Klaus Sutner

Classifying circular cellular automata

We introduce a hierarchy of linear cellular automata based on their limiting behavior on spatially periodic configurations. We show that it is undecidable to which class in the hierarchy a cellular automaton belongs. In particular, it is undecidable whether all spatially periodic configurations evolve to a fixed point. Furthermore, there is no computable bound on the period lengths of these configurations. Our arguments are based on a non-standard simulation of Turing machines on circular cellular automata.

Linear Cellular Automata and the Garden-of-Eden

Suppose each of the squares of an n x n chessboard is equipped with an indicator light and a button. If the button of a square is pressed, the light of that square will change from off to on and vice versa; the same happens to the lights of all the edge-adjacent squares. Initially all lights are off. Now, consider the following question: is it possible to press a sequence of buttons in such a way that in the end all lights are on? We will refer to this problem as the All-Ones Problem. A moment s reflection will show that pressing a button twice has the same effect as not pressing it at all. Thus a solution to our problem can be described by a subset of all squares (namely a set of squares whose buttons when pressed in an arbitrary order will render all lights on) rather than a sequence. In fact a set X of squares is a solution to the All-Ones Problem if and only if for every square s the number of squares in X adjacent to or equal to s is odd. Consequently, we will call such a set an odd-parity cover. Trial and error in conjunction with a pad of graph paper will readily produce solutions for n ~ 4. A little more experimentation shows that an odd-parity cover shou ld one exist--is difficult to construct even for n = 5 o r 6 . The brute-force approach to the problem, namely exhaustive search over all subsets of {1 . . . . n} x {1 . . . . n}, presents 2 n2 candidates, and the search becomes infeasible for moderate values of n even with the help of a computer. A less brute-force method would be to try to solve the system terpreted as a matrix over GF(2)) and 1 is the vector with all components equal to 1. This method, which involves n 2 equations, again becomes unwieldy for small values of n. For a similar approach to a game related to the All-Ones Problem, see [3]. In any case, Figure 1 shows odd-parity covers for n = 4, 5, 8. Several questions come to mind. For which n does a solution to the All-Ones Problem exist? More generally, how many odd-parity covers are there for an n x n board? What happens if the adjacency condition is changed--say , to an octal array (where a cell in the center has eight neighbors)? Can one replace an n x n rectangular grid by some other arrangement of sites and still obtain a solution? To answer some of these questions, we first rephrase the problem in terms of cellular automata.

On the Computational Complexity of Finite Cellular Automata

We study the computational complexity of several problems with the evolution of configurations on finite cellular automata. In many cases, the problems turn out to be complete in their respective classes. For example, the problem of deciding whether a configuration has a predecessor is shown to be NLOG-complete for one-dimensional cellular automata. The problem is NP-complete for all dimensions higher than one. Similarly, the question whether a target configuration occurs in the orbit of a source configuration may be P-complete, NP-complete or PSPACE-complete, depending on the type of cellular automaton.

The σ-game and cellular automata

1. SummaryIn an article in this journal Don Pelletier discussed the mathematics involved in a little battery operated toy called Merlin (see [3], and also the “Addenda” in this Monthly, Dec. 1987, page 994). Several years ago Stephen Wolfram, in another article that appeared in the Monthly, analyzed a number of simple cellular automata and the fractal patterns generated by some of these automata (see [6]). In this article we point out the close connection between MERLIN-type games and a class of cellular automata related to the ones described by Wolfram. We introduce a game played on directed graphs and give a detailed analysis of the special case where the graph is a rectangular grid. Our analysis uses linear algebra as well as ideas from the theory of cellular automata.

Motion planning among time dependent obstacles

SummaryIn this paper we study the problem of motion planning in the presence of time dependent, i.e. moving, obstacles. More specifically, we will consider the problem: given a bodyB and a collection of moving obstacles inD-dimensional space decide whether there is a continuous, collision-free movement ofB from a given initial position to a target position subject to the condition thatB cannot move any faster than some fixed top-speedc. As a discrete, combinatorial model for the continuous, geometric motion planning problem we introduce time-dependent graphs. It is shown that a path existence problem in time-dependent graphs is PSPACE-complete. Using this result we will demonstrate that a version of the motion planning problem (where the obstacles are allowed to move periodically) is PSPACE-hard, even ifD=2, B is a square and the obstacles have only translational movement. ForD=1 it is shown that motion planning is NP-hard. Furthermore we introduce the notion of thec-hull of an obstacle: thec-hull is the collection of all positions in space-time at which a future collision with an obstacle cannot be avoided. In the low-dimensional situationD=1 andD=2 we develop polynomial-time algorithms for the computation of thec-hull as well as for the motion planning problem in the special case where the obstacles are polyhedral. In particular forD=1 there is aO(n lgn) time algorithm for the motion planning problem wheren is the number of edges of the obstacle.

The complexity of the residual node connectedness reliability problem

This paper considers a probabilistic network in which the edges are perfectly reliable but the nodes fail with some known probabilities. The network is in an operational state if the surviving nodes induce a connected graph. The residual node connectedness reliability

Computing residual connectedness reliability for restricted networks

R(G)

The Computational Complexity of Cellular Automata

of a network G is the probability that the graph induced by the surviving nodes is connected. This reliability measure is very different from the widely studied K-terminal network reliability measure. It is proven that the problem of computing the residual connectedness reliability is NP-hard by showing that the problem of counting the number of node induced connected subgraphs of a given graph is

The intractability of the reliable assignment problem in split networks

# {bf P}

The Ordertype of

-complete. The problem remains