τ = (cid:3) f , R(cid:4).

(cid:3) f , R(cid:4).

The PS, P, PW and P1 properties form strict hierarchies in both directions, as the following theorem shows.

Theorem 18.  The following relationships hold:
(1) PS↓ ⇒ P↓ ⇒ PW↓ ⇒ PL↓ ⇒ P1↓,
(2) P1↓ (cid:3) PL↓ (cid:3) PW↓ (cid:3) P↓ (cid:3) PS↓,
(3) PS↑ ⇒ P↑ ⇒ PW↑ ⇒ PL↑ ⇒ P1↑,
(4) P1↑ (cid:3) PL↑ (cid:3) PW↑ (cid:3) P↑ (cid:3) PS↑.

3 While it is possible to give these two conditions different names and also deﬁne their symmetric counterparts, there seems little use for this since the 
two conditions are introduced only to make the DPP criterion comparable to our concepts.

13

C. Bäckström and P. Jonsson

Artiﬁcial Intelligence 302 (2022) 103608
A-framework-for-analysing-state-abstraction-metho_2022_Artificial-Intelligen