and

a) one of the constraints (I1 before I2) or (I3 before I4) should hold. Thus, the Cartesian product of label sets associated to the disjunctive constraints in (Ii ¬ecij Ij) is a set of I-L-Sets: {Rb2, Rb4} is an I-L-Set, b) only one of the constraints (I1 before I2) or (I3 before I4) should hold. Thus, the label set associated to the mutual fulfillment of constraints in (Ii ecij Ij) is an I-L-Set: {Rb1, Rb3} is an I-L-Set.
Thus:
{(I1 before I2) ∨ (I3 before I4)} ≡
(I1
+{(0 ∞){Rb1} (-∞ 0]{Rb2}} I2
-), (I3
+{(0 ∞){Rb3} (-∞ 0]{Rb4}} I4
-),
and {Rb2, Rb4}, {Rb1, Rb3} are I-L-Sets.

| Ii ecij Ij             | Ii ecij Ij             | Ii ¬ecij Ij                                                                         | Ii (ecij ∨ ¬ecij) Ij               |
|------------------------|------------------------|-------------------------------------------------------------------------------------|------------------------------------|
| - {(-∞ 0){Rd1}} I2 -   |                        |                                                                                     |                                    |
| (I1 during I2)         | I1                     | - {[0 ∞){Rd3}} I2 - )                                                               | - {(-∞ 0){Rd1}  [0 ∞){Rd3}} I2 -   |
| (I1                    | I1                     |                                                                                     |                                    |
| I1 + {(0 ∞){Rd2}} I2 + | ∨                      | (I1 + {(-∞ 0]{Rd4}} I2 + )                                                          | I1 + {(0 ∞){Rd2} (-∞ 0]{Rd4}} I2 + |
| (I3 starts I4)         | I3 - {[0 0]{Rs1}} I4 - | (I3 - {(0 ∞){Rs3} (-∞ 0){Rs4}} I4 - ) I3 - {[0 0]{Rs1} (0 ∞){Rs3} (-∞ 0){Rs4}} I4 - |                                    |
| I3 + {(0 ∞){Rs2}} I4 + | ∨                      | (I3 + {(-∞ 0]{Rs5}} I4 + )                                                          | I3 + {(0 ∞){Rs2} (-∞ 0]{Rs5}} I4 + |

Table 6: Point-based constraints for (I1 during I2) and (I3 starts I4)
In a similar way (Table 6), (I1 during I2) ∨ (I3 starts I4) ≡
(I1
-{(-∞ 0){Rd1} [0 ∞){Rd3}} I2
-), (I1
+{(0 ∞){Rd2} (-∞ 0]{Rd4}} I2
+),
(I3
-{[0 0]{Rs1} (0 ∞){Rs3} (-∞ 0){Rs4}} I4
-), (I3
+{(0 ∞){Rs2} (-∞ 0]{Rs5}} I4
+),
and {Rd1 Rd2 Rs1 Rs2} and the Cartesian product {Rd3 Rd4} X {Rs3 Rs4 Rs5} are I-L-Sets.

Therefore, logical relations on elemental constraints can be represented by a set of I-L-Sets. Thus, a labeled TCN (and the set of I-L-Sets) can represent a *special type of and/or* TCN. These types of non-binary constraints enrich the expressiveness of language and allow for the modeling of more complex problems (Meiri, 1996). Stergiou and Koubarakis (1996) and Jonsson and Bäckström (1998)
show that Disjunctions of Linear Constraints (DLR) are also able to represent these non-binary constraints. However, Pujari and Sattar (1999) remark that general methods from linear programming should then be applied for DLR management, such that specific temporal concepts (like the ones detailed in Section 2) are not considered in these general methods. In the proposed model, management of these non-binary constraints are performed by the proposed reasoning methods without increasing their computational complexity. The added functionality is of interest in several temporal reasoning problems, including planning, scheduling and temporal constraint databases
(Barber et al., 1994; Gerevini & Schubert, 1995; Brusoni et al., 1997; Stergiou & Koubarakis, 1998; etc.) where no general solutions are provided in the specific temporal reasoning area.

In addition, the proposed reasoning algorithms obtain a globally *labeled-consistent* TCN
(*Theorem 11*). This feature allows us to manage *hypothetical queries*, which is an important requirement in query processes on temporal constraint databases (Brusoni et al., 1997). Thus, queries