Lecture 20
Variations on the Turing
Machines
COT 4420
Theory of Computation
Section 10.1, 10.2
Slides from Prof. Busch
The Standard Model
Infinite Tape
☐☐
a a b a b b c a c a ☐☐☐
Read-Write Head
(Left or Right)
Control Unit
Deterministic
Standard Turing Machine
M = <Q, Σ, Γ, δ, q0, ☐, F>
•
•
•
•
•
•
•
Q: a finite set of internal states
Σ: is the input alphabet ( Σ ⊆ Γ-{☐} )
Γ: finite set of symbols called the tape alphabet
δ: transition function (Q × Γ  Q × Γ × {L, R} )
q0: the initial state (q0 ∈ Q)
☐: is a special symbol called blank (☐ ∈ Γ)
F: a set of final/accepting states (F ⊆ Q)
Variations of the Standard Model
Turing machines with:
• Stay-Option
• Semi-Infinite Tape
•Off-line
• Multitape
• Multidimensional
• Nondeterministic
Same Power of two machine classes:
both classes accept the
same set of languages
We will prove:
each new class has the same power
with Standard Turing Machine
(accept Turing-Recognizable Languages)
Same Power of two classes:
• Two automata are equivalent if they accept
the same language. For any machine M1 of
first class there is a machine M2 of second
class such that:
L(M1) = L(M2)
and vice versa.
Simulation:
A technique to prove same power.
Simulate the machine of one class
with a machine of the other class
Configurations in the original machine M1 have
corresponding configurations in the simulation
machine M2
If machine M1 has the sequence of
computations:
d0 ⊢ d1 ⊢ … ⊢ dn
Then M2 simulates this computation:
*
*
*
d’0 ⊢ d’1 ⊢ … ⊢ d’n
Turing Machines with Stay-Option
The head can stay in the same position
☐☐
a a b a b b c a c a ☐☐☐
Left, Right, Stay
L,R,S: possible head moves
Example:
☐☐
a a b a b b c a c a☐☐☐
q1
q1
☐ ☐b
q2
a → b, S
q2
a b a b b c a c a ☐☐☐
Theorem:
Stay-Option machines
have the same power with
Standard Turing machines
Proof – part 1
• Stay-option machines simulate Standard
Turing machines
• Proof: A Standard machine is also a Stayoption machine (that never uses the S move).
Proof – part 2
• Standard Turing machines simulate Stayoption machines
Proof:
We need to simulate the stay head option with
two head moves, one left and one right
For transitions with L and R moves, nothing changes
Stay-Option Machine
q1
a → b, L
q2
Simulation in Standard Machine
q1
a → b, L
Similar for Right moves
q2
Stay-Option Machine
q1
a → b, S
q2
Simulation in Standard Machine
q1
a → b, L
x → x, R
For every possible tape symbol
q2
x
Example
Stay-Option Machine:
b
S
a
,
→
q1
q2
1
aaba
☐
☐
q1
☐
2
baba
☐
q2
Simulation in Standard Machine:
☐
1
aaba
q1
☐
☐
2
baba
q3
☐
☐
3
baba
q2
☐
Standard Machine – Multiple Track
Tape : a useful trick
One Tape
☐☐
☐☐
a b a b
b a c d
☐
track 1
☐
track 2
One head
One symbol ( a, b)
It is a standard Turing machine, but each tape alphabet
symbol describes a pair of actual useful symbols
☐ ☐
☐☐
a b a b
b a c d
☐
track 1
☐
track 2
☐
track 1
☐
track 2
q1
☐☐
☐☐
a c a b
b d c d
q2
q1
(b, a ) → (c, d ), L
q2
Semi-Infinite Tape
The tape extends infinitely only to the right
a b a c
☐ ☐
.........
• Initial position is the leftmost cell
• No left move is permitted when the read-write
head is at the boundary
Theorem:
Semi-Infinite machines have the
same power as standard Turing
machines
Proof: 1. Standard Turing machines
Simulate semi-infinite machines
2. Semi-Infinite Machines
simulate Standard Turing machines
1. Standard Turing machines simulate
Semi-Infinite machines:
Trivial
2. Semi-Infinite tape machines simulate
Standard Turing machines:
.........
Standard machine
.........
Semi-Infinite tape machine
.........
Squeeze infinity of both directions
to one direction
Standard machine
......... ☐ a
b c
d e
☐ ☐
.........
reference point
Semi-Infinite tape machine with two tracks
Right part
Left part
# d e ☐
# c b a
☐ ☐
☐ ☐
.........
Standard machine
q1
q2
Semi-Infinite tape machine
Left part
q1L
q2L
Right part
q1R
q2R
Time 1
.........
Standard machine
☐
a b c d e
☐ ☐
.........
q1
Semi-Infinite tape machine
Right part
Left part
# d e ☐
# c b a
L
q1
☐ ☐
☐ ☐
.........
Standard machine
q1
a → g, R
q2
Semi-Infinite tape machine
Right part
R ( a, x ) → ( g , x ), R
q1
Left part
L
q1
( x, a ) → ( x, g ), L
For all tape symbols
x
R
q2
L
q2
Time 2
.........
Standard machine
☐
g b c d e
☐ ☐
.........
q2
Semi-Infinite tape machine
Right part
Left part
# d e ☐
g
c
b
#
L
q2
☐ ☐
☐ ☐
.........
At the border:
Semi-Infinite tape machine
Right part
R (# , # ) → (# , # ), R
q1
L
q1
Left part
L (# , # ) → (# , # ), R
q1
R
q1
Right part
Left part
Semi-Infinite tape machine
Time 1
# d e ☐
# c b g
☐ ☐
.........
☐ ☐
L
q1
Right part
Left part
Time 2
# d e ☐
# c b g
R
q1
☐ ☐
☐ ☐
.........
The Off-Line Machine
Input File
a b c
read-only (once)
Control Unit
Tape
☐ ☐
read-write
g d e
☐ ☐
Theorem:
Proof:
Off-Line machines
have the same power with
Standard Turing machines
1. Off-Line machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Off-Line machines
1. Off-line machines simulate Standard Turing
Machines
Off-line machine:
1. Copy input file to tape
2. Continue computation as in
Standard Turing machine
2. Standard Turing machines simulate Off-Line
machines:
Use a Standard machine with a four-track tape to
keep track of the Off-line input file and tape
contents
Off-line Machine
Tape
Input File
☐ ☐
a b c d
e f g
☐
Standard Machine -- Four track tape
# a
# 0
e
0
b
0
f
1
c d
1 0
g
0
Input File
head position
Tape
head position
Reference point
#
#
#
#
a
0
e
0
b
0
f
1
(uses special symbol # )
c d
1 0
g
0
Input File
head position
Tape
head position
Repeat for each state transition:
1. Search track 2 for the marked cell
2. Remember the symbol in the corresponding cell of track 1
3. Search track 4 for the head position
4. Remember the symbol in the corresponding cell of track 3
5. Change all 4 tracks according to the transition
6. Return to the standard starting position
Multi-tape Turing Machines
Control unit
(state machine)
Tape 1
☐
a b c
Tape 2
☐
Input string
Input string appears on Tape 1
☐
e f g
☐
Tape 1
☐
Time 1
a b c
Tape 2
☐
☐
e f g
q1
q1
Tape 1
☐
Time 2
a g c
☐
Tape 2
☐
e d g
q2
q2
q1
☐
(b, f ) → ( g , d ), L, R
q2
☐
Theorem:
Proof:
Multi-tape machines
have the same power with
Standard Turing machines
1. Multi-tape machines
simulate Standard Turing machines
2. Standard Turing machines
simulate Multi-tape machines
1. Multi-tape machines simulate Standard Turing
Machines:
Trivial: Use one tape
2. Standard Turing machines simulate Multi-tape
machines:
Standard machine:
• Uses a multi-track tape to simulate
the multiple tapes
• A tape of the Multi-tape machine
corresponds to a pair of tracks
Multi-tape Machine
Tape 1
Tape 2
☐
a b c
☐
☐
e f g h
☐
Standard machine with four track tape
a
0
e
0
b
1
f
0
c
0
g h
1 0
Tape 1
head position
Tape 2
head position
Reference point
#
#
#
#
a
0
e
0
b
1
f
0
c
0
g h
1 0
Tape 1
head position
Tape 2
head position
Repeat for each Multi-tape state transition:
1.
2.
3.
4.
Return to reference point
Find current symbol in Track 1 and update
Return to reference point
Find current symbol in Tape 2 and update