# 10/13/11 Solution: Minimax with Alpha-Beta Pruning and Progressive Deepening

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1 10/1/11 Solution: Minimax with Alpha-Beta Pruning and Progressive Deepening When answering the question in Parts C.1 and C. below, assume you have already applied minimax with alpha-beta pruning and progressive deepening on the corresponding game tree up to depth. The value shown next to each node of the tree at depth is the respective node s static-evaluation value. Assume the procedure uses the information it has acquired up to a given depth to try to improve the order of evaluations later. In particular, the procedure reorders the nodes based on the evaluations found up to depth in an attempt to improve the effectiveness of alpha-beta pruning when running up to depth. We want to know in which order the nodes/states A, B, C, and D in the game tree are evaluated when the procedure runs up to depth, after running up to depth and reordering. Part C.1: Game Tree I ( points) Choose the order in which the nodes/states A, B, C and D in game tree I above are evaluated when running minimax with alpha-beta pruning and progressive deepening after running up to depth and reordering. (Circle your answer) a. A B C D b. D A B C c. B A D C d. C D A B e. D C B A

2 Part C.: Game Tree II ( points) Choose the order in which the nodes/states A, B, C and D in game tree II above are evaluated when running minimax with alpha-beta pruning and progressive deepening after running up to depth and reordering (Circle your answer) f. A B C D g. D A B C h. B A D C i. C D A B j. D C B A

3 10/1/11 Problem : Games In the game tree below, the value below each node is the static evaluation at that node. MAX next to a horizontal line of nodes means that the maximizer is choosing on that turn, and MIN means that the minimizer is choosing on that turn. Part A (10 points) Using minimax without Alpha-Beta pruning, which of the three possible moves should the maximizer take at node A? What will be the final minimax value of node A?

4 Part B (10 points) Mark suggests that Alpha-Beta pruning might help speed things up. Perform a minimax search with alpha-beta pruning, traversing the tree, and list the order in which you statically evaluate the nodes (that is, you would start with E). Write your answer below. Note that there is, at the end of this quiz, a tear-off sheet with copies of the tree. Part C (10 points) Tom thinks that he might save some trouble by calculating static values at depth and then using those static values to reorder the tree for the alpha-beta search that goes all the way to the leaf nodes. Thus, Tom is attempting to deploy alpha-beta search in a way that will improve pruning. Suppose the static evaluator Tom uses at depth produces exactly the minimax values you found in Part A. Tom uses those numbers to reorder BC&D, EF, GH, and IJ. Note that Tom knows nothing about how the tree branches below depth at this point. Draw the reordered tree down to depth, the EFGHIJ level. Explain the reasoning behind your choices:

5

6 . Converting problems into constraint propagation form Paul, John, and Ringo are musicians. One of them plays bass, another plays guitar, and the third plays drums. As it happens, one of them is afraid of things associated with the number 1, another is afraid of Apple Computers, and the third is afraid of heights. Paul and the guitar player skydive; John and the bass player enjoy Apple Computers; and the drummer lives in an open penthouse apartment 1 on the thirteenth floor. What instrument does Paul play? How can we solve this problem? Try it yourself first, by any means you care, then we ll see how to do it by ta-da! the magic of constraint propagation! (You might want to think about constraints when you solve it, and what the constraints are.) What are the constraints? How might they be represented? We want to use the facts in the story to determine whether certain identity relations hold ore are excluded. Here is our notation: assume X(Peter, Guitar Player) means the person who is John is not the person who plays the guitar. Further, this relation is symmetrical, so that if we have X(Peter, Guitar Player) then we also have, X(Guitar Player, Peter). In this notation, the facts become the following (of course all the symmetrical facts hold as well): 1. X(Paul, Guitar Player). X(Paul, Fears Heights). X(Guitar Player, Fears Heights). X(John, Fears Apple Computers). X(John, Bass Player). X(Bass Player, Fears Apple Computers) 7. X(Drummer, Fears 1) 8. X(Drummer, Fears Heights) Now we can represent the possible relations implicitly by means of entries in tables, and use constraint propagation. An X entry in a table denotes that the identity relation is excluded, and an I denotes that the identity relation actually holds. We can then use three tables, one to represent the possible identities between people and instrument players; one to represent the possible identities between people and fears; and a third to represent the possible relationships between instrument players and fears.

7 1. X(Paul, Guitar Player). X(Paul, Fears Heights). X(Guitar Player, Fears Heights). X(John, Fears Apple Computers). X(John, Bass Player). X(Bass Player, Fears Apple Computers) 7. X(Drummer, Fears 1) 8. X(Drummer, Fears Heights) Instrument Player Bass Player Guitar Player Drum Player Paul X X I John X I X Ringo I X X Fears 1 Apple Computers Heights Paul X I X John I X X Ringo X X I ` Fears 1 Apple Computers Heights Bass player X X I Guitar player I X X Drum Player X I X How do we do constraint propagation in this system? Note that we can deduce rules like the following to fill in the three tables: 1. If all but one entry in a row are X, then the remaining entry is I.. If you know I(x,y) and X(y,z) then you may conclude X(x,z). If two names pick out the same thing (are identical), then they must share all the same properties. What other rules do we need?. If you know X(x,y) & X(z,y) then you may conclude X(x,z). If you know X(x,y) & X(x,z) then you may conclude X(y,z)

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13 10/1/11 Constraint Propagation You just bought a -sided table (because it looks like a benzene ring) and want to hold a dinner party. You invite your best friends: McCain, Obama, Biden and Palin. Luckily a moose wanders by and also accepts your invitation. Counting yourself, you have guests for seats labeled Your guests have seven seating demands: Palin wants to sit next to McCain Biden wants to sit next to Obama Neither McCain nor Palin will sit next to Obama or Biden Neither Obama nor Biden will sit next to McCain or Palin The moose is afraid to sit next to Palin No two people can sit in the same seat, and no one can sit in seats. McCain insists on sitting in seat 1 10

14 Part A (10 points) You realize there are ways to represent this problem as a constraint problem. For each below, run the domain reduction algorithm and continue to propagate through domains reduced to one value. That is, cross out all the impossible values in each domain without using any search. Variables: You, Moose, McCain, Palin, Obama, Biden Domains: Seats 1- Constraints: I-VII You: 1 Moose: 1 McCain: 1 Palin: 1 Obama: 1 Biden: 1 Variables: Seats 1- Domains: You, Moose, McCain, Palin, Obama, Biden Constraints: I-VII 1: You Moose McCain Palin Obama Biden : You Moose McCain Palin Obama Biden : You Moose McCain Palin Obama Biden : You Moose McCain Palin Obama Biden : You Moose McCain Palin Obama Biden : You Moose McCain Palin Obama Biden Part B (1 points) For now, you decide to continue using seats as variables and guests as domains (the nd representation). You decide to see how a depth-first search with no constraint propagation works, but as you run the search you see you are doing a lot of backtracking. 9

15 You break ties by choosing the left-most in this order: You (Y), Moose (M), McCain(Mc), Palin (P), Obama (O), Biden (B) Show the partial search tree only up to the first time you try to assign seat. Begin with the reduced domains from the previous page. Use only the constraints supplied; use no commensense beyond what you see in the constraints. You might want to work this out on the tear off sheet at the end of the quiz first. 1 Mc Y M P 11

16 {Mc} { Y, M, P} {Mc} {Mc}

17 Part C (1 points) Now you try to use depth-first search and constraint propagation through domains reduced to size 1. You break ties by choosing the left-most in this order: You (Y), Moose (M), McCain(Mc), Palin (P), Obama (O), Biden (B) Show the the full search tree (up until you find a solution), starting with the same prereduced domains. You might want to work this out on a copy at the end of the quiz.. 1 Mc Y M P 1

18 {Mc} { Y, M, P} {Mc} {Mc}

19 {Mc} { Y, M, P} {Mc} {Mc}

20 {Mc} { Y, M, P} {Mc} {Mc}

21 {Mc} { Y, M, P} {Mc} {Mc}

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