Arithmetic Operations on Self-Replicating Cellular Automata Enrico Petraglio Logic Systems Laboratory

Summary  Motivations  Introduction to the CA  Theoretical Notions  Binary Addition  Binary Multiplication  Implementation on the CA  Conclusions

Motivations  Von Neumann’s Universal Constructor:  Universal Construction  Universal Computation Is useful computation with self-replicating automata necessarily associated with high complexity?

Self-Replication

1-bit Addition Processor CellLeft Data Cell Right Data Cell Result Computation Steiglitz, Squier, Jakubow (1994)

N-bit Binary Addition Processor Cell Left AddendRight Addend Steiglitz, Squier, Jakubow (1994)

Left MultiplicandRight Multiplicand First Collision 3x3 Binary Multiplication Steiglitz, Squier, Jakubow (1994)

3x3 Binary Multiplication Data cell + carry Steiglitz, Squier, Jakubow (1994)

3x3 Binary Multiplication Steiglitz, Squier, Jakubow (1994)

Collision Rules Before After

Implementation on Self-Replicating Loops Addition of four 7-bit numbers ( )

Addition + Multiplication (1+1+4) x (2+2 +3)

Conclusions  Too complex!!!  Self-replication not really necessary  Useful computation with self-replicating automata IS feasible  This approach will be useful for the development of self-replicating hardware