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Game of Life is an interesting Cellular Automata and I read at many places it has lambda of 0.273 roughly. Kindly help by answering how we actually calculate this value for game of life.

Zubair J.
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1 Answers1

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The calculation is very much implied by the definition: for the Game of Life, lambda is the fraction of states which leads to a non-quiescent state, ie: to the state 1.

As you said: there are three situations which cause a cell to be non-quiescent:

  • Alive cell with 2 alive neighbours

  • Alive cell with 3 alive neighbours

  • Dead cell with 3 alive neighbours

Or simplified:

  • Half of the states where we have two alive neighbours

  • All of the states with three alive neighbours

So if we sum the fractions of states the above two options represent, we have the total fraction of states leading to a non-quiescent state.

We can now use the calculations for binomial probabilities, where I'll write C(n,r) when I mean the binomial coefficient of N over R.

0.5 * C(8,2) * (0.5)^2 * (0.5)^6

C (8,3) * (0.5)^3 * (0.5)^5

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which more equals 0.219 + 0.055 ~= 0.273

IRBosman
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  • Game of Life is basically an Outer Totalistic Cellular Automata (OTCA) usually written as "B3/S23" (OTCA Rule 224). Therefore, a dead cell comes to life (non-quiescent state) if it has exactly 3 live cells in its moore neighbourhood (Nm) (8 cell neighbourhood). Similarly a live cell, with either 2 or 3 live cells in Nm survives to next generation. I still don't get what fraction will give me the lambda value for it. – Zubair J. May 21 '18 at 13:29
  • I'll update the original answer with the full explanation. – IRBosman May 22 '18 at 18:39