Heuristics and Symmetries in Classical Planning: Additional Proofs

Proof: We only need to show one direction, since the inverse element σ−1 is also a symmetry. Let π = o1 · . . . · ok be a plan for s and let s0, . . . , sk be the sequence of states such that s0 = s and for 1 ≤ i ≤ k, si = si−1JoiK. Let TΠ = 〈S,E〉. Then we have 〈si−1, si; oi〉 ∈ E for all 1 ≤ i ≤ k and therefore, by the definition of symmetries we have 〈σ(si−1), σ(si);σ(oi)〉 ∈ E and C(σ(oi)) = C(oi). Thus, σ(π) is a sequence of actions that is applicable in σ(s) and ends with the state σ(sk). Since sk is a goal state, by the definition of symmetries σ(sk) is a goal state. Furthermore,