The Computational Beauty of Nature — Spring 2020

Problem Set 8: Cellular Automata

Due by class time Thursday, May 7

For these exercises, please write up your answers in the form of a PDF or Microsoft Word document.

Reading

• Review the class material from week 12 and week 13.
• Read Chapter 13 of Complexity: A Guided Tour for next week.

Written Exercises

1. Submit your completed table of Elementary CA rules from Part 1 of the Lab 10: Cellular Automata worksheet.

2. What is the Wolfram number in base 10 of Elementary CA rule 10111011? What Wolfram class is it in? What is its lambda (λ) value?

3. How many different Elementary CAs are there with a λ value of 0.125 (= 1/8)? Explain.

4. Design an Elementary CA that "inverts" the state of the Universe on each time step by changing all ON cells to OFF, and vice versa. Show the rule table for this CA, and give its Wolfram number. You should assume a circular lattice (i.e., the Universe edges "wrap around"). For example:

   

5. Design an Elementary CA that shifts the current Universe configuration one cell to the right on each time step. Show the rule table for this CA, and give its Wolfram number. You should assume a circular lattice. For example:

   

Extra Credit Problem (optional)

6. Recall that the rules for the Game of Life can be summarized as:

   • A live cell with fewer than 2 neighbors dies.
   • A live cell with more than 3 neighbors dies.
   • A live cell with 2 or 3 neighbors stays alive.
   • A dead cell with exactly 3 neighbors becomes alive.

   The first few rows of the rule table for the Game of Life are shown below, with neighborhoods in order from all cells OFF to all cells ON. Cell positions in a neighborhood are numbered from 1 to 9, and the ON/OFF state of each cell is indicated by a 1 or 0 at the corresponding cell position.

                      Cell#        New state
                    123456789      of cell #5
                    000000000   ->    0
                    000000001   ->    0
   Neighborhood     000000010   ->    0
      1 2 3         000000011   ->    0
      4 5 6         000000100   ->    0
      7 8 9         000000101   ->    0
                    000000110   ->    0
                    000000111   ->    1
                    000001000   ->    0
                       ...
                    111111111   ->    0
   

   1. How many rows are there in the rule table for the Game of Life? Explain.

   2. Calculate the value of λ for the Game of Life, and explain clearly how you arrived at your answer. Hint: the mathematical choose function may come in handy here, which counts the number of different ways that a subset of k elements can be chosen from a set of n elements: