The Pk properties form an inﬁnite hierarchy between PW and P1 in each direction.

Theorem 19. For all k, m (0 ≤ m < k),
(1) Pk↓ ⇒ Pm↓
Proof. (1,3) Trivial. (2) Example 17(1) is P1↓ but not P2↓. (4) Analogous to (2) by reversing the example. (cid:2)

(3) Pk↑ ⇒ Pm↑

(4) Pm↑ (cid:3) Pk↑

(2) Pm↓ (cid:3) Pk↓

However, the PL properties are incomparable with these latter hierarchies for all k > 1.

Theorem 20. 
(1) P2↓ (cid:3) PL↓,

(2) PL↓ (cid:3) P2↓,

(3) P2↑ (cid:3) PL↑,

(4) PL↑ (cid:3) P2↑.
A-framework-for-analysing-state-abstraction-metho_2022_Artificial-Intelligen