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Life without Death:
some elementary interactions

This applet is a companion to the research article Life without Death is P-complete by David Griffeath and Cris Moore. We demonstrate six of the basic ladder interactions used to show that the LwoD rule is capable of emulating Boolean circuits. See our web version of the paper for background, or the downloadable version for details.

To watch the demos, simply select each of the six available images, load them, and start the dynamics. Edge conditions default to Free so that ladders stop gracefully at the edge of the screen. By changing to Wrap you can unleash various complex scenarios, some of which fill most of the lattice with lava eventually. In the process, parasitic shoots are sometimes observed moving along the edges of ladders at twice the speed of their hosts.

1. Birth
Simply hit Start to see the genesis of a ladder traveling to the right from a small seed. The pentomino on the right gets the recursive structure started, while the remaining cells of the seed block progress to the left.

2. Death
Choose and load the second image, Death. Observe how a ladder can be stopped by a single cell.

3. Turn 1
The third image shows one interaction with a small constellation of occupied cells which causes a ladder to turn 90 clockwise without giving rise to an uncontrollable flow of lava.

4. Turn 2
Another, more elegant 90 clockwise turn.

5. Blocks
Here are two spatio-temporal phases for collision between a ladder and the side of another which produce clean blocks. Other phases give rise to parasites and eruptions of lava.

6. Shift
A key step in our argument for circuit emulation is to show that ladders can always be shifted by a local interaction to ensure clean blocking as in the previous experiment. We accomplish this by combining Experiments 3 and 4 above, then applying a little group theoretic reasoning to the space-time ladder shifts.

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