Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 207 — #215
7.2. Strings of Matched Brackets 207
These proofs illustrate the general principle:
The Principle of Structural Induction.
Let P be a predicate on a recursively defined data type R. If
P.b/ is true for each base case element b 2 R, and
for all two-argument constructors c,
ŒP.r/ AND P.s/ IMPLIES P.c.r; s//
for all r; s 2 R,
and likewise for all constructors taking other numbers of arguments,
then
P.r/ is true for all r 2 R:
7.2 Strings of Matched Brackets
Let f] ; [ g be the set of all strings of square brackets. For example, the following
two strings are in f] ; [ g :
[ ] ] [ [ [ [ [ ] ] and [ [ [ ] ] [ ] ] [ ] (7.1)
A string s 2 f] ; [ g is called a matched string if its brackets “match up” in
the usual way. For example, the left-hand string above is not matched because its
second right bracket does not have a matching left bracket. The string on the right
is matched.
We’re going to examine several different ways to define and prove properties
of matched strings using recursively defined sets and functions. These properties
are pretty straightforward, and you might wonder whether they have any particular
relevance in computer science. The honest answer is “not much relevance any
more.” The reason for this is one of the great successes of computer science, as
explained in the text box below.