Does not Compute 3: Awesomer Cellular Automata

1. Does not Compute 3:Awesomer Cellular Automata In which we consider how to upgrade our cellular automata A few key choices are considered, but eventually we decide there is no real choice but to buy the UtimateSuperdeluxeMegaAutomaton, which has a killer feature that puts them all to shame

2. After This Class… • You will be able to give an example of a cellular automata that really does some computation, and have an intuition for how it works • We will have considered the question of color and size, and know the definitive answer for which provides more computational power • The UtimateSuperdeluxeMegaAutomataarrives. But I won’t say more! • If we have time, we briefly mention the Game of Life, without which no discussion of cellular automata would be complete

3. You will be able to give an example of a cellular automata that really does some computation, and have an intuition for how it works: The Squaring Cellular Automaton

4. Questions • Questions…Part A - Colors • Your job is to prove that cellular automata with additional colors can be used to simulate cellular automata with greater amounts “looking”. • Figure out a way to make a 8-color cellular automata that looks only at 3 cells simulate a 2-color cellular automata that looks at 5 cells.

5. Hints • Realize that, because your cells have more states, you can represent the state of more than one cell of the 2-color automata within a single cell of your 8-color automata. • Also note that representing the state of 2 cells of a cellular automata makes things annoying because it’s not odd length. Go ahead and represented 3 cells.

6. Where We Are… • We used the Squaring Automaton as an example of a CA that really does some computation, and hopefully you have an intuition for how it works • We have considered the question of color and size, and know that (strangely enough) these things are just two sides of the same coin. This gives us hints of a new method of proving… NEXT The UtimateSuperdeluxeMegaAutomaton arrives.

7. The UtimateSuperdeluxeMegaAutomaton

8. What Does This Mean? • Cellular Automata can do a lot more than we might initially think • While we might imagine that computational power increases as new features do…at least in the case of cellular automata, there is a “plateau” after which more features do not gain you more power • One of the primary ways we will want to prove things about computational power is by getting machines to simulate each other. Proof by simulation!

9. The “Plateau” of CA Power More Computational Power ? More Colors

10. Conway’s Game of Life • For a space that is 'populated': Each cell with one or no neighbors dies, as if by loneliness. Each cell with four or more neighbors dies, as if by overpopulation. Each cell with two or three neighbors survives. • For a space that is 'empty' or 'unpopulated' Each cell with three neighbors becomes populated. • This (should you doubt it even for a second) is just a ‘ordinary’ 2D cellular automata

11. What Happened in Class • We used the Squaring Automaton as an example of a CA that really does computation • We have considered the question of color and size, and talked about the universal 1-D Cellular automata. What you ought to remember: • The method of proving by simulation, which we have used in two different ways (how?) • The surprising discovery of a computational plateau for 1-D cellular automata

12. What is to Come • A new machine! Very different from the old machine! • And with this machine we will actually be able to find problems that it really cannot compute the answers to PS…although we will not actually use it in class (at least at the moment) you owe it to yourself to check out the Game of Life. NetLogo has a simulation (just called “Life”). You can also download a simulator here: http://www.bitstorm.org/gameoflife/ Even if you don’t download a simulator yourself, the Life Lexicon will blow your mind: http://www.bitstorm.org/gameoflife/lexicon/