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Relating two blocks world theories

 

Our previous blocks world theory T1 uses tex2html_wrap_inline479 . Our less approximate new theory T2 uses tex2html_wrap_inline481 where d is a displacement of b1 from being centered on b2. Since T2 has another parameter for On, many situations tex2html_wrap_inline341 can correspond to a single situation s in T1.

We may have the relations

  equation189

Here tex2html_wrap_inline495 is the T1-situation corresponding to tex2html_wrap_inline341 . 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

  equation205

in T1, but the real world facts are

  equation214

where

  equation227

The goal is tex2html_wrap_inline499 , which might also be written as tex2html_wrap_inline501 . Anyway the robot infers that the appropriate action is tex2html_wrap_inline503 and infers that

  equation236

where we are omitting various qualifications.

In T2, the form of an action is tex2html_wrap_inline505 , and the effect of a move action is given by

  equation244

The translation of a move action in T1 to a move action in T2 may be given by

  equation251

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

  equation261

The success of multi-step plans would be expressed by longer correspondence formulas.

These are commutativity relations.


next up previous
Next: Temporary entities; wealth and Up: What kinds of approximate Previous: The Blocks World as

John McCarthy
Wed Feb 2 15:59:04 PST 2000