### Abstract

The Another Solution Problem (ASP) of a problem Π is the following problem: for a given instance x of Π 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 polynomial-time 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. They used this property to show the NP-completeness of ASP of Nonogram, a sort of puzzle. Following it, Seta considered the problem to find another solution when n solutions are given. (We call the problem n-ASP.) He proved the NP-completeness of n-ASP of some problems, including Cross Sum, for any n. In this thesis we establish a rigid formalization of n-ASPs to investigate their characteristics more clearly. In particular we introduce ASP-completeness, the completeness with respect to the reductions satisfying the properties mentioned above, and show that ASP-completeness of a problem implies NP-completeness of n-ASP of the problem for all n. Moreover we research the relation between ASPs and other versions of problems, such as counting problems and enumeration problems, and show the equivalence of the class of problems which allow