Why emulate nature?

2. Why emulate nature? • Rich dynamics • Capture essence of phenomena • Practical/ultimate algorithms • Physical models of computation • Semiclassical models of physics

3. Physical Worlds CA’s already have space, time, locality and finite-state in an exactly computable dynamics • Add reversibility and conservations • Add arbitrary complexity • Still missing? • Conclusions

4. Reversibility & other conservations • Reversibility is conservation of information • Why does exact conservation seem hard? The same information is visible at multiple positions For rev, 1/n-th of the neighbor info must be left at the center

5. Adding conservations • With traditional CA’s, conservations are a non-local property of the dynamics. • Simplest solution: redefine CA’s so that conservation is a manifestly local property • Regularity in space and time  CA • Regularity in time: repeated sequence of steps • Regularity in space: repeated structure

6. Diffusion rule Even steps:rotate cw or ccw a b c a b d or c d d b a c Odd steps:rotate cw or ccw a b c a b d or c d d b a c Use 2x2 blockings. Use solid blocks on even time steps, use dotted blocks on odd steps. We “randomly” choose to rotate blocks 90-degrees cw or ccw (we actually use a fixed sequence of choices for each spot).

7. Diffusion rule Even steps:rotate cw or ccw a b c a b d or c d d b a c Odd steps:rotate cw or ccw a b c a b d or c d d b a c We “randomly” choose to rotate blocks 90-degrees cw or ccw (we actually use a fixed sequence of choices for each spot).

8. Diffusion rule Even steps:rotate cw or ccw a b c a b d or c d d b a c Odd steps:rotate cw or ccw a b c a b d or c d d b a c We “randomly” choose to rotate blocks 90-degrees cw or ccw (we actually use a fixed sequence of choices for each spot).

9. a b c a c d d b TM Gas rule Even steps:rotate cw Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc Use 2x2 blockings. Use solid blocks on even time steps, use dotted blocks on odd steps.

10. a b c a c d d b TM Gas rule Even steps:rotate cw Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc

11. a b c a c d d b TM Gas rule Even steps:rotate cw Odd steps:rotate ccw a b b d c d a c Except:2 ones on diag, nc

12. TM Gas rule

13. TM Gas rule

14. Lattice gas hydrodynamics Six direction LGA flow past a half-cylinder, with vortex shedding. System is 2Kx1K.

15. Dynamical Ising rule Red/Blue checkerboard Even steps: update blue sublattice Odd steps: update red sublattice We divide the space into two sublattices, updating the blue on even steps, red on on odd. A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel.

16. Dynamical Ising rule Even steps: update blue sublattice Odd steps: update red sublattice A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel.

17. Dynamical Ising rule Even steps: update blue sublattice Odd steps: update red sublattice A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel.

18. Bennett’s 1D rule Red/Blue 1D lattice Even steps: update blue sublattice Odd steps: update red sublattice At each site in a 1D space, we put 2 bits of state. We’ll call one the “red” bit and one the “blue” bit. We update the blue bits on even steps, and the red bits on odd steps. A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel.

19. Bennett’s 1D rule Even steps: update blue sublattice Odd steps: update red sublattice A spin is flipped if exactly 2 of its 4 neighbors are parallel to it. After the flip, exactly 2 neighbors are still parallel.

20. 3D Ising with heat bath If the heat bath is initially much cooler than the spin system, then domains grow as the spins cool.

21. 2D “Same” rule Red/Blue checkerboard Even steps: update blue sublattice Odd steps: update red sublattice We divide the space into two sublattices, updating the blue on even steps, red on on odd. A spin is flipped if all 4 of its neighbors are the same. Otherwise it is left unchanged.

22. 2D “Same” rule Even steps: update blue sublattice Odd steps: update red sublattice A spin is flipped if all 4 of its neighbors are the same. Otherwise it is left unchanged.

23. 2D “Same” rule Even steps: update blue sublattice Odd steps: update red sublattice A spin is flipped if all 4 of its neighbors are the same. Otherwise it is left unchanged.

24. 3D “Same” rule

25. Reversible aggregation rule Red/Blue checkerboard Even steps: update blue sublattice g x xh x Odd steps: update red sublattice xh x g x We update the blue sublattice, then let gas and heat diffuse, then update red and diffuse. When a gas particle diffuses next to exactly one crystal particle, it crystallizes and emits a heat particle. The reverse also happens. for more info, see cond-mat/9810258

26. Reversible aggregation rule Even steps: update blue sublattice g x xh x Odd steps: update red sublattice xh x g x When a gas particle diffuses next to exactly one crystal particle, it crystallizes and emits a heat particle. The reverse also happens. for more info, see cond-mat/9810258

27. Conservations summary To make conservations manifest, we employ a sequence of steps and partitions in which: • Some data bits may be shifted uniformly. • Disjoint blocks of data may be updated independently. • We can partition data into overlapping blocks as long as shared data doesn’t change.

28. Arbitrary complexity If you can build logic elements and connect them, then you can construct computers ( arb complexity). This kind of universality is a basic property of physics. • It doesn’t take much. • Can construct rules that support logic. • Can discover logic in existing rules (eg. RA). A microprocessor simulation in a gate-array-like CA dynamics.

29. Famous example: Life • One can build signals, wires, and logic out of patterns of bits in the Life CA. • Is the evolution of complexity a robust property of Life? • Can we do better? Glider guns in Conway’s “Game of Life” CA. Streams of gliders can be used as signals in Life logic circuits.

30. Famous example: Life • One can build signals, wires, and logic out of patterns of bits in the Life CA. • Is the evolution of complexity a robust property of Life? • Can we do better? Life started from a random pattern of ones and zeros, running on a 2K x 2K space. Activity dies out quickly.

31. The “Critters” rule This rule is applied both to the even and the odd blockings. We show all cases: each rotation of a case on the left maps to the corresponding rotation of the case on the right. Note that the number of ones in one step equals the number of zeros in the next step. Use 2x2 blockings. Use solid blocks on even time steps, use dotted blocks on odd steps.

32. The “Critters” rule This rule is applied both to the even and the odd blockings. We show all cases: each rotation of a case on the left maps to the corresponding rotation of the case on the right. Note that the number of ones in one step equals the number of zeros in the next step.

33. Billiard Ball Logic

34. BBM CA rule 2x2 blockings. The solid blocks are used at even time steps, the dotted blocks at odd steps. BBMCA rule. Single one goes to opposite corner, 2 ones on diagonal go to other diag, no other cases change. A BBMCA collision:

35. “Critters” is universal Critters “glider” collision: A BBMCA collision:

36. Beyond “Critters” Many fundamental physical properties have not yet been added to CA’s, which may be essential for a robust Darwinian evolution. Notably, • Relativistic invariance would allow large-scale structures to move! • Requires us to reconcile forces and conservations with invertibility and universality.

37. Adding forces irreversibly becomes: Particles six sites apart along the lattice attract each other. 3D momentum conserving crystallization.

38. Adding forces irreversibly Crystallization using irreversible forces (Jeff Yepez, AFOSR)

39. Conclusions • Reversible systems last longer, and have a realistic thermodynamics. • Conservations lead to robust “gliders” and interesting macroscopic properties & symmetries. • Universality is a low threshold that separates triviality from arbitrary complexity. • More of the richness of physical dynamics can be captured by adding physical properties. for more info, see comp-gas/9811002