Our previous blocks world theory T1 uses
. Our less approximate new theory
T2 uses 
where d is a
displacement of b1 from being centered on b2. Since T2 has
another parameter for On, many situations 
can correspond to
a single situation s in T1.
We may have the relations
Here 
is the T1-situation corresponding to . For
simplicity we are assuming that every T2-situation has a corresponding
T1-situation.
T2 is a tiny step from T1 in the direction of the real world.
Suppose a robot uses T1 as a theory of the blocks world and takes actions accordingly, but the real world corresponds to T2. This is quite a simplification, but maybe it has enough of the right formal properties.
The simplest case is where there are two blocks, and the initial situation is represented by
in T1, but the real world facts are
where
The goal is 
, which might also be
written as 
. Anyway the
robot infers that the appropriate action is 
and infers
that
where we are omitting various qualifications.
In T2, the form of an action is 
, and the effect of a
move action is given by
The translation of a move action in T1 to a move action in T2 may be given by
The key point is that the move in T2 corresponding to a move in T1 depends on the blocks being moved and also on the situation.
The success of the one step plan worked out in T1 in the less approximate world T2 is expressed by
The success of multi-step plans would be expressed by longer correspondence formulas.
These are commutativity relations.
