

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

- **Competing Needs** Effect competing needs mutexes change for cross-world mutexes because two effects ϕ i(a) and ϕ j(a
′), in worlds S and S
′respectively, could be competing. Formally, a cross-world competing needs mutex (ˆℓk(ϕ i(a**)) =** S, ˆℓk(ϕ j(a
′**)) =** S
′) exists between ϕ i(a)
and ϕ j(a
′) in worlds S and S
′if:
∃l∈ρ i(a),l′∈ρ j (a
′)(
ˆℓk(l) = S, ˆℓk(l
′) = S
′)

- **Induced** Induced mutexes change slightly for cross-world mutexes. The worlds where one effect induces another, remains the same, but the mutex changes slightly.
If ϕ j(a) in ˆℓk(ϕ j(a)) is mutex with ϕ h(a
′) in ˆℓk(ϕ h(a
′)), and ϕ i(a) induces effect ϕ j(a)
in the possible worlds described by ℓk(ϕ i(a)) ∧ ℓk(ϕ j(a)), then there is an induced mutex between ϕ i(a) in ˆℓk(ϕ j(a)) ∧ ℓk(ϕ i(a)) and ϕ h(a
′) in ˆℓk(ϕ h(a
′)) (see Figure 12).

Literal Mutexes: The literal mutexes can now hold for literals that are supported in different possible worlds.

- **Inconsistent Support** changes for cross-world mutexes. A mutex (ˆℓk(l) = S, ˆℓk(l
′) = S
′)

holds for l in S and l
′in S
′if ∀ϕ i(a), ϕj(a
′) ∈ Ek−1 where l ∈ ε i(a), l′ ∈ ε j(a
′), there is a mutex ˆℓk−1(ϕ i(a**)) =** S, ˆℓk−1(ϕ j(a
′**)) =** S
′).