Shape

269 Classifying Rules with Parts
square fi square þ inscribed square
that I’ve been using all along. There’s one square in the left side of the rule, and there
are two squares in the right side. And in fact, the rule worked perfectly in this fashion
when I used it to calculate before. I applied the rule twice in this way
and three times so
with a redundant step that repeats the preceding shape. Moreover, the final shape in
both series matches the string ( ) ( ) ( ), where each pair of parentheses indicates
another square, although the palindrome ((( ))) looks to be a better fit. (Parenthesis
strings are paradigmatically context free. To see this, modify the two rules in the
introduction—S fi ðSÞ for h i fi hh ii and S fi SS for hifi h i h i—add the third
rule S fi ð Þ, and start with the symbol or ‘‘axiom’’ S.)
Other descriptions of my rule for squares, however, may be more appropriate.
What if I want to use it together with a rule that finds triangles? The identity
shouldn’t change anything. An identity
A fi A
in a generative grammar is totally useless, either because it doesn’t do anything or because
A isn’t a symbol I’ve used before. But any rule I add when I calculate with shapes
will automatically work as intended with any rule I’ve already used. The part relation
lets me recognize triangles—or uppercase K’s and lowercase k’s—after I’ve combined
squares. So it might be necessary to look at the rule
under the alternative description