by a conjunctive set of disjunctive point-based metric constraints (fourth column of Table 1). This conjunctive set of point-based constraints expresses the 'fulfillment or nonfulfillment' (ec ∨ ¬ec) of the elemental symbolic constraint ec.

iv) A disjunctive set cij={ecij.k} of elemental symbolic constraints between Ii and Ij is represented by:
- A conjunctive set of disjunctive point-based metric constraints between the time-points Ii
+, Ii
-, Ij
+and I
-
j. This conjunctive set is composed by the constraints in the fourth column of Table 1 for each elemental constraint in {ecij.k}.

- A set of I-L-Sets that expresses the logical relation among elemental symbolic constraints in {ecij.k}. That is, 'one and only one elemental symbolic constraint in {ec*ij.k*}
should hold':
iv.a) Only one elemental constraint in {ecij.k} should hold. This condition does not need to be represented since the different sets of point-based constraints that correspond to fulfillment of different elemental symbolic constraints (second column of Table 1) are already mutually exclusive.

iv.b) One of the elemental symbolic constraints in {ecij.k} should hold. Let S be the label sets, where each label set corresponds to the point-based constraints which are related to the non-fulfillment of each elemental symbolic constraint in {ecij.k}
(third column of Table 1). Thus, the Cartesian product among the label sets in S is a set of I-L-Sets.
For instance, I1 {b m s di} I2 can be represented as:

(I1
-{ (0 ∞){R0}} I1
+), (I2
-{ (0 ∞){R0}} I2
+),
I1 {b ¬b} I2 ⇒ (I1
+{(0 ∞){Rb1} (-∞ 0]{Rb2}} I2
-),
I1 {m ¬m} I2 ⇒ (I1
+{[0 0]{Rm1} (0 ∞){Rm2} (-∞ 0){Rm3}} I2
-),
I1 {s ¬s} I2 ⇒ (I1 -{[0 0]{Rs1} (0 ∞){Rs3} (-∞ 0){Rs4}} I2 -) ∧ (I1 +{(0 ∞){Rs2} (-∞ 0]{Rs5}} I2 +), I1 {di ¬di} I2 ≡ I2 {d ¬d} I1 ⇒ (I2 -{(-∞ 0){Rd1} [0 ∞){Rd3}} I1 -) ∧ (I2 +{(0 ∞){Rd2} (-∞ 0]{Rd4}} I1 +).
Moreover, one of the symbolic constraints in {b, m, s, di} should hold. Thus (according to Point iv.b of the method), the Cartesian product of the associated labels related to the non-fulfillment of each elemental symbolic constraints in {b, m, s, di}. That is:
{{Rb2}Χ{Rm2, Rm3}Χ{Rs3, Rs4, Rs5}Χ{Rd3, Rd4}
should be explicitly included in the set of I-L-Sets.

By applying this method, qualitative interval-based constraints can be fully integrated in the proposed labeled point-based constraints. In this case, the interpretation domain for time-points {Ii
-
Ii
+} can be restricted to only three values ({D}={(-∞, 0), [0 0], (0 ∞)}), such that, l=3. Therefore, the computational cost of reasoning algorithms is bounded by O(n 2 32e).

To illustrate the proposed method, let's show a typical example on symbolic interval-based constraints (Figure 11.a), which was given by Allen (1983). This example shows how interval-based constraints can be represented and managed by means of disjunctive metric point-based constraints and a minimal IA-TCN can be obtained.