Mathematics for Computer Science

(avery) #1

8.1. Divisibility 247


theb-jug is big enough:


.0;0/!.a;0/ fill first jug
!.0;a/ pour first into second
!.a;a/ fill first jug
!.2ab;b/ pour first into second (assuming2ab)
!.2ab;0/ empty second jug
!.0;2ab/ pour first into second
!.a;2ab/ fill first
!.3a2b;b/ pour first into second (assuming3a2b)

What leaps out is that at every step, the amount of water in each jug is a linear
combination ofaandb. This is easy to prove by induction on the number of
transitions:


Lemma 8.1.5(Water Jugs).In theDie Hardstate machine of Section 5.4.4 with
jugs of sizesaandb, the amount of water in each jug is always a linear combination
ofaandb.


Proof. The induction hypothesis,P.n/, is the proposition that afterntransitions,
the amount of water in each jug is a linear combination ofaandb.


Base case(nD 0 ):P.0/is true, because both jugs are initially empty, and 0 aC
0 bD 0.


Inductive step: Suppose the machine is in state.x;y/afternsteps, that is, the little
jug containsxgallons and the big one containsygallons. There are two cases:


 If we fill a jug from the fountain or empty a jug into the fountain, then that jug
is empty or full. The amount in the other jug remains a linear combination
ofaandb. SoP.nC1/holds.

 Otherwise, we pour water from one jug to another until one is empty or the
other is full. By our assumption, the amountxandyin each jug is a linear
combination ofaandbbefore we begin pouring. After pouring, one jug is
either empty (contains 0 gallons) or full (containsaorbgallons). Thus, the
other jug contains eitherxCygallons,xCya, orxCybgallons, all
of which are linear combinations ofaandbsincexandyare. SoP.nC1/
holds in this case as well.

SinceP.nC1/holds in any case, this proves the inductive step, completing the
proof by induction. 

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