REASONING ON INTERVAL AND POINT-BASED DISJUNCTIVE METRIC CONSTRAINTS IN TEMPORAL CONTEXTS
can be represented as:
i) A *conjunctive set of constraints* between (ni, nj) and between (nk, nl), where, ¬(lecx) can be represented by means of the complementary domain of (lecx):
(ni {(lecij.1){Rij.1}, (lecij.2){Rij.2}, ...., (lecij.p){Rij.p}, ¬{(lecij.1){Rij.1}, (lecij.2){Rij.2}, ...., (lecij.p){Rij.p}}} nj) ∧ (nk {(leckl.1){Rkl.1}, (leckl.2){Rkl.2}, ..., (leckj.q){Rkl.q}, ¬{(leckl.1){Rkl.1}, (leckl.2){Rkl.2}, ..., (leckj.q){Rkl.q}}} nl) ≡{(ni {(lecij.1){Rij.1}, (lecij.2){Rij.2}, ..., (lecij.p){Rij.p},(¬lecij.1){R'ij.1}, (¬lecij.2){R'ij.2}, ..., (¬lecij.p){R'ij.p}} nj) ∧ (nk {(leckl.1){Rkl.1}, (leckl.2){Rkl.2}, .., (leckj.q){Rkl.q}, (¬leckl.1){R'kl.1}, (¬leckl.2){R'kl.2}, ..., ( ¬leckl.q){R'kl.q} } nl)}
ii) A set of I-L-Sets to represent the *mutually exclusive* disjunction of lcij and lckl (they cannot simultaneously hold):

ii.a) One of the constraints lcij or lckl should hold: The Cartesian product of label sets from complementary domains of lcij and lckl, {R'ij.1,R'ij.2, ....,R'ij.p}Χ{R'kl.1, R'kl.2, ...., R'kl.q},
are I-L-Sets.

ii.b) Only one of the constraints lcij or lckl should hold: The Cartesian product of label sets from lcij and lckl, {Rij.1,Rij.2, ....,Rij.p}Χ{Rkl.1, Rkl.2, ...., Rkl.q} are I-L-Sets.
Thus, *disjunctive* and *conjunctive* sets of disjunctive constraints between points can be represented and managed by means of a *conjunctive* set of disjunctive constraints and a set of I-L-Sets. For example:
(ti {[5 5]{R1} [10 10]{R2}} tj) ∨ (tk {[0 0]{R3} [20 20]{R4}} tl) ≡
(ti {[5 5]{R1} [10 10]{R2} (-∞ 5){R5} (5 10){R6} (10 ∞){R7}} tj) ∧
(tk {[0 0]{R3} [20 20]{R4} (-∞ 0){R8} (0 20){R9} (20 ∞){R10}} tl),
and
(ii.a) since (ti {[5 5]{R1}, [10 10]{R2}} tj] or [tk {[0 0]{R3}, [20 20]{R4}} tl] should hold:

{R5 R6 R7}Χ{R8 R9 R10} are I-L-Sets,
(ii.b) since only one constraint (ti {[5 5]{R1} [10 10]{R2}} tj) or (tk {[0 0]{R3} [20 20]{R4}} tl) should hold:
{R1 R2}Χ{R3 R4} = {R1 R3}, {R1 R4}, {R2 R3}, {R2 R4} are I-L-Sets.

| Ii ecij Ij          | Ii ecij Ij              | Ii ¬ecij Ij                        | Ii (ecij ∨ ¬ecij) Ij   |
|---------------------|-------------------------|------------------------------------|------------------------|
| + {(0 ∞){Rb1}} I2 - | I1 + {(-∞ 0]{Rb2}} I2 - | I1 + {(0 ∞){Rb1} (-∞ 0]{Rb2}} I2 - |                        |
| + {(0 ∞){Rb3}} I4 - | I3 + {(-∞ 0]{Rb4}} I4 - | I3 + {(0 ∞){Rb3} (-∞ 0]{Rb4}} I4 - |                        |

Table 5: Point-based constraints for (I1 before I2) and (I3 before I4)
Similarly, *disjunctions of interval-based constraints* between different pairs of intervals can also be represented. For instance, from Table 1 and Table 5, {(I1 before I2) ∨ (I3 before I4)} can be represented as:
(I1
+{(0 ∞){Rb1} (-∞ 0]{Rb2}} I2
-), (I3
+{(0 ∞){Rb3} (-∞ 0]{Rb4}} I4
-),