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.

Describe the rules that transform A to B and C to each of (1,2,3,4,5).

Evaluate a matching similarity between the transformation A to B and those of C to (1,2,3,4,5).

Select the most similar.

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)

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:

Transition	Description 2	Description 3
AŽB	above(t,s)	leftof(t,s)
CŽ1	above(d,c)	below(d,c)
CŽ2	above(d,c)	rightof(d,c)
CŽ3	above(d,c)	leftof(d,c)

Note that there is no reason initially to associate dot (d) with triangle (t). We choose CŽ3 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:

Transition	Description 4
AŽB	rotate(s,p/8)
CŽ1	rotate(+,p/8)
CŽ2	Nothing
CŽ3	magnify(+,0.5),rotate(+,\fracp8)

Again it is the description matching which guides us to CŽ1.

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

Most similar	Nothing
	magnify
	rotate
	rotate and magnify
Ż	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.

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