Chuzo Iwamoto

Constructible functions in cellular automata and their applications to hierarchy results

We investigate time-constructible functions in one-dimensional cellular automata (CA). It is shown that (i) if a function t(n) is computable by an O(t(n)n)-time Turing machine, then t(n) is time constructible by CA and (ii) if two functions are time constructible by CA, then the sum, product, and exponential functions of them are time constructible by CA. As an application, it is shown that if t1(n) and t2(n) are time constructible functions such that limnt1(n)/t2(n)=0 and t1(n)n, then there is a language which can be recognized by a CA in t2(n) time but not by any CA in t1(n) time.

On Time-Constructible Functions in One-Dimensional Cellular Automata

In this paper, we investigate time-constructible functions in one-dimensional cellular automata (CA). It is shown that (i) if a function t(n) is computable by an O(t(n) - n)-time Turing machine, then t(n) is time-constructible by CA and (ii) if two functions are time-constructible by CA, then the sum, product, and exponential functions of them are time-constructible by CA. As an example for which time-constructible functions are required, we present a time-hierarchy theorem based on CA. It is shown that if t1(n) and t2(n) are time-constructible functions such that limn→∞t1(n)/t2(n)=0, then there is a language which can be recognized by a CA in t2(n) time but not by any CA in t1(n) time.

Finding Hamiltonian circuits in arrangements of Jordan curves is NP-complete

Abstract Let A={C1, C2,…, Cn} be an arrangement of Jordan curves in the plane lying in general position, i.e., every properly intersects at least one other curve, no two curves touch each other and no three meet at a common intersection point. The Jordan-curve arrangement graph of A has as its vertices the intersection points of the curves in A, and two vertices are connected by an edge if their corresponding intersection points are adjacent on some curve in A. We further assume A is such that the resulting graph has no multiple edges. Under these conditions it is shown that determining whether Jordan-curve arrangement graphs are Hamiltonian is NP-complete.

Embedding a Logically Universal Model and a Self-Reproducing Model into Number-Conserving Cellular Automata

A number-conserving cellular automaton (NCCA) is a cellular automaton (CA) such that all states of cells are represented by integers and the total number of its configuration is conserved throughout its computing process. It can be thought as a kind of modelization of the physical conservation law of mass or energy. Although NCCAs with simple rules are studied widely, it is quite difficult to design NCCAs with complex transition rules. We show a condition for two-dimensional von Neumann neighbor NCCAs with special symmetric rules and we construct a logically universal NCCA and a self-reproducing NCCA by employing this condition.

A faster parallel algorithm for k-connectivity☆

Abstract A new parallel algorithm for graph k -connectivity is shown. It runs in O( k log log k log n ) time using ( n + k 2 ) k ( C ( n , m ) + kn ) processors on the ARBITRARY CRCW PRAM model, where C ( n , m ) is the number of processors required to compute the connected components in logarithmic time. The previous best algorithm runs in ( Ok 2 log n ) time using ( n + k 2 ) kC ( n , m ) processors (Khuller et al., 1991). When k = log n for example, our new algorithm runs in time O( log 2 n log log log n ) against O( log 3 n ) of Khullers algorithm, i.e., almost an improvement of a factor of log n . The number of processors does not change (under the big-O notation) if m ⩾ kn log n , since C ( n , m ) is at least (m + n) log n . If we use ( n + k 2 ) k ( C ( n , m ) + k 1 + e n ) processors, the bound can be further decreased to O( k log n ).

Yosenabe is NP-complete

Yosenabe is one of Nikolis pencil puzzles, which is played on a rectangular grid of cells. Some of the cells are colored gray, and two gray cells are considered connected if they are adjacent vertically or horizontally. A set of connected gray cells is called a gray area. Some of the gray areas are labeled by numbers, and some of the non-gray cells contain circles with numbers. The object of the puzzle is to draw arrows, vertically or horizontally, from all circles to gray areas so that (i) the arrows do not bend, and do not cross other circles or lines of other arrows, (ii) the number in a gray area is equal to the total of the numbers of the circles which enter the gray area, and (iii) gray areas with no numbers may have any sum total, but at least one circle must enter each gray area. It is shown that deciding whether a Yosenabe puzzle has a solution is NP-complete.

Lower Bound of Face Guards of Polyhedral Terrains

We study the problem of determining the minimum number of face guards which cover the surface of a polyhedral terrain. We show that ⌊(2n-5)/7⌋ face guards are sometimes necessary to guard the surface of an n-vertex triangulated polyhedral terrain.

Hierarchies of DLOGTIME-uniform circuits

We present complexity hierarchies on circuits under two DLOGTIME-uniformity conditions. It is shown that there is a language which can be recognized by a family of

Computational Complexity in the Hyperbolic Plane

U_{\mbox{\tiny E}}

a-connectivity: a gradually nonparallel graph problem

-uniform circuits of depth