Making Rule 90 Injective

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In summary, Wolfram Rule 90 is a type of reversible cellular automata where each cell's value is computed using the XOR operation with its two neighbors. However, a small modification of this rule, specifically for the cells on the edges, can make it easier to determine the predecessor configuration but also changes the behavior and patterns that emerge from the rule. This highlights the importance of understanding and following the original rules when studying and experimenting with cellular automata.
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gendou2
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http://en.wikipedia.org/wiki/Rule_90

Wolfram Rule 90 is a type of cellular automata.
Each cell's value is computed as the XOR of its two neighbors in the predecessors generation.
Rule 90 cannot be reversed, because a given configuration has 4 possible predecessor configurations.

However, I've found that a small modification will cause the rule set to become reversible!

This modification applies to the cells on left and right edges (assuming your matrix is a finite, and so has edges).
It makes no sense to perform an XOR operation, since cells on the edge only have one neighbor.
Instead, these cells can inherit the value of their lone neighbor, without modification.
This way, 2 cells (on the edges) are always known, and we find one unique predecessor.

This may seem obvious to some people, but I thought the discovery worth sharing. :)
 
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Hello there,

Thank you for sharing your findings about modifying Wolfram Rule 90 to make it reversible. It is always exciting to see new developments in cellular automata and I appreciate your curiosity and experimentation.

I agree with your observation that the modification you propose does make the rule set reversible. However, I would like to point out that this modified version of Rule 90 is no longer following the original rule set proposed by Stephen Wolfram. In the original rule, all cells, including those on the edges, are supposed to follow the XOR operation with their two neighbors.

While your modification does make it easier to determine the predecessor configuration for a given state, it also changes the behavior and patterns that emerge from the rule. This is because the edges now behave differently from the rest of the cells, which can lead to different outcomes and patterns in the automata.

I believe that this is a great example of how small modifications can have a significant impact on the behavior of cellular automata. It also highlights the importance of understanding and following the original rules when studying and experimenting with these systems.

Thank you again for sharing your findings and sparking a discussion on the reversibility of Rule 90. I hope to see more of your explorations in the future.

 

1. What is Rule 90?

Rule 90 is a cellular automaton rule that determines the behavior of a one-dimensional array of cells based on the values of its neighboring cells. It is often used as a model for complex systems and has been studied in the field of computer science and mathematics.

2. What does it mean to make Rule 90 injective?

To make Rule 90 injective means to modify the rule in a way that each input produces a unique output. In other words, there are no two different inputs that result in the same output when using the modified version of Rule 90.

3. Why is making Rule 90 injective important?

Making Rule 90 injective is important because it allows for the creation of a one-to-one mapping between the inputs and outputs, which is useful in various fields such as cryptography and data compression. It also helps in understanding the behavior of complex systems and can lead to new insights and discoveries.

4. How can Rule 90 be made injective?

There are several ways to make Rule 90 injective, but one common method is to add a random number generator to the rule. This random element ensures that each input produces a unique output, making the rule injective. Other methods involve changing the rule's conditions or adding additional rules.

5. What are the limitations of making Rule 90 injective?

While making Rule 90 injective has many benefits, it also has some limitations. One of the main limitations is that it can be computationally expensive, especially for larger arrays of cells. Additionally, creating a truly injective version of Rule 90 may not be possible in some cases, leading to approximations or compromises in the modified rule.

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