# Characterizing Configuration Spaces of Simple Threshold Cellular Automata

@inproceedings{Tosic2004CharacterizingCS, title={Characterizing Configuration Spaces of Simple Threshold Cellular Automata}, author={Predrag T. Tosic and Gul A. Agha}, booktitle={ACRI}, year={2004} }

We study herewith the simple threshold cellular automata (CA), as perhaps the simplest broad class of CA with non-additive (that is, non-linear and non-affine) local update rules. We characterize all possible computations of the most interesting rule for such CA, namely, the Majority (MAJ) rule, both in the classical, parallel CA case, and in case of the corresponding sequential CA where the nodes update sequentially, one at a time. We compare and contrast the configuration spaces of arbitrary…

## 22 Citations

### PARALLEL vs . SEQUENTIAL THRESHOLD CELLULAR AUTOMATA : Comparison and Contrast

- Computer Science
- 2005

It is shown that there are 1D CA with very simple node update rules that cannot be simulated by any comparable SCA, irrespective of the node update ordering, and the granularity of the basic CA operations turns out to be not fine enough.

### On Convergence Properties of One-Dimensional Cellular Automata with Majority Cell Update Rule

- Computer Science
- 2011

This paper focuses on the convergence properties of a very simple kind of totalistic CA, namely, those defined on one-dimensional arrays where each cell or node updates according to the Boolean Majority function: the new state of a cell becomes 1 if and only if a simple majority of its inputs are currently in state 1, and it becomes 0 otherwise.

### Cellular Automata Communication Models: Comparative Analysis of Parallel, Sequential and Asynchronous CA with Simple Threshold Update Rules

- Computer ScienceInt. J. Nat. Comput. Res.
- 2010

The author argues that the classical CA model must be modified in several important respects to become a relevant model for large-scale MAS, and proposes genuinely asynchronous CA and discusses main differences between genuinely asynchronousCA and various weakly asynchronous sequential CA models found in the literature.

### 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.

### Phase Transitions in Possible Dynamics of Cellular and Graph Automata Models of Sparsely Interconnected Multi-Agent Systems

- Computer ScienceAAMAS
- 2017

This paper shows the combinatorics behind determining the total number of fixed point configurations for simple threshold CA and discusses a stark contrast with respect to intractability of counting for the related classes of Boolean graph automata with the same restrictions on the node update rules.

### Counting Fixed Points and Gardens of Eden of Sequential Dynamical Systems on Planar Bipartite Graphs

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2005

It is proved that the problems of exactly counting fixed points, gardens of Eden and two other types of S(y)DS configurations are all #P-complete, even if the SDSs and SyDSs are defined over planar bipartite graphs, and each of their nodes updates its state according to a monotone update rule given as a Boolean formula.

### Computational Complexity of Some Enumeration Problems About Uniformly Sparse Boolean Network Automata

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2006

It is proved that exactly enumerating FPs in such SDSs and SyDSs remains #P-complete even when no node degree exceeds = 3, which implies intractability of determining the exact memory capacity of discrete Hopfield networks with uniformly sparse and nonnegative integer weight matrices.

### On Complexity of Counting Fixed Point Configurations in Certain Classes of Graph Automata

- Mathematics, Computer Science
- 2005

It is proved that counting FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) is computationally intractable, even when each node is required to update according to a symmetric Boolean function.

### On Complexity of Counting Fixed Point Configurations in Certain Classes of Graph Automata

- Mathematics, Computer Science
- 2005

It is proved that counting FPs in Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively) is computationally intractable, even when each node is required to update according to a symmetric Boolean function.

### Cellular automata for distributed computing: models of agent interaction and their implications

- Computer Science2005 IEEE International Conference on Systems, Man and Cybernetics
- 2005

It is shown that the possibility of "looping" in parallel threshold CA is solely due to the unrealistic assumption of perfect inter-agent communication synchrony, and it is argued that thus generalized CA can capture many important MAS properties at the level of agent ensembles and their long-term global behavior patterns.

## References

SHOWING 1-10 OF 29 REFERENCES

### Gardens of Eden and Fixed Points in Sequential Dynamical Systems

- MathematicsDM-CCG
- 2001

This work addresses some questions concerning two special types of the SDS configurations, namely Garden of Eden and Fixed Point configurations and identifies several classes of SDSs for which the FPE problem can be solved efficiently.

### Computation theory of cellular automata

- Computer Science
- 1998

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.

### 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.

### Elements of a theory of computer simulation I: Sequential CA over random graphs

- Mathematics, Computer ScienceAppl. Math. Comput.
- 1999

### Reachability problems for sequential dynamical systems with threshold functions

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2003

### Cellular automata and complex systems

- Computer Science
- 1999

This work presents a uniform simulation of Turing Machines by Cellular Automata of Natural Processes using Interfaces S. Lyapunov Spectra in Spatially Extended Systems and shows good consistency in the properties of these systems.

### A new kind of science

- Computer Science
- 2002

A New Kind of Science, written and published by Stephen Wolfram, is the outcome of the studies he conducted systematically upon cellular automata, a class of computer model which may be visualized as a set of memory locations, each containing one bit.