= R2( f (I)) are  the  ground  and  abstract  state  spaces  reachable  from  I
and  f (I) respectively. We can then construct the restricted STGs G
and restrict  τ accordingly. 
Our framework can then be directly applied to this restriction. It is also common to additionally specify a set  G of goal 
(cid:10)
states. We may then in some cases wish to deﬁne  S
2, i.e. we 
consider only the space of states that are on a path from  I to G. We conclude that our choice of deﬁning the framework 
using STGs rather than speciﬁc structures is more ﬂexible. However, an interesting case here is GIDL abstraction, which is 
known to be admissible, but does not even satisfy the P1↑ property. Consider a SAS+ instance P1 = (cid:3)V 1, D1, A1, I1, G1(cid:4) and 
its corresponding GIDL transformation P2 = (cid:3)V 2, D2, A2, I2, G2(cid:4). Suppose there is a path s0, s1, . . . , sn in G(P1) such that 
s0 = I1 and G1 ⊆ sn. Then there is a path t0, t1, . . . , tn in G(P2) such that t0 = I2 and si
A-framework-for-analysing-state-abstraction-metho_2022_Artificial-Intelligen