Working through an analogy

It is handy to have a structured way to summarize and present analogies.  We'll use the following conventions for doing this.  A mapping will be displayed in terms of three tables:

  1. Entity correspondences.  The column on the left contains the base entity, the column on the right contains the corresponding target entity.
  2. Expression correspondences.  Like #1, except that the left contains the base expressions and the right contains the corresponding target expressions.
  3. Candidate inferences.  A single column or a list of what the mapping suggests about the target.

An example will make this clearer.  Consider the following description of a simple water flow:

(cause (> (pressure beaker) (pressure vial))
       (flow beaker vial water pipe))
(> (diameter beaker) (diameter vial))
(clear beaker)
(flat-top water)
(liquid water)

A very simple description of heat flow is:

(flow coffee ice-cube heat bar)
(> (temperature coffee) (temperature bar))
(flat-top coffee)
(liquid coffee)

(Notice that these descriptions are slightly more complicated than we used in class.)  Feeding these descriptions into SME, with simple water flow as the base and simple heat flow as the target, yields two mappings.  Here is how they look given the conventions above:

Mapping 1

Entity correspondences:

pipe bar
water heat
beaker coffee
vial ice cube

Expression correspondences:

(flow beaker vial water pipe) (flow coffee ice-cube heat bar)
(> (pressure beaker) (pressure vial)) (> (temperature coffee) (temperature ice-cube))

Candidate inferences:

(cause (> (temperature coffee) (temperature ice-cube)) (flow coffee ice-cube heat bar))
(liquid heat)
(flat-top heat)
(clear coffee)
(> (diameter coffee) (diameter ice-cube))

Mapping 2

Entity correspondences:

beaker coffee
vial ice cube

Expression correspondences:

(> (diameter beaker) (diameter vial)) (> (temperature coffee) (temperature  ice-cube))

Candidate inferences:

(cause (> (pressure coffee) (pressure  ice-cube))
          (flow coffee ice-cube (skolem heat) (sketching bar)))
(clear coffee)

A few observations about this method of working through an analogy:

  1. Notice that we are using the physical structure of the table to enforce the 1:1 constraint.  If you find yourself repeating the same element in either column, there is a problem with the mapping.
  2. While we used predicate calculus statements in this example, one can also use natural language statements in the tables, if formal representations aren't necessary for the problem at hand.
  3. When filling out a mapping in English, instead of saying "(skolem X)" one says "something like X", putting parentheses around it if necessary.

Last edited 10/31/02, by KDF.