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. ., an in states A1, . . ., An, respectively. Aswith basic transition schematics, the object a must be in state A prior to, or at least nolater than, the start of p; and bi must be in state Bi or must begin to be in state Bi after thestart of p. The possible variability of starting and ending points for the bi is indicated bythe use of dotted lines in the general instantiation pattern pictured in Figure 3-57.Instance of PB1 b1AaB2 b2•••Bn bnFigure 3-57General Semantics of Figure 3-56These logical schematics also have “converses”—specifically, where * is O, X, or &,Figure 3-58 is also a schematic.UOB/P•••B*Figure 3-58Converse SchematicThe semantics in each case will be exactly the converse of the correspondingschematic above.68Finally, it is possible that a transition can involve complex logic at both the beginningand end of the overarching process.

For instance, it might be that objects in states A1 andA2 can transition either to state B1 or B2 in the course of a process P. The general syntaxfor characterizing such transitions is depicted in Figure 3-59, where * and # are any twoof the logical symbols O, X, or & (possibly the same symbol).UOB /PA1B1A2B2#*••••••AnBnFigure 3-59Using Multiple Junction Symbols to Display Complex Transition LogicSchematics 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 statesA1 and A2 can both transition to either B1 or B2, or that a1 transitions to state B1 and a2 toB2, and so on. Such ambiguities can be resolved in the elaboration form for the link.Hiding Object State InformationAs with composition and classification schematics, it is possible to hide informationin object schematics.

That is, for certain purposes, it may often prove useful to collapsecomplex state transition information about a given object into a single object state. Forexample, a series of state transitions involved in the process of heating water fromfreezing to boiling is depicted in Figure 3-60.69UOB/Melt iceWater:FrozenUOB/Heat to 40ÞUOB/Heat to100Þ CCWater:ColdWater:HotWater:WarmWater:BoilingFigure 3-60Object Transitions in a Heating ProcessIf, from a certain perspective, the intermediate transitions from ice to boiling waterare irrelevant, then these transitions can be hidden in a single state in which the onlyrelevant state is the coarse-grained Water being heated as depicted in Figure 3-61.

Againa double circle is used; in this case an ‘S’ indicates that the type of information hidden isstate transition information:UOB/Melt iceUOB/Heat to 40ÞCSIceWaterbeingheatedBoilingwaterFigure 3-61Hiding State Transition InformationThe procedure for generating a coarse-grained schematic from a finer-grainedschematic is not quite algorithmic. In the example, the state symbol for Water beingheated can be thought of as directly replacing the “schematic” of Figure 3-60, consistingof the middle three kind symbols and their connecting links. However, the instantaneoustransition schematic in Figure 3-60 had to be replaced by an ordinary state transitionschematic, and an appropriate label had to be found for the attached process box.

Theexact nature of this alteration had to be determined by the nature of the representedprocess, and is, in general, a nonalgorithmic modeling decision.70Enhanced Transition SchematicsIn the course of describing an object transition, it is often highly useful to be able toprovide surrounding contextual information that, while not intrinsic to the actualtransition, is nonetheless closely related to it.

Cataloging these context-setting objectsand relations may not only be useful, but necessary. To provide this capacity, a variety ofconstructs from the IDEF5 ontology capture method are made available in the IDEF3object schematic language. These constructs are entirely optional. If an analyst wishes todescribe only transitions, there is no need to delve into the additional constructs discussedhere. However, familiarity with these constructs provides an analyst with a good dealmore expressive power.

In the following subsections, the additional constructs will bepresented independent of transition schematics. The integration of the two will then bedemonstrated.First-Order SchematicsIndividual objects (i.e., individuals) are of a different logical type than the propertiesof those individuals. Properties are the abstract, general features that are shared bydistinct individuals, the respects in virtue of which distinct individuals are the same. In asimilar way, relations are the general associations which can be shared by distinct pairs(triples, etc.) of individuals.

Properties and relations are identified by abstractingparticular 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 andrelations of first-order objects as first-order properties and relations.

The transitions-torelation is a typical example of a first-order relation.Displaying first-order relations between objects involves connecting two objectsymbols with a first-order relation symbol, as shown in Figure 3-62.<Kind Label><Relation Label><Kind Label>Figure 3-62General Form of a Basic First-Order SchematicSuch schematics need a default semantics (i.e., an accepted meaning that can beassumed in the absence of any further clarification in the elaboration language). For thispurpose, consider the concrete example in Figure 3-63.71Spark PlugPart-ofEngineFigure 3-63Example of a Basic First-Order SchematicRoughly, the default meaning of this construct is a type specification for the part-ofrelation; that is, it specifies that spark plugs and engines are the sorts of things that canlegitimately stand in that relation.

It is not saying, for example, that every spark plug is apart of some engine, or that every engine has spark plugs; there may be loose spark plugsor plugless engines in the domain in question. Rather, in its basic, default meaning, it issimply documenting the fact that a Spark plug is the kind of thing that can be Part-of anEngine. If one wishes a stronger reading, it can be specified in the IDEF3 elaborationlanguage.As an alternative syntax for the schematics illustrated above, it is permissible (andoften preferable) to replace the two connecting symbols and the relation symbol with asingle arrow labeled by the same relation label, as illustrated in Figure 3-64.

There issome potential for confusion here with transition schematics, but using an “open” ratherthan “closed and filled” arrowhead together with other particulars of the schematic shouldprevent ambiguity.Spark PlugPart-ofEngineFigure 3-64Example Illustrating Alternative Syntax for Basic First-Order SchematicsRelations like part-of that hold between two entities are often referred to as 2-placerelations, indicating that the number of arguments in the relation, or the “arity” of therelation, is two. However, there is no theoretical bound on the “arity” of a relation; therelation between, for instance, holds between three objects. More artificial butnonetheless useful relations can easily be defined with four or more arguments.The semantics for first-order schematics involving 2-place (first-order) relationsymbols generalizes to schematics involving n-place relation symbols.

So, for example,Figure 3-65 indicates only that an instance of the Conveys-to relation can involve aConveyer, a Car body, and a Paint primer vat.72Car body2Conveyer1Conveys-to3Paintprimer vatFigure 3-65Example of a Basic 3-Place First-Order SchematicThe numbers (optionally) attached to the spokes generalize the arrows on connectingsymbols in the 2-place case. Specifically, they indicate that Conveyer, Car body, andPaint primer vat are to be associated with the first, second, and third argument places ofthe Conveys-to relation, respectively, as they occur in the natural English reading of thelabel: 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 cansimply be used, as in Figure 3-66.

In this document, this notation will generally bepreferred.Car body2Conveyer13Conveys-toPaintprimer vatFigure 3-66Alternative Syntax for Figure 3-65Though they are somewhat uncommon, relations of “arity” four and greater can beexpressed in a similar fashion.The use of individual symbols eliminates some of the indefiniteness of the schematicsin Figure 3-66. For instance, the situation depicted by Figure 3-66 permits multiple paint73primer vats. However, it might be desirable in some situations to focus on one particularvat, and to represent it explicitly by an individual symbol as in Figure 3-67.Car body2Conveyer1Conveys-to3PPV-1Figure 3-67Example Illustrating the Use of an Individual SymbolThis schematic now expresses that a conveyer can convey a car body to paint primervat PPV-1, as indicated by the individual symbol, providing a more definite propositionthan the one expressed in Figure 3-66.Indefiniteness is eliminated completely if only individual symbols are used.

Thus, theschematic 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 paintprimer vat PPV-1.6CB-J27S1212Conv-2PPV-11Conveys-to3Figure 3-68Fully Particularized Example6That is, in terms of the elaboration language, Figure 3-68 translates to (conveys-to Conv-2 CB-J27S121 PPV-1).74Multiple circles can be connected to the same circle by different arrows to createcomplex schematics.

In general, complex object schematics that do not involve transitionlinks are essentially just conveniences; they simply enable one to reuse graphical elementsand enable one to make several assertions in the language by means of a single complexschematic. Thus, for instance, if one wished to express both that spark plugs can be partsof engines and that engines can be parts of cars, there is no need for two circlesrepresenting the kind engine. Rather, the two facts in question can be expressed moresuccinctly, as in Figure 3-69.Spark PlugPart-ofPart-ofEngineCarFigure 3-69Small Complex SchematicSimilarly, one might want to add the information that, in the given domain, cars canbe made in Detroit and be shipped from there to dealers. This information isconveniently expressed in Figure 3-70.23DetroitMade-inDealerShipped-tofrom1Spark PlugPart-ofPart-ofEngineCarFigure 3-70Complex Schematic Involving Multiple RelationsAt the same time, an object schematic may involve only one type of relation.

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