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UOB/

P

B1

B2

•

•

•

Bn

Figure 3-55

Exclusive Disjunctive State Transition Schematic

By the same token, a conjunctive schematic is introduced to indicate a transition from a given state to all of several subsequent states, as illustrated in Figure 3-56.

UOB/

P

B1

B2

•

•

•

Bn

Figure 3-56

Conjunctive State Transition Schematic

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The semantics for schematics including logical junctions is a generalization of the semantics for basic transition schematics. Intuitively, once again, the schematic in Figure 3-56 represents a type of situation in which there is a transition involving the process P, from an object a in state A to objects a1, . . ., an in states A1, . . ., An, respectively. As with basic transition schematics, the object a must be in state A prior to, or at least no later than, the start of p; and bi must be in state Bi or must begin to be in state Bi after the start of p. The possible variability of starting and ending points for the bi is indicated by the use of dotted lines in the general instantiation pattern pictured in Figure 3-57.

Figure 3-57

General Semantics of Figure 3-56

These logical schematics also have “converses”—specifically, where * is O, X, or &, Figure 3-58 is also a schematic.

UOB/

P

Figure 3-58

Converse Schematic

The semantics in each case will be exactly the converse of the corresponding schematic above.

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Finally, it is possible that a transition can involve complex logic at both the beginning and end of the overarching process. For instance, it might be that objects in states A1 and A2 can transition either to state B1 or B2 in the course of a process P. The general syntax for characterizing such transitions is depicted in Figure 3-59, where * and # are any two of the logical symbols O, X, or & (possibly the same symbol).

Figure 3-59

Using Multiple Junction Symbols to Display Complex Transition Logic

Schematics of this sort are generally ambiguous; for instance, letting n = m = 2, if * is & and # is O, Figure 3-59 could mean that, in an instance of P, a1 and a2 objects in states A1 and A2 can both transition to either B1 or B2, or that a1 transitions to state B1 and a2 to B2, and so on. Such ambiguities can be resolved in the elaboration form for the link.

Hiding Object State Information

As with composition and classification schematics, it is possible to hide information in object schematics. That is, for certain purposes, it may often prove useful to collapse complex state transition information about a given object into a single object state. For example, a series of state transitions involved in the process of heating water from freezing to boiling is depicted in Figure 3-60.

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Figure 3-60

Object Transitions in a Heating Process

If, from a certain perspective, the intermediate transitions from ice to boiling water are irrelevant, then these transitions can be hidden in a single state in which the only relevant state is the coarse-grained Water being heated as depicted in Figure 3-61. Again a double circle is used; in this case an ‘S’ indicates that the type of information hidden is state transition information:

Figure 3-61

Hiding State Transition Information

The procedure for generating a coarse-grained schematic from a finer-grained schematic is not quite algorithmic. In the example, the state symbol for Water being heated can be thought of as directly replacing the “schematic” of Figure 3-60, consisting of the middle three kind symbols and their connecting links. However, the instantaneous transition schematic in Figure 3-60 had to be replaced by an ordinary state transition schematic, and an appropriate label had to be found for the attached process box. The exact nature of this alteration had to be determined by the nature of the represented process, and is, in general, a nonalgorithmic modeling decision.

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Enhanced Transition Schematics

In the course of describing an object transition, it is often highly useful to be able to provide surrounding contextual information that, while not intrinsic to the actual transition, is nonetheless closely related to it. Cataloging these context-setting objects and relations may not only be useful, but necessary. To provide this capacity, a variety of constructs from the IDEF5 ontology capture method are made available in the IDEF3 object schematic language. These constructs are entirely optional. If an analyst wishes to describe only transitions, there is no need to delve into the additional constructs discussed here. However, familiarity with these constructs provides an analyst with a good deal more expressive power. In the following subsections, the additional constructs will be presented independent of transition schematics. The integration of the two will then be demonstrated.

First-Order Schematics

Individual objects (i.e., individuals) are of a different logical type than the properties of those individuals. Properties are the abstract, general features that are shared by distinct individuals, the respects in virtue of which distinct individuals are the same. In a similar way, relations are the general associations which can be shared by distinct pairs (triples, etc.) of individuals. Properties and relations are identified by abstracting particular features of individuals and, hence, are often characterized as being of a higher (i.e., roughly, more abstract) logical type than the individuals that exemplify them. Individuals are thus frequently referred to as first-order objects, and properties and relations of first-order objects as first-order properties and relations. The transitions-to relation is a typical example of a first-order relation.

Displaying first-order relations between objects involves connecting two object symbols with a first-order relation symbol, as shown in Figure 3-62.

Figure 3-62

General Form of a Basic First-Order Schematic

Such schematics need a default semantics (i.e., an accepted meaning that can be assumed in the absence of any further clarification in the elaboration language). For this purpose, consider the concrete example in Figure 3-63.

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Car body

Figure 3-65

Example of a Basic 3-Place First-Order Schematic

The numbers (optionally) attached to the spokes generalize the arrows on connecting symbols in the 2-place case. Specifically, they indicate that Conveyer, Car body, and Paint primer vat are to be associated with the first, second, and third argument places of the Conveys-to relation, respectively, as they occur in the natural English reading of the label: a Conveyer conveys a Car body to a Paint primer vat.

As in the 2-place case, the relation symbol can be omitted and labeled links can simply be used, as in Figure 3-66. In this document, this notation will generally be preferred.

Car body

Figure 3-66

Alternative Syntax for Figure 3-65

Though they are somewhat uncommon, relations of “arity” four and greater can be expressed in a similar fashion.

The use of individual symbols eliminates some of the indefiniteness of the schematics in Figure 3-66. For instance, the situation depicted by Figure 3-66 permits multiple paint

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primer vats. However, it might be desirable in some situations to focus on one particular vat, and to represent it explicitly by an individual symbol as in Figure 3-67.

Figure 3-67

Example Illustrating the Use of an Individual Symbol

This schematic now expresses that a conveyer can convey a car body to paint primer vat PPV-1, as indicated by the individual symbol, providing a more definite proposition than the one expressed in Figure 3-66.

Indefiniteness is eliminated completely if only individual symbols are used. Thus, the schematic in Figure 3-68 is taken to express that the particular car body CB-J27-S121 is (as opposed to only can be) at some time conveyed by conveyer Conv-2 to the paint primer vat PPV-1.6

CB-J27-

S121

Figure 3-68

Fully Particularized Example

6That is, in terms of the elaboration language, Figure 3-68 translates to (conveys-to Conv-2 CB-J27- S121 PPV-1).

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Multiple circles can be connected to the same circle by different arrows to create complex schematics. In general, complex object schematics that do not involve transition links are essentially just conveniences; they simply enable one to reuse graphical elements and enable one to make several assertions in the language by means of a single complex schematic. Thus, for instance, if one wished to express both that spark plugs can be parts of engines and that engines can be parts of cars, there is no need for two circles representing the kind engine. Rather, the two facts in question can be expressed more succinctly, as in Figure 3-69.

Figure 3-69

Small Complex Schematic

Similarly, one might want to add the information that, in the given domain, cars can be made in Detroit and be shipped from there to dealers. This information is conveniently expressed in Figure 3-70.

Figure 3-70

Complex Schematic Involving Multiple Relations

At the same time, an object schematic may involve only one type of relation. In such a 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 3- 71.

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Connected-to Mouse

Figure 3-71

Peripheral Connections to a Personal Computer

Composition Schematics

Because the part-of relation is so common in design, engineering, and manufacturing ontologies, the “part-of” label and associated axioms are explicitly included in the IDEF5 languages. In particular, this capability enables users to express facts about the composition of a given kind of object. Bills of Material (BOM) are common examples of this form of expression. In general, expressing composition relations among objects is achieved by means of schematics of the form illustrated in Figure 3-72.

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