1. Variants of Turing Machines
The Chinese University of Hong Kong
Fall 2011
CSCI 3130: Formal languages and automata theory
Variants of Turing Machines
Andrej Bogdanov

2. The Church-Turing Thesis
All arguments [for the CT Thesis] which can be given are bound to be, fundamentally, appeals to intuition, and for this reason rather unsatisfactory mathematically.
The arguments which I shall use are of three kinds:
1. A direct appeal to intuition A proof of the equivalence of two definitions (In case the new definition has greater intuitive appeal) Giving examples of large classes of numbers [languages] which are computable.
1936:
“On Computable Numbers, with an Application to the Entscheidungsproblem”
Section 9. The extent of the computable numbers

3. The multitape Turing Machine
control …
tape 1 1 … 1
tape 2 … 1
tape 3
The transition may depend on the contents of all the cells
Different tape heads can be moved independently

4. The multitape Turing Machineq7
0/1R ☐/1R
0/0L … 1 … 1 … 1 q3 … 1
Multiple tapes are convenient, e.g. one can serve as temporary storage

5. How to argue equivalence
Multitape Turing Machines are equivalent to single-tape Turing Machines
easy
single tape
multiple tapes
describe
a simulation

6. The multitape Turing Machine
Multitape Turing Machines are equivalent to single-tape Turing Machines M … 1
G = {0, 1, ☐} S … 1 # 
G’ = {0, 1, ☐, 0, 1, ☐, #}

7. Simulating a multitape TM
We show how to simulate a multitape TM on an single tape TM
To be specific, let’s do a 3-tape TM … x1 x2 xa xi y1 y2 yb yj z1 zc zk
#x1x2...xa...xi#y1y2...yb...yj#z1z2...zc...zk# 
Single-tape TM
Multitape TM

8. Simulating a multitape TM
Single-tape TM: Initialization
w1w2...wn
#w1w2...wn#☐#☐#  … 1 … 1 #  S:
On input w1...wn:
Replace tape contents by
#w1w2...wn#☐#☐#  1.
Remember that M is in state q0

9. Simulating a multitape TM
Single-tape TM: Simulating Multitape TM moves
Suppose Multitape TM wants to move like this: … 1 … 1 q3
0/1R ☐/1R
0/0L q7
We simulate move on single-tape TM like this: 1 #  1 # 

10. Simulating a multitape TM
On input w1...wn:
Replace tape contents by
#w1w2...wn#☐#☐#  1.
Remember (in state) that M is in state q0 2.
To simulate a step of M:
Make a pass over tape to find x, y, z 
#x1x2...x...xi#y1y2...y...yj#z1z2...z...zk#  qi
x/x’A y/y’B
z/z’C qj
If M is in state qi and has transition
update state / tape accordingly. 3.
If M reaches accept (reject) state, accept (reject).

11. Doing simulations To simulate a model M by another model N: 1.
Say how the state and storage of N is used
to represent the state and storage of M 2.
Say what should be initially done to convert
the input of N (if anything) 3.
Say how each transition of M can be implemented
by a sequence of transitions of N

12. What does a computer look like?
CPU
instruction memory
0000 PC
arithmetic logical unit
data memory
1010
0100
0000
registers R0 R1 R2 R3

13. What does a computer look like?
CPU
instruction memory PC
0000 load 0001
0001 write R3
0010 store R5
0011 add R5
0100 jpos 0011
...
0000
arithmetic logical unit R0
0100 R1
1010
data memory
registers R2
0000
0000
0001
0010
0011
0100 R3
0000
0110
0110
0110
0110
0110

14. Instruction set load x Put the value x into R0 load Rk
Copy the value of Rk into R0
store Rk
Copy the value of R0 into Rk
read Rk
Copy the value at memory location Rk into R0
write Rk
Copy the value of R0 into memory location Rk
add Rk
Add R0 and Rk, and put result in R0
jump n
Set PC to n
jzero n
Set PC to n, if R0 is zero
jpos n
Set PC to n, if R0 is positive

15. Random access machines
instruction meaning
program counter PC 1 2 3 4 5
load -7 R0 := -7
write R2 M[R2] := R0
store R1 R1 := R0
add R1 R0 := R0 + R5
jzero 3 if R0 = 0 then PC := 3
accept
registers R0 R1 R2 … M 2 1 2 2
memory 1 2 3 4
It has registers that can store integer values, a program counter, and a random-access memory

16. Random access machines
instruction meaning PC 1 2 3 1 2 3 4 5
load 7 R0 := 7
write R2 M[R2] := R0
store R1 R1 := R0
add R1 R0 := R0 + R1
jzero 3 if R0 = 0 then PC := 3
accept 4 5 R0 7 14 R1 7 R2 … M -7 1 2 3 4
The instructions are indexed by the program counter
Initially, the input is in the first k memory cells, all registers and PC are set to 0

17. Random access machines
Random access machines are equivalent to Turing Machines
Simulating a Turing Machine on a RAM: M … 2 1
head
tape
blank PC 2 R0 1 M …
G = {0, 1, 2, ..., k}

18. Simulating a TM on a RAM program M … q0 1/2R q1 1 2 3 6 7 8 9 101 2 3 6 7 8 9 10
store R1 handle for state q0
store R1 handle for state q1 …
accept handle for state qacc 30
200
store R1 save head position read R1 read tape contents x
add -1
jzero 6 if x = 1 goto line 6
load 2 new value of cell
write R1 write in memory
load R1 recall head position
add 1 move head to right
jump 30 go to state q1 … 1 2 1 PC R0 1 M 2 R1

19. Simulating a TM on a RAM program M … (head) … (tape) q0 1/2R q1 1 2 31 2 3 6 7 8 9 10
store R1 handle for state q0
store R1 handle for state q1 … 30
store R1 save head position read R1 read tape contents x
add -1
jzero 6 if x = 1 goto line 6
load 2 new value of cell
write R1 write in memory
load R1 recall head position
add 1 move head to right
jump 30 go to state q1 … 1 2 1 2 3 2 1
(head) R0 2 2 … R1 M 1 2 1 2
(tape)

20. Simulating a RAM on a Turing Machine
The configuration of a RAM consists of
Program counter
Contents of registers
Indices and contents of all nonempty memory cells PC 14
configuration = (14, 3, 17, 5, (0, 2), (2, 1), (3, 2)) R0 3 R1 17 R2 5 … M 2 1 2 1 2 3 4

21. Simulating a RAM on a 2-tape TM
The TM has a simulation tape and a scratch tape
The simulation tape stores RAM configuration
The TM has a set of states corresponding to each program instruction of the RAM
The TM tape is updates according to RAM instruction M
(14,3,17,5,(0,2),(2,1),(3,2))

22. Simulating a RAM on a 2-tape TM
Initialization
TM input:
122
RAM initial state: PC R0
... M 1 2 2 S:
On input w1...wn: 1.
Replace tape contents by
(0, 0, 0, ..., 0, (0, w1), (1, w2), ..., (n-1, wn))

23. Simulating a RAM on a 2-tape TM
Example: load R1 c
(14,3,17,5,(0,2),(2,1),(3,2))
1. Copy R1 to scratch tape 17 s
(14,1,17,5,(0,2),(2,1),(3,2)) 17 c s
2. Write R1 to conf tape
(14,1,17,5,(0,2),(2,1),(3,2) ) c
(14,1 ,17,5,(0,2),(2,1),(3,2)) .
Make more space as needed
(14,17,17,5,(0,2),(2,1),(3,2)) c
3. Erase scratch tape
4. Update PC
(15,17,17,5,(0,2),(2,1),(3,2)) c

24. Simulating a RAM on a 2-tape TM
On input w1...wn: 1.
Replace tape contents by
(0, 0, 0, ..., 0, (0, w1), (1, w2), ..., (n-1, wn)) 2.
Simulate instruction in RAM program
by a sequence of TM transitions
See notes for details. 3.
If RAM instruction is accept, go to accept state.
If RAM instruction is reject, go to reject state.