

![0_image_0.png](0_image_0.png)

Consistency-Test (lcij, lc'ij) =
If (lcij ⊕lc lc'ij) = {∅}
Then Return (False)
Else If ∃lecij.k∈lcij, ∃lecij.p∈lc'ij / lecij.k ⊕lc lecij.p={∅}
Then I-L-Sets ← I-L-Sets ∪ ({labelij.k}∪{labelij.p}),
If ∃lecij.k∈lcij / lecij.k ⊕lc lc'ij = {∅}
Then I-L-Sets ← I-L-Sets ∪ {labelij.k},
End-If Return (True)
End- Consistency-Test Figure 3: Consistency-Test function For example, Consistency-Test ({([0 10]{R1}), ([20 25]{R2}), ([100 110]{Ra})}, 
{([0 30]{R3}), ([40 50]{R4}), ([-50 -40]{Rb})}) = True since
{{([0 10]{R1}), ([20 25]{R2}), ([100 110]{Ra})} ⊕lc {([0 30]{R3}), ([40 50]{R4}), ([-50 -40]{Rb})} =
{([20 25]{R3 R2}), ([0 10]{R3 R1})} ≠ {∅}.

In this function, the label sets {R4 R2}, {R4 R1} and {Ra} are detected as I-L-Sets and should be added to the current set of I-L-Sets, since:
{[20 25]{R2}} ⊕lc {[40 50]{R4}}={∅}, {[0 10]{R1})} ⊕lc {[40 50]{R4})}={∅},
{([100 110]{Ra})} ⊕lc {([0 30]{R3}), ([40 50]{R4}), ([-50 -40]{Rb})}={∅}.

Note that {Rb} does not need to be detected as an I-L-Set, since the label Rb is not included in the final constraint {([20 25]{R3 R2}), ([0 10]{R3 R1})} to be added to the TCN.

Any superset of an I-L-Set is also an I-L-Set (Theorem 1). Moreover, note that {R4 R2}, {R4 R1}
do not need to be added to the set of I-L-Sets, since the label R4 is not included in the final constraint.

Therefore, the following simplifications can also be performed each time a new I-L-Set is added to the current set of I-L-Sets. These simplifications do not modify the results of reasoning processes, but minimize the size of the set of I-L-Sets and improve its management efficiency.

i) No new I-L-Set that is superset of an existing I-L-Set is added to the set of I-L-Sets.

ii) If an existing I-L-Set is superset of the new I-L-Set, then the existing I-L-Set is removed.

iii) No new I-L-Set that contains a label of lc'ij, which does not appear in the labeled constraint
(lcij ⊕lc lc'ij) to be added to the TCN, should be added to the set of I-L-Sets.
Let's see an example of the updating and consistency-test processes. Let's take the labeled-TCN
that results from Example 1 once the following constraints have been updated and closured: