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A Multitrack Turing machine is a specific type of multi-tape Turing machine .
In a standard n-tape Turing machine, n heads move independently along n tracks. In a n-track Turing machine, one head reads and writes on all tracks simultaneously. A tape position in an n-track Turing Machine contains n symbols from the tape alphabet. It is equivalent to the standard Turing machine and therefore accepts precisely the recursively enumerable languages .
Formal definition [ edit ]
A multitrack Turing machine with
n
{\displaystyle n}
-tapes can be formally defined as a 6-tuple
M
=
⟨
Q
,
Σ
,
Γ
,
δ
,
q
0
,
F
⟩
{\displaystyle M=\langle Q,\Sigma ,\Gamma ,\delta ,q_{0},F\rangle }
, where
Q
{\displaystyle Q}
is a finite set of states;
Σ
⊆
Γ
∖
{
b
}
{\displaystyle \Sigma \subseteq \Gamma \setminus \{b\}}
is a finite set of input symbols , that is, the set of symbols allowed to appear in the initial tape contents;
Γ
{\displaystyle \Gamma }
is a finite set of tape alphabet symbols ;
q
0
∈
Q
{\displaystyle q_{0}\in Q}
is the initial state ;
F
⊆
Q
{\displaystyle F\subseteq Q}
is the set of final or accepting states ;
δ
:
(
Q
∖
F
×
Γ
n
)
→
(
Q
×
Γ
n
×
{
L
,
R
}
)
{\displaystyle \delta :\left(Q\backslash F\times \Gamma ^{n}\right)\rightarrow \left(Q\times \Gamma ^{n}\times \{L,R\}\right)}
is a partial function called the transition function .
Sometimes also denoted as
δ
(
Q
i
,
[
x
1
,
x
2
.
.
.
x
n
]
)
=
(
Q
j
,
[
y
1
,
y
2
.
.
.
y
n
]
,
d
)
{\displaystyle \delta \left(Q_{i},[x_{1},x_{2}...x_{n}]\right)=(Q_{j},[y_{1},y_{2}...y_{n}],d)}
, where
d
∈
{
L
,
R
}
{\displaystyle d\in \{L,R\}}
.
A non-deterministic variant can be defined by replacing the transition function
δ
{\displaystyle \delta }
by a transition relation
δ
⊆
(
Q
∖
F
×
Γ
n
)
×
(
Q
×
Γ
n
×
{
L
,
R
}
)
{\displaystyle \delta \subseteq \left(Q\backslash F\times \Gamma ^{n}\right)\times \left(Q\times \Gamma ^{n}\times \{L,R\}\right)}
.
Proof of equivalency to standard Turing machine [ edit ]
This will prove that a two-track Turing machine is equivalent to a standard Turing machine. This can be generalized to a n-track Turing machine. Let L be a recursively enumerable language. Let
M
=
⟨
Q
,
Σ
,
Γ
,
δ
,
q
0
,
F
⟩
{\displaystyle M=\langle Q,\Sigma ,\Gamma ,\delta ,q_{0},F\rangle }
be standard Turing machine that accepts L. Let M' is a two-track Turing machine. To prove
M
=
M
′
{\displaystyle M=M'}
it must be shown that
M
⊆
M
′
{\displaystyle M\subseteq M'}
and
M
′
⊆
M
{\displaystyle M'\subseteq M}
.
M
⊆
M
′
{\displaystyle M\subseteq M'}
If the second track is ignored then M and M' are clearly equivalent.
M
′
⊆
M
{\displaystyle M'\subseteq M}
The tape alphabet of a one-track Turing machine equivalent to a two-track Turing machine consists of an ordered pair . The input symbol a of a Turing machine M' can be identified as an ordered pair
[
x
,
y
]
{\displaystyle [x,y]}
of Turing machine M . The one-track Turing machine is:
M
=
⟨
Q
,
Σ
×
B
,
Γ
×
Γ
,
δ
′
,
q
0
,
F
⟩
{\displaystyle M=\langle Q,\Sigma \times {B},\Gamma \times \Gamma ,\delta ',q_{0},F\rangle }
with the transition function
δ
(
q
i
,
[
x
1
,
x
2
]
)
=
δ
′
(
q
i
,
[
x
1
,
x
2
]
)
{\displaystyle \delta \left(q_{i},[x_{1},x_{2}]\right)=\delta '\left(q_{i},[x_{1},x_{2}]\right)}
This machine also accepts L.
References [ edit ]
Thomas A. Sudkamp (2006). Languages and Machines, Third edition. Addison-Wesley. ISBN 0-321-32221-5 . Chapter 8.6: Multitape Machines: pp 269–271
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Multi-track_Turing_machine&oldid=1227182275 "
C a t e g o r y :
● T u r i n g m a c h i n e
H i d d e n c a t e g o r i e s :
● W i k i p e d i a a r t i c l e s t h a t a r e t o o t e c h n i c a l f r o m J u n e 2 0 1 8
● A l l a r t i c l e s t h a t a r e t o o t e c h n i c a l
● T h i s p a g e w a s l a s t e d i t e d o n 4 J u n e 2 0 2 4 , a t 0 6 : 4 9 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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