1. Derived Variables A derived variable v is simply a function
that associate value with state.
Die Hard Example: total amount of
water in the two jugs
v ::= b + l

2. For GCD, already have variables
Derived Variables
For GCD, already have variables
x, y.

3. Derived Variables Proof of GCD termination:
Show that variable y is strictly decreasing and natural number valued.

4. Derived Variables Termination follows by least number principle:
y must take a least value, and thus the
algorithm is stuck.

5. Strictly Decreasing Variable16 12 8 4
State

6. Weakly Decreasing Variable16 12 8 4
State

7. Weakly Decreasing Variable
We used to call weakly decreasing
variables “nonincreasing” variables.
OK terminology but remember:
nondecreasing is
NOT EQUAL to “not decreasing.”

8. There are 5 girls and 5 boys
Stable Marriage
There are 5 girls and 5 boys 1 A 2 B 3 C 4 D 5 E

9. Stable Marriage Preferences: Boys Girls 1: CBEAD A: 35214
2 : ABECD B : 52143
3 : DCBAE C : 43512
4 : ACDBE D : 12345
5 : ABDEC E : 23415

10. Stable Marriage Preferences: Marry 1 with C: Boys 1: CBEAD 2 : ABECD
3 : DCBAE
4 : ACDBE
5 : ABDEC
Marry 1 with C: 1 C

11. Stable Marriage Only 4 boys and 4 girls left. Boys 2 : ABED 3 : DBAE
4 : ADBE
5 : ABDE
A B D E

12. Stable Marriage A problem: Rogue couples 1 C 4 B
Girl C likes like boy 4 better
Boy 4 like girl C better

13. Stable marriage The mating algorithm – day by day
Morning: boys serenade favorite girls
Afternoon: girl rejects all but best Serenader
Evening: rejected boys cross off girls

14. Stable marriage Termination condition: stop when no
rejection is possible.
The algorithm terminates.
Everybody ends up married.
The resulting marriages are stable.

15. Infinitely Wide Trees … …

16. Infinitely Wide Trees … …

17. Infinitely Wide Trees

18. Infinitely Wide Trees

19. Tutorial Problem 1