Copyright © 2005 The Institute of Electronics, Information and Communication Engineers
Special Section on Discrete Mathematics and Its Applications -- Papers |
Finding Yozume of Generalized Tsume-Shogi is Exptime-Complete
1 The authors are with the Department of Computer Science, Graduate School of Information Science and Technology, the University of Tokyo, Tokyo, 113-0033 Japan. E-mail: yato{at}is.s.u-tokyo.ac.jp
Generalized Tsume-Shogi (GTS) is Tsume-Shogi on the board of size n x n for arbitrary n. The problem to decide the existence of a winning sequence of moves (where the attacker must always check) on an instance of GTS was proved to be exptime-complete by Yokota et al. (2000). This paper considers the complexity of yozume problem of GTS, which is, roughly speaking, the problem whether a given position of GTS has a winning sequence other than given sequences (though the actual rule of yozume is more complicated). The detection of yozume is an important issue in designing Tsume-Shogi problems, since the modern designing rule strongly prohibits it. We define a function problem of GTS appropriately to formulate yozume problem as its Another Solution Problem (ASP; the problem to decide the existence of solutions other than given ones). Moreover, we extend the existing framework for investigating ASPs so that it can be applied to exptime-complete problems. In particular, since the decision of correctness of given winning sequences is not easy, we establish a framework to treat ASP of function problems with promises. On the basis of these results, we prove that the decision version of yozume problem of GTS is exptime-complete as a promise problem using the existing reduction which was constructed by Yokota et al. to prove the exptime-completeness of GTS.
Key Words: computational complexity, generalized Tsume-Shogi, yozume, combinatorial games, another solution problem
Manuscript received August 19, 2004. Final manuscript received January 17, 2005.