∃ac(S̄t axc ∧ s ≡t a + x + c))
f29 C4t xs ↔ (∃abc(St abcx ∧ s ≡t a + b + c + x) ∨ ∃ac(S̄t axc ∧ s ≡t a + x + c))
This problem is prevented in the case of systems having one “reference” component c, i.e.,
all the other components can be identified just by taking into account the relation they have
with c. For instance, consider the specification in Fig. 4 where the reference component C4 is
characterized only by F4, i.e., it does not have a relational characterization, and where C1 is
characterized as in (f30) (similarly for C2 and C3). A typical example of a reference component is
a frame: the “position” in the frame is enough to identify all the other components. One can then
assume that the system can persist the replacement and/or the loss of all the components except
the frame. By relaxing (f30) as in (f31) one could take into account some cases of C1-components
that are malfunctioning because they lack some features required by the specification.
f30 C1t xs ↔ F1t x ∧ ∃a(F4t a ∧ R1t xa ∧ x + a ≤t s)
f31 C1t xs ↔ ∃a(F4t a ∧ R1t xa ∧ x + a ≤t s)
Let us now consider the interaction between partial compliance and the approach based on
s-components previously discussed. A first possibility is to assume that, during its whole lifetime,
a system (of a given kind) maintains its essential s-components, but can in principle lose (for
11 Note that we cannot just write K x ↔ ∃abc(S̄ abc ∧ a + b + c ≤ x) otherwise it would be possible to have
t t t
K-systems with five or more components or with four components that however do not satisfy S.
some periods) optional s-components. In this case we need to modify (f24) taking into account
only essential s-components. Optional components are then, in general, intermittent entities, a
view embraced by West in a 4D-setting.
A second possibility is to allow for “empty” s-components, which we will argue for in the
following. When experts talk about components or systems, they seem to refer sometimes to
physical individuals and some other times to conceptual or abstract ones. In a sense, components
and systems seem to have both a physical and a conceptual dimension, a sort of “dual nature”.
To capture this duality, one could allow s-components to be “empty”, i.e., to lack a physical
realization: when an s-component is physically realized, experts refer to this realization while
when it is empty, they refer to the s-component itself. Systems would be therefore static with
respect to their s-components while optional components would reduce to possibly empty s-
components. However, this notion of s-component does not match the one discussed at the end of
Sect. 3.3 where s-components are intended as physical entities that always have a constituent, i.e.,
given an s-component c, the constitution relation ◁ usually satisfies EXt c → ∃x(x ◁t c).
We sketch a possible way to capture this duality where s-components have a conceptual
nature. Consider the approach based on the reification of properties discussed in Sect. 3.2
but now consider the reifications of the Ci-properties specialized to a given K-system s: cis
is the reification of Cits x ↔ Cit xs, i.e., x ::t cis ↔ xs ::t ci (these properties are sort of saturated
roles as introduced in [14]). By assuming that the instances at time t of a concept all exist at
t (i.e., x1 . . . xn ::t c → EXt x1 ∧ . . . ∧ EXt xn ), then ∃x(x ::t cis ) → EXt s holds for all the components
while the vice versa does not hold for optional components, i.e., one can have empty cis .
Furthermore, when replacement is allowed, x ::t cis ∧ y ::t̄ cis → x = y holds in general only when
t = t̄. However, differently from the previous s-components, the cis are not part of the system
s. Our idea is to introduce the conceptual counterpart ds of the system s by defining it as
ds = c1s + c2s + c3s + c4s . By guaranteeing that both EXt s → s ::t ds and x ::t ds → x ≡t s hold,
ds becomes a sort of definite description of s, i.e., it is a concept that individuates a single system.
In this way, both the conceptual (via ds and the cis ) and the physical (via s and, when present, the
instances of the cis ) perspectives on a system can be represented. Note however that, following
the discussion about (f28) and (f29) and assuming that some Ci-components are optional for s,
this strategy can be pursued only when the definition of the other C j-components (or at least
some of them) can be satisfied also when the Ci-components are not present.12 Otherwise the
loss of (some of) the Ci-components would imply the loss of all the other components.
Let us consider now the problem of the missing component discussed in Sect.1. One can
approach the problem at the conceptual level. The specification and the explicit conditions
required to (fully or partially) comply with it allow talking about the characteristics of any
realization of a system-kind. From the characterization of K-systems (f14) and of their Ci-
components (e.g., (f13)), together with additional (background) knowledge on the world shared
by engineers and (if the system is at least partially realized) the information about the physical
properties of the actual components, some properties of K-systems and Ci-components can be
inferred. For instance, consider again the example of the frame, assuming it is an essential
component of the whole system. In this case, we can use the information about its spatial
12 As discussed, this could be problematic when the absence of C1-components causes the loss of the individuation
role of other components, and could be prevented, e.g., by using a component as “reference”.
localization or its specific temperature to infer additional information concerning the specific
system of which the physical frame is a component. Note however, that only the approach based
on saturated roles allows having an individual in the domain of quantification that can be used to
refer to the component even though it is missing.
4. Discussion and Conclusion
As we have seen in the previous section, the identified engineering modeling problems can be
tackled from different ontological perspectives. The result of our analysis is that none of these
views is clearly more advantageous than the others. Therefore, there is no definite preferable
approach. We gained however some insights that can hopefully help knowledge engineers to
choose an approach to develop their models on the basis of formal implications along with, for
those interested, philosophical principles (e.g., 3D vs 4D).
To sum up, in all cases we have seen, engineering physical systems are discussed in relationship
with design specifications; the match between the two establishes the degree of systems com-
pliance. In particular, design specifications represent the properties (in terms of features Fi and
relations Ri) of physical systems. In a standard FOL setting, properties correspond to predicates
and FOL predication models the ontological notion of instantiation. This formal mechanism
can be tuned to both 3D and 4D views (see Sect. 3.2). In addition, there is the option to reify
properties to model them as FOL individual constants (see (f17)). This move, which is coherent
with both the 3D and 4D, complicates the formal representation, since new predicates are needed
to characterize the new constants (see, e.g., (f14) and (f17)).
The notion of compliance can be treated in different ways. A physical system is compliant
with a specification when it satisfies all properties established by the latter (see, e.g., (f14) for 3D,
(f16) for 4D, and (f17) for the reified variant of 3D). However, there are cases where a weaker
sense of compliance, which we called partial compliance, is preferable. To capture this notion,
one can introduce the distinction between essential and optional systems’ properties such that the
lack of the latter does not compromise systems’ identity (see Sect. 3.4).
For the representation of components’ replacement, in West’s 4D-framework [3], a whole
system’s component is a 4D worm formed by all temporal slices that at different times have been
part of it. In this sense, having a system’s component replaced at a certain time implies that, from
that time, the whole worm has as part a temporal slice of a different object. This view requires
however the specification of some rules, possibly motivated from an engineering perspective, to
guarantee the continuity of the 4D worm. Otherwise, it is not clear how (possibly) disconnected
temporal slices come to form a unique whole. In a standard 3D approach, we showed how to
overcome the identity problem of an object whose components are all replaced by relying on the
notion of stable components, namely, objects that – at different times – can change constituents
while remaining the same entities (see (f23)–(f25)). Hence, in this perspective, to replace a
system’s component means to change its constituent.
Finally, we have seen that engineers talk of systems and their components referring sometimes
to specific physical entities and some other times to their conceptual counterparts. In both 3D
and 4D approaches, reference to entities that have not been physically realized or are simply
absent at some time can be made meaningful by referring to the reification of the tags in the
specification standing for possibly empty components, i.e., the things engineers refer to as if
they were fully-fledged physical entities. Overall, the approach based on reification makes the
modeling framework logically more complex but it allows making sense of engineers talks about
the missing component problem.
Further work is necessary to complete the analysis we presented here. First, other approaches,
(e.g., see [1] for a preliminary proposal based on possible objects), should be explored and added
to the comparison. Second, one should assess the identified ontological views in more specific
and varied engineering requirements, a larger set of requirements may help to better identify the
dis-/advantages of the views in more articulated scenarios.
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