We present a tight time-hierarchy theorem for nondeterministic cellular automata by using a recursive padding argument. It is shown that, if t2(n) is a time-constructible function and t2(n) grows faster than t1(n+1), then there exists a language which can be accepted by a t2(n)-time nondeterministic cellular automaton but not by any t1(n)-time nondeterministic cellular automaton.

A disentanglement puzzle consists of mechanically interlinked pieces, and the puzzle is solved by disentangling one piece from another set of pieces. A cast puzzle is a type of disentanglement puzzle, where each piece is a zinc die-casting alloy. In this paper, we consider the generalized cast puzzle problem whose input is the layout of a finite number of pieces (polyhedrons) in the 3-dimensional Euclidean space. For every integer k ≥ 0, we present a polynomial-time transformation from an arbitrary k-exponential-space Turing machine M and its input x to a cast puzzle c1 of size k-exponential in |x| such that M accepts x if and only if c1 is solvable. Here, the layout of c1 is encoded as a string of length polynomial (even if c1 has size k-exponential). Therefore, the cast puzzle problem of size k-exponential is k-EXPSPACE-hard for every integer k ≥ 0. We also present a polynomial-time transformation from an arbitrary instance f of the SAT problem to a cast puzzle c2 such that f is satisfiable if and only if c2 is solvable.

Kurotto and Juosan are Nikoli's pencil puzzles. We study the computational complexity of Kurotto and Juosan puzzles. It is shown that deciding whether a given instance of each puzzle has a solution is NP-complete.

The Building puzzle is played on an N×N grid of cells. Initially, some numbers are given around the border of the grid. The object of the puzzle is to fill out blank cells such that every row and column contains the numbers 1 through N. The number written in each cell represents the height of the building. The numbers around the border indicate the number of buildings which a person can see from that direction. A shorter building behind a taller one cannot be seen by him. It is shown that deciding whether the Building puzzle has a solution is NP-complete.

Shisen-Sho is a tile-based one-player game. The instance is a set of 136 tiles embedded on 817 rectangular grids. Two tiles can be removed if they are labeled by the same number and if they are adjacent or can be connected with at most three orthogonal line segments. Here, line segments must not cross tiles. The aim of the game is to remove all of the 136 tiles. In this paper, we consider the generalized version of Shisen-Sho, which uses an arbitrary number of tiles embedded on rectangular grids. It is shown that deciding whether the player can remove all of the tiles is NP-complete.

This paper investigates relationships among deterministic, nondeterministic, and alternating complexity classes defined in the hyperbolic space. We show that (i) every t(n)-time nondeterministic cellular automaton in the hyperbolic space (hyperbolic CA) can be simulated by an O(t4(n))-space deterministic hyperbolic CA, and (ii) every t(n)-space nondeterministic hyperbolic CA can be simulated by an O(t2(n))-time deterministic hyperbolic CA. We also show that nr+-time (non)deterministic hyperbolic CAs are strictly more powerful than nr-time (non)deterministic hyperbolic CAs for any rational constants r 1 and > 0. From the above simulation results and a known separation result, we obtain the following relationships of hyperbolic complexity classes: Ph= NPh = PSPACEh EXPTIMEh= NEXPTIMEh = EXPSPACEh , where Ch is the hyperbolic counterpart of a Euclidean complexity class C. Furthermore, we show that (i) NPh APh unless PSPACE = NEXPTIME, and (ii) APh EXPTIME h.