The document discusses variations of Turing machines and proves they have the same computational power as standard Turing machines. It introduces Turing machines with additional capabilities like staying in place, semi-infinite tapes, multiple tapes, and multiple dimensions. It then proves that each variation can simulate a standard Turing machine and vice versa through techniques like tracking multiple tape positions or dimensions on a single tape. This establishes that all the machine variations accept the same formal languages.

1. © 2020 Wael Badawy
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Turing’s Thesis
1
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3. © 2020 Wael Badawy

4. © 2020 Wael Badawy
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Turing’s thesis (1930):
Any computation carried out
by mechanical means
can be performed by a Turing Machine

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Algorithm:
An algorithm for a problem is a
Turing Machine which solves the problem
The algorithm describes the steps of
the mechanical means
This is easily translated to computation steps
of a Turing machine

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When we say: There exists an algorithm
We mean: There exists a Turing Machine
that executes the algorithm

7. 7
Variations
of the
Turing Machine
7

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Read-Write Head
Control Unit
àà a a c àà àb a cb b a a
Deterministic
The Standard Model
Infinite Tape
(Left or Right)

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Variations of the Standard
Model
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• Stay-Option
• Semi-Infinite Tape
• Multitape
• Multidimensional
• Nondeterministic
Turing machines with:
Different Turing Machine Classes

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We will prove:
each new class has the same power
with Standard Turing Machine
Same Power of two machine classes:
both classes accept the
same set of languages
(accept Turing-Recognizable Languages)

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Same Power of two classes means:
for any machine of first class1M
there is a machine of second class2M
such that: )()( 21 MLML =
and vice-versa

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A technique to prove same power.Simulation:
Simulate the machine of one class
with a machine of the other class
First Class
Original Machine
1M 1M
2M
Second Class
Simulation Machine
simulates 1M

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Configurations in the Original Machine
have corresponding configurations
in the Simulation Machine
nddd !"!! 10Original Machine:
Simulation Machine: nddd ¢¢¢
***
!"!! 10
1M
2M
1M
2M

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the Simulation Machine
and the Original Machine
accept the same strings
fdOriginal Machine:
Simulation Machine:
fd¢
Accepting Configuration
)()( 21 MLML =

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Turing Machines with Stay-
Option
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The head can stay in the same position
àà a a c àà àb a cb b a a
Left, Right, Stay
L,R,S: possible head moves

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Example:
àà a a c àà àb a cb b a a
Time 1
àà b a c àà àb a cb b a a
Time 2
1q 2q
1q
2q
Sba ,®

17. © 2020 Wael Badawy
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Stay-Option machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Stay-Option Machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Stay-Option machines

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1. Stay-Option Machines
simulate Standard Turing machines
Trivial: any standard Turing machine
is also a Stay-Option machine

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2. Standard Turing machines
simulate Stay-Option machines
We need to simulate the stay head option
with two head moves, one left and one right

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1q 2q
Sba ,®
1q
Lba ,®
2q
Rxx ,®
Stay-Option Machine
Simulation in Standard Machine
For every possible tape symbol x

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1q 2q
Lba ,®
1q 2q
Lba ,®
Stay-Option Machine
Simulation in Standard Machine
Similar for Right moves
For other transitions nothing changes

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example of simulation
à àa a b a
1q
Stay-Option Machine:
1
à àb a b a
2q
2
1q 2qSba ,®
Simulation in Standard Machine:
à àa a b a
1q
1
à àb a b a
2q
2
à àb a b a
3q
3
END OF PROOF

23. © 2020 Wael Badawy
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A useful trick: Multiple Track Tape
à
à
à
à
à
à
b
d
a
b
b
a
a
c
track 1
track 2
One symbol ),( ba
One head
helps for more complicated simulations
One Tape
It is a standard Turing machine, but each tape alphabet symbol
describes a pair of actual useful symbols

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à
à
à
à
à
à
b
d
a
b
b
a
a
c
track 1
track 2
1q 2q
Ldcab ),,(),( ®
1q
à
à
à
à
à
à
b
d
a
b
c
d
a
c
track 1
track 2
2q

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Semi-Infinite Tape
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.........a b a c à à
• When the head moves left from the border,
it returns back to leftmost position
The head extends infinitely only to the right
• Initial position is the leftmost cell

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Semi-Infinite machines
have the same power with
Standard Turing machines
Theorem:
Proof:
2. Semi-Infinite Machines
simulate Standard Turing machines
1. Standard Turing machines
simulate Semi-Infinite machines

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1. Standard Turing machines simulate
Semi-Infinite machines:
a. insert special symbol
at left of input string
#
a b a c à à#
b. Add a self-loop
to every state
(except states with no
outgoing transitions)
R,##®
Standard Turing Machine
àà
Semi-Infinite machine modifications
..................

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2. Semi-Infinite tape machines simulate
Standard Turing machines:
Standard machine
.........
Semi-Infinite tape machine
..................
Squeeze infinity of both directions
to one direction

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Standard machine
.........
Semi-Infinite tape machine with two tracks
..................
reference point
#
#
Right part
Left part
à à àa b c d e
ac b
d e à à
à
à
à

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1q
2q
R
q2
L
q1
L
q2 R
q1
Left part Right part
Standard machine
Semi-Infinite tape machine

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1q 2q
Rga ,®
Standard machine
L
q1
L
q2
Lgxax ),,(),( ®
R
q1
R
q2
Rxgxa ),,(),( ®
Semi-Infinite tape machine
Left part
Right part
For all tape symbols x

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Standard machine
..................
à à àa b c d e
1q
.........
Semi-Infinite tape machine
#
#
Right part
Left part ac b
d e à à
à
à
à
L
q1
Time 1

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Time 2
à à àg b c d e
2q
#
#
Right part
Left part gc b
d e à à
à
à
à
L
q2
Standard machine
..................
.........
Semi-Infinite tape machine

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L
q1
R
q1
R),#,(#)#,(# ®
Semi-Infinite tape machine
Left part
At the border:
R
q1
L
q1
R),#,(#)#,(# ®Right part

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.........
Semi-Infinite tape machine
#
#
Right part
Left part gc b
d e à à
à
à
à
L
q1
.........#
#
Right part
Left part gc b
d e à à
à
à
à
R
q1
Time 1
Time 2
END OF PROOF

36. © 2020 Wael Badawy
Multi-tape Turing Machines
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àà a b c àà e f g
Control unit
Tape 1 Tape 2
Input string
Input string appears on Tape 1
(state machine)

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àà a b c àà e f g
1q 1q
àà a g c àà e d g
2q 2q
Time 1
Time 2
RLdgfb ,),,(),( ®
1q 2q
Tape 1 Tape 2
Tape 1 Tape 2

38. © 2020 Wael Badawy
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Multi-tape machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Multi-tape machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Multi-tape machines

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1. Multi-tape machines simulate
Standard Turing Machines:
Trivial: Use one tape

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2. Standard Turing machines simulate
Multi-tape machines:
• Uses a multi-track tape to simulate
the multiple tapes
• A tape of the Multi-tape machine
corresponds to a pair of tracks
Standard machine:

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àà a b c hà e f g à
Multi-tape Machine
Tape 1 Tape 2
Standard machine with four track tape
a b c
e f g
0 0
0 0
1
1
Tape 1
head position
Tape 2
head position
h
0

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Repeat for each Multi-tape state transition:
1. Return to reference point
2. Find current symbol in Track 1 and update
3. Return to reference point
4. Find current symbol in Tape 2 and update
a b c
e f g
0 0
0 0
1
1
Tape 1
head position
Tape 2
head position
h
0
#
#
#
#
Reference point
END OF PROOF

43. © 2020 Wael Badawy
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( steps)
}{ nn
baL =
Standard Turing machine:
Go back and forth times)( 2
nO
2-tape machine:
1. Copy to tape 2n
b
2. Compare on tape 1
and on tape 2
)(nO
n
b
n
a
to match the a’s with the b’s
( steps))(nO
)( 2
nO time
)(nO time
Same power doesn’t imply same speed:

44. © 2020 Wael Badawy
Multidimensional Turing
Machines
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x
y
a
b
c
2-dimensional tape
HEAD
Position: +2, -1
MOVES: L,R,U,D
U: up D: down
à
à
à
à

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Multidimensional machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Multidimensional machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Multi-Dimensional machines

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1. Multidimensional machines simulate
Standard Turing machines
Trivial: Use one dimension

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2. Standard Turing machines simulate
Multidimensional machines
Standard machine:
• Use a two track tape
• Store symbols in track 1
• Store coordinates in track 2

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x
y
a
b
cà
à
à
à
a
1
b
#
symbol
coordinates
2-dimensional machine
Standard Machine
1 # 2 # 1-
c
# -
1q
1q
1

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Repeat for each transition followed
in the 2-dimensional machine:
1. Update current symbol
2. Compute coordinates of next position
3. Find next position on tape
Standard machine:
END OF PROOF

50. © 2020 Wael Badawy
Nondeterministic Turing
Machines
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Lba ,®
Rca ,®
1q
2q
3q
Allows Non Deterministic Choices
Choice 1
Choice 2

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a b cà à
1q
Lba ,®
Rca ,®
1q
2q
3q
Time 0
Time 1
b b cà à
2q
c b cà à
3q
Choice 1
Choice 2

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Input string is accepted if
there is a computation:
w
yqxwq f
*
!0
Initial configuration Final Configuration
Any accept state
There is a computation:

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Nondeterministic machines
have the same power with
Standard Turing machines
Theorem:
Proof: 1. Nondeterministic machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Nondeterministic machines

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1. Nondeterministic Machines simulate
Standard (deterministic) Turing Machines
Trivial: every deterministic machine
is also nondeterministic

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2. Standard (deterministic) Turing machines
simulate Nondeterministic machines:
• Stores all possible computations
of the non-deterministic machine
on the 2-dimensional tape
Deterministic machine:
• Uses a 2-dimensional tape
(equivalent to standard Turing machine with one tape)

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All possible computation paths
Initial state
Step 1
Step 2
Step i
Step i+1
acceptreject
infinite
path

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The Deterministic Turing machine
simulates all possible computation paths:
•with breadth-first search
•simultaneously
•step-by-step
depth-first may result getting stuck at exploring
an infinite path before discovering the accepting path

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a b cà à
1q
Lba ,®
Rca ,®
1q
2q
3q
Time 0
NonDeterministic machine
Deterministic machine
a b c
1q
# # # # ##
#
#
# # #
#
#
# #
current
configuration

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Lba ,®
Rca ,®
1q
2q
3q
b b c
2q
# # # # ##
#
# #
#
# #
Computation 1
b b cà à
2q
Choice 1
c b cà à
3q
Choice 2
c b c
3q ##
Computation 2
NonDeterministic machine
Deterministic machine
Time 1

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Repeat
For each configuration in current step
of non-deterministic machine,
if there are two or more choices:
1. Replicate configuration
2. Change the state in the replicas
Deterministic Turing machine
Until either the input string is accepted
or rejected in all configurations

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If the non-deterministic machine accepts
the input string:
The deterministic machine accepts and halts too
The simulation takes in the worst case
exponential time compared to the
shortest length of an accepting path

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If the non-deterministic machine does not
accept the input string:
1. The simulation halts if all paths
reach a halting state
OR
2. The simulation never terminates
if there is a never-ending path (infinite loop)
In either case the deterministic machine
rejects too (1. by halting or 2. by simulating the infinite loop)
END OF PROOF