A classic problem in all Test Your Own IQ Books is that of geometrical analogy: A is to B as C is to ...?, as in the diagram below.
Describing the transformation rules proceeds in four parts. We will use the
example in the diagram below
It description is
1 | Label subshapes | s,c,t |
2 | Describe A | above(c,t), above(c,s), inside(s,t) |
3 | Describe B | leftof(s,t) |
4 | Describe subshape changes | delete(c), enlarge(s), shrink(t) |
We won't bother how to determine the subshapes (s,c,t or whatever), nor how we determine inside() etc (see Winston p26 if you are interested) but instead concentrate on matching rules 2 and 3. We pick out, therefore, a problem with no rules under item 4 - see the diagram
Its description is:
Transition | Description 2 | Description 3 |
A®B | above(t,s) | leftof(t,s) |
C®1 | above(d,c) | below(d,c) |
C®2 | above(d,c) | rightof(d,c) |
C®3 | above(d,c) | leftof(d,c) |
Now consider a case with only ``type 4'' matching:
Its description is:
Transition | Description 4 |
A®B | rotate(s,p/8) |
C®1 | rotate(+,p/8) |
C®2 | Nothing |
C®3 | magnify(+,0.5),rotate(+,\fracp8) |
To perform the description we obviously need some similarity measures. One might propose the following ordering:
Most similar | Nothing |
magnify | |
rotate | |
rotate and magnify | |
¯ | reflect |
reflect and magnify | |
reflect and rotate | |
reflect, magnify and rotate | |
Least similar | Delete |
Such similarity ordering can also help in reducing ambiguity in forming descriptions, as in the diagram.