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H. F. Restrepo – Logic Systems Laboratory
A Self-ReplicatingA Self-ReplicatingUniversal Turing MachineUniversal Turing Machine
H. F. Restrepo – Logic Systems Laboratory
A specialized Turing machine (1)A specialized Turing machine (1)
Binary counter:A machine which writes out the binary numbers 1, 10, 11, 100, 101, etc.
0 00 1 0
DATA
Statetable 1(ST1)
FSM
TAPE
… …
H. F. Restrepo – Logic Systems Laboratory
A specialized Turing machine (2)A specialized Turing machine (2)
0 00 1 0
DATA
Statetable 1(ST1)
FSM
TAPE
… …STATE TABLE 1 (ST1)
Q =
Q+, S+ S=0 S=1
Q = 10 0,0
0,1
1,1
1,0
H. F. Restrepo – Logic Systems Laboratory
Universal Turing machine (UTM)Universal Turing machine (UTM)
…
SQ S+
PROGRAM STATE TABLE 1
Q+
1 1 1 0 1 0 10SQ S+Q+
Q =
Q+, S+ S=0 S=1
Q = 10 0,0
0,1
1,1
1,0
0 00 1 0
DATA
Statetable 2(ST2)
FSM
TAPE
… …
H. F. Restrepo – Logic Systems Laboratory
UTM: From state table to PICOPASCAL programUTM: From state table to PICOPASCAL program
STATE TABLE 1
Q =
Q+, S+ S=0 S=1
Q = 10 0,0
0,1
1,1
1,0
PROGRAM
ADDR DATA PROGRAM00 5 if (Q)01 5 if (S)02 A do 0 (S)03 9 do 1<- (Q)04 4 else05 B do 1 (S)06 8 do 0-> (Q)07 6 endif08 4 else09 5 if (S)0A B do 1 (S)0B 9 do 1<- (Q)0C 4 else0D A do 0 (S)0E 8 do 0-> (Q)0F 6 endif10 6 endif11 2 end
H. F. Restrepo – Logic Systems Laboratory
UTM: Towards a cellular implementation (1)UTM: Towards a cellular implementation (1)
DATA
PROGRAM
HEAD
PROGRAM
STATE TABLE
H. F. Restrepo – Logic Systems Laboratory
UTM: Towards a cellular implementation (2)UTM: Towards a cellular implementation (2)
0 0 1 0 0
if if do do else endif end…
DATA TAPE(shift right, shift left, hold)
PROGRAM TAPE(Picopascal)
FIXED HEAD
……
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
QL0 = 0 QL1 = 0
QC = 1 QR0 = 0
…
…QR1 = 0
STACK ST1 = 1 ST2 = 0 ST3 = 0
Q = 0 S = 1
…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
QC = 1 QR0 = 0 …
QL0 = 0QL1 = 0 …
QR1 = 0
STACK ST1 = 1 ST2 = 0 ST3 = 0
Q = 0 S = 1
…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
if
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
QC = 1 QR0 = 0 …
QL
0 =
0Q
L1
= 0
...
QR1 = 0
STACK ST1 = 1 ST2 = 0 ST3 = 0
Q = 0 S = 1
…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
QC = 1 QR0 = 0 …
QL0 = 0
QL1 = 0
…
QR1 = 0
STACK ST1 = 1 ST2 = 0 ST3 = 0
Q = 0 S = 1
…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
QC = 1 QR0 = 0 …QL0 = 0QL1 = 0 QR1 = 0
STACK ST1 = 1 ST2 = 0 ST3 = 0
Q = 0 S = 1
…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
QR0:3 …QCQL3:0
STACK ST1 = 1 ST2 = 0 ST3 = 0
Q = 0 S = 1
…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
STACK ST1 = 1 ST2 = 0 ST3 = 0
Q = 0 S = 1
…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
QR0:3 …QCQL3:0…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
STACK
Q = 0 S = 1
ST1 = 1ST2 = 0
ST3 = 0…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
QR0:3 …QCQL3:0…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
STACK
Q = 0 S = 1
ST1 = 1
ST2 = 0
ST3 = 0
……Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
QR0:3 …QCQL3:0…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
STACKQ = 0 S = 1
ST1 = 1
ST2 = 0
ST3 = 0
…
…Q, S
REGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
QR0:3 …QCQL3:0…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
STACK…
Q, SREGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
Q = 0 S = 1ST1 = 1ST2 = 0ST3 = 0…
QR0:3 …QCQL3:0…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
STACK …
Q, SREGISTER
DATA TAPE
FIXED HEAD
PROGRAM TAPE
Q SST1ST2ST3…
QR0:3 …QCQL3:0…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif
…
DATA TAPE
FIXED HEAD
PROGRAM TAPE
Q SST1ST2ST3…
QR0:3 …QCQL3:0…
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif PROGRAM TAPE
Q SST1ST2ST3 QR0:3QCQL3:0
STACK REGISTER DATA TAPE
H. F. Restrepo – Logic Systems Laboratory
UTM: Ideal cellular implementationUTM: Ideal cellular implementation
…
…if
end endif if
elseif PROGRAM
TAPE
Q SST1ST2ST3 QR0:3QCQL3:0
STACK REGISTER DATA TAPE
H. F. Restrepo – Logic Systems Laboratory
UTM: Actual cellular implementationUTM: Actual cellular implementation
…
if
end endif if
elseif
endif
do 0 do 1
do 0
else
else
do 1
do 1
do 0
do 1
endif
QL3:0 QC QR0:3Q S
do 0
ST1ST3 ST2
STACK REGISTER DATA TAPE
PROGRAMTAPE
H. F. Restrepo – Logic Systems Laboratory
The Mictree artificial cellThe Mictree artificial cell
RAM
PROGRAM
BDM REG3:0
Y3:0X3:0
SB
H. F. Restrepo – Logic Systems Laboratory
UTM: Embryonic actual implementation (1)UTM: Embryonic actual implementation (1)
Y
X
progif
progendif
progelse
progdo 0
progdo 1
progdo 1
progdo 0
progendif
progendif
progdo 0
progdo 0
progelse
progdo 1
progif
progif
progendif
progelse
progdo 1
ST30
ST20
ST11
Q0
S1
QL3:00000
QR0:30000
QC1
Gene(subprogram)
Initialcondition
1 2 3 4 5 6 7 8 9
1
2
3
H. F. Restrepo – Logic Systems Laboratory
UTM: Embryonic actual implementation (2)UTM: Embryonic actual implementation (2)self-replication
Y
X
self-replication
self-repair
progif
progendif
progelse
progdo 0
progdo 1
progdo 1
progdo 0
progendif
progendif
progdo 0
progdo 0
progelse
progdo 1
progif
progif
progendif
progelse
progdo 1
ST30
ST20
ST11
Q0
S1
QL3:00000
QR0:30000
QC1
1 2 3 4 5 6 7 8 9
1
2
3
Mother
progif
progendif
progelse
progdo 0
progdo 1
progdo 1
progdo 0
progendif
progendif
progdo 0
progdo 0
progelse
progdo 1
progif
progif
progendif
progelse
progdo 1
ST30
ST20
ST11
Q0
S1
QL3:00000
QR0:30000
QC1
1
2
3
Daughter
H. F. Restrepo – Logic Systems Laboratory
UTM: Embryonic actual implementation (3)UTM: Embryonic actual implementation (3)self-repair
progif
progendif
progelse
progdo 0
progdo 1
progdo 1
progdo 0
progendif
progendif
progdo 0
progdo 0
progelse
progdo 1
progif
progif
progendif
progelse
progdo 1
ST30
ST20
ST11
Q0
S1
QL3:00000
QR0:30000
QC1
1 2 3 4 5 6 7 8 9
1
2
3
7
Faulty column
progif
progendif
progelse
progdo 0
progdo 1
progdo 1
progdo 0
progendif
progendif
progdo 0
progdo 0
progelse
progdo 1
progif
progif
progendif
progelse
progdo 1
ST20
ST11
Q0
S1
QL3:00000
QR0:30000
QC1
1
2
3
1 2 3 4 5 6 8 9 97
ST30
Y
X
self-replication
self-repair
Spare cells
H. F. Restrepo – Logic Systems Laboratory
Actual implementation (2) Actual implementation (2) (self-replication)(self-replication)
First organismFirst organism
Second organismSecond organism
H. F. Restrepo – Logic Systems Laboratory
ConclusionConclusion
• 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.
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