# 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} }

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…

## 77 Citations

### Computational classification of cellular automata

- Computer ScienceInt. J. Gen. Syst.
- 2012

It is seen that a large variety of properties relating to the short-term evolution of configurations are decidable in principle, but questions related to the long-term Evolution are typically undecidable.

### Computing aggregate properties of preimages for 2D cellular automata.

- Computer ScienceChaos
- 2017

The incremental aggregation algorithm is demonstrated on two problems from the two-dimensional binary Game of Life cellular automaton: precursor count distributions and higher-order mean field theory coefficients.

### A brief history of cellular automata

- Computer ScienceCSUR
- 2000

A history of cellular automata from their beginnings with von Neumann to the present day is traced, mainly on topics closer to computer science and mathematics rather than physics, biology or other applications.

### Computational complexity of finite asynchronous cellular automata

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### On the Complexity of the Stability Problem of Binary Freezing Totalistic Cellular Automata

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### Constrained Eden

- Computer Science, MathematicsComplex Syst.
- 2009

C-EDEN(CA) is a finitary variant of the Garden of Eden problem, providing the first examples of NP-complete problems associated with cellular automata over a twoelement alphabet.

### Constrained Eden

- Computer Science, Mathematics
- 2009

C-EDEN(CA) is a finitary variant of the Garden of Eden problem, providing the first examples of NP-complete problems associated with cellular automata over a twoelement alphabet.

### SAT-Based Analysis of Cellular Automata

- Computer ScienceACRI
- 2004

An encoding of the evolution of Cellular Automata into formulas in propositional logic is defined that can be used to test forward and inverse reachability problems for the original cellular Automata in a modular way.

### On the complexity of enumerating possible dynamics of sparsely connected Boolean network automata with simple update rules

- Computer ScienceAutomata
- 2010

It is proved that counting the Fixed Point (FP) configurations and the predecessor and ancestor configurations in two classes of network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems.

## References

SHOWING 1-10 OF 17 REFERENCES

### Computation theory of cellular automata

- Computer Science
- 1984

The sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages and it is suggested that such undecidability is common in these and other dynamical systems.

### The Constructibility of a Configuration in a Cellular automaton

- Computer ScienceJ. Comput. Syst. Sci.
- 1973

### Algebraic properties of cellular automata

- Computer Science, Mathematics
- 1984

Algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata, and the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities.

### On the predictability of coupled automata: an allegory about chaos

- MathematicsProceedings [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 division…

### De Bruijn Graphs and Linear Cellular Automata

- Computer ScienceComplex Syst.
- 1991

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.

### Classifying the Computational Complexity of Problems

- Computer ScienceJ. Symb. Log.
- 1987

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.

### The monotone and planar circuit value problems are log space complete for P

- MathematicsSIGA
- 1977

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.

### Undecidability of CA Classification Schemes

- Computer ScienceComplex 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.

### Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures

- MathematicsJ. Comput. Syst. Sci.
- 1972