Describe and Match

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.



  1. Describe the rules that transform A to B and C to each of (1,2,3,4,5).
  2. Evaluate a matching similarity between the transformation A to B and those of C to (1,2,3,4,5).
  3. Select the most similar.

Describing the transformation rules proceeds in four parts. We will use the example in the diagram below



It description is

1Label subshapess,c,t
2Describe Aabove(c,t), above(c,s), inside(s,t)
3Describe Bleftof(s,t)
4Describe subshape changes delete(c), enlarge(s), shrink(t)
Notice that although we describe the subshape changes explicitly, we describe the overall organizational changes implicitly as a difference between descriptions 2 and 3.

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:

TransitionDescription 2Description 3
A®Babove(t,s)leftof(t,s)
C®1above(d,c)below(d,c)
C®2above(d,c)rightof(d,c)
C®3above(d,c)leftof(d,c)
Note that there is no reason initially to associate dot (d) with triangle (t). We choose C®3 on the basis of the best transformation matching. This in turn leads us to associate d with t and so on.

Now consider a case with only ``type 4'' matching:



Its description is:

TransitionDescription 4
A®B rotate(s,p/8)
C®1 rotate(+,p/8)
C®2Nothing
C®3magnify(+,0.5),rotate(+,\fracp8)
Again it is the description matching which guides us to C®1.

To perform the description we obviously need some similarity measures. One might propose the following ordering:

Most similarNothing
magnify
rotate
rotate and magnify
¯reflect
reflect and magnify
reflect and rotate
reflect, magnify and rotate
Least similarDelete

Such similarity ordering can also help in reducing ambiguity in forming descriptions, as in the diagram.

<<<Previous Next>>>