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

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

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

4. Self-Replication

5. 1-bit Addition Processor CellLeft Data Cell Right Data Cell Result Computation Steiglitz, Squier, Jakubow (1994)

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

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

8. 3x3 Binary Multiplication Data cell + carry Steiglitz, Squier, Jakubow (1994)

9. 3x3 Binary Multiplication 1 100 Steiglitz, Squier, Jakubow (1994)

10. Collision Rules 01 00 00 01 1 1 1 0 1 01 1 Before After

11. Implementation on Self-Replicating Loops Addition of four 7-bit numbers (13+16+10+37+19)

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

13. 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