Chapter 6 Conclusion and Future Work
6.2 Future Work
There are still some open issues regarding our problem domain. The optimal strategies of deductive games with much higher dimensions, which are called m×n AB games while m ≥ 4, are still unknown. It is interesting to investigate them because they may become NP-complete problems or harder problems if the value of m is getting larger constantly. Then, the boundary value of m is significant as well. Besides the original versions of much higher dimensions, other variants of deductive games are also worth studying such as static deductive games or deductive games with multiple unreliable responses. From the progress of research, 3×n deductive games in the expected case and 4×n deductive games in the worst case may be solved completely in the near future.
There are other important problems such as the Renyi-Ulam game and the counterfeit coin problem, whose styles are similar to deductive games. In fact, the Renyi-Ulam game has been widely surveyed in the fault-tolerance area and
meanwhile, the counterfeit coin problem has been discussed constantly in the information-theory area as well. However, there are still a lot of open issues about the two significant problems. These open questions are likewise worth studying in further detail for discovering their solutions.
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Appendix A. Equivalence Transformations for AB Game at the Second Query
The following equivalence transformations for the second query of AB game transform the 209 codes into their corresponding representatives.
Table 14. Equivalence transformations
Order Representative Each
query Equivalence transformations
1 0123 - -
2301 ⎟⎟⎠
3456 ⎟⎟⎠
4103 ⎟⎟⎠
4253 ⎟⎟⎠
3405 ⎟⎟⎠
3415 ⎟⎟⎠
0314 ⎟⎟⎠
4130 ⎟⎟⎠
2034 ⎟⎟⎠
2014 ⎟⎟⎠
Appendix B. Optimal Strategy for AB Game in the Expected Case
Prior to introducing the optimal strategy of AB game, its representation will be illustrated first. The lower-case alphabets, a, b, c, …, m, n, represent the 14 responses (hints), as shown in Table 15.
Table 15. The mapping between responses and representative letters
Response Representative letter Response Representative letter
[4, 0] a [1, 1] h
[3, 0] b [1, 0] i
[2, 2] c [0, 4] j
[2, 1] d [0, 3] k
[2, 0] e [0, 2] l
[1, 3] f [0, 1] m
[1, 2] g [0, 0] n
Three kinds of tokens will appear in the strategy. The first kind is four-digit Arabic numerals, which means the query made by the codebreaker. The second one is lower-case letters mentioned above, which indicate the responses. The last kind is parentheses. The tokens in parentheses refer to the optimal tactic of the state. In other words, it is an optimal game tree of that state. The tactic is constructed with a recursive form and can be treated as a game tree. For example, suppose that a game tree depicted in Figure 21 is given. Then its corresponding representation will be
“4872 ( j 7248 ( a ) f 4287 ( f 8274 ( a ) a ) a )”. Furthermore, it is easy to reconstruct the game tree from its representation with depth-first ordering.
j f a 4872
7248 4287
8274 a
7248
a
a f {7248} {4287, 8274}
{8274}
4872 ( j 7248 ( a ) f 4287 ( f 8274 ( a ) a ) a ) {4287, 4872, 7248, 8274}
4287
8274
4872
Figure 21. The transformation between the game tree and its corresponding representation
The derived optimal strategy of AB game in the average case is shown partially as follows due to space restrictions. In order to clarify the levels, we use an indent structure. We have established a website (http://www.csie.ntnu.edu.tw/~linss/
ABgame/optimal_strategy.html) that includes the full text of the optimal strategy.
0123 ( n 4567 ( l 5689 ( l 7498 ( j 8974 ( c 9874 ( a ) a )
f 8794 ( f 9748 ( a ) a )
c 7894 ( f 7948 ( a ) j 9478 ( a )
a ) a )
k 6948 ( l 8795 ( c 7895 ( a ) f 9875 ( a ) a )
k 8495 ( l 9876 ( a ) j 9854 ( a ) h 7896 ( a ) e 8796 ( a ) a )
j 8496 ( a )
h 8975 ( j 9758 ( a ) a )
g 8954 ( j 9845 ( a ) f 9458 ( a ) e 8976 ( a ) a )
f 6894 ( f 9846 ( a ) a )
e 7958 ( a )
d 6798 ( l 8945 ( a )
a )
c 6498 ( j 8946 ( a ) a )
b 6978 ( a ) a )
// The full text of the optimal strategy is included at http://www.csie.ntnu.edu.tw/
~linss/ABgame/optimal_strategy.html.
Appendix C. Publication List
(a) Referred Papers:
As a PhD student
[a1] Huang, L. T., and Lin, S. S. (2009), “Optimal analyses for 3×n AB games in the worst case,” to appear in Lecture Notes in Computer Science series.
[a2] Huang, L. T., Chen, S. T., Huang, S. J., and Lin, S. S. (2007), “An efficient approach to solve Mastermind optimally,” ICGA Journal, Vol.
30, No. 3, pp. 143–149.
[a3] Chen, S. T., Lin, S. S., and Huang, L. T. (2007), “A two-phase optimization algorithm for Mastermind,” Computer Journal, Vol. 50, No.
4, pp. 435–443.
[a4] Chen, S. T., Lin, S. S., Huang, L. T., and Hsu, S. H. (2007), “Strategy optimization for deductive games,” European Journal of Operational
Research, Vol. 183, No. 2, pp. 757–766.
As a master’s student
[a5] Huang, L. T., Chen, S. T., and Lin, S. S. (2006), “Exact-bound analyses and optimal strategies for Mastermind with a lie,” Lecture Notes in
Computer Science, Advances in Computer Games 11, Vol. 4250, pp.
195–209.
[a6] Chen, S. T., Lin, S. S., Huang, L. T., and Wei, C. J. (2004), “Towards the Exact Minimization of BDDs — An Elitism-Based Distributed Evolutionary Algorithm,” Journal of Heuristics: Special Issue on New
Advance on Parallel Meta-Heuristics for Complex Problems, Vol. 10, No.
3, pp. 337–355.
(b) Submitted Paper:
[b1] Huang, L. T., Chen, S. T., and Lin, S. S. (2009), “Optimal Algorithm for AB Game in the Expected Case,” submitted to IEEE Transactions on
Computational Intelligence and AI in Games.
(c) Conference Papers:
As a PhD student
[c1] Huang, L. T., and Lin, S. S. (2009), “Optimal analyses for 3×n AB games in the worst case,” The 12th
conference on Advances in Computer Games
(ACG12), Pamplona, Spain.[c2] Huang, L. T., Chen, S. T., Huang, S. J., and Lin, S. S. (2007), “An efficient approach to solve Mastermind optimally,” COMPUTER GAMES
WORKSHOP 2007, Amsterdam, The Netherlands.
[c3] Chen, S. T., Lin, S. S., Chang, S. W., and Huang, L. T. (2006), “A two-phase search algorithm for the set covering problem”，第十一屆人工 智慧與應用研討會，國立高雄應用科技大學，台灣，中華民國。
As a master’s student
[c4] Huang, L. T., Chen, S. T., and Lin, S. S. (2005), “Exact-bound analyses and optimal strategies for Mastermind with a lie,” The 11th
Advances in Computer Games Conference (ACG11), Taipei, Taiwan.
(d) Technical Reports:
[d1] 陳善泰、黃立德、張書維、劉耀才、江漢昇、胡淑瓊，2005，國科會 研究報告：演繹競局問題最佳化策略及其應用於容錯系統之研究 (2/2)，NSC93-2213-E-003-001。
[d2] 陳善泰、黃立德、張書維、劉耀才、江漢昇、胡淑瓊，2004，國科會 研究報告：演繹競局問題最佳化策略及其應用於容錯系統之研究 (1/2)，NSC92-2213-E-003-006。