 In this lecture, we will look into certain variants of a Turing machine. You see, when we have discussed in case of deterministic finite automaton, there also we have looked at certain variants and which motivates you to a Turing machine particularly. You know, we have talked about two way Turing machine, two way finite automata and then more milli type of machines which essentially having certain features of the Turing machine that is how we have discussed certain variants there. Here, the variants of Turing machine what we are going to discuss is essentially some sort of extensions and having certain features which you can use in various places to construct Turing machines very easily using those features. Among which, I will now state few of them that is two way infinite type Turing machine with k type Turing machine and you know there are certain others like something like two dimensional etcetera. And something which is very important is non deterministic. Now, let me explain what are all these variants of these Turing machines what kind of extensions are there. Now, you see when we are working on the standard Turing machine, we have left justified right side infinite type. Now, whenever we are working you know our working nature is starting from here and we keep on checking whether the symbols and depending on the type of input what we are receiving. We are always keeping of course, as a special symbol this blank always at the left end or depending on the situation whatever the way that we are computing. We are always cross checking carefully whether we have received to this end or not otherwise what will happen the machine will hang. Now, to avoid such situation and you know sometimes when you are computation when we are looking into each time you are cross checking and you are you wanted to go back and forth this way. You may get better flexibility and constructing Turing machine very easily you know by taking a tape which is both sides infinite. That is what is essentially the feature of two way infinite tape Turing machine because here the tape we may assume that I do not have any worry that I have to just verify that I am reaching to the left end. I can simply look at the computation as long as I am having you know I wanted to cross check something without worrying I will simply keep on moving to left there is no question of hanging. Such a feature when we are including such Turing machine you know we may call that as two way infinite tape Turing machine and depending on your sort of like configurations and those things when we are defining we have we are not worried to look at this. So, where you are starting you may say sell them that is 0, 1, 2, 3 etcetera and you can put some negative say minus 1, minus 2 sell numbering you can make appropriately whatever we wanted. So, this is what so called this two way infinite tape Turing machine and it is all working nature is almost same here only worry you like you do not have to worry about what is going to happen when during a computation whether this will hang or not that kind of care you need not worry. You can formally define it because if you look at the standard Turing machine when we have defined m is equal to we have written q sigma delta q naught quadruple state set sigma input alphabet delta the transitions you know and this q naught of course, sigma has a special symbol blank here also you can consider the same thing and the transitions are also the same only thing is when you are talking about the computation you know you can have better facilities exactly like the standard Turing machine you can give, but that is all, but when we come to the second one the k tape Turing machine what is this how it will be this will be like you know you will have I can write this k tape separately or I can put it like a grid whatever you want let me put for the time being separately. So, say some if I consider two tapes three tapes whatever like this I may put and here the finite control again only one control you will have because for a one Turing machine you have only one control it is not that it will correspond to each tape, but here you will put you know you will have heads like this connecting to each of this tape you will have k heads. So, reading and writing head the finite control is essentially one only and the states you will have in this the internal states one set of states only, but what is the working nature we can fix some convention you know I can always take the input on the first tape and if I have to give some output if it is you know for a purpose of Turing computable function if it is computing a function you require an output I can expect always an output on the same first tape. And now what is the use of other tapes the other tapes can be used for some rough work essentially you know depending on the type of problem that we are considering the other tapes can be considered for that purpose and I will work on this k tape Turing machine likewise. And you see for certain examples you can have very nice facility when you are doing some multiplication or whatever it is a certain amount of information if you wanted to keep here what we use it to do we use it to go to right hand and we keep using those cells and making all that this thing for example, if you wanted to have say multiplication you do not have to worry to once again come back here and copy come back and copy that is what we have discussed. Now, what you can do see for example, if you are given two numbers in your format let me take that on the first tape say for example, i power n say blank i power m say two numbers when you are given you are starting here what you can first do is this one of the numbers that you will copy to this tape right say for example, i power m I have copied here. Now, what I can simply do even i power n if you want to copy to another tapes you copy that. Now, this candidate when you are reading from one side to another you can simply type on the first tape i power m once then you can mark simultaneously here once again you when you are reading this simultaneously you can type once again i power m. So, second i you can cross here and i power m likewise say for example, if you have number three here three times this i power m can be copied very quickly. So, you see the complexity will reduce because earlier what we are doing you have only one tape you have to go back and forth to see that this number of i's are to be copied and how many times this number of i's say for example, i power m is getting copied how many times that you have to copy you have to check. So, that means several times you have to go back and forth and see like whether this many times I have copied it or not. But now what you do you simply you can simultaneously do because you have various heads here similarly there are many other examples that very important one you know with respect to universal language we will be constructing a universal Turing machine that is a three tape Turing machine that is a very good example and you see by giving such a facility you can in fact construct very nice simpler Turing machines which looks very simple the simulations that you can show very easily. So, such Turing machine can be constructed with the extended feature now how to define this formulae. So, here again internal states you can have the same this thing you have a one set and input alphabet again the same thing now the transition map delta and q naught how should be this transition map because this transition map is from states earlier what we are writing in case of standard Turing machine the transition map is q cross sigma 2 you know you can have halting state also and then when we are looking at either you can print a symbol or move left or right that is what is the standard Turing machine we are writing. But here you see what is the input input can be from the first tape can I mean input the input to the Turing machine we are always keeping in the first tape. But in a transition you will be reading this simultaneously you can read what is here simultaneously you can read what is here that means here this sigma you can take k copies of this because in all the on all the tapes what is the information that you have that you will read and then you will change the state right. So, this is to q union this the halting you may get halting state and then what is here in each of this tape each of this tape either you know you can put some symbol you can print some symbol and another tape simultaneously you can move may be right or in another tape you can move to left. So, that means whatever that we have here sigma union l r so this candidate in each tape individually you can do whatever that you want I mean this k I am putting this k times and one one is the each of this is corresponding to one of these tapes. So, we can formally define a k tape Turing machine this way and there is another variant I have not listed there let me put that you know k hat Turing machine k hat Turing machine how this k hat Turing machine because I will you will be having one tape, but you can put k number of hats on this tape like standard Turing machine I may have one tape here always finite control is only one internal states you have certain states here this is a pointing here you can have some hats simultaneously you see these are all virtual machines right this is this kind of machine you do not have in your laboratory you will essentially a simulator this one now you do not have any problem that you know whether one hat will go on the other hat will get jammed this kind of situation will not occur. So, there is no problem of that kind of thing now by having this many hats what kind of flexibility or what kind of extension what kind of facility that you can have you see for example, if I wanted to look at the language set of all those strings x is equal to set of all those strings palindromes for example, x is equal to x power r whether a string is reversal of itself or not. Now, if you consider the input on this you given x to cross check what do you do one hat you put it here other hat you can first move in the first place to this. And then you just come from this side one cell and from this side also you come to one cell and see in the internal memory you just cross check whether these two symbols are same or not otherwise what we are doing you are going till this end of the tape and reading the symbol and then you are moving till end of the other side and cross checking by storing this in the internal memory through some state. And then you are cross checking whether these two symbols are same or not you see the complexity you have to go back and forth on the entire tape twice you have to go to that end and come back to this end to cross check whether these two symbols that is what in every loop that we are doing now if you have two you know if you have k hats. Now, I can choose two hats k equal to 2 and then simultaneously I will be able to cross check and see that this can be easily pursued. Similarly, it is a multiplication also when I do not require even that many tapes if I am allowed this many hats like in the previous example what I did we have copied those the input number m and n on different tapes or whatever and then we have generated i power m n we can generate i power m n on the first tape as output. So, but here what do what do we do if you have for example, say three hats if I consider one is tracking this i power n other can track i power m and then other hat can take care of the printing part. So, that way this kind of extension will be helpful in giving a Turing machine very quickly. So, now you can try formally defining such a Turing machine k hat Turing machine the way that I have mentioned for k tape Turing machine. Now, you can try formally defining k hat Turing machine. Now, what is two dimensional Turing machine? So, here the situation is like you know in your argon plane first quadrant you know a grid of tapes all the sites you know this is an infinite tape you can think like this the cells and now you may start you will have one hat to this. Now, you have a possibility of moving this hat to up left right down. So, this kind of features will be given now the input you may fix that I will be taking the input on this first grid this sort of tape and then you will be using this entire grid here all the sites you have infinite because this is left justified of course and of course, this is bottom justified now the grids you see the cells this is something like your first quadrant argon first quadrant of the argon plane you know we are marking the cells it will look like this. And in this two dimensional Turing machine I have to this is what two dimensional Turing machine you can now use these cells and appropriately you can use these cells for the memory and the computation can be reduced. So, this kind of two dimensional Turing machine is another variant and you can also give a formal definition of this kind of variant. Now, let us come to the last variant that I have listed here non deterministic you see this non determinism is already familiar with you in the context of finite automata push down automata. So, non determinism is essentially in case of deterministic Turing machine once again let me write this this is delta q cross sigma that means given a state and a symbol what we are doing we are giving a state including a holding state or I print or move the head to left or right that is how we are having this assignment. But, here what do we do this will be like you know the standard tape only here we do not have any change of that, but only thing is that the simulation that computation will consider non determinism here that means if you write here m equal to q sigma let me put this delta the big one q naught. Now, what is this delta this delta given a state and a symbol in determinism what we are doing we are giving a next state and something we are doing corresponding to that state here we give finitely many possibilities. That means this delta will become a relation it is a finite subset of q cross sigma cross you know you can give a next state one among the holding states or this cross that sigma union l r and I am writing like this you see given a state and a symbol now I can assign once you know this corresponding to say q a for example I can choose some p and I can say for example print some b that is one possibility I can give this is a relation. And then for q a I can also give say some p prime and ask it to go to left and for q a I can also take some p double prime and I can ask it to go to r I can also q a I can take say some p triple prime and I can print some other symbol. So, you can give various possibilities among this. So, that means here I am considering this as a subset of this that means a relation right. So, the non-determinism can be given as a relation and in that is only difference, but of course which makes a lot of difference that means non-deterministically you can choose one of the possibilities that you are going to assign among the possibilities given to a particular situation that q a this is what is non-determinism. Now, you see the facility will automatically increase right I have explained in case of k type turing machine or 2 a infinite type turing machine in non-determinism you know already in case of finite automata how flexibility that you have how quickly that you can give certain finite automata. So, similarly here also non-determinism will help you to give turing machines for some of the languages very nicely very quickly very easily. Now, what how does the computational power among this variance among this extensions would that vary if you look at that question the point the answer is there is no difference in the computational power between the standard turing machine and in any of this variance. You know in case of finite automata we have that situation in case in case of finite automata whatever the language actually deterministic finite automata it is same thing as the languages accepted by non-deterministic finite automata they are essentially regular languages. Similarly, in case of turing machines also of course, turing machines is not just as a language accepted we have this as a computational device that means you can have certain functions can be computed using turing machines right. Now, the same thing here if you have language accepted by standard turing machine that can be of course accepted by any of this variance because that can be easily covered in any of this variance because these are extensions and now the converse if you have a language accepted by any of this variance can we have a standard turing machine to accept that the answer is yes that means as far as the language acceptance concern these are not something like you know you can have a some bigger class of language accepted by this automata. Similarly, computation computable functions any function which can be computed using any of this variance can be sort of like done by and computed by a standard turing machine that means as far as the computational power concern all this variance are equally you know they are they are now more powerful than standard turing machine or the standard turing machine is equally powerful comparing to any of this variance. Now, what I will do I will just give way how do we look at for at least 1 or 2 cases that how we can really simulate the standard turing machine with this kind of behavior extended behavior. Let me start with maybe the 2 a infinite type 2 a infinite type turing machine simulating or a standard turing machine. So, assume you are given a 2 a infinite type turing machine that means let me call that as say m 1 is say q 1 sigma you can call sigma 1 delta 1 say the initial state here q naught does not matter. Now, this mission we can whatever this is doing this 2 a infinite type machine is doing we will do it by a standard turing machine. Let me write s t m for standard turing machine how do we do that this tape I will write it formally little late, but let me because there is nothing like you know tape width is fixed because this is the missions are not something physical right. What do we do whatever the input given to you on 2 a infinite type turing machine let me parallely write this also the given one is this you have cells. So, the input once it is given to you like this a 1 a 2 and so on say a n this is how probably we are working and going back and forth and doing the computation on this. Here what do you do that a 1 a 2 a n whatever is there first I will change the input in the format that I can see 2 tracks on the standard tape except the first cell in the first cell I will print a dollar symbol and then I will make this tape I will see this tape making means I will see this tape into 2 tracks. So, that my input will be seen this way a 1 a 2 a 3 and so on a n and all others are just blanks. Now, you see what is the meaning of this because whatever that tape content that I have I am now dividing it into the 2 tracks that means what I will do when I am to do this the tape is just one tape only I will have one cell only here, but this cell content I will now fill with a pair that a 1 comma blank this kind of pair I will type it here whenever the reading head is moving it will it will read the entire cell content content it is not that you know reading head when it is moving it will read only this portion and it will not it will not be able to because we have the standard tape whenever the reading head is moving you know the reading head will read the entire this cell content. So, essentially what we are going to do is what are the symbol that you have and the 2 way infinite tape Turing machine now we are changing this cell information that to a 1 blank of course what I have suggested is in the standard tape first when the input is given a 1 a 2 a n you to make this kind of thing what I have to do I have to first shift the entire input to 1 cell right or you know when you are considering with the blank here I will put a dollar here that is all and therefore, a 1 a 2 a n those are there only right and this a 1 we are now changing it to the symbol a 1 blank this is you treat it as a new symbol you may call it as b 1 what is b 1 symbol that is a 1 blank a pair similarly what are that a 2 the cell content that you have that I will print it as a 2 blank likewise you know what are the input we have that input now it is transform to pay so that I can see it is a 2 track now once I have converted this to this kind of situation what are the simulation that this guy is doing here on 2 way infinite tape Turing machine we will simulate that this here parallely how do we do that when we are taking a left move on this when you are coming to a n I will also come to the left take the left move of that head, but what I have to look at here you know I will concentrate on this input, but I will be reading the entire thing I will be concentrating on this if I if it keep on coming to this I can always take the left move the left move on the first track to read the first component because I will be sort of like looking into or changing whatever that we are doing on the first step if I take a left move I will take a left move from here. So, let me call this is cell 0 this is cell 1 cell 2 likewise and then I will call this is cell minus 1 cell minus 2 cell minus 3 likewise. So, what do I have here 1 let me call this is 0 2 3 and so on this will be minus 1 minus 2 likewise what I will do as long as I am in this positive number cells on the 2 way infinite tape if I am going left here I will also go on this standard Turing machine left here if I am going right on the 2 way infinite machine I will also go on the right on this. Now, if I reach to this 0 thing I will come to this dollar now from dollar I have to use the option of changing track that means when I am in the 2 way infinite tape machine if the reading head comes to this 0 cell from this cell if it is going to write that means on the positive number cell then I have to go on the upper track if it is going left then I will now consider the lower track that means now I will concentrate on reading on the cells which are on the lower track or changing the cells here if it is a printing step if it is going for example, from here if it is going to left if it is L then what I have to do on this tape I have to go right it is there is no left because it will get hanged. So, I have to go right and on this cell if it is going to write say for example, when you reach to this position if it is going to write then what I have to do because if you are going to write then I have to go to left because when I am coming to say for example, minus 5 the cell minus 5 that means somewhere I am here minus 3 minus 4 minus 5 when it is going to write on this lower tape I have to read the symbols which are on the left side of this therefore, I have to come to left this kind of simulation that we will have on standard tape. So, that whatever this exactly the 2 a infinite tape machine is doing exactly the same thing can be simulated on this standard machine. So, now let me give you a formal way of representing this let m 1 equal to say q 1 sigma say delta let me call it as q does not matter because I am writing m 1 m q naught 2 a infinite machine 2 a infinite tape machine that you have I may write m prime here I require states the states now. So, let me write it as q prime what should be this because here the states essentially look for the states have to consider all the things you know the state should be able to distinguish what is there on the left and what is there on the upper track what is there in the lower track. So, states now we will divide with this this q prime q cross 1 comma 2 by this what I am going to get in this q union sigma also sorry halting state also because in the upper track if I am getting halting state that will also should also be covered. This states cross 1 comma 2 in the sense that we are going to have whatever the simulations here of course the halting state will come otherwise in this and sigma now you see what do I require in case of sigma sigma we require dollar a new symbol which is not there earlier union the original symbols of sigma that you require union you know the symbols which you should be able to accommodate in the board the tracks that means sigma cross sigma this also you would require and then I take some special symbols which will record the halting that means let me call it a sigma bar if it is halting on the lower track I will record through this union sigma bar cross sigma I will tell you what is the sigma bar this sigma bar I mean you will just consider what are the symbols of sigma with a different representation that is all for each a belongs to sigma. How do I use this see for example you know when I am having say for example a here blank and if the machine is halting at this kind of situation that means on the first step if I am having a what I will now look at is this kind of situation because I have h 1 this is the current when it is halting that we will make it h 1. So, that we will understand that this is on the first step it is getting halted and then this symbol in the tape symbol also we have to look into because this is not a halting state now because h 1 we will not consider a halting state because for each storing machine h is the halting state. So, when I have h 1 this tape content you know where finally it is halting when I convert it back to a I should be able to distinguish. So, for that matter what we are writing this is a bar will be the symbol that we will be considering. So, just to distinguish wherever exactly it is halting you know we make this special symbols. So, this is what is now the alphabet that we consider and now let me write delta prime that I will explain you and then the initial state that we consider as it is. So, where we are starting you are starting in the first step. So, you can declare that to be the initial state. So, you will be starting there now the simulation part the simulation part is now first what we have to do you have to change the tape into this and tracks that printing you should be able to have and now this delta prime how it works if you have this q 1 that is on the first step whatever the input that you have a 1 a 2 that means you are on the first step we will change it to the state p 1 left if you know. So, this I am writing in connection to in case of 2 way tape machine whatever is there that means if the original this thing assume delta of q comma a 1 is equal to say p comma b then if b is equal to l then we take l and this is p 2 we will take in case of this is the first step. So, if it is going to if b is r we take going to write now with this phenomena I will explain you like how formally we can design a standard Turing machine corresponding to given 2 way infinite tape Turing machine. So, if you are given 2 way infinite tape Turing machine q sigma delta q naught we will consider you know this m prime some q prime sigma prime delta prime and this the states here we consider corresponding to you know the states of this which are which can be manipulated on the on both the tapes on both the tracks and this sigma prime is essentially looking at this special symbol and the elements of this. So, to accommodate the symbols a space like both upper track and lower track and then some more symbols that we include that I will explain you to look at this sigma that sigma bar I am writing. So, tapes on the first the first component is on the first track second component on the second track the sigma bar cross sigma. Now, what is this sigma bar the sigma bar is essentially taking the symbols of sigma we just denote a bar the purpose of this is whenever the machine is halting I have the state when we am the first track I will have h 1 and when I am the second track I will have h comma 2 this kind of state I will halter, but the halting state of a Turing machine is essentially h just when we are when we wanted to halt at the same place I will record the symbol and then go to that particular place and make the halt. So, to make it we will have this kind of special symbol sigma bar which is 1 symbol each corresponding to each symbol of sigma. So, now how do we simulate it the simulation portion is this you take any transition in 2 way infinite machine that say q comma for example, a if it is p b now depending on b and where you are depending on that we will be doing this simulation. That means, this delta prime if this state when we are considering on the first track what do we do we will now consider corresponding to this a and any symbol a say for example, a prime should I write a prime or may be any symbol say some c on the lower track what do we do we will give that the p 1 the state p 1 moving to left if this b is left if b is right of course, the change state is this on the first step we will ask it to go to right this is the pair we will assign. Now, for example, if b is an element of sigma that means, if it is a printing step the state which is getting changed is p 1, but now on the first track only this change of symbol will take place that means, b on the first track on the second track there is no change the same c will be there. Now, so corresponding to the transition in the two way infinite type machine in the standard machine we will give this transitions this way and if this is in the lower track if this information is in the lower track that means, the current state q and if the symbol is in the lower track that means, the symbol is on the cell which is left to the 0 then what I will do I will now give transitions for the lower track that means, the state will be this q 2 and for any symbol on the upper track that is c and a we will give like this if b is l as I had mentioned we should move in this track to right. If b is right then in the lower track that means, the state component is 2 we have to move to left and if it is an element of sigma if it is an element of sigma what I have to do in this the first component that is an upper track there is no change, but here we are going to print b here is this definition clear. So, once again if I have to explain let me put 0 here 1 2 the cells like this and on this after we have converted this to you know 2 track when I am reading a here on the first track I will always have internal state corresponding to this q here. Suppose I am reading the second cell in the state q this first the take the cell will be on this entire cell of course, the reading head will be, but in the internal state through internal state we will recognize that if I have the internal state q 1 I mean that I am reading the first track I feel that I consider that I am reading the first track if the internal state having the second component say for example, if it is q 2 I consider the input that on the lower track of upper track and the lower track that is what is the convention. Now depending on what is happening in this 2 way infinite type machine we will be updating this tape. So, as I had explained you earlier now if a is the symbol which is on the right side of the tape that means the internal state if it is q 1. So, the first component will get affected if it is printing step or movement will be considered on the first track with respect to the first track and the second track information will not change in the printing step. And now what else we have to do when I am coming to this dollar symbol I have to look at that you see I have to change the track that means when I am coming to the dollar from the upper track then I will change to the lower track and so on. So, that means how do I consider that delta of this q 1 if I get dollar on the tape then what I will this you change to the second track that means the internal state will be represented by this and ask it to go to right. Similarly, if delta of if I am in q 2 if I am receiving dollar this thing that means I am from the second tape lower track I have come to this dollar then you ask it to go to right. So, that means this q 1 or you ask because whenever I am in dollar the track change will be considered this way and then now you look at the situation will be like this you have made till this point the tracks. So, this 0 1 2 and so on and till this point that you have considered minus 1 minus 2 minus 3 and so on you have made it this way, but there after you have cells like this only. That means whenever you come to a blank cell what to do this blank cell first now we make it into 2 blanks that means if it is on the upper track that means in any state q in the upper track that means this is the internal state if I am receiving a blank symbol this delta prime what we have to do in the first we are in the first track the same state, but first we will convert this input to 2 blanks that means upper track blank lower track blank. Similarly, this if you are in the lower track also when I am reading the blank symbol then you will continue to the same state and you continue to the do the computation in the lower track, but you will first convert it into 2 blanks so that the tape can be extended to 2 tracks. So, for example, if I am here first I will not just see this way I will now consider this like this. So, this is 1 and now this head recording that the position the halting position recording related to that what do we do if I am coming to the halting state when I am simulating on the upper track. So, I have something a 1 a 2 what I will do I will now convert it into this halt state, but this a 1 bar a 2 and similarly this when I am halting that in the lower track now you see this for example, some input a 1 a 2 that will now this is the head recording position this halting state that a 1 a 2 bar. So, to indicate that where exactly this is getting halt now you see once it is coming to a halting position I do not have to now we are we do not have any possibility of continuing only thing is it is just recording the position. So, that you know if you wanted to record this position where exactly this is halting so that will be shown up here with this symbol whether it is in the upper track and the lower track for that matter only we are distinguishing this symbols. So, with this kind of construction with this kind of simulation what we can do whatever the 2 way infinite type mission you are doing the job that can be simulated on a standard mission as well. Similarly, by considering this kind of tracking business that means what are the standard type that you have we have now converted the input by a space now if I make this as triplet you know that means say for example, for the symbol a 1 in one cell if you have I can put in any symbol any complex symbol I can put in one particular cell say for example, a 1 a 2 a 3 I am putting it vertically then I will see that I am dividing it into 3 tracks. So, this is not something like you know type physically is getting changed only the input on that we are we are changing so that we can track the simulation whatever that we want to what kind of extension that we wanted to have. Now, the similar mechanism can be adopted to simulate the even other higher machines higher versions what are the extended versions. So, for example, if you are looking for k type Turing machine. So, I have k types like this and I have k heads also I have say various heads what do you do in standard type now you divide this type into first cell this is one cell only for just to write to accommodate this many lines I am just writing it bigger. So, I will put this special symbol like this the input is always we will now make it into first track and odd number track will be considered will be seen as the tapes that you have here say for example, I have 3 tapes here then I will divide this into 6 tracks then 1 and 2 for the first tape 3 and 4 for the second 5 and 6 for the third what I will do here why I am making this because I have to record the head position also. So, the first the odd number track will contain the elements of sigma this third one fifth one now what is in between ones the even number ones there I will print either 0 or 1 from here. If I put if I have 0's here that means the reading head is not present here wherever the reading head is present there I will put 1 that means when I am reading. So, I will convert first the input given to me on to the upper track I mean the first track and the remaining cells to start with they may be blanks, but the reading head wherever it is there I will have 1 and other cells we have 0's. So, that is how we will first change the what is called the standard tape and then we start simulating and how do I simulate see for example, if I am doing some movement with the first head then I have to consider that. So, where is the head head will be denoted by this symbol say for example, 1 now first I will pursue this here I can do it simultaneously first tape I can simultaneously consider with this second I can consider this simultaneously third I can simultaneously, but here what I will do first I will pursue 1 because I have only one reading head on this tape and then see like I will transform this change then I will pursue whatever has been pursued by the second head. So, by considering where exactly now because there is only one head now when I am working on the first tape I will now first look at the where is the head position. So, that I will know where is the what is the input that I have to pursue when I am concentrating on the second head to pursue the matter then I will now first look at and the fourth track where is 1 and then the particular input will be pursued and when I am concentrating on the third one then I will look at where is the current head and then pursue the concerning input. So, that means what is happening here whatever that simultaneously there I am doing here I will do it sequentially because I have only one head, but the when I am moving from one track to another that means if I am working with the first tape and then with the second tape then I have to know where exactly the reading and writing head was. So, that is essentially denoted by this 0 1s. So, once I know that the reading head was left here you just go to that position and do the appropriate changes go back and forth. So, you may have to do so many steps for example for one step here you may have to consider so many steps here because you have to completely you may have to come from a particular position of the head to the position to search for where exactly was the head in a for a corresponding track and to consider this simulation. So, this kind of extension can also be simulated the k type Turing machine can also be simulated by a standard tape Turing machine and see that there is no difference in the computational power. Of course, the complexity may vary because whatever that you can do it easily in case of this higher variance for standard machine you may have to put little more time you may have to give more steps to do the job. So, the complexity will may vary, but as far as the computational power concern they are all equivalent. So, that is about you know some of the variants of this Turing machine and regarding non determinism we will be studying little more in other lectures.