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A.4.3 Basic Situation Theoretic Relations

Situation theory is highly typed in the sense that the world it describes is partitioned into a number of different semantic categories; most notably, objects, first-order properties and relations, infons, situations, courses-of-events, object types, situation types, processes, and temporal intervals. To capture these distinctions, the theory of situations developed within the elaboration language defines terms that denote each of these categories. In addition, a variety of terms are defined that signify a class of special relations, along with axioms that express precisely what categories of objects can stand in these relations. With these terms at his or her disposal, a user is able to clearly express any additional information or constraints not expressible in terms of the IDEF3 schematic language.

Specifically, then, the supports and denies relations between situations and infons were discussed at length above. The occurrence-of relation holds between a situation s and a UOB U just in case s is an instance of U. The activation-of relation holds between a course-of-events c and the process P just in case c is an activation of P. The occurs-in relation holds between a situation s and courses-of-events c just in case s occurs in c. The activity-in relation mirrors this relation at the type level—it holds between a UOB U and a process P just in case U is among the situation types that constitute P. The of-type relation holds between a situation s and a UOB U just in case s is an instance of U. The object-in relation holds between an object b and either a situation s or a UOB U just in case b occurs in s or in instances of U.

A variety of temporal relations are needed to describe the temporal structure of complex processes. The only primitive relation required is meets, where, intuitively, one interval i meets another j just in case the endpoint of i is the starting point of j. Further relations—e.g., precedes, starts, finishes, overlaps, during—can be defined in terms of meet, as illustrated in Section A.5.4 below. Note that intervals are treated as first-order objects, thus the temporal relations are all first-order relations. Temporal relations are used to define a variety of corresponding temporal relations among situations.

A.5 A Formal Language for IDEF3 Elaborations

In this section the core elaboration language is extended with definitions that introduce the basic semantic categories of situation theory along with appropriate defining axioms. To aid comprehension, axioms are usually first given in English, and are formatted in italics to enhance readability. Note that this extension of the elaboration language core is not a formalization of full blown situation theory. Rather, it is a specification of the basic constructs needed to express situation-based IDEF3 elaborations, as illustrated in the examples above and in Section 3, “IDEF3 Process Description Language,” of this report. Note also that no formal model theory is provided here. Since the purpose of this report is to enable enterprise modelers and knowledge engineers to effectively use IDEF3, informal, intuitive characterizations of the semantics of the language have been provided instead.

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A.5.1 Extending the Core Elaboration Language

The first task to be addressed is to extend the core elaboration language to the full IDEF3 elaboration language, including a new class of infon terms. This is accomplished by adding the following clause:

<infonterm> ::= (<term> <term>+) |

(and <infonterm> <infonterm>+) | (or <infonterm> <infonterm>+) | (<quant> <var> <infonterm>)

Infon terms of the form (<term> <term>+) are known as atomic infon terms.

In addition, the category <compterm> is modified to include the category <infonterm>. For a more robust version, the explicit set theoretic apparatus included in KIF could be added at this point, but this is unnecessary for purposes here; see (Genesereth & Fikes, 1992) for details of the set theory included in KIF.

A.5.2 Basic Situation Theoretic Semantic Categories

In this section the basic semantic categories of situation theory are introduced and characterized.

(define-relation infon (?x))

(define-relation situation (?x))

(define-relation UOB (?x))

(define-relation COE (?x))

(define-relation process (?x))

(define-relation interval (?x))

(define-relation polarity)

A single axiom expresses that the basic semantic categories are all disjoint; e.g., no individual is an infon, function, or relation.

(=/ ?x ?y))

(forall ?z (not (and (?x ?z) (?y ?z))))))

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A.5.2.1 First-order Relations

In the version of situation theory developed here, infons are constructed only out of firstorder relations and individuals. Hence, the notion of a first-order relation needs to be defined explicitly and axiomatized with an appropriate schema.

(define-relation FO-relation (?rel)

:=> (relation ?rel))

A first-order relation is a relation that can only be true of n-tuples of individuals.

(forall (?rel : (FO-relation ?rel) (= (arity ?rel) n))) (forall (var1 . . . varn : (?rel var1 . . . varn))

(and (individual var1) . . . (individual varn))))

Recall that change in the extension of a typical first-order relation over time is captured by including a parameter for temporal intervals among its arguments. For example, the relation walks is taken to be a 2-place relation that holds between an individual b and an interval t just in case b walks throughout the interval t. Because intervals are themselves individuals, this is accommodated by the above definition.

A.5.2.2 Axioms for Infon Terms

Given the notion of a first-order relation, axioms can be given that express what is required for an infon term to denote a legitimate infon (as opposed to the null object).

A basic infon term denotes an infon if and only if it consists of a term denoting an n-place first-order relation followed by n terms denoting individuals and a term denoting a polarity.

(forall (?rel var1 . . . varn ?pol) (<=> (infon (?r var1 . . . varn ?pol))

(and (FO-relation ?r) (individual var1)

. . .

(individual varn) (polarity ?p))))

A conjunctive (disjunctive) infon term denotes an infon if and only if each conjunt (disjunct) denotes an infon.

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(and (infon ?inf1) (infon ?inf2))))

Analogous axioms for quantified infons must be stated as a schema. Let Q be any quantifier, var any variable, and infterm be any infon term. If var occurs free as the first term in any atomic infon in infterm, then

(<=> (infon (Q var infterm))

(Q (var : (FO-relation var)) (infon infterm)))).

Similarly, if var does not occur free as the first term in any atomic infon term in infterm (i.e., occurs bound, or occurs free elsewhere than as the first term in an atomic infon term in infterm, or does not occur at all in infterm), then

(<=> (infon (Q var infterm))

(Q (var : (individual var)) (infon infterm)))).

A.5.3 Basic Situation Theoretic Relations

In this section, the basic situation theoretic relations are introduced and, by means of defining axioms, their legitimate argument types are declared. Note that, as with the basic categories of the elaboration language core, “legitimate” here is understood semantically, and enforced axiomatically. Any terms whatsoever can be used to construct syntactically correct sentences involving the relation terms introduced below. However, such sentences can be true only if the axiomatized constraints are satisfied.

(define-relation supports (?x ?y ?z))

:=> (and (situation ?x) (infon ?y) (interval ?z)))

(define-relation denies (?x ?y)

:=> (and (situation ?x) (infon ?y) (interval ?z)))

(define-relation occurrence-of (?x ?y) :=> (and (situation ?x) (UOB ?y)))

(define-relation activation-of (?x ?y) :=> (and (COE ?x) (process ?y)))

(define-relation occurs-in (?x ?y)

:=> (and (situation ?x) (COE ?y)))

(define-relation activity-in (?x ?y) :=> (and (UOB ?x) (process ?y)))

(define-relation of-type (?x ?y)

:=> (and (situation ?x) (UOB ?y)))

(define-relation object-in (?x ?y)

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:=> (and (individual ?x) (or (situation ?y) (UOB ?y)))

A.5.4 Basic Temporal Relations

In this section, some basic temporal relations between intervals are introduced. The only primitive relation needed for characterizing the temporal intervals used in IDEF3 is the meets relation, which can be true only of intervals, as indicated in the following partial definition.

(define-relation meets (?x ?y)

:=> (and (interval ?x) (interval ?y)))

Intuitively, as noted, one temporal interval meets another just in case the end point of the first is identical with the starting point of the second. A logic for the meets relation as found in (Allen & Hayes, 1987) is assumed [see also (van Bentham, 1983)].

A variety of useful temporal relations can be defined in terms of meets. The first is stronglyprecedes, where one interval strongly precedes another just in case the first meets an interval that meets the second.

(define-relation strongly-precedes (?x ?y)

:= (exists (?z ?w : (/= ?z ?w)) (and (meets ?x ?z) (meets ?z ?w) (meets ?w ?y))))

Note that, because points are intervals, we need to put two distinct intervals between ?x and ?y. In fact, a point can be defined as an interval that meets itself (this is a divergence from Allen and Hayes).

(define-relation point (?x) := (meets ?x ?x))

One temporal interval i starts another j just in case both are met by a given interval but j meets an interval which is met by an interval met by i:

(define-relation starts (?x ?y)

:= (exists ?z (and (meets ?z ?x) (meets ?z ?y)

(exists ?w (and (meets ?y ?w) (strongly-precedes ?x ?w)))))

Similarly, a temporal interval i finishes another j just in case both meet a given interval but i is met by an interval that starts j:

(define-relation finishes (?x ?y)

:= (exists (?z ?w : (starts ?w ?y)) (and (meets ?x ?z) (meets ?y ?z) (meets ?w ?x))))

i overlaps j just in case some interval that finishes i starts j:

(define-relation overlaps (?x ?y)

:= (exists ?z (and (finishes ?z ?x) (starts ?z ?y))))

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i is during j just in case some interval that starts j meets i and i meets some interval that finishes j:

(define-relation during (?x ?y)

:= (exists (?z ?w) (and (starts ?z ?y) (meets ?z ?x) (finishes ?w ?y) (meets ?x ?w)))

Other useful relations can be defined in a similarly straightforward fashion.

A.5.5 The Interval Over Which a Situation Occurs

It is very important in describing processes and their activations to be able to talk about the interval of time over which a given situation occurs. For this reason, a function is defined that, when applied to a given situation, yields exactly that interval:

(define-function interval-of :=> (forall (?sit ?t)

(=> (= interval-of ?sit) ?t)

(and (situation ?sit) (interval ?t)))))

Given the temporal relations, by defining the interval over which a situation occurs, a variety of useful temporal relations among situations can be defined in terms of corresponding temporal relations between the intervals over which they occur. See (Menzel & Mayer, forthcoming) for details.

A.5.6 Using Sorted Variables

Note that the IDEF3 elaboration language proper is completely untyped; in particular, there is only one sort of variable. However, the informal examples shown in Section A.4 and in Section 3 use a wide variety of sorted variables whose possible values are restricted to various semantic categories—e.g., UOBs, situations, infons, etc. This practice can be viewed as simply a convenient use of alternative notation, as any sentence in a so-called many-sorted language with many different sorts of restricted variables can be translated directly into a sentence of a single sorted language such as the elaboration language. The trick is simply to use the terms in the single sorted language that denote the semantic categories to which the sorted variables are restricted in the many-sorted language. For example, suppose “?sit” is a sorted variable ranging only over situations and “?ind” is a sorted variable ranging only over individuals. Then the sentence “(forall (?sit ?ind : (FOO ?sit)) (BAR ?ind ?sit)))” says that, for any situation s and individual b, if s is a FOO, then b bears BAR to s. Clearly, however, this can be expressed in the strict, single-sorted elaboration language as “(forall (?x ?y : (situation ?x) (individual ?y) (FOO ?x)) (BAR ?y ?x))”. The use of sorted variables is therefore innocuous, and indeed, encouraged, as their use typically decreases significantly the length of a sentence written in single sorted notation. For a rigorous account of the relation between many-sorted and single-sorted languages, see Chapter 4, §4.3 of (Enderton, 1972). In any case, regardless of whether one is using a many-sorted or single-sorted language, it is generally good practice to choose variables

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that reflect their intended semantic categories—e.g., “?ind”, “?rel”, “?f”, “?sit”, “?inf”, “?uob”, “?coe”, and so on.18

18 Although this practice is adhered to in informal discussions in this document, it is not always followed in the statement of the formalization above in order to drive home the fact that the elaboration language is itself single-sorted, and that typing distinctions are introduced and enforced axiomatically.

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