 Good afternoon everyone, and our friends online, can you hear me okay? Hopefully yes, shout out to not hear us. And it's a very great pleasure for me to introduce this afternoon's seminar speaker, Professor Hasan Akin. Professor Akin is in his MSc in Disney from 100 University in Turkey, and worked for many years at Howard University and Summit University in Turkey. He has an impressive list of publications in fields ranging in between statistical mechanics and the robotic theory and dynamical systems. And in particular one of his interest is cellular automata, and he will give us a brief introduction to the subject today. Thank you very much. First of all, I would like to thank you. I think maybe you can remove your mask. All for your participation in my seminar. Also, I would like to express my sincere to the Institute of International Education, Scholar-Rescue-Form, and the Sims Foundation for their support. In addition, I would like to thank ICT for support and hosting. This seminar will be an introduction to cellular automata. So I think that the most of you are hearing this subject first time, maybe. Therefore, I want to explain the simple concept and the introduction to cellular automata. If the time is not enough, maybe I would like to give an additional seminar. In this seminar, I will talk about the algorithmic properties of cellular automata. And firstly, I give some definition, invertibility, one-day cellular automata, especially. You know that there are higher-day cellular automata. Now in this seminar, especially, I focused on the one-day cellular automata, some algorithmic properties, kind of entropy, and other properties. First, I will give some invertibility, one-day cellular automata, then algorithmic properties, one-day cellular automata. One-day cellular entropy, topological entropy, directional major theoretical entropy. And lastly, I will talk about the direction of topological entropy, one-dimensional, infinite cellular automata. Cellular automata, first used in the field of physics and biology. One-dimensional cellular automata, first used by Ulam and Fanluman. Another mathematical point of view, the Hadron defined the cellular automata, especially Hadron defined the leading block mapping, first definition. You know that study of the dynamics told that cellular automata is a remarkable attention, especially in the last few years. Also, cellular automata investigated in a lot of disciplines, for example, physics, chemistry, biology, and neuroscience, a lot of different poses. In this section, I will talk about the invertibility of one-dimensional cellular automata, especially in the order ring ZM. Lastly, the Manziner-Margara defined the, sorry, studied and given an extended formula of the computation inverse of linear cellular automata over ring ZM. Of course, if the cellular automata is invertible, then related to global transition, also we check the linear cellular automata, because in this case we should check the global state. Okay, let us consider ZM, the ring ZM. Also, ZM via the sequence of all double infinite segments. You know that ZM powers the topology of direct product, and also ZMZ is a totally disconnected space, compact space. Also we can study to find the shift map, you know, shift map. Sorry, what was that? Shift map, we can say that this is a shift by sigma. Also, the shift map is a more and more compact space, CMZ. We can define the cellular automata. Cellular automata is generally generated by the local rule. What is the local rule? For example, we can close the sequences, especially we used the notation. Is this an example you're having now? No, no, definition. So, we say that the double sequences, and we define the local rule. You know that the shift map is shifting the sequence, and in this case, cellular automata, is reporting with any finite flow. So, we obtain the new one. For example, left, right, okay. In general, we use the summation. Can you give us an example? Yes, this is an example? Yes. We have defined the local rule F, they have the state local rule F, linear, if and only if, it can be right as this one. Sorry, I just don't understand the first example. Can you give us just a very simple example, you say it's a map commute? Yeah, I can give a simple example. For example, let's consider field minus 1, 1, c, a, for example, x minus 1, x0, x1 equals 2, x minus 1 plus 1x. So, we can define the cellular automata by means of this local rule. In this case, we obtain the new sequences and iterations. For example, this is sequence, double or we find a sequence, we obtain the new x, for example. So does this commute to the shift map? Yeah. What functions exactly commute to the shift? Not for this. Okay. In this case, we say that T minus 1, 1, c, a, dF minus 1, 1, c, a. Close shift map, shift things, all right on the left. So, what is the map T defined? T defined? Yes. Okay. That's the map F. F, yeah, okay. So what's the map T? What's F minus 1? T, F, F is a local rule. So how is the map T defined? What is the image of the sequence? Yes. The first line, second line. Yes, the coordinate I. Is F applied to that? Oh, I see. For every coordinate, you take the two next to it. Okay. Of course, you can define the higher dimension. At each coordinate, you take a few on this side and you take some average. For example, you can define the higher dimension. Okay. This map T commutes with the shifts. It's reasonable if the map F is local rule is linear. In general, but is that necessary? No, no, no. There are a lot of linear non-linear cell auto-automata, but linear non-linear cell auto-automata is very difficult problems. In general, I studied linear once. But there are examples where the local map is known. Yeah. So the linear ones, they always commute dramatically. Of course. Yes. Okay. Sorry. Okay, in this case, I should define the permutative or permutative. Permutative, it's very important for the studies. For example, invertibility, for example, you know, a very properties and other properties in general, use the permutative or permutative cell auto-automata. Local rule F is permutative in X, Y. If for any given sequences, in this case, we have ZF. The local rule F, the local rule of linear cell auto-automata permutative, this position, if I, if and only if, it satisfies the question. Okay. Now we can define the way we can study the invertibility of one-dimensional cell auto-automata. First, I want to investigate the M cos P over K. In this case, for example, we can, if we can write the formal power series, sorry, formal power series F, we can write, if we say, in this case, we obtain the inverse of the F. And related to, also we can find the corresponding cell auto-automata. For example, that if F of X is given as question 3, then the local rule F associated formal power series can write and follow this case. Also, so we need to follow the theorem on concept problem, invertibility, finite power series. So, this is the theorem given by Marguerite Manzlin and Marguerite. Let F of X be denoted invertible, finite formal power series over Z F P. H and F to define the S3. That's A, G such that this case, then we obtain the inverse of the formal power series F. For example, in this case, for K equal to 2, and this problem is very simple. Okay, let us give an example. Assume that the local rule is defined as this one. Okay. And related to corresponding power series is one. Of course, we can compute the worst formal power series. Sorry, how did you get the power series from the rule again? Yes. This is a rule, a hard rule. This is formal power series. I can't remember. How did you get one from the other? There's a formula from one to the other. In general, three minus hope. Change it. Minus plus minus. In this case, but do you do that also with, when you use I, I mean, this is also for any I, right? The formula holds one, two and three are just examples, right? Usually we have I minus one, I and I plus one. And then in the formal policies, you also have I minus one. I and I plus one. Yes. Okay, so I'm just trying to understand. So we can check this formula for therefore we complete that the local corresponding F or my spot. So we can complete the inverse. Convolution consistency in this section I give some properties, the finisher, the versatility. Now I want to give the property. For a new is a sigma invariant mission. The following properties are worth first thing. This is a, this is a cylinder set. Also, this is cylinder set is a say that the new is a probability measure. Also non-negativity and this properties and this properties say that the total consistency theorem satisfied. In this case, the mistake is the theorem. So this theorem, the proof of the theorem type can really check it for Bernoulli and the Markov musules. Bernoulli measure, you know that Bernoulli measure. Here we define the probability vector. So Bernoulli measure of cylinder set C is defined. Define these equations where the high is probability vector and you know that the summation of one and the non-negation. Another measure is Markov. In order to define the Markov measure, we should define the stochastic matrix and also we should define the probability vector. This satisfies the equation. In this case, we can compute the Markov measure of any cylinder set, such as in this case, Ip011 elements and stochastic matrix. An example of this several automata are the ones of Wolfram, right? Yeah, Wolfram, for example. All of these several automata are the ones of Wolfram. Yeah, in general, Wolfram defines the elementary several automata. What is the elementary several automata? Elementary several automata defines all the C of the C2. A lot of the problems, very important problems, Wolfram. Stop on Wolfram, I know. So not all of them are important, right? No, I don't know. Maybe sometime, in this case, if you study in finite, maybe you can define the invertibility. But in this case, in finite case, if you take the Zp prime number, of course, there isn't an invertibility. In invertibility, we study the e-comp. In the elementary several automata case, in general, we study the finite several automata. So finite means on a finite alphabet? Yes, no, no. For example, in this case, we define the... Finite means it's a lot of load. Sorry, double in finite sequence, okay? Z, Z, for example. But in this case, the finite case, we, in general, we study the below, only this one, okay? In general, the applied, for example, physics, chemistry and other science, sorry, scientists investigate the... What do you mean by we study? You mean that this local rule depends only on a finite number of coordinates? I'm not sure I understand. The definition of several automata is always on a doubly infinite shift, is that right? So what is it that makes it finite? What do you mean by finite? For example, in the finite case, you show the finite sequence. You do what with the finite sequence? For example? Yeah, yeah, but what about the finite sequence? It's one, two. Only we consider this one. Only we consider... Finite sequence. So what do you mean by consider? Sorry, the... You consider it for what purposes? I mean, what do you mean by you consider? In general, in this case, we define the matrix, for example, in this case, the finite case, that we not study the matrix term. You know that these theorems can find both books. Let X be a major space, and let A be a semi-algebra that generates B. Let T, from X to XB, measure-preserving transformation, then you know that T is ergodic, if and only if, or A, B, M, and B, satisfy this equation. Secondly, the T is a beat mixing, if and only if, A, B, M, and B, satisfy this equation. And lastly, T is a stroke mixing, if and only if, for A, B, and M, B, satisfy this equation. Sorry, it wasn't different between the first and only the second. You know that, in this case, you know that if any transformation is stroke mixing, then the weak mixing and ergodic part, the university in general, in the second case, the second case you're taking the absolute value of the difference. Ah, the sum of the absolute value of the difference. It's a little bit excessive. You know that, you know that if T is stroke mixing, the second is not necessary. Then T is, I mean, I have not seen weak mixing value of the second, T is ergodic. University, generally, not good. Shivani and the rogers, who that all into absolute, onto absolute automata are invariant and stroke mixing with respect to harm use, harm use. Then Shershersky studied some strong ergodic properties of using natural extension of a major theoretical and no more. Also, he's answered the sum of questions in Reis-Sinai and Shivani. He also defined the anti-interaction of permutative, servo-automatic. Okay, we can give the lemmas. For example, this is very important for us. Another lemmas. Shershersky defined the given lemmas. We will use the permutative, suppose that servo-automata defined equation seven that F is a right and left permutative. In this case, and Markov measure, the final eight. The dinar cell of automata is uniform Markov measure preserving transformation. Also, this question is Bernoulli Markov, sorry, Bernoulli measure preserving transformation. I prove that this case, suppose that at least one of the conditions satisfied, then TFR, mixing with respect to uniform Bernoulli measure. Of course, you can study the uniform Bernoulli measure. Just to make sure I understand. The permutative, servo-automata is a map on the shift space. Similar shift, not shift. Yes, it was defined as a map, commute with the shift. But it's a map on the symbolic space. On the space. You can define the block and the coding by coding. It's a map on the same space on which the shift is a map. Yes, similar to shift map. It's similar to the shift map, but it's another map on the same matrix space. On the same matrix. On the same matrix. Yes. So you're saying that, so you're studying the robotic properties of this map. Yes. So when you talked about invertibility, did you mean actual invertibility as a map on the symbolic space? Yeah. That's what you mean by invertibility. That the map is invertible on the space. Yeah. Okay. Is that what you mean? It's just my question. Yeah. Just invertible as a map. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Note that that . B Ice goblin s vida L Sister oh Yeah. Okay. Sure. All right. So I an internet North respect to Bernoulli Museum or of course you can study the Markov Museum and for example Gipsy Museum. Also I studied the invertible stones mixing cello-automata under some conditions they have proved that every invertible one... Sorry, I just saw some questions. Okay, sorry they say they cannot see what's on the board. I'm not sure how we can resolve that. It says can you zoom in what the speaker writes on the platform? Oh I see. No problem. Okay, I'm sorry. Now I can study the major torque entropy. Vision torque entropy, a concept where it has been extensively studied in the very the variety of the disciplines for you, you know that in physics, chemistry, information theory, coding theory, and so on. Visit for different poses entropy. In ergodic theory, there are a lot of kind of entropy. For example to measure torque entropy, direction entropy, topological entropy, algebraic entropy, especially in the last years, this is a very popular subject in the ergodic theory, entropy. For example, in the theory to probability, the entropy is the measure of the insertion. Information theory, entropy is no corresponding entropy, carries the maximum amount of information in system. In ergodic theory, entropy infers that capacity of cannot come. There are a lot of types and kinds of entropy. In this case, we in general, in general, we defined three steps. First steps, quantity, quantity, each new alpha defined is one. It's called entropy of the partition. In general, we take the base to the variety, maybe you can change. In general, information theory, we use the variety. Secondly, let alpha be a partition of the finite entropy. Then the question, sort of quantity, is called the entropy of key with respect to alpha. So in this case, if we change the alpha partition, then we obtain the different entropy, quantity entropy. Also, if you can change the measure, then the entropy, lastly, the quantity, quantity, each new key is called the measure of entropy key with respect to new, in this case, new exchange, change the entropy values of entropy change. So, you know that with the help of the partition or sub-algebras, it is difficult to calculate the entropy line system from the definition, because we need to calculate for all partitions. In this case, in general, we use the Kolmogorov-Sneitor. Kolmogorov-Sneitor, in order to study the Kolmogorov-Sneitor, we need to define the strong generator. So, strong generator definition, Kolmogorov-Sneitor, if you can see the focus of books, and alpha, the measure of the partition of the Kolmogorov-Sneitor system, the partition is called the strong generator, if satisfied with the definition. So, in this case, we can define, we can compute the Kolmogorov-Sneitor in case it is a measure for the transformation, but not necessarily invertible of the measure probability space. And if A is a finite sub-algebra key, we can case, we have this one. Yes, that is alpha, that is space. Sorry, sorry, you are, this alpha, please, we write alpha, the seed of alpha, sorry. So, A should be alpha. Yes, yes, yes. Sorry, I misread it. I hope that the measure can drop the additive sub-automata generated by, by permutative local rule with respect to uniform Bernoulli-Mission. And let us consider the finite field and then correspond the local rule of radius R is depermutative in this case. Of course, we can define the Bernoulli-Uniform Bernoulli-Mission, such as this case, we define the partitions. See, by means of partitions, we obtain the slope generator for one-dimensional sub-automata defined by f. Then that new p pi be the uniform Bernoulli-Mission. And for each element at p, then this case, then the entropy of the measure present time, present time system is equals to my first result. For example, let us consider the F local rule. For example, modulo 19. Then we compute, can compute measure total entropy, 5 over 5. What does modulo 19 mean? You've got a shift on 19 elements, your space has 19 elements. We put in half of it. Yeah. I see. Another theorem, we obtained another theorem. We generalize the four ZM remits. We obtain this equation and Markov-Mission find. Also, I give another theorem that new p pi p Markov-Mission given by the stochastic matters and the probability vector. And assume that positive integers satisfy these correlations. And also f be a one-dimensional sub-automata, then we can compute the entropy, measure total entropy. Now I can study the topological entropy. In this section, I will investigate the important numerical quantity called the topological entropy of the continuous map, find only compact metric space, topological space. Firstly, the heart showed that topological entropy cannot be calculated algorithmically. Then, Damico had all computed the entropy of two important class and taking into account approached by Damico. We studied the quantity entropy of one-dimensional sub-automata or ZM-Z. Let us consider this, sorry, we say that space-time evaluation or space-time theorem, the theorem v and the height t. In this case, the number of these rectangles of v, w, and height t occurring in the space diagram, then we cannot reduce, reduce of the left local rule f associated with tf. And it is easy to see that this this inequality is satisfactory. So, definition of topological entropy in case sub-automata, the topological entropy is given this formula, this formula defined by Damico at all. So, for example, another theorem given by Damico at all, that's tf, tv, one-dimensional sub-automata, and the prime factor of the composition of m. So, we can define the ti set and le le. So, in this case, we can compute the topological entropy by means of le and le. For example, let us consider the local rule, what do you do, 36. And let us, we can see that pi 2, pi 3 is 2, 2, 3. And we obtain the l2, l2 and so on. So, we can compute the topological entropy. For example, then I obtain another theorem and we define the case and the local rule 1. If t is invertible, then we can write this equation. You know that in the algorithm the topological entropy, t equals topological entropy, invertible. Similar, model theorem, any corollary we can obtain this. For example, if we obtain this condition, then we can obtain the topological entropy. Now, I can give the direction entropy, firstly, measure topological entropy and lastly, I will give the direction entropy. Milner, John Milner, you know, John Milner finds the notation of the entropy. Also, Sinai obtain some smaller problems, direction entropy, measure topological aspect. And the condition and the part to find the direction for z2 action, actions on the on the log that explains. And then Kaminsky and Corvich investigate the metric direction entropy, z action of z2 can generate five local rules and shift them. We can sum the fine, give some time. For example, let's see the z2 action on the measure space for a subset R2 and the p is a partition. Then we can write this equation okay. In this case, we, in general, we define the topological, sorry, direction entropy by means of the block mapping or the cellular automata and shift map, two maps. We have two maps, shift maps and shift map and cellular automata. In this case, you can see that the direction of entropy, measure topological aspect, is equals one. And he can define this notation. And the quantity f, sorry, hf vector, we call the direction mean entropy, z2 with respect to partition in direction v. And this is another definition. The quantity hv, c is once, it's called the direction entropy of c in the direction v. And this definition given by Corvay and Cummings. Okay, so I just say that you see the cellular automata. Maybe I can change one. In this case, from the definition of entropy, we obtained this one, also another property. So we can summarize this results obtained, obtained Corvay and Cummings, one is case, second case. And if the local rule is different by derivative, then the direction entropy, measure topological aspect, this equals zero. It is similar entropy. I have obtained some results with respect to non-limiter and the Markov vision. In general, in this case, I have studied the upper bones, measure, measure topological entropy. You know that entropy is positive or negative. So I can, or I can study the upper bones. So my other result, let's f be like the nearest local rule, find this one. And we obtain the probability vector, then we have upper bones. For example, let us give an example, minus one, zero, one, this is a local rule. Then we obtain the formal power series. And we obtain the inverse of the formal power series. And related to the minus, sorry, local rule is one. Okay, in this case, we obtain the measure topological entropy with respect to Markov vision. I obtain another result. So in this case, I consider the Markov vision. Generally, I take the uniform value of measure, then we obtain the measure topological direction entropy, upper bones. Now I can give the definition of direction topological entropy. In this section, I studied the quantity of topological direction entropy, the issue of action similar to algorithm dynamical at all. Then I presented an algorithm complicating topological direction entropy generated by shift map and pseudo-automation. But let's consider this case, the windows. In the topological entropy, we consider the rectangles. In this case, we consider the parallel outcomes. For example, the direction change, then entropy change. So in this case, the definition of topological or directional topological entropy is given by similar to topological entropy. But in this case, we consider the numbers of the parallel outcomes. This definition is by C9. The formula of directional entropy of depended to subautomata, you mean that the Bernhardt and Corbidge defined another aspect of the direction entropy, aspect of the topological. We can define this. We can give the lemma Corbidge that has presented the direction entropy by this equation. In this case, if it is very important in this case, if the theta pi over 2, the directional topological entropy takes much more than the equals to topological entropy, direction entropy, equals to direction entropy. Of course, the exchange. For example, you can see one example. For example, let us consider the finite local rule, order of 5. And then this is, of course, in this case, it becomes 5. We obtain the theta and so on. So we can you can draw the sector corresponding to angle theta minus 4 of this one. And by using the formula, we can compute the entropy related to the local rule, this one. Also, this is a graph. The direction entropy curves for B-permitted subautomata generated by Mukherjee even. Okay, my best theorem. Let the tf be a subautomata or wings. Here. And then we obtain this set and similar to the p pi, so p, and we obtain the entropy. Generalize the formula topological direction entropy. And in this case, sorry, I give the example, another example, let us consider this one, but small. The two case, in this case, two, three, four, we can obtain. And by means of these values, we can obtain the sectors, sectors. Of course, in this sector, the entropy exchange, the sector alone, the sector will change. So we obtain the formula. And you gave us a lot of results and theorems and formulas there. Sorry. That's okay. Thank you. Are there any questions? Yeah. So in the definition of the tf automata, the crucial difference between it and the shift map is that in the definition of the tf automata, it's emphasized that the alphabet is involved with the ring structure. I wonder what's how this comes to definition, why I made that point. So where the definition you also consider only that you have a ring of an alphabet? Of course, you know that the ring is changed, then the subautomata is changed. So if I understand correctly, it doesn't use the ring structure. Just of course, using the word because that's a lot of subautomata. For example, left, right changes and the ring and the fields change. I think Hansa's question is, do you use, so when you usually consider the shift map on a finite alphabet, you just talk about having a finite alphabet. Whereas in your case, you talk about this finite alphabet being a ring. This is your question, right? So do you use the fact that this finite alphabet has a, is it just a finite alphabet or is it actually, do you use in the definition of cellular automata the algebraic structure of the ring of this? Generally, we use the algebraic aspect. Okay. Okay. Well, let's thank the speaker again. Thank you very much.