# On the Computational Complexity of Finite Cellular Automata

@article{Sutner1995OnTC,
title={On the Computational Complexity of Finite Cellular Automata},
author={Klaus Sutner},
journal={J. Comput. Syst. Sci.},
year={1995},
volume={50},
pages={87}
}
• Klaus Sutner
• Published 1995
• Computer Science, Mathematics
• J. Comput. Syst. Sci.
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… Expand

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#### References

SHOWING 1-10 OF 17 REFERENCES
Computation theory of cellular automata
Self-organizing behaviour in cellular automata is discussed as a computational process. Formal language theory is used to extend dynamical systems theory descriptions of cellular automata. The setsExpand
The Constructibility of a Configuration in a Cellular automaton
• Takeo Yaku
• Computer Science, Mathematics
• J. Comput. Syst. Sci.
• 1973
A configuration is said to be with finite support if the states of all but fintely many cells in the array are quiescent and it is recursively unsolvable when d>=2, for a configuration c with finiteSupport in a d-dimensional cellular automaton. Expand
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 cellularExpand
Algebraic properties of cellular automata
• Mathematics
• 1984
Cellular automata are discrete dynamical systems, of simple construction but complex and varied behaviour. Algebraic techniques are used to give an extensive analysis of the global properties of aExpand
On the predictability of coupled automata: an allegory about chaos
• Mathematics, Computer Science
• Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science
• 1990
The authors show a sharp dichotomy between systems of identical automata with symmetric global control whose behavior is easy to predict and those whose behavior is hard to predict. The divisionExpand
De Bruijn Graphs and Linear Cellular Automata
Every recursive configuration that has a predecessor on a linear CA already has a recursive pr edecessor, and it is shown that it is in genera l impossible to comput e such a predecessor effect ively. Expand
Classifying the Computational Complexity of Problems
A branch of computational complexity theory is described which attempts to expose more structure within the decidable side of the boundary by placing upper bounds on the amounts of computational resources which are needed to solve the problem. Expand
Space-Bounded Reducibility among Combinatorial Problems
• N. Jones
• Computer Science, Mathematics
• J. Comput. Syst. Sci.
• 1975
Two versions of the polynomial time-reducibility of Cook and Karp are defined, by means of Turing machines and by bounded-quantifier formulas, and they are shown to be complete for nondeterministic (deterministic) log n space. Expand
The monotone and planar circuit value problems are log space complete for P
It is shown that Ladner's simulation of Turing mac]hines by boolean circuits seems to require an "adequate" set of gates, such as AND and NOT, but the same simulation is possible with monotone circuits using AND and OR gates only. Expand
Undecidability of CA Classification Schemes
• Computer Science, Mathematics
• Complex Syst.
• 1988
It is shown that it is undecidable to which class a given cellula r automaton belongs, even when choosing only between the two simp lest classes. Expand