Takumi Kasai

Classes of Pebble Games and Complete Problems

A “pebble game” is introduced and some restricted pebble games are considered. It is shown that in each of these games the problem to determine whether there is a winning strategy (two-person game) is harder than the solvability problem (one-person game). We also show that each of these problems is complete in well-known complexity classes. Several familiar games are presented whose winning strategy problems are complete in exponential time.

A characterization of time complexity by simple loop programs

There is general agreement that computations which are exponentially difficult in time are practically intractable. Thus, even the class L in the hierarchy of Meyer and Ritchie is still inclusive in the sense of practical computability. In this paper, we introduce the notion of simple loop programs, by which we characterize the class L which coincides with the class of Kalmars elementary functions. We also characterize two hierarchies B0⊆B1⊆B2⋯ and Eξp0⊆Eξp1⊆Eξp2⊆⋯ syntactically. Bk and Eξpk are the classes of languages which can be recognized in O(nκ) times and gκ(p(n)) time, respectively, where g0(n) = n, gκ+1(n) = 2gκ(n), p is a polynomial of n, and n is the length of input. Thus, Uk=0∞Bk=Eξp0 is the class of languages which can be recognized in polynomial time and Uk=0∞Eξpk is the class of elementary problems (i.e., L is in Uk=0∞Eξpk if and only if the characteristic function of L is in L2).

Low level complexity for combinatorial games

There have been numerous attempts to discuss the time complexity of problems and classify them into hierarchical classes such as P, NP, PSPACE, EXP, etc. A great number of familiar problems have been reported which are complete in NP (nondeterministic polynomial time). Even and Tarjan considered generalized Hex and showed that the problem to determine who wins the game if each player plays perfectly is complete in polynomial space. Shaefer derived some two-person game from NP complete problems which are complete in polynomial space. A rough discussion such as to determine whether or not a given problem belongs to NP is independent of the machine model and the way of defining the size of problems, since any of the commonly used machine models can be simulated by any other with a polynomial loss in running time and by no matter what criteria the size is defined, they differ from each other by polynomial order. However, in precise discussion, for example, in the discussion whether the computation of a problem requires O(n k) time or O(nk+l) time, the complexity heavily depends on machine models and the definition of size of problems. From these points, we introduce somewhat stronger notion of the reducibility.