Problem Set #5

  1. Look again at the cellular automaton introduction from the slides on Module 10.2 (slides 11 - 14). These slides show two different cellular automaton (CA) rules for CA where each cell has two neighbors. How many different possible rules are there for a two-neighbor CA? To answer this question, consider that each rule is defined by the second row in the little two-row table, the top row being merely a label for our convenience (as in Excel).

  2. Now consider a CA where each cell has four neighbors (e.g., a simple two-dimensional CA with N, S, E, and W neighbors, where each cell can only take on a single value). How many rules possible rules are there for such a CA?

  3. In general, what is the formula for the total number of rules in a CA where each cell has n neighbors?

  4. How many neighbors can there be before the number of possible rules exceeds the total number of atoms in the universe? Google can help with both the latter quantity and the conversion from powers of two to powers of ten (or you can just use the formula that 23 &cong 10).

  5. Look again at the Matlab automaton introduction from the slides on Module 11.2 (slides 17 - 20). Write a one-line Matlab statement to set the value of every cell in the grid to 2 if both its east and west neighbors or both its north and south neighbors are on fire. You can assume that the variables efire etc. are already defined.