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
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.