one of the following holds:
(1) ε 0(a) ∧ ρ e(a
′) =⊥
(2) ρ e(a) ∧ ε 0(a
′) =⊥
(3) ε 0(a) ∧ ε 0(a
′) =⊥
(4) ρ e(a) ∧ ρ e(a
′) =⊥
- **Competing Needs** Two actions **a, a**′ have competing needs in a world when a pair of literals from their execution preconditions are mutex in the world. The worlds where a and a
′are mutex because of competing needs are described by:

$$\ell_{k}(a)\wedge\ell_{k}(a^{\prime})\wedge\bigvee_{l\in\rho^{j}(a),l^{\prime}\in\rho^{j}(a^{\prime})}\hat{\ell}_{k}(l,l^{\prime})$$

In the above formula we find all worlds where a pair of execution preconditions l ∈ ρ e(a), l′ ∈
ρ e(a
′) are mutex and both actions are reachable.

Effect Mutexes: The effect mutexes are a set of labelled pairs of effects. Each pair is labelled with a formula that indicates the set of possible worlds where the effects are mutex. The possible reasons for mutex effects are associated action mutexes, interference, competing needs, or induced effects.

- **Mutex Actions** Two effects ϕ i(a) ∈ Φ(a), ϕj(a
′) ∈ Φ(a
′) are mutex in all worlds where their associated actions are mutex, ˆℓk(**a, a**′).

- **Interference** Like actions, two effects ϕ i(a), ϕj(a
′) interfere if (1) the consequent ε i(a) of one is inconsistent with the antecedent ρ j(a
′) of the other, or (2) vice versa. They additionally interfere if (3) both effect consequents ε i(a) and ε j(a
′) are inconsistent, or (4) both antecedents ρ i(a) and ρ j(a
′) are inconsistent. The mutex will exist in all possible world projections, so the label of the mutex is ˆℓk(ϕ i(a), ϕj(a
′**)) =** BSP . Formally, ϕ i(a) and ϕ j(a
′)
interfere if one of the following holds:
(1) ε i(a) ∧ ρ j(a
′) =⊥
(2) ρ i(a) ∧ ε j(a
′) =⊥
(3) ε i(a) ∧ ε j(a
′) =⊥
(4) ρ i(a) ∧ ρ j(a
′) =⊥

- **Competing Needs** Like actions, two effects have competing needs in a world when a pair of literals from their antecedents are mutex in a world. The worlds where ϕ i(a) and ϕ j(a
′) have a competing needs mutex are:
$\ell_{k}(\varphi^{i}(a))\wedge\ell_{k}(\varphi^{j}(a^{\prime}))\wedge\bigvee\ell_{k}(l,l^{\prime})$  $l\in\rho^{i}(a),l^{\prime}\in\rho^{j}(a^{\prime})$
In the above formula we find all worlds where a pair of execution preconditions l ∈ ρ i(a), l′ ∈
ρ j(a
′) are mutex and both actions are reachable.