Evolution of Hybrid Cellular Automata for Density Classification Problem
Abstract
:1. Introduction
- Density conserving—the initial density, corresponding to the input configuration of the system, should be conserved over time.
- Balanced rule—the rule table should demonstrate a density of 0.5 and maintain uniformity across the configuration space.
- An efficient hybrid CA-based solution for DCP that can be used in parallel and distributed systems and applied to many real-world decentralized problems that require a decision to be taken by separating clusters with different values is proposed.
- The technical details and the arguments that illustrate the capability of hybrid CA to combine the two necessary features, density conserving and translation of the information stored in the cells across the lattice, in order to obtain the solution to the DCP are provided.
- The CA-based method for DCP developed here is a general one and is relatively simple and efficient for implementation, as it can be scaled to a larger or a smaller number of cells.
- Experimental results, showcased across different working scenarios, confirm and offer relevant insights into how to improve the exploration of the CA search space and also the ways of creating solutions in the CA context.
2. Materials and Methods
2.1. Fundamentals of the Problem Statement
- Von Neumann model, radius r = 1, where, for establishing the next state of a central cell, three neighboring cells are used in the one-dimensional CA case (Figure 1a) and five neighboring cells, formed by the central cell and its four adjoining horizontal and vertical neighbors, in the two-dimensional CA case (Figure 2a).
- Moore model, radius r = 1, in which three neighboring cells are used in the one-dimensional CA case and nine neighboring cells, formed by the central cell and its eight adjoining neighbors, including diagonals, in the two-dimensional CA case (Figure 2b).
- In the case of one-dimensional CA with radius r = 1:
- 2.
- In the case of one-dimensional CA with radius r = 2:
- 3.
- In the case of two-dimensional CA with radius r = 1 and von Neumann model:
- 4.
- In the case of two-dimensional CA with radius r = 1 and Moore model:
- 5.
- In the case of two-dimensional CA with radius r = 2 and Extended Moore model:
- C stands for cell;
- i and j represent the position of a single cell in a two-dimensional lattice of cells;
- t represents the time step;
- Ci(t+1) represents, in the one-dimensional CA, the output state of the central cell at the time step t+1;
- Ci,j(t+1) represents, in the two-dimensional CA, the output state of the central cell at the time step t+1;
- f represents the evolution function of the CA.
- Rule 30:
- 2.
- Rule 90:
- 3.
- Rule 204:
- 4.
- Rule 226:
2.2. The Hybrid CA-Based Algorithm for the Problem of Density Classification
- UseRules (Rule64_Rule1_Rule4)—can be used to move the 1′s values to the bottom-right direction in the matrix (see explanations in previous paragraph).
- UseRules (Rule256_Rule1_Rule16)—can be used to move the 1′s values to the bottom-left direction in the matrix.
- UseRules (Rule16_Rule1_Rule256)—can be used to move the 1′s values to the top-right direction in the matrix.
- UseRules (Rule128_Rule1_Rule8)—can be used to move the 1′s values to the bottom direction in the matrix.
- UseRules (Rule32_Rule1_Rule2)—can be used to move the 1′s values to the right direction in the matrix.
3. Implementation of the Proposed Hybrid CA Solution
- If (A=1) then use rule 1 and if (A=0) then use rule B
- If (the states of the three neighborhood cells in the translation direction of rule B is 011) then use rule 1. That means density conserving.
- Else if (the states of the three neighborhood cells in the translation direction of rule B is 110) then use rule C. This means that 0 s will be moved in the translation direction of rule C.
- Else (That means the states of the three neighborhood cells in the translation direction of rule B is 110) then use rule B. This means that 1 s will be moved in the translation direction of rule B.
4. Results
5. Discussion
6. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
References
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Rules 1 | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000 |
---|---|---|---|---|---|---|---|---|
30 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 |
90 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 |
204 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
226 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
Left Neighbors of the Central Cell | Top and Bottom Neighbors of the Central Cell | Right Neighbors of the Central Cell |
---|---|---|
64 Shifting of information towards the bottom right. | 128 Shifting of information towards the bottom. | 256 Shifting of information towards the bottom left. |
32 Shifting of information towards the right. | 1 1 The densities of 0 s and 1 s remain unchanged, yet no translation occurs. | 2 Shifting of information towards the left. |
16 Shifting of information towards the top right. | 8 Shifting of information towards the top. | 4 Shifting of information towards the top left. |
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Anghelescu, P. Evolution of Hybrid Cellular Automata for Density Classification Problem. Symmetry 2024, 16, 599. https://doi.org/10.3390/sym16050599
Anghelescu P. Evolution of Hybrid Cellular Automata for Density Classification Problem. Symmetry. 2024; 16(5):599. https://doi.org/10.3390/sym16050599
Chicago/Turabian StyleAnghelescu, Petre. 2024. "Evolution of Hybrid Cellular Automata for Density Classification Problem" Symmetry 16, no. 5: 599. https://doi.org/10.3390/sym16050599
APA StyleAnghelescu, P. (2024). Evolution of Hybrid Cellular Automata for Density Classification Problem. Symmetry, 16(5), 599. https://doi.org/10.3390/sym16050599