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.

The Another Solution Problem (ASP) of a problem is the following problem: for a given instance x of and a solution s to it, find a solution to x other than s. The notion of ASP as a new class of problems was first introduced by Ueda and Nagao. They also pointed out that parsimonious reductions which allow polynomial-time transformation of solutions can derive the NP-completeness of ASP of a certain problem from that of ASP of another. In this paper we consider n-ASP, the problem to find another solution when n solutions are given, and formalize it to investigate its characteristics. In particular we consider ASP-completeness, the completeness with respect to the reductions satisfying the properties mentioned above. The complexity of ASPs has a relation with the difficulty of designing puzzles. We prove the ASP-completeness of three popular puzzles: Slither Link, Cross Sum, and Number Place. Since ASP-completeness implies NP-completeness, these results can be regarded as new results of NP-completeness proof of puzzles.