1. CS 171 Discussion Session Week 3
Jia Chen

2. Quiz 1 return First letter of your last name
A - H:  Kyoungwon Kim (Office hours Tuesday 9-10am in ICS 424A) I - P:  Zephyr Yao (Office hours Monday 9-10am in ICS 424F) Q - Z:  Dongxu Zhao (Office hours Friday 9-10am in ICS 424F) If time conflicts, send to the Reader to which you are assigned, and to a Reader with whom you do not have a time conflict, asking to be reassigned.  

3. Schedule
Games (Minimax, Alpha-beta pruning)
CSP

4. Game playing as a search problem
Idea: we cannot predict the behavior of the opponent, so we consider the worst case (the opponent makes the “optimal” move)
Generate game tree
Apply utility function to leaves
Bottom up calculation
At root, choose node leading to the highest value

5. Minimax

6. Can we do better?
When we visit this green node, we know the value of its parent is min(2, some other numbers), which cannot be larger than 2

7. Alpha Beta Pruning

8. At root
alpha = -Inf, beta = Inf,
MIN Node
MIN tests if new value is less than current value, so v = Inf [any number is < Inf]
MIN updates beta (and use alpha from its parent MAX to prune children)
MIN can prune if alpha >= beta
MAX Node
MAX tests if new value is greater than current value, so v=-Inf [any number is > -Inf]
MAX updates Alpha (and use beta from its parent MIN to prune children)
MAX can prune if alpha >= beta

9. alpha=-Inf
beta=Inf
v=-Inf (always less than any number)
MAX

10. v=-Inf
alpha=-Inf
v=+Inf
beta=Inf
MIN

11. v=-Inf
v=+Inf
v=-Inf
alpha=-Inf
beta=Inf
MAX
Compare 1 and v=-Inf → 1 > v=-Inf → update v=1, alpha=1
Test if alpha >= beta; 1 >=Inf? No
Compare 2 and v=1 → 2 > v=1 → update v=2, alpha=2
Test if alpha >= beta; 2 >=Inf? No

12. v=-Inf
alpha=-Inf
beta=2
v=2
v=2
MIN
Compare 2 and v=+Inf → 2 < +inf → update v=2, alpha=-Inf, beta = 2

13. v=-Inf
v=2
v=-Inf
alpha=-Inf
beta=2
MAX
Compare 3 and v=-Inf → 3 > -Inf → update v = 3, update alpha = 3
Test if alpha >= beta → 3 >=2 Yes! Prune
“MAX: Even if I return 3 or something greater than 3, MIN will enforce me to get only 2. I should check if there is a possibility of getting
Lower score than 2 under this branch”

14. v=-Inf
v=2
alpha=-Inf
beta=2
v=3
Min
Compare new value 3 and current best 2
3 > 2 ? No → keep current v=2 and beta = 2

15. v=-Inf
v=2
v=-Inf
alpha=-Inf
beta=2
MAX
Compare 5 and v=-Inf
5 > v=-Inf, update v=5 and update alpha=5
Test if v alpha >= beta → 5 >=2
Prune!

16. v=-Inf
v=2
alpha=-Inf
beta=2
v=5
MIN
Compare 5 and v=2
Keep v=2

17. v=2
alpha=2
beta=Inf
v=2
MAX
Compare 2 (from MIN) and v=-Inf
2 > -Inf? → yes → update v=2, update alpha=2

18. v=2
alpha=2
beta=Inf
alpha=2
v=Inf
beta=Inf
MIN

19. v=2
alpha=2
beta=Inf
alpha=2
v=Inf
beta=Inf
alpha=2
v=7
beta=Inf
MAX
Compare 1 and v=-Inf
1 > -Inf → update v=1, update alpha? 1 > 2 No → alpha is 2
Test if alpha >= beta( Inf) ? No
Compare 7 and v=1
7 > 1 → update v=7, update alpha 7 > 2 Yes→ alpha is 7
Test if alpha >=Inf? No

20. v=2
alpha=2
beta=Inf
alpha=2
v=7
beta=7
v=7
MIN
Compare 7 from MAX and v=Inf
7 < INF → update v=7, update beta = 7
Test if alpha >= beta? 2 >=7 No
(if MAX branch was (1,2) then beta would become 2 Then alpha >= 2? Yes)
MIN updates beta from INF to 7
If beta <= alpha then Prune not!! 7 <=2

21. v=2
alpha=2
beta=Inf
alpha=2
v=7
beta=7
v=7
MIN
MIN player wishes to search for some score which is less than or equal to 2 from the search space below (B)

22. v=2
alpha=2
beta=Inf
v=7
v=7
v=8
alpha=8
MAX
update v=8; alpha=8
Test if alpha >=beta; 8 >=7 Yes,
Because…. MIN (B) will choose 7 anyway
beta=7

23. v=2
alpha=2
beta=Inf
v=7
v=7
v=8
v=8
alpha=2 (from B)
MAX
v =3; alpha=3
Test if alpha >= beta; 3 >=7 No
MAX player cannot prune
v=8; alpha = 8
Could test… but no more nodes..
beta=7

24. v=7
alpha=7
beta=Inf
v=INF
alpha=7
beta=Inf
v=7
v=8
v=8
MIN

25. v=7
alpha=7
beta=Inf
v=INF
alpha=7
beta=Inf
v=1
v=7
v=8
v=8
alpha=7
beta=Inf
MAX
Update v as 0; alpha remains 7;
Alpha >= beta? 7>= INF? No
Update v as 1; alpha remains 7;

26. v=7
alpha=7
beta=Inf
v=INF
alpha=7
beta=1
v=1
v=7
v=8
v=8
alpha=7
MIN
Update v as 1; update beta as 1
Test if alpha >= beta; 7>=1
Yes, Prune
beta=Inf

27. CSP

28. CSP

31. TFFF

33. NW BC SA all 2 values remaining

34. B [from 3 possible choice R,G,B]
R (ON 1 NU1 SA 1 total 3)
G (ON 1 NU 1 SA 1 total 3)
B (ON 1 NU 1 SA 0 total 2)