TC+ CON(TC*) ⊢ P!=NP, P=BPP, What is the value of this statement?

(Could we remove CON(TC*) in the above deduction? TC- P!=NP, P=BPP)

Farzad Didehvar

Amir-Kabir University of Tehran

**Abstract.** Here, we try to answer some questions about the project "Fuzzy Time and the possible impacts of it on science", mainly proposed by people around the author. Beside all

Show how to remove CON(TC*) in TC+ CON(TC*) ⊢ P!=NP.

**Keywords.** TC, TC* (Theory of Fuzzy Time Computation), P!=NP, P=BPP, scope*, Negative time model, Fuzzy time model

**Introduction.** Based on some discussions with my PhD student, friends and the questions which proposed, the author writes this paper. Section one, starts with a dialogue and the questions and answers are not necessarily exactly the questions which were proposed.

In section2, by removing Con(TC*), we give a way to prove TC- P!=NP, P=BPP.

**Section 1. The first questions**

In this section we try to answer the questions which enlighten the previous
discussions.

1. **Q:** What is the difference between these two statements ?

TC+ There exists a random generator ⊢ P!=NP, P=BPP

TC+ CON(TC*) ⊢ P!=NP, P=BPP

**A:** In the first statement, we should have at least one random generator, in the real world, something that is severely doubtful. Even the existence of Psudo randomgenerators as a weeker and modified concept of random generator, are under the question.
In the second statement, we have a model in which we have random generators, actually this model is not necessarily a model of real world, (nevertheless here there is a similarity between this world and real world, any way mathematically, similarity is not necessary). The proofs of the second are in [1], [7]. In [7] by applying scope* the author gives the first proof, later in [1] it is shown that it is possible to omit scope* from proof. For example, in this world one of the three conditions in below hold

1. Either we have a fair coin in this world (A coin that we know it as a fair coin)

2. Or we have a coin by exactly the probability p, 1-p. (A coin that we know the probability of it distributed p,1-p) as generalization of first condition.

3. Or we have a function $f:R→R$ such that at each time t belongs to R, the probability of tossing coin is $f(t)$, $1-f(t)$, which $f(t)$ is sufficiently enough near to $\frac{1}{2}$(or any p between 0 and 1).

4. We know there is a coin and a function f such that ...

In probability and statistic we have the same situation. It means no real coin could be considered fair or satisfies the above descriptions, but in an imaginary and abstract space we have a faire coin, which our real coins are somewhat approximation of that. Anyway, in these theories we have an imaginary or absteact coin, which satisfies one of the three above conditions. Actualy in probalitty theory we have more hypothesis, like below

"Not only we have such a world but also this imaginay world is sufficinttly enough closed to the real world".

As a conclusion, the above hypothesis CON(TC*) is a usual hypothesis as similar hypothesis in many fields of science. Therefore, not only this hypothesis is not surprising, but it is essential to promote the subjects, as we see in statistics. So scientists has this hypothesis as a background of their thoughts and in their practical purposes. In other words and as a consequence, practically by our major deduction, we accept P!=NP.
**Q:** Why fuzzy time? We need only existence of one random generator. We are able to consider a simpler model, for example by considering one abstract fair coin.

To have this conclusion and a model with random generator we needn't to have fuzzy time, having a faire coin seems enough.

**A:** You are right, but we use this model not only here and for this conclusion. There are two other reasons for that

1. Our goal is not simply solving these two problems, we have the other problems which we wish to solve them, as MA*, AM*.

2. Unifying quantum computation and classical computation as it is explained in [1].

**Q:** A more personal question, How and when do you start to study Logic?

**A:** The first book which I see the logical topics come back to my childhood, a turquoise book (blue, Firoozi) about Aristotelian logic. I don't remember the title. It is certain that, one of the major reasons of our interest to logic was the discussions and debates in the atmosphere of Iranian Revolution. Specially, the problem of Idealism and Materialism and mind and body problem.

This Philosophical problem, interests to Mathematical structures and conditions, the rule and the restrictions of environment are three factors shaped the future way.

In 9th years of our education in Iran we studied simple symbolic logic to be used for Mathematics, mainly based on Gholamhossein Mosaahab written book, about Logic. Then a book of Mir publisher written by Ershov and Palutin "Mathematical Logic", a book translated by M.Ardeshir about Gödel Theorem and a book titled by "History of Logic" by Makovoleski translated in Persian (I did not find it in net). Enderton book "A Mathematical Introduction To Logic" was translated in that time, by Gholamreza Baradaran Khosroshahi, Mohammad Rajabi Tarkhorani.
By the above books, it was the first time which I see the concepts like
Model, proof systems,... etc.

After high school, in university we face a mixture of Philosophical and
logical discussions. In that time and later S.Shahshahani was
interested in Philosophical subjects of Mathematics. Under influence
of J.Larijani, logical courses proposed more and more in Sharif
university. H.Vahid presents some very useful lectures in
Philosophical Logic. After collapse of Soviet Union, Russian Logicians
mainly from Moscow and Novosibirsk, met Iran. (Yuri Ershov,
Lubetski, E. Palutin, Sergey Goncharov, Andrei Morozov, Marat
Arslanov), the author taken part in all of their courses. Besides them,
Ali Enayat give some courses in Model Theory and Set Theory.

M.Ardeshir has a crucial impact on Non classical Logic and
Mathematics in Iran. The major role of Zia Movahed, his books,
seminars and lectures shouldn't be forgotten. Author took part in
these courses. My PhD subject was about Complexity Theory and
Model Theory on a problem which was proposed by Bruno Poizat (A
model theorist from France). I read his book, A course in Model
Theory besides Chang & Kiesler book Model Theory.

The books which I studied in Computability Theory: Some of them
Computability Theory: Cutland book, roger book, barry cooper book,
odifreddi book in Computability Theory,
Complexity Theory: Michel Sipser book..., Oded Goldreich..., Sanjeev
arora book.

I teach some courses in the subjects of, Complexity Theory,
Computability Theory, Logic and foundation of Mathematics.

This is about the author, later the author see the other attempts in
growing the logical subjects in different universities and in different
subjects. Much later, I met people from the other disciplines in logic.

The above answer is an excuse to show a part of progress of logic in
nowadays Iran. This picture should be completed... .

**Section2. To put aside the second condition**
In this section we give a suggestion to reach TC+ CON(TC*) ⊢ P!=NP, P=BPP from TC+ CON(TC*) ⊢ P!=NP, P=BPP.

**Q:** In TC+ CON(TC*) ⊢ P!=NP, P=BPP, is it possible to remove CON(TC*)?

**A:** All is based on the following conclusion.

**Conclusion 1.** TC ⊢ CON(TC *)

**Proof.** In TC, we have ZF axioms. By ZF axioms we construct the fuzzy time model of Turing machines as it is in [1]. Actually, by this construction we have ZF ⊢ CON(TC *). Consequently, TC ⊢ CON(TC *).

**Conclusion 2.** TC ⊢ P!=NP, P=BPP

In [13], we give another proof but it needs a correction...

**Important Remark.**

By applying the method used to prove TC +con(TC*) ⊢ P!=NP, P=BPP, we have a world in which time is fuzzy but the computations happen as it happen in classical world. In this world we have random generator, so we have one way function. The same function exists in classical model of computation. "*So one way function exists*".

It answers one of the questions of some friends.

**Connections to the other field of Science (Model Theory)**

In a general view, these discussions are based on the thesis of the author [2] which is in streamline of a logical approach which in return, it was abstraction of an approach to redefine the concepts of Theory of Computation to be useful for Numerical Analysis [3], [4]. As a brief of the history of evolution of this theory

We have the following
## 1. Precedents

1. An approach of computability based on the suggestion of smale [3], [4] (<1989,2000<).

1.b. The logical method proposed by B.Poizat [5] based on the abstraction of the above concepts, also Pascal Koiran et al works. (~1990-2010)

## 2. The works by the author

2.a The thesis work, in the streamline 1.b. (Trying to continue it, 1999-2015) [2]

2.b. Changing the strategy, to have another logical approach(2007-2009, 2018).

2.c. Introducing Fuzzy Time to solve these problems(2018,2019-2023,2024) [10], [14], [8], [9].

## 3. Trying to work with the others (As the summer and the end of summer 2024 till ...)

The motto is:

Seemingly, the approach in this series (Fuzzy Time and the possible impacts of it on science), as a continuation of the above discussions could impact different scopes of Science.

Till now, we mainly speak about “Theory of computation” and Physics.

Although, from the abstraction level, Model Theory and Model Theorists (Bruno Poizat) was in these discussions and the general concepts of Model theory is used here, but technically the theory is far from to consider having a large commonality
with Model Theory till now. Nowadays, In Iran Model theory is studied.

At least, we have two possibilities which could make the above picture different.

1. Continuation of the above discussion and $\mathfrak{RG}$, as it is explained in ....

2. Bi Theory Method [6]

**A brief of the main article[1]**

**A. How do you classify the subjects in the main articles?**

**Q. As a classification we have.**

1. Modeling Fuzzy Time

2. TC* (Definitions and some Theorems)

3. The impact of TC* on TC (TC+ CON(TC*) F P!=NP, P=BPP)

4. Unifying TC & TC*

5. Generalizations (Polynomial hierarchy and Quantum Computation)

**Q. What is impact of Philosophical debates and discussions on this subject?**

**A. In general, to determine and recognize the line of research needs Philosophical type of thinking.**

Usually people with high standards in mathematical education and Problem solving are trying to solve these types of problems. But some type of Philosophical thinking is necessary too. Usually in arguing or computation we make some
mistakes, if we have Philosophical reasons for what we did and
do, we will not be disappointed but without it, people usually
quit the subject and attempting on that.

Here, the general view is a logical approach to solve P vs NP
problem. The basis are Gödel Theorem and Unexpected
Hanging Paradox. The philosophical impact of specific culture
here, is to think about the problem in fuzzy type of thinking.

**Q.** A simple question. Where do you define NP*, BPP* in the
main article?

**A.** NP* is defined in page 9, just after definition4. In definition 4, for NP classes, we should put Fuzzy Turing Machine instead of Turing Machine. Fuzzy Turing Machine is just Fuzzy time model of Turing machine.

About BPP*, like above we consider the Fuzzy time model of
Turing machine instead of standard model of Turing Machine.

**Q.** Do you find any mistake in the article?

We find an important mistake in the article, Theorem 4 (but
not with me by S.Fa, when we read it again). First of all it
didn't harm main stream of proofs, since we will not use this
proof later.

**Q.** Have we the other type of model instead of Fuzzy time model?

**A.** There is another type of model which is introduced recently,
"Negative time model".[12]
This model could be explained by Fuzzy time model. Since we have backward and forward time in this model the computing model of both are similar.

**Q.** And final question, what is the point in the proof? What will be the result for the theory?

**A.** In appearance, the solution of these problems should be find in an Probabilistic-Combinatorics approach. But it is shown that the solution is in an Logical-Physical approach with a philosophical taste. It might be the reason why these problems didn't solved yet.

The type of this problems was hidden. The major success and finding is to recognize how to see these types of problems, and discovering what really these problems are.

Usually in theories we start from simple problems and gradually we reach to harder and harder problems. We know this as a natural situation. The real situation in Theory of Computational Complexity is not like this. We have too many hard problems from first. By these processes the theory find a different face, much near to the usual situation of theories. We call it naturalization of theory!

The above shows why we need the process which is passed for this solution [2],[8],[9],[10],[1],[6]. We need this process to understand the type of problem. This is based on some previous works [3], [4], [5].

As a final result, in this study we point the Achilles heel of this area of science, Complexity Theory.
**Acknowledgement.** I should thank my friends, my young friends and my Ph.D student because of their useful questions.

This article is somewhat different to the previous articles since it is mainly to answer the questions of some friends, specialists in the different fields, across the globe.

References.

1. F. Didehvar, Theory of Fuzzy Time Computaion (TC* vs TC, TQC), hal-0433081

2. F.Didehvar, Some Concepts and Problems in Theory of Complexity in Structures, published as "Generalization of Some Concepts Theory in Structures", Ph.D Thesis, 2002.

3. L. Blum, M.Shub, S. Smale, "On a Theory of Computation and Complexity over the real numbers: NP-Comleness, Recursive Functions and Universal Machines, Bull.Amer.Mth.Soc.(N.S.) 21 (1): 1-46 (July 1989).

4. L. Blum, F.Cucker, M.Shub, S. Smale, Complexity and Real Computation, Spinger, 1998

5. B.Poizat, Les petit cailloux. Une approach modele-theorique de l'Algorithmie, Nur al-Mantiq wal-Ma'rifah, no.3.Ale'as, Lyon 1995, 2 pp.

6. F. Didehvar, Bi-Theory Method & Navier Stokes Problems, hal, 18/7/2024

7. F. Didehvar, Theory of Fuzzy Time Computaion (2), (TC+ CON(TC*) ⊢ P!=NP), Philpapers.org

8. F. Didehvar, Is classical Mathematics appropriate for Theory of Computation?, 2017, Vixra, Philpapers 2019

9. F. Didehvar, Zeno Paradox, Unexpected Hanging Paradox (Modeling of Reality & Physical Reality, a Historical-Philosophical View), 2022, SSRN

10. F. Didehvar, Fuzzy Time, from Paradox to Paradox, December 2019, Philpapers, The article is presented in 50th Annual Iranian Conference of Mathematics.

11. F. Didehvar Some Concepts and Problems in Theory of Complexity in Structures, PhD Thesis, IPM, 2002

12. D. Angulo et al, Experimental evidence that a photon can spend a negative amount of time in an atom cloud, 5 sep 2024
13. F. Didehvar, TC+ CON(TC*) F P!=NP, P=BPP, What is the value of this statement? (Could we remove CON(TC*) in the above deduction? TC- P!=NP, P=BPP), Philpapers, 2025

14. F.Didehvar, [TC+CON(TC*) deduce P is not equal to NP](https://philpapers.com/article/14-F.Didehvar_TC+CON(TC*)_deduce_P_is_not_equal_to_NP), Philpapers, 2020