Commitments due to mereology

Mereology is a branch of logic that uses part-whole relationships to describe entities. For example, in a conventional ontology, one might model an automobile as having the property of having four wheels; in a mereological perspective, a relation is defined between the automobile and the wheels themselves. Obviously, in engineering design, both the conventional and the mereological approaches are relevant. Surprisingly, little work appears in the recent literature outside Europe on mereology in AI, and there is almost no work on mereology as such in the engineering literature.

The basic problem of mereology as a field of study is that there appear to be various, often inconsistent, semantics associated with the term ``part of''. For example, consider the following three statements.

1.

A piston is a part of an engine.

2.

An engine is a part of an automobile.

3.

An automobile is a part of a fleet.

Each statement, on its own, is perfectly reasonable. Furthermore, from the first two statements, we can reasonably deduce that ``A piston is a part of an automobile.'' But from all three statements, can we reasonably deduce that ``A piston is a part of a fleet''? The problem is that there are two different meanings of the term ``part of,'' and that transitivity is not preserved between them. Mereology's main concern is establishing an overall structure to reason reliably with all the possible part-whole relationships.

There appear to be two schools of thought regarding the treatment of mereology. One school advocates a single, universal, and transitive part-of relation, based on the assumption that all distinctions about types of parts are really conceptualizations and are not rooted in reality. In order to address the paradoxes that result, first-order predicate calculus is used to introduce sufficient predicates to distinguish between kinds things. This approach is taken by the developers of Ontolingua and KIF [17] and the logics of Lesniewski [30].

The other school of thought contends the cognitive distinctions must be represented; in other words, a proper mereology must handle the transitivity problem directly by admitting distinctions between different part-of relations. This approach is supported by the work of Artale et al. [4,3] and Simons [28]. In this approach, different part-of relations are explicitly defined to handle different conceptualizations (e.g. assembly/component versus space/region), and transitivity is not preserved across them. Also, the part-of relation is seen as complex, rather than primitive, which requires the development of specialized logics that integrate mereology with topology and morphology as in, for example, [6].

A fundamental problem with this approach is that there is no way to enumerate all the different ``primitive'' part-of relations. For example, in [34] six primitive part-of relations are defined; they are summarized in Table 2. It has been shown ([4]) that (a) it is impossible to decide if these constitute a complete set of part-of primitive relations, and (b) some of these relations (such as stuff/object) are more linguistic artifacts than actual cognitive or other constructs of knowledge.

 

Table: Summary of part-of relations in [34].

*main::open_tags

 

RELATION

EXAMPLE component/integral-object

Rather than siding with one school or the other, the author proposes a new mereological framework, wherein the part-of relation is a well-defined function mapping triplets of arguments to the boolean values. That is:

$$P (p, W, \pi) \Rightarrow \left\{ \cal{T}, \cal{F} \right\}$$ (1)

where p is a part, W is a whole, and $\pi$ is a property or properties used by the P part-of function.

This approach is based on the observation that parthood is related to some sort of overlapping between the values of at least one property of a part on the values of the same properties of a whole. For example, to establish a part-of relation for regions of a space, $\pi$ is the set of properties defining the size and position of a spatial region. P, then, asserts that p is wholly contained by W if its volume is contained in W's volume.

This approach addresses a variety of open issues. First, by ``deferring'' uniqueness of different part-of relations to the properties $\pi$, P itself remains a single universal, ternary predicate, which is logically elegant. Second, primitive part-of relations can be defined as those whose properties $\pi$ are fundamental in AIM-D (e.g. properties of length, mass, time, etc.); complex part-of relations are constructed by composing primitive ones; this suggests an abstraction hierarchy of part-of relations which would be useful for automated reasoning processes, such as case-based reasoning and decision support. Third, transitivity is preserved to the degree that different properties are used in different part-of relations.

This last point deserves some explanation. Transitivity is preserved entirely in reasoning processes where different instances of P use the same properties $\pi$. On the other hand, different instances of P that use properties that have no commonality in the abstraction hierarchy are not transitive at all. These two cases correspond to the typical behavior of other approaches. For example, transitivity is preserved over different instances of the group/member relation, but not between a group/member relation and an assembly/component relation.

However, the author's approach allows partial transitivity to be recognized. In the example at the beginning of this section, it was shown that one may reason that a piston is a part of a rental fleet of automobiles if there is only one part-of relation. While there is clearly something wrong with such a conclusion for most conventional uses, there is still a certain sense in which it is reasonable. The author believes that this ``partial'' sense of the conclusion results from the partial subsumption of the properties with which part-of is used in the example. Being able to represent this kind of partial parthood opens the possibility of substantially different reasoning processes that can be automated in a KBS, and should allow for a richer representation of design knowledge.

Specific mereological axioms using the formalism presented above are currently under development for the next ``version'' of AIM-D. The current version of AIM-D [27] contains only an implied notion of mereology as captured by the four levels of product composition defined therein. That is, AIM-D has specific axioms for the construction of assemblies from parts, parts from features, and features from quantities. Equation 2 gives an example: the axiom relating parts and assemblies. It states that a part p consists of features f that are in the set of all features F, and that satisfy a predicate $\phi$, which is taken to be any possible mereological relation between features f.

$$\exists p \left[ \forall f \left[ (f \in p) \equiv (f \in F) \bullet \phi (f) \right] \right]$$ (2)

Each axiom implies a different parthood relation between the whole (e.g. assemblies) and its parts (e.g. parts). The next version of AIM-D will have a more explicit formulation of parthood relations based on the material presented herein.