Algor it hm 3 Propagate-Forward(beliefs b, plans M)
1: for all plans X E M do b t 1 (X, ¬ block) ← 0.0 2:
b ε 1 (X, block) ← 0.0 3:
out z ← 0.0 4:
10 :
11: 12:
13:
14: 15:
16:
17: 18: 19:
20:
21:
22:
nz ← 0.0 5:
6 : for all plans X E M in post-order do {children in temporal order before parents}
if X is a leaf then 7:
out z ← b t ( X , ¬ block )(1 − e − λ x ) {calculate probability of X terminating at time t }
8:
else {X is a parent}
9:
out z is known { because post-order guarantees all children set it in line 20}
for all temporal outgoing transitions T z → y from X do nz + nz + (1 - pzy)Tzy if n 2 > 0 then {some transition can be taken}
for all temporal outgoing transitions T z → y from X do p ← out z (1 - µ z y ) π z y if T z → leads to a successor plan Y then b t +1 (Y, ¬block) ← b t +1 (Y, ¬block) + ρ PROPAGATE-DOWN(Y, p, b, M)
else {T z → y is a terminating transition}
outparent(z) {{ outparent(z) + (1 - fizy)Tzy {parent's outgoing probability is its children's b t + 1 (X, block) ← b t + 1 (X, block) + outz  - η z b t + 1 (X, ¬ block) ← b t + 1 (X, ¬ block) - out z Determining the non-blocked outgoing probability η x (lines 11–12).

The probability, η x is the sum over all possible values of the numerator in Equation 3 (i.e., over all temporal outgoing transitions originating in X ), as illustrated in the derivation. As we see in line 21, η x is critical for calculating the belief that the agent has terminated execution of X, but has not yet begun execution of a successor (i.e., the belief bi+1(X, Wock ) that the agent is blocking).

Propagating 7x along temporal outgoing transitions (lines 13–20).  This is the key component in the propagation. For every temporal outgoing transition Tx → y, OVER-
SEER calculates ρ , a temporary variable that holds the probability mass corresponding to OVERSEER's belief in the joint event of (i) the agent having completed execution of X , (ii)
the agent taking the transition T X → Y , and (iii) the agent doing so without sending out an observable message. The calculation of ρ is derived as follows:
p =
Probability that X is done A no message was observed A agent chose T x → y

l Pr(done(X,t)|X,t) Pr(¬msg|X,t),done(X,t)) Pr(Y,t)),done(X,t),¬msgt)
l outx * ηx * (1 - μ xy)π xy
n outx * (1 - µxy)πxy
(4)
If the transition T x → y leads to a successor plan Y (lines 16–18), then ρ is added to Y's future state (at time t + 1) as temporal incoming probability.  Since decomposition is assumed to be immediate, this incoming probability is propagated (added) to Y's first