Derived Variables
Mathematics for Computer Science
MIT 6.042J/18.062J
A derived variable, v, is a function
giving a “value” to each state:
v: Q → Values.
If Values = `, we’d say v was State Machines II:
Derived Variables,
Stable Marriage
Copyright © Albert R. Meyer, 2005.
October 24, 2005
“natural-number-valued,” or
“`-valued.”
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Derived Variables
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Derived Variables
Robot on the grid example:
States Q = `2.
Another derived variable:
π ::= σ (mod 2).
π is {0,1}-valued.
Define the sum-value, σ, of a state:
σ(⟨x,y⟩) ::= x+y
An `-valued derived variable.
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Derived Variables
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Derived Variables
Called “derived” to distinguish from actual variables that appear in a program. For robot
Actual: x, y
Derived: σ, π
Copyright © Albert R. Meyer, 2005.
Copyright © Albert R. Meyer, 2005.
For GCD, have (actual) variables x, y.
Proof of GCD termination:
y is strictly decreasing and
natural number-valued.
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Copyright © Albert R. Meyer, 2005.
October 24, 2005
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1
Derived Variables
Strictly Decreasing Variable
Termination followed by
Well Ordering Principle:
y must take a least value –
and then the algorithm is stuck.
N
16
8
4
0
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Down or constant
after each step
12
8
4
0
Q0
Q1 Q2
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Q3 Q4
Q5
Q6
Q7
Q8
State
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Weakly Decreasing Variable
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Q1 Q2
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Q3 Q4
Q5
Q6
Q7
Q8
State
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σ: up & down all over the place –
neither increasing nor decreasing.
π: is constant –
both increasing & decreasing
(weakly)
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Team Problem
(We used to call weakly
decreasing variables
“nonincreasing” variables.)
Copyright © Albert R. Meyer, 2005.
Q0
σ, π for the Diagonal Robot
Weakly Decreasing Variable
N
16
Goes down at
every step
12
Problem 1
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Copyright © Albert R. Meyer, 2005.
October 24, 2005
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2
Stable Marriage
Stable Marriage
Preferences:
A Marriage Problem
1
2
3
4
5
Boys
1: CBEAD
2 : ABECD
3 : DCBAE
4 : ACDBE
5 : ABDEC
Boys
Girls
A
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B
C
D
E
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Preferences
Copyright © Albert R. Meyer, 2005.
Preferences
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Copyright © Albert R. Meyer, 2005.
1
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Copyright © Albert R. Meyer, 2005.
1
C
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Stable Marriage
Marry Boy 1 with Girl C
(his 1st choice)
2 : ABE D
3 : D BAE
4 : A DBE
5 : ABDE
Marry Boy 1 with Girl C
(his 1st choice)
1: CBEAD
2 : ABECD
3 : DCBAE
4 : ACDBE
5 : ABDEC
Try “greedy”
strategy for boys
Stable Marriage
Preferences
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Stable Marriage
Stable Marriage
1: CBEAD
2 : ABECD
3 : DCBAE
4 : ACDBE
5 : ABDEC
Copyright © Albert R. Meyer, 2005.
Girls
A : 35214
B : 52143
C : 43512
D : 12345
E : 23415
Preferences
Marry Boy 1 with Girl C
(his 1st choice)
2 : ABED
3 : DBAE
4 : ADBE
5 : ABDE
C
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Copyright © Albert R. Meyer, 2005.
1
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C
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3
Stable Marriage
Preferences
1
C
Next:
Marry Boy 2 with Girl A:
(best remaining choice)
2 : ABED
3 : DBAE
4 : ADBE
5 : ABDE
Copyright © Albert R. Meyer, 2005.
Stable Marriage
1 C
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1
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Stable Marriage
Copyright © Albert R. Meyer, 2005.
1
C
4
B
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Rogue Couple
C
4
B
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Stable Marriage
and vice-versa
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C
4
B
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1
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Boy 4 likes Girl C better than his wife.
C
4
5 E
Stable Marriage
Trouble!
1
Copyright © Albert R. Meyer, 2005.
3 D
2 A
4 B
A
2
Stable Marriage
Copyright © Albert R. Meyer, 2005.
Final “boy greedy” marriages
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Copyright © Albert R. Meyer, 2005.
B
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4
Stable Marriage
Stable Marriage
Let’s Try it!
?Volunteers:
5 Boys & 5 Girls
Stable Marriage Problem:
Marry everyone without
any rogue couples!
Copyright © Albert R. Meyer, 2005.
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Stable Marriage I.
5A
4C
3A
1E
4C
5B
2E
1D
Boy Optimal
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October 24, 2005
Stable Marriage II.
2B
3D
Copyright © Albert R. Meyer, 2005.
Copyright © Albert R. Meyer, 2005.
All Girls get 1st Choice
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Stable Marriage
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Stable Marriage
More than a puzzle:
• College Admissions
(original Gale & Shapley paper, 1962)
• Matching Hospitals & Residents.
• Matching Dance Partners.
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Copyright © Albert R. Meyer, 2005.
October 24, 2005
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5
Stable Marriage
Mating Algorithm
Morning: boy serenades favorite girl
The Mating Algorithm:
day by day
Ted
Alice
Bob
Copyright © Albert R. Meyer, 2005. All rights reserved.
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Mating Algorithm
Copyright © Albert R. Meyer, 2005.
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Mating Algorithm
Morning: boy serenades favorite girl
Afternoon: girl rejects all but favorite
Evening: rejected boy writes off girl
Morning: boy serenades favorite girl
Afternoon: girl rejects all but favorite
…
Alice
Ted
…
Ted
Alice
Bob
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Mating Algorithm
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Team Problem
Stop when no girl rejects.
Girl marries her
favorite suitor.
Copyright © Albert R. Meyer, 2005.
Copyright © Albert R. Meyer, 2005.
Problem 2
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Mating Algorithm
Mating Algorithm
Partial Correctness:
Model as State Machine
State q: Each boy’s set of “eligible”
1424
3 girls
• Everyone is married.
• Marriages are stable.
not crossed off
Termination:
q(Bob) = {Carole, Alice, …}
there exists a Wedding Day.
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Mating Algorithm: variables
Copyright © Albert R. Meyer, 2005.
October 24, 2005
Mating Algorithm: variables
Derived Variable
Derived Variable
serenading(Bob):
suitors(Alice):
Bob’s favorite eligible girl.
(“Top” on Bob’s list.)
::= max{q(Bob)}
using Bobs’ preference order.
all boys serenading Alice.
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Stable Marriage: Termination
::= serenading−1(Alice)
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October 24, 2005
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Stable Marriage: Termination
Derived Variable
boy’s-list-length:
boy’s-list-length:
strictly decreasing & N-valued.
total number of names not crossed
off boy’s lists
So ∃ Wedding Day.
::= Σb ∈ boys |q(b)|
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Copyright © Albert R. Meyer, 2005.
October 24, 2005
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7
Mating Algorithm: variables
Mating Algorithm
Derived Variable
favorite(Carole):
Carole’s preferred suitor.
::= max{suitors(Carole)}
using Carole’s preference order.
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Mating Algorithm: Girls improve
That is, favorite(G) is weakly
increasing for each G.
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Lemma: A boy’s 1st love tomorrow
will be no more desirable than today’s.
That is, serenading(B) is weakly
decreasing for each B.
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Lemma: A girl’s favorite tomorrow
will be at least as desirable as today’s.
…because today’s favorite will
stay until she rejects him for
someone better.
Mating Algorithm: Boys Get Worse
Copyright © Albert R. Meyer, 2005.
Copyright © Albert R. Meyer, 2005.
Mating Algorithm: Girls improve
Lemma: A girl’s favorite tomorrow
will be at least as desirable as today’s.
Copyright © Albert R. Meyer, 2005.
Different girls have different
favorites,
because boys serenade
one girl at a time.
(favorite: Girls→Boys
is an injection)
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Mating Algorithm: Boys Get Worse
Lemma: A boy’s 1st love tomorrow
will be no more desirable than today’s.
…because boys work straight
down their lists.
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Copyright © Albert R. Meyer, 2005.
October 24, 2005
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8
Mating Algorithm: Invariant
Stable Marriage: Termination
If G has rejected B, then she has
a better current favorite.
Proof:
favorite(G) is weakly increasing.
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Mating Algorithm: Everyone Marries
On Wedding Day:
• Each girl has 1 suitors
• Each boy is married, or
has no girls on his list
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Team Problem
Everyone is Married by Wedding Day
Proof: by contradiction.
If B is not married, his list is empty.
By Invariant, all girls have favorites
better than B -- so they do have a favorite.
Problem 3
That is, all girls are married.
So all boys are married.
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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Mating Algorithm
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Mating Algorithm is Optimal for
all Boys at once. Pessimal for all
Girls.
Girls’ suitors get better, and
boy’s sweethearts get worse, so
girls do better? No!
October 24, 2005
October 24, 2005
Mating Algorithm
Who does better, boys or girls?
Copyright © Albert R. Meyer, 2005.
Copyright © Albert R. Meyer, 2005.
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Copyright © Albert R. Meyer, 2005.
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9
Stable Marriage
More questions, rich theory:
Other stable marriages possible?
- Can be many.
Can a boy do better by lying? – No!
Can a girl do better by lying? – Yes!
Copyright © Albert R. Meyer, 2005.
October 24, 2005
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