Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 239 — #247
7.5. Induction in Computer Science 239
Problems for Section 7.3
Homework Problems
Problem 7.25.
One version of the the Ackermann function A W N 2 ! N is defined recursively by
the following rules:
A.m; n/ WWD 2n if m D 0 or n 1, (A-base)
A.m; n/ WWD A.m 1; A.m; n 1// otherwise: (AA)
Prove that if B W N 2 ! N is a partial function that satisfies this same definition,
then B is total and B D A.
Problems for Section 7.4
Practice Problems
Problem 7.26. (a) Write out the evaluation of
eval.subst.3x; x.x 1//; 2/
according to the Environment Model and the Substitution Model, indicating where
the rule for each case of the recursive definitions of eval.; / and ŒWD] or substitution
is first used. Compare the number of arithmetic operations and variable lookups.
(b) Describe an example along the lines of part (a) where the Environment Model
would perform 6 fewer multiplications than the Substitution model. You need not
carry out the evaluations.
(c) Describe an example along the lines of part (a) where the Substitution Model
would perform 6 fewer multiplications than the Environment model. You need not
carry out the evaluations.
Class Problems
Problem 7.27.
In this problem we’ll need to be careful about the propositional operations on truth
values and connective symbols that appear in formulas. We’ll restrict ourselves to
formulas with connective symbols And and Not, since we know every propositional