Proc. 12th Int'l Conf. on Engineering Design (ICED 99)
Birkhofer, Meerkamm, and Vajna, eds., Munich, 1999
pages 1125-1130.
Keywords: formal logic, mereology, product model
However, the related field of mereology, which treats parts and wholes, has been neglected. The distinction between taxonomy and mereology may seem subtle but is fundamental; in a conventional taxonomy, a car would typically be represented with the property of having four wheels; mereologically, however, a relation would exist between the car and its wheels. In a purely merelogical perspective, the most fundamental predicate is part-of (part, whole), a binary relation mapping part/whole pairs to boolean values, whereas in a taxonomic perspective, the fundamental predicate is is-a(instance, type), which maps instance/type pairs to boolean values.
While it might seem natural for engineers and engineering researchers to be interested in mereology as well as taxonomy, there is no work on the subject in the recent engineering literature of which the authors are aware. Most of the reported work appears to have been carried out by philosophers, cognitive scientists, and artificial intelligence researchers. While this research is important, its focus has been on representing common sense notions of parthood that the average non-engineer might hold. Although engineers will claim common sense as a valuable tool, it is a common sense resulting from years of technical education and experience, and is quite dissimilar from that of the lay person. Therefore, it seems reasonable that a mereology developed specifically for engineering will differ substantially from those developed for other domains. Indeed, this sentiment is also expressed by Artale et al [1].
There are many different kinds of part/whole relations; some examples include:
This kind of transitivity paradox leads to invalid, or at least undecidable, reasoning systems that are of limited use. Representative efforts from the recent literature to treat this problem are surveyed in the next section. The holy grail of all these research programs is the development of a single part-of relation that exhibits the basic required properties of being reflective, asymmetric, and transitive, but without leading to paradoxes.
The goal of the authors' work in this area is to establish a mereology of engineered products that takes advantage of the advanced technical knowledge of engineers, such that the resulting formalism can be integrated with the authors' other work in product modeling. This paper will overview the problems of formalizing parthood, review the most relevant available literature, and report on the progress made to date and what remains to be done.
The second school of thought, as suggested by the work of Borgo et al. [3], Artale et al. [1], and the logics of Lesniewski [10], is that it is vital to represent the distinctions between categories of parts as humans understand them. As such, parthood cannot be reduced to a primitive predicate, and there are many different kinds of part-of relations that must all be treated consistently. This school of thought seems more interested in representing cognitive structures than those that exist in reality; though clearly more relevant to representing knowledge, it should also be expected that incorrect or inconsistent knowledge could be represented as well, to the detriment of the system and its use.
The first school's approach is an elegant logical solution, but does not address easily the problem of intransitivity between different kinds of part-of relations. Also, the use of first-order predicates to distinguish between categories of parts implies that all such discriminating predicates must be known a-priori, which is not the case in design domains. On the other hand, the second school's approach treats the transitivity problem adequately, but it suffers from another problem: it is impossible to enumerate all the possible primitive part-of relations exhaustively.
The goal of the current authors' work is to develop a universal formal theoretical framework for the description of designed products. The theory, the Axiomatic Information Model of Design (AIM-D) presents an interpretation of axiomatic set theory [4] as such a framework. Currently, the theory supports only a naïve notion of a single, universal part-of relation. The work reported herein extends AIM-D to support various different kinds of part-of relations in a consistent manner. AIM-D is discussed in detail elsewhere [8].
From this observation, the authors hypothesize that: all part-of relations can be distinguished given sufficient knowledge about characteristics of the items, and the context in which the relations are asserted. Our formalism assumes this hypothesis to be true. Whereas all other research of which the authors are aware define a binary part-of predicate, we propose a single, universal, ternary part-of predicate - part-of (p, W, C), where p is a part, W is a whole, and C is a collection of the characteristics of p and of W that are relevant to that particular relation.
Our part-of predicate is logically complex, representable by a well-formed formula: an item p is a part of a whole W if and only if for each characteristic c in C, the set of values of c in p are entirely contained by the set of values of c in W. Logically, this can be written as:
part-of (p, W, C) =df FORALL [(c IN C) (c(p) < c(W)]
NOTE: english capitalized words substituted for some logical symbols.
where a characteristic c is taken to be representable here by a function mapping an item to the values that the characteristic takes in that item.
Different part-of relations have different C parameters; this allows an infinite variety of specialized relations to be derived from one general form. Thus it is not necessary to enumerate all the possible part-of relations exhaustively, and yet compare independently developed relations by comparing their C parameters.
Transitivity between instances of part-of relations exists only when the discriminating characteristics C are the same in the instances. If the characteristics are dissimilar, then the result of a comparison will be the unknown value (discussed below). In these latter cases, there is only a partial correspondence between C parameters. The authors believe this indicates partial transitivity. This term is used by the authors to denote the naïve or partial sense that arose in various examples given previously (such as a person being a part of a dwelling). In this way, our approach is uniquely able to distinguish between true and partial parthood relations - a necessary step needed to avoid the problems associated with intransitive part-of relations in artificial intelligence applications (such as case-based reasoning).
It is noted that the characteristics C are not properties in the typical sense of 1st order logic. If they were, then this formalism would constitute a 2nd order logic, which is undesirable for various reasons (see [8] for details), including a loss of formal rigor. The elements of C must thus be items in the domain of the theory; in our case, C is currently constituted of arbitrary complex combinations of quantities (pairs of values and dimensional metrics) as defined in AIM-D. This is sufficient for establishing Cs based on structural characteristics of products. The exact nature of the constituents of C is still being investigated by the authors; clearly, just structural characteristics are not enough.
Computationally, it is relatively easy to develop an algorithmic form of Equation 1; the only special case is that of a characteristic not being present in the part or in the whole. There are various ways to address such degenerate cases; for various reasons that are beyond the scope of this paper, the authors consider that any function or predicate may return a special, unknown value in such circumstances. Thus, in cases where a comparison of values of characteristics cannot be carried out because the characteristic is not defined in the part or in the whole, no conclusions can be drawn at all regarding parthood and the unknown value results. Indeed, it seems clear to the authors that at least a 3-valued logic is needed to properly represent incomplete or in-progress designs, for exactly this reason. Further details on this matter will appear in a future paper.
The role of contexts becomes evident if multiple part-of relations are allowed, which are only partly treated by allowing different labels for different part-of relations. In fact, the particular nature of a part-of relation depends on the nature of the reasoning task being carried out, on domain-specific knowledge, and on agent-specific knowledge. Subtle contextual differences in part-of relations intended to be the same can lead to flawed designs. Such problems are more than just naming problems. The formalization of context is a matter of interest in the AI community. An overview of current research is available in [2]. The authors are currently studying the approach of McCarthy and Buvac [7], wherein operators are defined to provide comparison and validation services between terms with multiple definitions, as a framework for incorporating context logic into our theory. It will be used for, among other reasons, permitting the existence of multiple part-of relations, and for comparing those relations for various reasoning tasks.
The axiomatization of our mereology within AIM-D is on-going, but the work's current status is promising. Our approach consists of using the axiom of separation of set theory to define sets based on other existing sets and the part-of predicate. For example, in order to distinguish parts from assemblies, the authors have developed the following axiom:
FORALL A EXISTS S [FORALL p (p IN S) -> (p IN P) AND
part-of (p, A, Ca)]
NOTE: english capitalized words substituted for some logical symbols.
where A is an assembly, S is a set of parts, P is the set of all parts, and Ca is the set of characteristics that define a physical part/assembly relation. This axiom states that an assembly contains a set of parts such that the parts are related to the assembly via Ca. Clearly, establishing the nature of Ca is fundamental to this formalism. The set of parts S cannot appear in its definition or paradoxes will arise; this means something other than a product's topology must be used to determine Ca. The authors have found that function or purpose of a part is a good characteristic in this case. If we assume that every part of a product serves some purpose in the product, and that product functions can be categorized into logically valid structures, then the connection between a product's function and form can be used to identify parts of a particular assembly for the axiom given in Equation 2.
This raises a number of interesting points. Firstly, AIM-D does not currently provide function modeling capabilities, but it must in order to completely specify the information needed in Equation 2; this is a current research area. Secondly, there are axioms in AIM-D that relate parts to features, and features to quantities, that are similar in form to the axiom in Equation 2. Clearly, however, the characteristic of the part-of relation in these other axioms must differ from the one given here. It is unclear whether function modeling will be of use for the other axioms. If it can be used, it will obviously have to differ somehow from its use here; if some other characteristic is needed, the authors are currently unsure what it might be. Thirdly, once function modeling is integrated into AIM-D, there will be a new axiom for the construction of complex functions from simple ones, similar in form to Equation 2. It is not yet clear exactly what characteristic will satisfy the part-of relation in that axiom
Some of the literature discusses various kinds of part-of relations (e.g. [1]). These relations are all representable under the authors' formalism, in various ways. For example, a member/group relation can be used to represent batches of parts. In our formalism, the characteristic for this relation could be the function of grouping elements for some purpose (such as transportation from one machine to another) or the physical contact of the batch elements to the container or tray used to batch them, or even both. A part/assembly relation could be represented using the function played by the part in the assembly, or on product assembly processes, or on the duration of the relation (as in storage tanks for liquids). The relation between a piece and a whole (e.g. a certain length of a stock rod) can be represented using characteristics between features of the piece and of the whole (e.g. length or mass).
The authors believe specific instances of these kinds of mereological relations are dependent on specific situations and design environments. However, we also believe that a hierarchy of characteristics rooted with a relatively small set of primitives is also possible. Further details of this aspect of the formalism, and more detailed examples, will be presented in a future paper.