hv functions but one are the identity function, i.e. it only abstracts the domain of one compound variable.
} and  let  (cid:9)i be  the 
(cid:4) as ⊗G∈(cid:9)i G. We 
result of applying operation opi on (cid:9)i−1 for all i (1 ≤ i ≤ m). For each (cid:9)i, deﬁne the STG G A
= G(F ).
can then treat the sequence G A
m as a hierarchy of abstractions of G A
For arbitrary i (1 ≤ i ≤ m), let G1, . . . , Gp be the STGs in (cid:9)i−1. Then G A
= G1 ⊗ G2 ⊗ . . . ⊗ Gp. There are three cases 
0
i
1. Suppose opi is a merge operation of G j and Gk in  (cid:9)i−1, where  j (cid:7)= k. Without losing generality, assume  j = 1 and 
i−1,  since  synchronous 
i as  the  identity  function,  i.e. 
⊗ Gvk ) = L(Gv j ) = L(Gvk ) and that L(Gv j ) = L(Gvk ) = L(G(F )), as previously noted, 
i ) as the identity relation, i.e. Ri ((cid:2), (cid:2))
i ). Then deﬁne τi = (cid:3) fi, Ri(cid:4), which is an M↑R(cid:14)C(cid:14) transformation since  fi is the identity function, 
(cid:4) and shrinking function
A-framework-for-analysing-state-abstraction-metho_2022_Artificial-Intelligen