1. H. F. Restrepo – Logic Systems Laboratory A Self-Replicating Universal Turing Machine

2. H. F. Restrepo – Logic Systems Laboratory A specialized Turing machine (1) Binary counter: A machine which writes out the binary numbers 1, 10, 11, 100, 101, etc. 00010 DATA State table 1 (ST1) FSM TAPE …… 

3. H. F. Restrepo – Logic Systems Laboratory A specialized Turing machine (2) 00010 DATA State table 1 (ST1) FSM TAPE ……  STATE TABLE 1 (ST1) Q = Q +, S + S=0 S=1 Q = 11 00 0 ,0 0 ,1 1 ,1 1 ,0

4. H. F. Restrepo – Logic Systems Laboratory Universal Turing machine (UTM) … SQS+S+ PROGRAM  STATE TABLE 1 Q+Q+ 11 1 11 0 11 01 00 SQS+S+ Q+Q+ Q = Q +, S + S=0 S=1 Q = 11 00 0 ,0 0 ,1 1 ,1 1 ,0 00010 DATA State table 2 (ST2) FSM TAPE …… 

5. H. F. Restrepo – Logic Systems Laboratory UTM: From state table to PICOPASCAL program STATE TABLE 1 Q = Q +, S + S=0 S=1 Q = 11 00 0 ,0 0 ,1 1 ,1 1 ,0 PROGRAM ADDR DATA PROGRAM 005 if (Q) 015 if (S) 02A do 0 (S) 039 do 1<- (Q) 044 else 05B do 1 (S) 068 do 0-> (Q) 076 endif 084 else 095 if (S) 0AB do 1 (S) 0B9 do 1<- (Q) 0C4 else 0DA do 0 (S) 0E8 do 0-> (Q) 0F6 endif 106 endif 112 end

6. H. F. Restrepo – Logic Systems Laboratory UTM: Towards a cellular implementation (1) DATA PROGRAM HEAD PROGRAM STATE TABLE

7. H. F. Restrepo – Logic Systems Laboratory UTM: Towards a cellular implementation (2) 00100 if do elseendifend … DATA TAPE (shift right, shift left, hold) PROGRAM TAPE (Picopascal) FIXED HEAD …… 

8. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if QL0 = 0QL1 = 0 QC = 1QR0 = 0 … … QR1 = 0 STACK ST1 = 1ST2 = 0ST3 = 0 Q = 0S = 1 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE

9. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if QC = 1QR0 = 0 … QL0 = 0 QL1 = 0 … QR1 = 0 STACK ST1 = 1ST2 = 0ST3 = 0 Q = 0S = 1 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE if

10. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if QC = 1QR0 = 0 … QL0 = 0 QL1 = 0... QR1 = 0 STACK ST1 = 1ST2 = 0ST3 = 0 Q = 0S = 1 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE

11. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if QC = 1QR0 = 0 … QL0 = 0 QL1 = 0 … QR1 = 0 STACK ST1 = 1ST2 = 0ST3 = 0 Q = 0S = 1 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE

12. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if QC = 1QR0 = 0 … QL0 = 0 QL1 = 0 QR1 = 0 STACK ST1 = 1ST2 = 0ST3 = 0 Q = 0S = 1 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE …

13. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if QR0:3 … QC QL3:0 STACK ST1 = 1ST2 = 0ST3 = 0 Q = 0S = 1 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE …

14. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if STACK ST1 = 1ST2 = 0ST3 = 0 Q = 0S = 1 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE QR0:3 … QC QL3:0 …

15. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if STACK Q = 0S = 1 ST1 = 1ST2 = 0ST3 = 0 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE QR0:3 … QC QL3:0 …

16. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if STACK Q = 0S = 1 ST1 = 1 ST2 = 0 ST3 = 0 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE QR0:3 … QC QL3:0 …

17. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if STACK Q = 0S = 1 ST1 = 1 ST2 = 0 ST3 = 0 … … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE QR0:3 … QC QL3:0 …

18. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if STACK … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE Q = 0S = 1 ST1 = 1 ST2 = 0 ST3 = 0 … QR0:3 … QC QL3:0 …

19. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if STACK … Q, S REGISTER DATA TAPE FIXED HEAD PROGRAM TAPE QS ST1 ST2 ST3 … QR0:3 … QC QL3:0 …

20. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if … DATA TAPE FIXED HEAD PROGRAM TAPE QS ST1 ST2 ST3 … QR0:3 … QC QL3:0 …

21. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if PROGRAM TAPE QS ST1 ST2 ST3 QR0:3 QC QL3:0 STACKREGISTERDATA TAPE

22. H. F. Restrepo – Logic Systems Laboratory UTM: Ideal cellular implementation … … if endendif if else if PROGRAM TAPE QS ST1 ST2 ST3 QR0:3 QC QL3:0 STACKREGISTERDATA TAPE

23. H. F. Restrepo – Logic Systems Laboratory UTM: Actual cellular implementation … if endendifif elseif endif do 0 do 1  do 0 else do 1 do 1  do 0  do 1 endif QL3:0QCQR0:3QS do 0  ST1ST3ST2 STACKREGISTERDATA TAPE PROGRAM TAPE

24. H. F. Restrepo – Logic Systems Laboratory The Mictree artificial cell RAM PROGRAM BDM REG3:0 Y3:0X3:0 SB

25. H. F. Restrepo – Logic Systems Laboratory UTM: Embryonic actual implementation (1) Y X prog if prog endif prog else prog do 0 prog do 1 prog do 1  prog do 0  prog endif prog endif prog do 0  prog do 0 prog else prog do 1  prog if prog if prog endif prog else prog do 1  ST3 0 ST2 0 ST1 1 Q 0 S 1 QL3:0 0000 QR0:3 0000 QC 1 Gene (subprogram) Initial condition 123456789 1 2 3

26. H. F. Restrepo – Logic Systems Laboratory UTM: Embryonic actual implementation (2) self-replication Y X self-repair prog if prog endif prog else prog do 0 prog do 1 prog do 1  prog do 0  prog endif prog endif prog do 0  prog do 0 prog else prog do 1  prog if prog if prog endif prog else prog do 1  ST3 0 ST2 0 ST1 1 Q 0 S 1 QL3:0 0000 QR0:3 0000 QC 1 123456789 1 2 3 Mother prog if prog endif prog else prog do 0 prog do 1 prog do 1  prog do 0  prog endif prog endif prog do 0  prog do 0 prog else prog do 1  prog if prog if prog endif prog else prog do 1  ST3 0 ST2 0 ST1 1 Q 0 S 1 QL3:0 0000 QR0:3 0000 QC 1 1 2 3 Daughter

27. H. F. Restrepo – Logic Systems Laboratory UTM: Embryonic actual implementation (3) self-repair prog if prog endif prog else prog do 0 prog do 1 prog do 1  prog do 0  prog endif prog endif prog do 0  prog do 0 prog else prog do 1  prog if prog if prog endif prog else prog do 1  ST3 0 ST2 0 ST1 1 Q 0 S 1 QL3:0 0000 QR0:3 0000 QC 1 123456789 1 2 3 7 Faulty column prog if prog endif prog else prog do 0 prog do 1 prog do 1  prog do 0  prog endif prog endif prog do 0  prog do 0 prog else prog do 1  prog if prog if prog endif prog else prog do 1  ST2 0 ST1 1 Q 0 S 1 QL3:0 0000 QR0:3 0000 QC 1 1 2 3 123456899 7 ST3 0 Y X self-replication self-repair Spare cells

28. H. F. Restrepo – Logic Systems Laboratory Actual implementation (1)

29. H. F. Restrepo – Logic Systems Laboratory Actual implementation (2) (self-replication) First organism Second organism

30. H. F. Restrepo – Logic Systems Laboratory Conclusion We presented a new ``multicellular'' automaton, in which every cell contains a complete copy of the genome. We showed that such an automaton is able to self-replicate and to self-repair. We showed that it is possible to embed a universal Turing machine in such a multicellular array.