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# No Perfect Two-State Cellular Automata for Density Classification Exists

Physical Review Letters, no. 25 (1995): 5148-5150

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Abstract

Recently there have been many attempts to evolve one-dimensional two-state cellular automata which classify binary strings according to their densities of l's and 0's. The current best-known approaches involve particle-based systems of information transfer. A proof is given that there does not exist a two-state cellular automata which per...More

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Introduction

- Cellular automata (CA) have proven to be an extremely austere yet powerful class of algorithmic specifications.
- No Perfect Two-State Cellular Automata for Density Classification Exists
- There have been many attempts to evolve one-dimensional two-state cellular automata which classify binary strings according to their densities of l's and 0's.
- A proof is given that there does not exist a two-state cellular automata which performs the task perfectly.

Highlights

- Since their inception, cellular automata (CA) have proven to be an extremely austere yet powerful class of algorithmic specifications
- Beyond simple understanding the limits of CA computation, a central motivation for this work concerns the potential of CA as models of biological phenomena such as ontogenesis
- A defining property of CA is the way in which their computations depend upon and affect an arbitrarily large global state field, despite the fact that all of their operations are defined over a finite, local neighborhood
- Given an arbitrary initial configuration of a one-dimensional two-state CA, the CA should converge to a state of all 1's if the initial configuration contains a density of 1's ~p, and to all 0's otherwise, for some p between 0 and 1
- Experiments of our own evolving CA for the density classification task using a different representation have resulted in an automaton that may be in some ways superior to, but is roughly comparable with, that found by Mitchell
- Our solution was correct 97.8% of the time on the uniform density distribution test described above, comparable to GKL
- ) that for a one-dimensional lattice of fixed size n, and for a fixed r 1, there exists no two-state CA rule defined over radius r which correctly classifies all possible initial configurations

Results

- A canonical example of such an emergent computation is to use a locally specified CA to determine the global density of bits in an initial state configuration.
- The best known solution to this problem for p = 1/2 is the Gaks-Kurdyumov-Levin (GKL) rule [5].
- Their best solution was correct about 95% of the time, compared to 97.8% for GKL, on lattices of size 149.
- ) that for a one-dimensional lattice of fixed size n, and for a fixed r 1, there exists no two-state CA rule defined over radius r which correctly classifies all possible initial configurations.
- P)], there does not exist a two-state CA rule of radius r which correctly classifies every configuration of size n for density p.
- First the authors will assume a perfect CA rule exists, and the authors will use lemmas to consider a sequence of perverse initial configurations which cannot all be handled correctly by it.
- Proof: Consider the special case of configuration o.
- Consider the complementary case 0' 1~, where j' = [pnJ + 1; there is exactly one more 1 and one less 0 than in the above configuration, and the density is greater than p.
- Applying R* to 0" '1 for a single cycle results in a string which has either zero 1's or more than one 1.
- The only other possibility is that 0" '1 is taken to a string with multiple 1's in a single cycle.

Conclusion

- In any configuration in which there is the substring 0 '10 ', the middle 2r + 1 cells will contain more than one 1 at the cycle.
- The best known solutions seem to correctly classify only about 80% of all possible initial configurations, with performance worsening for larger lattices.
- Informal experiments suggest that the GKL rule misclassifies only configurations whose densities are within some e of 0.5, with e decreasing as lattice size increases.
- Is there a number k, independent of the width of the lattice, such that some CA with k states can solve the problem for all cases on arbitrarily large lattices?

Summary

- Cellular automata (CA) have proven to be an extremely austere yet powerful class of algorithmic specifications.
- No Perfect Two-State Cellular Automata for Density Classification Exists
- There have been many attempts to evolve one-dimensional two-state cellular automata which classify binary strings according to their densities of l's and 0's.
- A proof is given that there does not exist a two-state cellular automata which performs the task perfectly.
- A canonical example of such an emergent computation is to use a locally specified CA to determine the global density of bits in an initial state configuration.
- The best known solution to this problem for p = 1/2 is the Gaks-Kurdyumov-Levin (GKL) rule [5].
- Their best solution was correct about 95% of the time, compared to 97.8% for GKL, on lattices of size 149.
- ) that for a one-dimensional lattice of fixed size n, and for a fixed r 1, there exists no two-state CA rule defined over radius r which correctly classifies all possible initial configurations.
- P)], there does not exist a two-state CA rule of radius r which correctly classifies every configuration of size n for density p.
- First the authors will assume a perfect CA rule exists, and the authors will use lemmas to consider a sequence of perverse initial configurations which cannot all be handled correctly by it.
- Proof: Consider the special case of configuration o.
- Consider the complementary case 0' 1~, where j' = [pnJ + 1; there is exactly one more 1 and one less 0 than in the above configuration, and the density is greater than p.
- Applying R* to 0" '1 for a single cycle results in a string which has either zero 1's or more than one 1.
- The only other possibility is that 0" '1 is taken to a string with multiple 1's in a single cycle.
- In any configuration in which there is the substring 0 '10 ', the middle 2r + 1 cells will contain more than one 1 at the cycle.
- The best known solutions seem to correctly classify only about 80% of all possible initial configurations, with performance worsening for larger lattices.
- Informal experiments suggest that the GKL rule misclassifies only configurations whose densities are within some e of 0.5, with e decreasing as lattice size increases.
- Is there a number k, independent of the width of the lattice, such that some CA with k states can solve the problem for all cases on arbitrarily large lattices?

Funding

- Solutions were tested by picking random initial configurations with a uniform distribution over densities. This eliminated the problem of the combinatorial explosion of cases with density close to 1/2. With this distribution, their best solution was correct about 95% of the time, compared to 97.8% for GKL, on lattices of size 149 (more recent work by Mitchell has achieved better results, but still no better than GKL [7])
- Our solution was correct 97.8% of the time on the uniform density distribution test described above, comparable to GKL
- The best known solutions seem to correctly classify only about 80% of all possible initial configurations (this is in contrast to the uniform den sity distribution test described earlier), with performance worsening for larger lattices

Study subjects and analysis

guarantees: 3

i to nO' "Pl' 2" y on the next cycle. Proof: Lemma 3 guarantees that cells r to i. —1 will be 0's and cells i + r to n —r —1 will be 1's

Reference

- 19 JUNE 1995
- 19 JUNE 1995 ior of R* on "solid" blocks of 1's or 0's, then demand that the perfect rule never causes the lattice density to cross the p threshold. We then focus on configurations of the general form 0'1~, where, of course, the most important
- [1] C. Langton, Physica (Amsterdam) 42D, 12—37 (1990).
- [3] Theory and Applications of Cellular Automata, edited by S. Wolfram (World Scientific, Singapore, 1986).
- P. Gaks, G. L. Kurdyumov, and L. A. Levin, Probl. Peredachi. Inform., 14, 92 —98 (1978).
- M. Mitchell, P. T. Hraber, and J. P. Crutchfield, Complex Systems 7, 89 —130 (1993).

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