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2.3.2 Formal definition of a Turing Machine Model
There are lots of variations of the basic definitions of a Turing machine in
the literature, differing in details and level of formalization. I will choose an
approach that is quite formal in order to remove as much of physics that is
22
possible, following [16]. See also [18] for a modern treatment. The parts of the
definition are all motivated by the intended semantics of Turing machines.
Program
Consider an alphabet A consisting of the following tokens
M
1. a finite non-empty set of symbols Σ = S ,S ,...,S , not containing
1 2 n
{ }
the special tape blank # or any other tape markers. Σ is called the input
alphabet.
2. a finite non-empty set of tape symbols Γ, one of which is the tape blank
S = used to mark tape ends. Γ also contains any other tape markers
0
⊔
such as # used to separate the input into tuples. Note that Σ Γ.
⊆
3. a finite non-empty set of internal configurations Q= q ,q ,...,q , also
1 2 n
{ }
called machine states.
4. a (small) finite set of halting configurations Q , where Q Q= .
h h
∩ ∅
5. a set of moves M = L,R 7
{ }
One of the machine states q is singled out as the start state. It is also
0
convenient to single out halt states. If the machine is programmed for compu-
tation problems, one halt state, q , suffices. If the machine is used for decision
h
problems, two halt states q ,q corresponding to the answers yes or no, are
y n
{ }
singled out. Note that the halting states are not in Q.
Anexpression isafinitesequenceoftokenschosenfromA . Aninstruction
M
is an expression having one of the following forms
q S S q R
i k l j
•
q S S q L
i k l j
•
The intuition is that, if the machine is in the configuration q scanning the
i
symbol S , it prints the symbol S , changes configuration to q and it makes a
k l j
move R or L.
A Turing machine M has a program P that is a finite non-empty set of
M
instructions. The programcan be thought of as defining a transition function
δ :Q Γ (Q Q ) Γ M. (2.2)
h
× → ∪ × ×
This definition makes explicit that there are no transitions from the halting
configurations.
As an example of the correspondence between instructions and the transi-
tion function, note that the instruction q S S q R corresponds to δ(q ,S ) =
i k l j i k
(q ,S ,R). If the transition function is undefined for a certain q S then there
j l i k
simply is no instruction in the program P with the first two symbols equal
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7Sometimesitisusefultoincludea”nomove”S(Stay).
23
to q S . In that case the machine gets stuck. This should be considered as a
i k
programming error, unless the configuration q is not one of the halting config-
i
urations.
No two instructions have the first two symbols q S the same. This means
i k
that at each step of the computation, the action of the machine is uniquely
determined. Therefore, what we have defined so far are deterministic Turing
machines. Removing this restriction leads to the classes of non-deterministic,
probabilistic and quantum Turing machines respectively. These will be consid-
ered in sections 2.7, 2.8 and 6.2.
To prepare the way for this generalization, the transition function can be
defined in a different way that will be useful when discussing generalizations of
the Turing machine concept. Define the function ∆
∆:Q Γ (Q Q ) Γ M 0,1 . (2.3)
h
× × ∪ × × →{ }
Clearly,this is afunction frominstructions tothe set 0,1 . Forallinstruc-
{ }
tionsinthe programP ,∆evaluatesto1. Otherwiseitevaluatesto0(i.e. the
M
instruction is not contained in the program).
This can be formalized somewhat further. Consider the set I of all possible
instructions
I =Q Γ (Q Q ) Γ M, (2.4)
h
× × ∪ × ×
This is a finite set and a programP is a subset of this set, or P I.8
M M