Skip to main content
MathOverflow

Return to Question

edited tags
Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286
added 31 characters in body
Source Link
Hans-Peter Stricker
Hans-Peter Stricker
  • 9.7k
  • 5
  • 53
  • 113

In Tables of cellular automata, p.542, Wolfram defines the density $\delta$ of a rule to be the asymptotic density of nonzero sites when the initial configuration has density $1/2$. Wolfram quotes rational numbers for densities "whenever analytical arguments yield exact results. In a few cases, the rigour of these arguments may be subject to question."

For rule 110 he quotes $\delta_{110} = 4/7$ (so seems to have an analytical argument).

For rule 54 he quotes $\delta_{54} = 0.49 \pm 0.01$ (so seems to have noneno argument for $\delta_{54} = 1/2$).

Note that rules 110 and 54 both belong to Wolfram's class IV.

My question is two-fold:

  1. I could not find Wolfram's analytical argument for $\delta_{110} = 4/7$. Can anyone give me a reference, please?

  2. The best rational approximation of $0.49$ with denominator $d \leq 32$ is $15/31 = (2^4 - 1)/(2^5 - 1)$. On the other hand we have $\delta_{110} = 4/7 = 2^2/(2^3 - 1)$. Might there be an analytical argument that yields $\delta_{54} = 15/31$? Or is the finding spurious and/or numerology?

In Tables of cellular automata, p.542, Wolfram defines the density $\delta$ of a rule to be the asymptotic density of nonzero sites when the initial configuration has density $1/2$. Wolfram quotes rational numbers for densities "whenever analytical arguments yield exact results. In a few cases, the rigour of these arguments may be subject to question."

For rule 110 he quotes $\delta_{110} = 4/7$ (so seems to have an analytical argument).

For rule 54 he quotes $\delta_{54} = 0.49 \pm 0.01$ (so seems to have none).

Note that rules 110 and 54 both belong to Wolfram's class IV.

My question is two-fold:

  1. I could not find Wolfram's analytical argument for $\delta_{110} = 4/7$. Can anyone give me a reference, please?

  2. The best rational approximation of $0.49$ with denominator $d \leq 32$ is $15/31 = (2^4 - 1)/(2^5 - 1)$. On the other hand we have $\delta_{110} = 4/7 = 2^2/(2^3 - 1)$. Might there be an analytical argument that yields $\delta_{54} = 15/31$? Or is the finding spurious and/or numerology?

In Tables of cellular automata, p.542, Wolfram defines the density $\delta$ of a rule to be the asymptotic density of nonzero sites when the initial configuration has density $1/2$. Wolfram quotes rational numbers for densities "whenever analytical arguments yield exact results. In a few cases, the rigour of these arguments may be subject to question."

For rule 110 he quotes $\delta_{110} = 4/7$ (so seems to have an analytical argument).

For rule 54 he quotes $\delta_{54} = 0.49 \pm 0.01$ (so seems to have no argument for $\delta_{54} = 1/2$).

Note that rules 110 and 54 both belong to Wolfram's class IV.

My question is two-fold:

  1. I could not find Wolfram's analytical argument for $\delta_{110} = 4/7$. Can anyone give me a reference, please?

  2. The best rational approximation of $0.49$ with denominator $d \leq 32$ is $15/31 = (2^4 - 1)/(2^5 - 1)$. On the other hand we have $\delta_{110} = 4/7 = 2^2/(2^3 - 1)$. Might there be an analytical argument that yields $\delta_{54} = 15/31$? Or is the finding spurious and/or numerology?

Source Link
Hans-Peter Stricker
Hans-Peter Stricker
  • 9.7k
  • 5
  • 53
  • 113

Asymptotic densities of rules of elementary cellular automata

In Tables of cellular automata, p.542, Wolfram defines the density $\delta$ of a rule to be the asymptotic density of nonzero sites when the initial configuration has density $1/2$. Wolfram quotes rational numbers for densities "whenever analytical arguments yield exact results. In a few cases, the rigour of these arguments may be subject to question."

For rule 110 he quotes $\delta_{110} = 4/7$ (so seems to have an analytical argument).

For rule 54 he quotes $\delta_{54} = 0.49 \pm 0.01$ (so seems to have none).

Note that rules 110 and 54 both belong to Wolfram's class IV.

My question is two-fold:

  1. I could not find Wolfram's analytical argument for $\delta_{110} = 4/7$. Can anyone give me a reference, please?

  2. The best rational approximation of $0.49$ with denominator $d \leq 32$ is $15/31 = (2^4 - 1)/(2^5 - 1)$. On the other hand we have $\delta_{110} = 4/7 = 2^2/(2^3 - 1)$. Might there be an analytical argument that yields $\delta_{54} = 15/31$? Or is the finding spurious and/or numerology?