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2.2.2. Implicit Pair-Subsumption

Subtyping Between Output Roles of the Inference Steps. This type of implicit pair-subsumption occurs when any pair of inference steps which have the same input roles lead to different output roles, among which there exists partial or complete subtyping. The inference step s is subsumed by another inference step iff:


FIGURE 4 Implicit pair-subsumption arising from subtyping between output roles.

An example in given in Figure 4, where the same input roles lead to different output roles, among which there exists the relationship z [inverted y] z’.

The counterpart anomaly in rule bases is called semantic conflict. An example is given below:

p(x) [wedge] q(x) —> r(x, a1’)

p(y) [wedge] q(y) —> r(y, a11’)

Although it is called a semantic conflict, there is not really a conflict. This case may be significant in some application domains, where the situation when an action is structurally subsumed may be sensible for the interpretation of the results. In decision-aid systems, for example, a general advise should not be given to the user when a more specific one can be suggested.

Subtyping Between Input Roles of the Inference Steps. This type of implicit pair-subsumption occurs when any pair of inference steps which have a complete subsumption relation between input roles, lead to different output roles among which there is no subsumption relationship. An inference step s is in conflict with another inference step iff:

An example is given in Figure 5. Between s and s’ there is a conflict relation rather than a subsumption relation. However, such a conflict may reveal incompleteness of specifications. A conflict situation may be the result of an attempt to prove some properties of the specification during the validation activities. Failure to prove a property may be caused by exactly the kind of situation described in Figure 5. The correction of such a conflict should be resolved by augmenting the specification with elements that allow the specific property to be proved. In this example, such an element would be either introducing an explicit subtyping relation between z and w if the situation requires so, or to relate z and w through an inference step (structure).


FIGURE 5 Implicit pair-subsumption based on subtyping between input roles.

The counterpart anomaly in rule bases also causes a conflict. An example is given below:

p(x, a11) [wedge] q(x, a21) —> r(x, b1)

p(x, a12) [wedge] q(x, a21) —> r(x, b1)

There is no conflict between these two rules, as a11 and a12 are children of a1’. However, if we introduce the following rule

p(x, a1’) [wedge] q(x, a21) —> r(x, b2)

a conflict occurs when substituting a1’ with any of its children.

Subtyping Between Output Roles of the Inference Steps. This type of implicit pair-subsumption occurs when any pair of inference steps that have a complete subsumption relation between input roles, lead to different output roles among which there exists partial or complete subsumption relationship.

Two situations can be identified. The first one where the input roles in the first inference of the pair are completely subsumed by the input roles in the second inference, and the output roles of the first inference is partially or completely subsumed by the output roles of the second one. An inference step s is subsumed by another inference step iff:

An example is given in Figure 6. The situation does not lead to any conflict anomaly, but rather is reduced to the case where the inference step s is redundant as is subsumed by s’.


FIGURE 6 Implicit pair-subsumption based on subtyping between external roles.


FIGURE 7 Implicit pair-subsumption based on (inverse) subtyping between external roles.

The second situation may occur where the input roles in the first inference of the pair are completely subsumed by the input roles in the second inference, and the output roles of the first inference partially or completely subsume the output roles of the second inference. This is expressed as follows:

An example is given in Figure 7.

This situation may lead to conflict, especially in situations where knowledge roles are not allowed to take multivalues.


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