Farzad Didehvar

در تکوین نظریه محاسبات از اوایل قرن بیستم پارادکسها و خود ارجاعی نقش ویژه ای را بازی کرده اند. هر چند نظریه محاسبات عام بر اساس تعریف ماشین تورینگ، فرض تورینگ_چرچ و کاربردهای آن بنا شده ،اما از همان ابتدا تا به امروز منطق و حوزه های مختلف این علم در ارتباط تنگاتنگ با این تیوری و در ابتدا نظریه محاسبات خاص بوده و این ارتباط روز به روز گسترده و گسترده تر گشته است. از تاثیر پارادوکس دروغگو و پارادکس بری در تمایز میان مفهوم محاسبه پذیر و محاسبه پذیر شمارایانه، قضایای گودل و قضیه چیتین همه جا شاهد این تاثیر و تاثر متقابل هستیم. در این مقال می خواهیم بعد از معرفی اجمالی مطالب باال، تاثیر یک پارادوکس دیگر، پارادوکس اعدام غیر منتظره، را بر این نظریه بررسی نماییم.

In our research about Fuzzy Time and modeling time, "Unexpected Hanging Paradox" plays a major role. Here, we compare this paradox to the Zeno Paradox and the relations of them with our standard models of continuum and Fuzzy numbers. To do this, we review the project "Fuzzy Time and Possible Impacts of It on Science" and introduce a new way in order to approach the solutions for these paradoxes. Additionally, we have a more general discussion about paradoxes, as Philosophical back ground of the subject and in this way, we introduce the concepts of General View, the first picture, BBF-General View.

Here, the author tries to build the structure of the Theory of computation based on considering time as a fuzzy concept. In fact, there are reasons to consider time as a fuzzy concept. In this article, the author doesn’t go to this side but note that Brower and Husserl views on the concept of time were similar [8]. Some reasons have been given for it in [3]. Throughout this article, the author presents the Theory of Computation with Fuzzy Time. Given the classic definition of Turing Machine, the concept of Time is modified to Fuzzy time. This new term calls as Theory TC* [2] and this type of computation “Fuzzy time Computation”. We have relatively large number of fundamental unsolved problems in Complexity Theory. In the new theory, some of the major obstacles and unsolved problems have been solved [2]. It should be noted that in this article, the writer considers fuzzy number associated to instants of time as a symmetric one. The point about the symmetry is in the proof of Lemma 3, although it is generalizable. In particular, the new classes of complexity Theory, P*, NP*, BPP* in the TC* analogues to the definitions of P, NP, BPP defines as their natural alternative definition. Here, we will see P*≠ NP*, P*= BPP*. Finally, we have Theorem 4.

Here, we try to build the structure of a Theory of computation based on considering time as a fuzzy concept. Actually, there are some reasons to consider time as a fuzzy concept. In this article, we don’t go to this side but we remind that Brower and Husserl ideas about the concept of time were similar [14]. Throughout this article, we present the Theory of Computation with Fuzzy Time. Considering the classical definition of Turing Machine we change and modify the concept of Time to Fuzzy time. We call this new Theory TC* and this type of computation “Fuzzy time Computation”. We have relatively large number of fundamental unsolved problems in Complexity Theory. In the new Theory some of the major obstacles and unsolved problems are solved. It should be mentioned that in this article, we consider fuzzy number a symmetric one. The point about the symmetry is in the proof of Lemma 3, although we are able to generalize it. More specifically, we define the new classes of complexity Theory, P*, NP*, BPP* in TC* analogues to the definitions P, NP, BPP as their natural substituted definition. We show P*≠ NP*, P*= BPP*. Finally, we have Theorem 4, P≠NP .

The reason ability of considering time as a fuzzy concept is demonstrated in [7],[8]. One of the major questions which arise here is the new definitions of Complexity Classes. In [1],[2],…,[11] we show why we should consider time a fuzzy concept. It is noticeable to mention that that there were many attempts to consider time as a Fuzzy concept, in Philosophy, Mathematics and later in Physics but mostly based on the personal intuition of the authors or as a style of Fuzzifying different various of the concepts. Consequently, fuzzifying time doesn’t go to be popular. In the new attempts we are trying to show why we are somewhat forced to consider time as a Fuzzy concept. It is mostly based on the “Unexpected Hanging Paradox” introduced by a Swedish Mathematician Lennart Ekbom. Our question is:” what will be the impact of it in Theory of Computation and Physics?”. Here, we discuss about the impact of fuzzifying time on Theory of Computation.

(THIS PAPER NEEDS A CORRECTION) Satisfaction is a complex concept which has a key role in each individual’s everyday life and impacts their behavior. Abraham Maslow (1943) suggested a framework [1] to study human motivation, which was a starting point towards developing the quality of life(QOL) theory. On that article, he described a hierarchy of human needs, that is generally consist of fundamental needs which are required for human survival, and environment dependent ones, like society, safety and etc. In this study, we are trying to represent a tangible mathematical model for assessing the sense of need for water in terms of the available source of water. On this very subject, M. Joseph Sirgy [2] states that ”From a human developmental perspective, QOL goals can be defined as satisfaction of human developmental needs in a community or society”. Accordingly, in the first section we propose a model for the water needs of an individual. In the second section, we continue the discussion by checking the behavior of the model on a given population, and each time by assuming that we have two out of three involved factors, we try to suggest an estimation for the other one.

The major point in [1] chapter 2 is the following claim: “Any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction.” So, in the case we wish to save Classical Logic we should change our Computational Model. As we see in chapter two, the mentioned contradiction is about and around the concept of time, as it is in the contradiction of modified version of paradox. It is natural to try fabricating the paradox not by time but in some other linear ordering or the concept of space. Interestingly, the attempts to have similar contradiction by the other concepts like space and linear ordering, is failed. It is remarkable that, the paradox is considered either Epistemological or Logical traditionally, but by new considerations the new version of paradox should be considered as either Logical or Physical paradox. Hence, in order to change our Computational Model, it is natural to change the concept of time, but how? We start from some models that are different from the classical one but they are intuitively plausible. The idea of model is somewhat introduced by Brouwer and Husserl [3]. This model doesn’t refute the paradox, since the paradox and the associated contradiction would be repeated in this new model. The model is introduced in [2]. Here we give some more explanations.

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics.

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time.

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic and continuous logic, and prove effective versions of some related theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; we show that provability degree of a formula with respect to a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory is decidable.

Throughout this paper, by representing some paradoxes and their associated proofs and arguments, we try to show the cases which proving some assertions doesn’t conclude the truth of them. In the next step, we try to find out Which proofs could be considered as reliable in a way that it shows the Truth of their related assertion, specially We claim that math- metical proofs could be considered as reliable ones in this sense. Nevertheless, we claim that the validation of the previous assertion is comparable to validation of Church Thesis in Computability Theory. So, we could use it and we consider these proofs as reliable ones but we should be doubtful about their truth. At last we give a possible explanation for this subject (based on local arguments and global arguments, the concepts which we introduce them throughout this paper), and we investigate the validity of this explanation. What we should bold it here is: In this approach we know the proofs in these paradoxes as correct proofs, but the point which we want to stress on that would be: the proof doesn’t show the truth..