1. Automatic Generation of Alternative Starting Positions for Simple Traditional Board Games Umair Z. Ahmed (IIT Kanpur) Krishnendu Chatterjee (IST Austria) Sumit Gulwani (MSR, Redmond) AAAI-15 1

2. Simple Traditional Board Games Board games such as Tic-Tac-Toe and Connect-4 Default start state is the empty board The traditional research question: who is the winner and optimal strategies from the empty starting state AAAI-15 2

3. Novel Problem: Alternative Starting Positions Customizing hardness level of starting state Default start state can be biased (Tic-Tac-Toe 4x4) Even if unbiased not conducive for novice players Generating multiple fresh start states Strategies can be memorized Customizing length of play Long plays can be boring. AAAI-15 3

4. Alternative Starting Positions Automatically generate interesting starting states Our results: Theoretical : A search strategy that applies to all graph games. Experimental : For simple board games \ Source: www.shredderchess.com AAAI-15 4

5. Alternating Graph Game AAAI-15 5

6. Our Approach AAAI-15 6

7. Two Issues AAAI-15 7

8. Solution Ideas AAAI-15 8

9. Board Games Variants Variable parameters 1. Board Size : 3x3, 4x4, 4x5, … 2. Winning Condition : RCD, RC, CD, RD 3. Gravity : None, Partial, Full AAAI-15 9 OX XOX XO OXO Tic-Tac-Toe X OX O Bottom-2 OO XXX XOX XXOXO OOOXO Connect-4

10. Experimental Results AAAI-15 10 Connect-4 5x5 with 5000 Random Sampling, against k 2 = 3 j State Space Win Cond # States |W j | k 1 = 1 k 1 = 2 k 1 = 3 EMHEMHEMH 2 6.9x10 7 RCD1.2x10 6 *184215*141129*00 8.7x10 7 RC1.6x10 6 *81239*70186*00 1.0x10 8 RD1.1x10 6 *106285*15182*00 9.5x10 7 CD5.3x10 5 *364173*20996*00 3 6.9x10 7 RCD2.8x10 5 *445832*397506*208211 8.7x10 7 RC7.7x10 5 *328969*340508*111208 1.0x10 8 RD8.0x10 5 *3981206*464538*179111 9.5x10 7 CD1.5x10 5 *14673*171110*12072

11. Experimental Results AAAI-15 11 Existence of vertices of different hardness levels GameSize Player-1 depth k 1 = 1 k 1 = 2 k 1 = 3 Tic-Tac-Toe3x3Only RCD Tic-Tac-Toe4x4 Bottom-23x3Only RD Bottom-24x4Except RCD CONNECT-34x4 CONNECT-45x5

12. Experimental Results AAAI-15 12 Existence of vertices of different hardness levels Number of interesting vertices are rare – negligible fraction of huge state space Interesting games now possible in games with heavily biased starting states, like Tic-Tac-Toe 4x4

13. Auto Generated Starting States Positions Player-1 ( X ) can win in j = 2 turns, which are difficult against k 2 = 3 AAAI-15 13 OX XOX XO OXO Tic-Tac-Toe RC, k 1 =1 X OX O Bottom-2 RCD, k 1 =1,2 OO XXX XOX XXOXO OOOXO Connect-4 RCD, k 1 =1,2

14. Conclusion Novel problem definition: automatic generation of interesting starting states Search technique: Utilizing symbolic methods and iterative simulation Present results for simple traditional board games Future work: Whether can be extended to more complicated games AAAI-15 14

15. Acknowledgment AAAI-15 travel partially supported by AAAI-2015 student scholarship Microsoft Research India travel grant Indian Institute of Technology (IIT) Kanpur Contact email: umair@iitk.ac.in AAAI-15 15

16. Binary Decision Diagrams (BDD) AAAI-15 16

17. BDD Operations EPre (Existential Predecessor) … … AAAI-15 17 OOX XOO XX OX XOO XXO O OX XOX XXO X OO OOX XOO XXX OX OX XOX XXO X OXO XOX XOO XXO

18. BDD Operations APre (Universal Predecessor) … … AAAI-15 18 OX XOO XX OX XO X O OX XX XXO X O OOX XOO XX OX XOO XXO O OX XOX XXO X OO

19. Search Strategy 1. Symbolic Methods : Represent game symbolically using variables and compute starting states using BDD operations 2. Iterative Simulation : Determine hardness (Easy, Medium or Hard) of the starting states, using Min- Max AAAI-15 19

20. 1. Symbolic Methods AAAI-15 20

21. 2. Iterative Simulation: Min-Max AAAI-15 21