Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 228 — #236
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Chapter 7
Recursive Data Types
(b) Let rev.s/ be the reversal of the string s. For example, rev.001/ D 100. A
palindrome is a string s such that s D rev.s/. For example, 11011 and 010010
are palindromes.
Give a cat-OK definition of the palindromes.
(c) Give a cat-OK definition of the set P of strings consisting solely of 0’s whose
length is a power of two.
Problems for Section 7.2
Practice Problems
Problem 7.16.
Define the sets F 1 and F 2 recursively:
F 1 :
F 2 :
– 5 2 F 1 ,
– if n 2 F 1 , then 5n 2 F 1 .
– 5 2 F 2 ,
– if n; m 2 F 1 , then nm 2 F 2 .
(a) Show that one of these definitions is technically ambiguous. (Remember that
“ambiguous recursive definition” has a technical mathematical meaning which does
not imply that the ambiguous definition is unclear.)
(b) Briefly explain what advantage unambiguous recursive definitions have over
ambiguous ones.
(c) A way to prove that F 1 D F 2 , is to show first that F 1 F 2 and second that
F 2 F 1 . One of these containments follows easily by structural induction. Which
one? What would be the induction hypothesis? (You do not need to complete a
proof.)
Problem 7.17. (a) To prove that the set RecMatch, of matched strings of Definition
7.2.1 equals the set AmbRecMatch of ambiguous matched strings of Defini-