h. f. restrepo – logic systems laboratory a self-replicating universal turing machine

H. F. Restrepo – Logic Systems Laboratory

A Self-ReplicatingA Self-ReplicatingUniversal Turing MachineUniversal Turing Machine


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

… …


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


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

… …


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


UTM: Towards a cellular implementation (1)UTM: Towards a cellular implementation (1)

DATA

PROGRAM

HEAD

PROGRAM

STATE TABLE


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

……


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



…

…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



…

…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



…

…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



…

…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

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

…



…

…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…



…

…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…



…

…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…



…

…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…



…

…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…



…

…if

end endif if

elseif

STACK …

Q, SREGISTER

DATA TAPE

FIXED HEAD

PROGRAM TAPE

Q SST1ST2ST3…

QR0:3 …QCQL3:0…



…

…if

end endif if

elseif

…

DATA TAPE

FIXED HEAD

PROGRAM TAPE

Q SST1ST2ST3…

QR0:3 …QCQL3:0…



…

…if

end endif if

elseif PROGRAM TAPE

Q SST1ST2ST3 QR0:3QCQL3:0

STACK REGISTER DATA TAPE



…

…if

end endif if

elseif PROGRAM

TAPE

Q SST1ST2ST3 QR0:3QCQL3:0



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


PROGRAMTAPE


The Mictree artificial cellThe Mictree artificial cell

RAM

PROGRAM

BDM REG3:0

Y3:0X3:0

SB


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


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


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


Actual implementation (1)Actual implementation (1)


Actual implementation (2) Actual implementation (2) (self-replication)(self-replication)

First organismFirst organism

Second organismSecond organism


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.

h. f. restrepo – logic systems laboratory a self-replicating universal turing machine

Documents

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st2 fsm tape slide

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