Case Study of the Proof of Cook's theorem - Interpretation of A(w)

04/30/2019 ∙ by Yu Li, et al. ∙ Laboratoire des Glucides 0

Cook's theorem is commonly expressed such as any polynomial time-verifiable problem can be reduced to the SAT problem. The proof of Cook's theorem consists in constructing a propositional formula A(w) to simulate a computation of TM, and such A(w) is claimed to be CNF to represent a polynomial time-verifiable problem w. In this paper, we investigate A(w) through a very simple example and show that, A(w) has just an appearance of CNF, but not a true logical form. This case study suggests that there exists the begging the question in Cook's theorem.

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1 Introduction

Cook’s theorem [1] is now expressed as any polynomial time-verifiable 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 time-verifiable 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 time-verifiable problem and Turing Machine

A polynomial time-verifiable 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 time-verifiable 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 .

Figure 1: The computation on input .
Figure 2: The computation on input .

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 time-verifiable problem can be reduced to the SAT problem, but any polynomial time-verifiable 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. p151-158 (1971)

  • [2] Garey Michael R., David S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and company (1979)