Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 200 — #208
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Chapter 6
State Machines
To prove this, we begin with a definition: a state machine has commuting transitions
if for any states p; q; r,
.p ! q AND p ! r/ IMPLIES 9t: q ! t AND r ! t:
(a) Verify that the Flexible Mating Ritual has commuting transitions.
(b) Now conclude from Problem 6.29 that the Flexible Mating Ritual always terminate
with the same set of stable marriages as the day-by-day Ritual.
Exam Problems
Problem 6.31.
Four unfortunate children want to be adopted by four foster families of ill repute.
A child can only be adopted by one family, and a family can only adopt one child.
Here are their preference rankings (most-favored to least-favored):
Child
Bottlecap:
Lucy:
Dingdong:
Zippy:
Family
Grinches:
Hatfields:
Scrooges:
McCoys:
Families
Hatfields, McCoys, Grinches, Scrooges
Grinches, Scrooges, McCoys, Hatfields
Hatfields, Scrooges, Grinches, McCoys
McCoys, Grinches, Scrooges, Hatfields
Children
Zippy, Dingdong, Bottlecap, Lucy
Zippy, Bottlecap, Dingdong, Lucy
Bottlecap, Lucy, Dingdong, Zippy
Lucy, Zippy, Bottlecap, Dingdong
(a) Exhibit two different stable matching of Children and Families.
Family Child in 1st match Child in 2nd match
Grinches:
Hatfields:
Scrooges:
McCoys:
(b) Examine the matchings from part a, and explain why these matchings are the
only two possible stable matchings between Children and Families.
Hint: In general, there may be many more than two stable matchings for the same
set of preferences.