Finding Yozume of Generalized Tsume-Shogi is Exptime-Complete

Takayuki YATO
Takahiro SETA
Tsuyoshi ITO

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E88-A    No.5    pp.1249-1257
Publication Date: 2005/05/01
Online ISSN: 
DOI: 10.1093/ietfec/e88-a.5.1249
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
Category: 
Keyword: 
computational complexity,  generalized Tsume-Shogi,  yozume,  combinatorial games,  another solution problem,  

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Summary: 
Generalized Tsume-Shogi (GTS) is Tsume-Shogi on the board of size n 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.

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