Linear Algebra of Cellular Automata

  Cellular automata, being discrete computational models, can be integrated with linear algebra to create equations that describe their behavior and evolution. Here are five equations that merge cellular automata with linear algebra:

1. Cellular Automata State Transition Equation:

$\mathrm{\mid }{�}_{\text{new}}⟩=�\mathrm{\mid }{�}_{\text{old}}⟩$

In this equation, $\mathrm{\mid }{�}_{\text{old}}⟩$ and $\mathrm{\mid }{�}_{\text{new}}⟩$ represent the state vectors of a cellular automaton before and after a computation step, respectively. $�$ is a matrix operator representing the local rules of the cellular automaton. This equation demonstrates the evolution of cellular automaton states using a linear transformation.

2. Cellular Automata Rule Matrix Equation:

In this equation, $�$ represents the matrix of transition rules for a cellular automaton with $�$ states. Each element ${�}_{��}$ in the matrix indicates how the state $�$ transforms into state $�$. This matrix illustrates the discrete nature of cellular automata rules using linear algebraic representation.

3. Cellular Automata Neighborhood Sum Equation:

$\mathrm{\mid }{�}_{\text{sum}}⟩=�\mathrm{\mid }{�}_{\text{old}}⟩$

Here, $\mathrm{\mid }{�}_{\text{sum}}⟩$ represents the vector obtained by summing the neighboring states of each cell in the cellular automaton configuration represented by $\mathrm{\mid }{�}_{\text{old}}⟩$. $�$ is a matrix operator that defines the neighborhood structure. This equation showcases the neighborhood summation process within cellular automata using linear algebra.

4. Cellular Automata Evolution Matrix Equation:

$�={�}_1\otimes {�}_2\otimes \cdots \otimes {�}_{�}$

In this equation, $�$ represents the evolution matrix of a multi-dimensional cellular automaton. ${�}_{�}$ are the transition matrices for each dimension, and $\otimes$ denotes the Kronecker product. This equation demonstrates the construction of the higher-dimensional evolution matrix using linear algebra, emphasizing the discrete and parallel nature of cellular automata.

5. Cellular Automata Density Matrix Equation:

$�=\frac{1}{�}{\sum }_{�=1}^{�}\mathrm{\mid }{�}_{�}⟩⟨{�}_{�}\mathrm{\mid }$

In this equation, $\mathrm{\mid }{�}_{�}⟩$ represents the state vector of the $�$th configuration of a cellular automaton, and $�$ is the density matrix representing the ensemble average over $�$ configurations. This equation illustrates the computation of the density matrix, emphasizing the discrete probabilistic nature of cellular automata using linear algebra.

These equations demonstrate the integration of cellular automata with linear algebra, capturing the discrete and computational essence of cellular automata models.

6. Cellular Automata Convolution Equation:

$\mathrm{\mid }{�}_{\text{new}}⟩=�\ast \mathrm{\mid }{�}_{\text{old}}⟩$

In this equation, $\mathrm{\mid }{�}_{\text{old}}⟩$ and $\mathrm{\mid }{�}_{\text{new}}⟩$ represent the state vectors of the cellular automaton before and after a convolution operation, respectively. $�$ represents the convolution kernel, which is a matrix defining the weights of neighboring cells. This equation demonstrates the convolutional nature of cellular automata using linear algebraic operations.

7. Cellular Automata Transition Probability Equation:

$�({�}_{�}\to {�}_{�})={�}_{��}$

Here, $�({�}_{�}\to {�}_{�})$ represents the transition probability from state ${�}_{�}$ to state ${�}_{�}$ in the cellular automaton. ${�}_{��}$ is an element of the transition matrix $�$, indicating the probability of transition. This equation illustrates the discrete probabilistic transitions within a cellular automaton represented using linear algebraic elements.

8. Cellular Automata State Projection Equation:

$\mathrm{\mid }{�}_{\text{proj}}⟩=�\mathrm{\mid }{�}_{\text{old}}⟩$

In this equation, $\mathrm{\mid }{�}_{\text{proj}}⟩$ represents the projected state vector obtained by applying a projection matrix $�$ to the current state $\mathrm{\mid }{�}_{\text{old}}⟩$. Projection matrices can represent specific cellular automata rules or filtering operations, highlighting their discrete nature using linear algebra.

9. Cellular Automata Eigenstate Equation:

$�\mathrm{\mid }�⟩=�\mathrm{\mid }�⟩$

In this equation, $�$ represents the transition matrix of the cellular automaton, $\mathrm{\mid }�⟩$ represents an eigenstate, and $�$ represents the corresponding eigenvalue. This equation showcases how specific states ($�$) evolve deterministically under the action of the cellular automaton's transition matrix, demonstrating the discrete, computational nature of eigenstates within cellular automata.

10. Cellular Automata Probabilistic Rule Equation:

$�({�}_{�}\to {�}_{�})=\frac{{�}_{��}}{{\sum }_{�=1}^{�}{�}_{��}}$

Here, $�({�}_{�}\to {�}_{�})$ represents the probabilistic transition probability from state ${�}_{�}$ to state ${�}_{�}$, considering the probabilities defined by the transition matrix $�$. This equation emphasizes the normalization of transition probabilities, ensuring that the probabilities sum to 1 for each cell, demonstrating the discrete and probabilistic nature of cellular automata using linear algebra.

These equations further illustrate the integration of cellular automata with linear algebra, capturing the discrete, probabilistic, and computational aspects of cellular automaton behavior.

11. Cellular Automata Transition Dynamics Equation:

$\mathrm{\mid }�(�+1)⟩=�(�)\mathrm{\mid }�(�)⟩$

In this equation, $\mathrm{\mid }�(�)⟩$ represents the state vector of the cellular automaton at time $�$, and $�(�)$ represents a time-varying transition matrix. This equation demonstrates the dynamic evolution of cellular automata states, capturing changes in behavior over time through linear algebraic transformations.

12. Cellular Automata Global State Equation:

$\mathrm{\mid }{�}_{\text{global}}⟩={\sum }_{�=1}^{�}\mathrm{\mid }{�}_{�}⟩$

Here, $\mathrm{\mid }{�}_{\text{global}}⟩$ represents the global state vector obtained by summing up the individual state vectors $\mathrm{\mid }{�}_{�}⟩$ of $�$ distinct regions or cells within the cellular automaton. This equation illustrates the composition of a global state from local cellular automata states, emphasizing the discrete nature of aggregation through linear algebra.

13. Cellular Automata Error Correction Equation:

$\mathrm{\mid }{�}_{\text{corrected}}⟩=(�-�)\mathrm{\mid }{�}_{\text{received}}⟩$

In this equation, $\mathrm{\mid }{�}_{\text{corrected}}⟩$ represents the corrected state vector, $\mathrm{\mid }{�}_{\text{received}}⟩$ represents the received state vector, $�$ represents the error matrix indicating error positions, and $�$ is the identity matrix. This equation demonstrates error correction in cellular automata through linear algebraic operations, highlighting the discrete nature of error recovery processes.

14. Cellular Automata Quantum Superposition Equation:

$\mathrm{\mid }{�}_{\text{superposition}}⟩=\frac{1}{\sqrt{�}}{\sum }_{�=1}^{�}\mathrm{\mid }{�}_{�}⟩$

In this equation, $\mathrm{\mid }{�}_{\text{superposition}}⟩$ represents the quantum superposition of $�$ distinct cellular automata states. This equation illustrates how quantum principles, such as superposition, can be represented within cellular automata using linear algebra, emphasizing the discrete and computational aspects of quantum-like behavior.

15. Cellular Automata Entropy Calculation Equation:

$�=-{\sum }_{�=1}^{�}{�}_{�}\log ({�}_{�})$

Here, $�$ represents the entropy of the cellular automaton's state distribution, and ${�}_{�}$ represents the probability of state $�$ in the distribution. This equation showcases the calculation of entropy, a measure of uncertainty, within cellular automata states, emphasizing the discrete probabilistic nature of cellular automata through linear algebraic expressions.

These equations further explore the integration of cellular automata with linear algebra, capturing diverse aspects of their computational dynamics, error correction, quantum-like behavior, and probabilistic nature.