Consider the following planning problem in PDDL

Init(At(P1,A),At(P2,B),At(P3,C),At(P4,D),At(P5,E),Pipe(P1,P3),Pipe(P3,P5),Pipe(P5,P2),Pipe(P2,P4),Pipe(P4,P1))

Goal(At(P1,D),At(P2,C),At(P3,A),At(P4,E),At(P5,B))

Action(Jump(x,from,to)

PRECOND:At(from,x)∧Pipe(from,to)
EFFECT:¬At(from,x)∧At(to,x))

At which level of the planning graph does the literal At(P5,A) appear for the first time?

Question

Consider the following planning problem in PDDL

Init(At(P1,A),At(P2,B),At(P3,C),At(P4,D),At(P5,E),Pipe(P1,P3),Pipe(P3,P5),Pipe(P5,P2),Pipe(P2,P4),Pipe(P4,P1))

Goal(At(P1,D),At(P2,C),At(P3,A),At(P4,E),At(P5,B))

Action(Jump(x,from,to)

PRECOND:At(from,x)∧Pipe(from,to)
EFFECT:¬At(from,x)∧At(to,x))

At which level of the planning graph does the literal At(P5,A) appear for the first time?

BlackTom AI · Accepted Answer

Let's restate the scenario to ground the analysis. The initial state includes At(P1,A) and a set of pipes: P1→P3, P3→P5, P5→P2, P2→P4, P4→P1. The goal asks when At(P5,A) first becomes true.

First, consider what it takes to move A from one location to another. The Jump action has precondition At(from,A) and Pipe(from,to), and its effect moves A to the destination, removing it from the origin. So we can only perform a jump for A along a pipe if A is currently at the source location.

From the initial state (S0): we have At(P1,A). Therefore, at level S1, we can apply the Jump action along pipes whose source matches P1. There is Pipe(P1,P3), so we can execute Jump(A, P1, P3), which yields At(P3,A) and ¬At(P1,A) in the next state. No other Jump actions from S0 source P1 exist that move A elsewhere in one step, so the only S1 literal directly derived about A is At(P3,A).

Now, to obtain At(P5,A), we need A to be at a location from which there is a pipe to P5. Indeed, there is Pipe(P3,P5). However, to use Jump(A, P3, P5), we must first have At(P3,A) available as a precondition. As established, At(P3,A) becomes available at S1 due to Jump(A, P1, P3).

Because planning graphs separate layers, the precondition At(P3,A) for the second Jump cannot be used in the same level (S1) to produce At(P5,A) in that same level. The Jump(A, P3, P5) action requires At(P3,A) which is produced by an action in S1, thus its effect At(P5,A) appears in the next level, S2.

Therefore, the literal At(P5,A) first appears at level S2, after the sequence: at S0, perform Jump(A, P1, P3) to get At(P3,A) in S1; then at S1, perform Jump(A, P3, P5) to move A to P5, yielding At(P5,A) in S2.

In summary, due to the dependency chain and the strict alternation of action effects between consecutive levels, At(P5,A) cannot appear at S0 or S1 but first at S2. This matches the provided option Level S2.

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