Notes on self-assembly of Turing machine simulations

1
Notes on self-assembly of Turing machine
simulations

• Day 9 of Comp Sci 480

2
What is a Turing machine?

• A Turing machine is an abstract mathematical
  model of a computer.
• A Turing machine is used to study the process of
  computation.

3
Intuitively

• A Turing machine consists of three components
• An infinite tape (infinite in both directions),
• a control module that maintains the current state
  of the machine, and
• a read/write head

q
4
Formal definition

• A Turing machine is a tuple M
  (Q,S,G,d,q0,qhalt), where
• Q is a finite set of states,
• S is a finite set of characters, usually 0,1,
• G is a finite set of characters such that S is a
  subset of G,
• dQG ? QGL,R is the transition function, and
• q0 and qhalt are the start and halting states,
  respectively

5
The transition function

• The transition function says what to do next,
  given the current state and symbol being read)
  and is stated as
• dQG ? QGL,R
• The previous notation is just a formal way of
  specifying a function with the following
  behavior
• Input a state and tape symbol
• Output a state, tape symbol and direction
• For example, d(p,a) (q,b,d) means If Im
  currently in state p, reading tape symbol a, then
  in the next step, I will go to state q, overwrite
  symbol a with b, and move in direction d (either
  left or right)

6
Turing machine behavior

• A configuration of a Turing machine is the
  current state of the machine, which tape symbol
  is currently being read and the current state of
  the tape
• The Turing machine starts in an initial
  configuration
• The next configuration is given by the transition
  function d
• The halting configuration, if ever reached, is
  the unique configuration that has no next
  configuration

7
Initial configuration

• An input string w w0w1 wk-1 is put
  somewhere on the tape, blank symbols ()
  everywhere else
• The state is set to q0
• The read/write head is positioned so that it is
  reading the leftmost character in the input,
  i.e., w0

q0
8
Next configuration

• Suppose the current state is p and the read/write
  head is reading tape symbol a
• If d(p,a) (q,b,d), then the Turing machine does
  the following
• Changes state from p to q
• Overwrites the tape symbol being read (a) with b
  (a and b need not be different)
• Moves the read/write head in direction d (either
  left or right)

9
Halting configuration

• The Turing machine is said to halt if it ever
  enters the halting state, i.e., if its in state
  q, reading tape symbol a, and d(q,a)
  (qhalt,___,___), for any tape symbol b and
  direction d
• There is no next configuration from the halting
  configuration
• A Turing machine need not ever reach a halting
  configuration (i.e., it runs forever)

10
A Turing machine example
11
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
12
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
q0
1




















13
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
q01



C0









q2
1




















14
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C1
1
q2











q01



C0









q2
1




















15
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C1
1
q2











q01



C0









q1
1



















1
16
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C2
q1









1

1
C1
1
q2











q01



C0









q1
1



















1
17
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C2
q1









1

1
C1
1
q2











q01



C0









q0
1


















1
1
18
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q0
1


















1
1
19
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q1
1

















1
1
1
20
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q1
1

















1
1
1
21
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q1
1

















1
1
1
22
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q1
1

















1
1
1
23
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q1
1

















1
1
1
24
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C6








1

1
1
q11
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q1
1

















1
1
1
25
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C6








1

1
1
q11
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q1
1

















1
1
1
26
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C7








1
1
1
q1
1
C6








1

1
1
q11
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q1
1

















1
1
1
27
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C7








1
1
1
q1
1
C6








1

1
1
q11
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q0
1
















1
1
1
1
28
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C8








1
1
1
q01
1
C7








1
1
1
q1
1
C6








1

1
1
q11
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q0
1
















1
1
1
1
29
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C8








1
1
1
q01
1
C7








1
1
1
q1
1
C6








1

1
1
q11
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q2
1
















1
1
1
1
30
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
C9








q21
1
1
1
1
C8








1
1
1
q01
1
C7








1
1
1
q1
1
C6








1

1
1
q11
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









q2
1
















1
1
1
1
31
Let M (Q,S,G,d,q0,qhalt), where Q
q0,q1,q2,qhalt, S 1, G 1,, and d is
defined in the following table
d 1
q0 q2,1,L q1,1,R
q1 q1,1,R q0,1,L
q2 qhalt q1,1,L
Chalt
C9








q21
1
1
1
1
C8








1
1
1
q01
1
C7








1
1
1
q1
1
C6








1

1
1
q11
C5








1

1
1
q11
C4








1

1
1
q11
C3
q0








1

1
1
C2
q1









1

1
C1
1
q2











q01



C0









qhalt
1
















1
1
1
1
32
Simulating a Turing machine with self-assembly

• Basic idea self-assemble the space-time
  configuration history of a Turing machine
• Each configuration is represented by a row of
  tiles, above the previous configuration

33
Seed row

• Hard code the initial configuration in a seed row
  by creating the following tile types

q0w0
w1
wk-1




q0w0
w1
wk-1
0
1
0
k-1
k-2
k-1
-1
-1
q0w0
w1


wk-1









34
Move right

• For all transitions of the form d(p,a) (q,b,R)
  where q is not the halting state
• create the following tile types, where c ? G

35
Move left

• For all transitions of the form d(p,a) (q,b,L)
  where q is not the halting state
• create the following tile types, where c ? G

36
Copy tape symbols

• For each tape symbol x ? G
• create the following tile types

37
Grow the tape

• If the read/write head tries to grow too far left
  or too far right, we need to expand the tape, so
  create the following tile types

38
Halt

• If the Turing machine reaches the halting state,
  the self-assembly should halt as well
• create the following tile type

39
How do we use a Turing machine simulation to make
a square?