1 Introduction
Cook’s theorem [1] is now expressed as any polynomial timeverifiable problem can be reduced to the SAT (SATisfiability) problem. The proof of Cook’s theorem consists in simulating a computation of
TM (Turing Machine)
by constructing a propositional formula that is claimed to be CNF (Conjonctive Normal Form) to represent the polynomial timeverifiable problem [1].In this paper we investigate whether this is a true logical form to represent a problem through a very simple example.
2 Example
2.1 Polynomial timeverifiable problem and Turing Machine
A polynomial timeverifiable problem refers to a problem for which there exists a Turing Machine to verify a certificat in polynomial time, that is, check whether is a solution to .
Let us study a very simple polynomial timeverifiable problem :
Given a propositional formula for which there exists a Turing Machine to verify whether a truth value of is a solution to .
The transition function of can be represented as follows:
0  1  
1  0  
1  1  
0  0 
where means that the tape head does not move, and means that the tape head moves to right; refers to the state where stops and indicates that is a solution to , and refers to the state where stops and indicates that is not a solution to .
2.2 Computation of Turing Machine
A computation of consists of a sequence of configurations: , where and is a polynomial function. A configuration represents the situation of at time where is in a state, with some symbols on its tape, with its head scanning a square, and the next configuration is determined by the transition function of .
Fig.1 and Fig. 2 illustrate two computations of on inputs : and .
3 Form of
According to the proof of Cook’s theorem [1][2], the formula is built by simulating a computation of , such as . is claimed to represent a problem .
We construct for the above example.
3.1 Basic elements
The machine possesses:

4 states : , where is the initial state, and , are two final states.

3 symbols : , where is the blank symbol.

2 square numbers : .

4 rules.

is the input size, ; is a polynomial function of , and .

3 times () et 2 steps to verify a certificat of , where corresponds to the time for the initial state of the machine.
3.2 Proposition symbols
Three types of proposition symbols to represent a configuration of :

for , , . is true iff at step the square number contains the symbol .

for , . is true iff at step t the machine is in state .

for , is true iff at step the tape head scans square number .
3.3 Propositions
1. , where represents the truth values of , and at time :

()); ())

and are determined by the transition function of
2. , where asserts that at time one and only one square is scanned :
3. , where asserts that at time there is one and only one symbol at each square. is the conjunction of all the .
:
:
:
4. , where asserts that at time the machine is in one and only one state.
5. , , and assert that for each time the values of the , and are updated properly.
, where is the conjunction over all and of , where asserts that at time the machine is in state scanning symbol , then at time is changed into , where is the symbol given by the transition function for .
:

, with the rule

, with the rule
:

, with the rule

, with the rule
, where is the conjunction over all and of , where asserts that at time the machine is in state scanning symbol , then at time the machine is in state , where is the state given by the transition function for .
:

, with the rule

, with the rule
:

, with the rule

, with the rule
, where is the conjunction over all and of , where asserts that at time the machine is in state scanning symbol , then at time the tape head moves according to the transition function for .
:

, with the rule

, with the rule
:

, with the rule

, with the rule
6. , asserts that the machine reaches the state or at time 3.
Finally, .
4 Conjunctive form of
We develop as a computation of for as input (see Fig. 1) in order to clarify the real sense of .
Let us define the configuration and the transition of configurations of :
: the truth values of , , and their constraints.
: is changed to according to the transition function of .
1. At , :

, representing the initial configuration where is in , the tape head scans the square of number 1, and a string is on the tape.

.

:


2. At , is obtained from .
is represented by , and at :
 , with the rule
 , with the rule
 , with the rule

, with , , , , and other proposition symbols concerning are assigned with 0.


:


3. At , is obtained from .
is represented by , and at :
, with the rule
, with the rule
, with the rule

, with , , , , and other proposition symbols concerning are assigned with 0.


:


Therefore, the computation of for as input can be represented as :
It can be seen that is just the conjonction of all configurations of to simulate a concret computation of for verifying a certificat of . Given an input ( or in this example), whether accepts it or not, is always true. Obviously, has just an appearance of conjunctive form, but not a true logical form.
5 Conclusion
In fact, a true CNF formula is implied in the transition function of corresponding to , , as well as , however the transition function of is based on the expressible logical structure of a problem.
Therefore, it is not that any polynomial timeverifiable problem can be reduced to the SAT problem, but any polynomial timeverifiable problem itself asserts that such problem is representable by a CNF formula. In other words, there exists the begging the question in Cook’s theorem.
Acknowledgements
Thanks to Mr Chumin LI for his suggestion to use this simple example to study .
References

[1]
Stephen Cook, The complexity of theorem proving procedures. Proceedings of the Third Annual ACM Symposium on Theory of Computing. p151158 (1971)
 [2] Garey Michael R., David S. Johnson, Computers and Intractability: A Guide to the Theory of NPCompleteness. W. H. Freeman and company (1979)
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