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In sucha case, to prevent needless clutter the analyst can omit labels and simply note the (single)meaning of the relation symbols at the bottom of the schematic, as illustrated in Figure 371.75PowerSupplySurgeProtectorTerminalServerCPU unitKeyboardConnected-toMouseFigure 3-71Peripheral Connections to a Personal ComputerComposition SchematicsBecause the part-of relation is so common in design, engineering, and manufacturingontologies, the “part-of” label and associated axioms are explicitly included in the IDEF5languages.
In particular, this capability enables users to express facts about thecomposition of a given kind of object. Bills of Material (BOM) are common examples ofthis form of expression. In general, expressing composition relations among objects isachieved by means of schematics of the form illustrated in Figure 3-72.76A1Pa rA2t-ofPart-ofB•••tParofAnFigure 3-72Composition SchematicThe default semantics of Figure 3-72 mean that A1’s (instances of A1) can be parts ofB’s, A2’s can be parts of B’s, .
. ., and Ai’s can be parts of B’s. However, in the contextof part-of, a stronger reading is often desired. For instance, in a BOM, one wishes to saynot simply that A1’s can be parts of B’s, and so on, but that every B does in fact consist ofan A1, an A2, and so forth. For example, one might wish to represent the componentstructure for a certain kind of ballpoint pen, as in Figure 3-73.77SpringCartridgeInkSupplyLowerLowerBodyBodyBallpointPenLowerBarrelButtonRetractionMech.Part-ofUpperBodyUpperBarrelFigure 3-73Composition SchematicTo capture this stronger meaning, one must resort to a note or to the elaborationlanguage.7 On this stronger semantics, then, the schematic in Figure 3-73 expresses that aballpoint pen in the domain in question has both an upper body and a lower body, that theformer consists of a button, a retraction mechanism, and an upper barrel, while the latterconsists of a lower barrel and a cartridge, which in turn consists of a spring and an inksupply.87Specifically, in the case of a kind B whose instances have three parts of kinds A1, A2, and A3, onewould add the elaboration language statement (forall ?x (-> (B ?x) (exists (?y1 ?y2 ?y3)(and (A1 ?y1)(A2 ?y2) (A3 ?y3) (part-of ?y1 x) (part-of ?y2 x) (part-of ?y3 x)))).8Adding junctions to composition schematics also serves to narrow the range of possibleinterpretations.
For example, using an ‘&’ junction to ‘join’ multiple part-of links precludes thepossibility of excluding one or more of the attached objects in the composition. In the absence of the78Second-Order SchematicsProperties and relations that hold among individuals are identifiable (albeit abstract)objects themselves. But because they are one level of abstraction above ordinary firstorder objects, they are said to be of a higher logical type and, hence, classified as secondorder objects. When treated as objects, first-order properties and relations canthemselves have properties.
Such properties are typically known as second-orderproperties, because they apply to second-order objects. Second-order objects can alsostand in relations with one another. Thus, kinds, properties, and relations that apply toindividual objects are commonly known as second-order objects, since they are of a“higher,” more general logical order than individuals, or first-order objects. Likeindividuals, second-order objects can stand in relations to other (first- or second-order)objects. A prominent example is the subkind-of relation that holds between kinds, whilea paradigm of a relation that holds between individuals and kinds (or properties generally)is the instance-of relation.A distinct type of arrow is needed to represent second-order relations because bothtypes of arrows connect circles, and because the associated semantics in the two cases arequite different.
The basic form of a second-order schematic looks just like that of a firstorder schematic, except for the presence of a so-called second-order relation arrow (asshown in Figure 3-74) instead of a first-order relation arrow.Kind LabelRelation LabelKind LabelFigure 3-74Basic Second-Order SchematicThe semantics for second-order schematics is much more definite than the semanticsfor most first-order schematics.
Specifically, second-order schematics are about theindicated kinds, rather than about their instances: in Figure 3-74 the kind represented bythe left-hand circle stands in the (second-order) relation indicated by the arrow with thekind represented by the right-hand circle. Furthermore, the default semantics are notqualified; unlike general first-order schematics, the semantics are not merely about howthings can be in the domain but about how two kinds are in fact related.above elaboration language statement for example, Figure 3-73 permits ballpoint pens withoutsprings and retraction mechanisms.
By adding junctions to the schematic, the analyst can indicatethat, for example, springs and ink supplies can be parts of cartridges for ballpoint pens but cartridgeswithout springs cannot exist, and so forth.79The schematic in Figure 3-75 expresses that there are more U.S. citizens thanCanadian citizens (i.e., more literally, that the kind U.S. Citizen has more instances thanthe kind Canadian Citizen).Has-moreinstances-thanU.S.CitizenCanadianCitizenFigure 3-75Example of a General Second-Order SchematicFigure 3-76 illustrates a schematic involving the second-order relation subkind-of.
Bythe semantics just given, the kind hex-headed bolt is a subkind of the kind fastener.9Hex-headedboltSubkind-ofFastenerFigure 3-76Example of a Second-Order Schematic with Subkind-ofClassification SchematicsBecause the subkind relation is so common, the default meaning of the second-orderrelation arrow with no associated label represents the subkind relation, thus permittingusers to avoid having to attach the label subkind-of repeatedly throughout a schematic.This choice is motivated by the observation that among the more common mechanismsfor representing knowledge are taxonomy diagrams (Brachman, 1985).
Domain expertsengaged in knowledge acquisition often make statements such as A is a B, A is a type ofB, or A is a kind of B. The cognitive activity involved in organizing knowledge in thisfashion is called classification. There are several identifiable varieties of classification.Two particularly prominent types of classification are description subsumption andnatural kind classification. In description subsumption, (1) the defining properties of the“top-level” kind K in the classification, as well as those of all its subkinds, constituterigorous necessary and sufficient conditions for membership in those kinds, and (2) thedefining properties of all the subkinds are “subsumed” by the defining properties of K inthe sense that the defining properties of each kind entail the defining properties of K; thedefining properties of K constitute a more general concept.9In terms of the elaboration language again, we have simply (subkind-of hex-headed-bolt fastener).80In natural kind classification, by contrast, it is not assumed that there are rigorouslyidentifiable necessary and sufficient conditions for membership in the top-level kind K,but that, nonetheless, there are some underlying structural properties of its instances that,when specialized in various ways, yield the subkinds of K.
The best examples of suchclassification schemes are, of course, genuine natural kinds such as metal, feline, and soforth, but the idea can be extended to artifactual kinds like automobile and NC machine.These two types of classification are illustrated in Figure 3-77.Natural KindClassificationDescriptionSubsumptionMetalPolygonTriangleAluminumRectangleCopperIronHexagonFigure 3-77Different Types of ClassificationClearly, with its central notion of a kind, a natural application for the general objectschematic language is the development of taxonomy diagrams, or as we shall call them,classification schematics.Classification is typically much more detailed than the examples suggest.
Mostclassification schemes will involve several levels of more specialized subkinds “below”more general kinds in the scheme. (Both the subkind-of and instance-of relations areoften ambiguously expressed by the relation “is-a” in semantic nets and other graphicallanguages. Such schemes are often called ‘is-a hierarchies,’ but the use of ‘is-a’ isstrongly discouraged; either the subkind-of relation or the instance-of relation should beused instead, depending on the intended meaning.) To illustrate, it is essential in projectplanning that one categorize the kinds of resources that will be needed for the project’ssuccess. Informally, a resource can be defined as an object that is consumed, used, orrequired to perform activities.
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