Welcome back to the school, like I said, it's Jaco here.  I think she's zenning out somewhere. I remember she's also a yoga teacher. Coach Georgie, are you ready? Oh, hey Jaco, yes, I'm ready to go.  So you could do with just a bit of space, maybe the size of a yoga mat. If you have a yoga mat, then awesome, but if you don't, don't worry, don't need it. The other thing you might find comes in handy as a block, but if you haven't got a block, you can also use books as a substitute. So what we're gonna do is focus on getting the body open and warm and then we're gonna play with some different variations from there. If you get any videos or photos throughout this class, then definitely fire them our way at the end. So to begin with, let's come onto into the space and we will start by warming up the wrists and just waking up those joints a little bit. So fingertips towards the knees, press the heels of the hands down into the ground and then just make some little circles. So just move around going one way, then changing direction, just working into that wrist extension. We're also gonna work into the backs of the hands into the wrist flexion. So we take the back of one hand onto the ground, maybe both together and to start with the elbows will probably flare out quite wide. So we're going to extend our arms by turning the armpits of the elbows forwards and then just gently rocking back and forth. If this feels too intense on the wrist, you can take one hand away and just do one at a time or just simply put more weight into your legs so you don't have to push down too hard. So just a few rocks going back and forth and we'll go back into that wrist extension, fingertips towards the knees. You can bring the hands closer to your body if this feels too much or take them further away if you want a little bit more. So in our balancing, well obviously the base is our hands and we're gonna be controlling our body weight. The wrist can get quite a battering throughout all of these sort of balances and inversions that we do. Therefore I want you to think about looking after your wrist, so number one, we're getting them ready but when we're actually balancing, we're gonna start to grip the ground a little bit more and actually bend these finger knuckles to help control the body. This will also protect the wrist, especially when I go into some of the balances where I'm in a really deep extended position. So just think about clawing and gripping the mat and that is gonna help you a lot in terms of your balance through some of these poses. Okay, so just a little bit more and then from there, we will bring the hands in, just make some fists and then just turn and make circles into the wrists, changing direction, working through any tight areas, might get a few clicks. Keep squeezing the fists as you do, keeping some tension into the body and then to get some more heat into the wrists and forearms, we're going to open and close the hands. And we're just gonna do this as fast as we can for as long as you can and eventually we're gonna start to feel them slow down. So see how long you can go. I'm obviously pretty poor because mine are already going already. So this is just gonna get some heat into the forearms. That's it, slowing right down now. And then from there, come on to the hands again. This time we're gonna raise up and down. Again, just getting some heat into the wrists and the forearms and then release, give them a shake out. So once you've warmed the wrists up, we're gonna warm up the spine. Come on to all fours and we're going through cat and cow, which most of you are probably familiar with. So hands are under the shoulders, knees are under the hips. We're gonna start from the base of the spine. So I'm gonna start by tucking the bum under, then slowly curling up the spine until I get super high on my shoulder blades of spreading across the upper back. From there, from starting from the base of the spine again, I tilt the pelvis forwards and then work the belly down towards the ground, working all the way through the spine. Once I've got into that extension through to the top of the back, I'm gonna tuck the shoulder blades down into the back, look forwards and then open the chest. And then again, tuck the bum rounding out, pressing the mat away, spreading those shoulder blades. Gently tuck the chin, tilt the pelvis forwards. And extend the spine.  So we create that tension.  And release. So just have a little wiggle about moving side to side, working to the body, see how it feels. And from here, we are going into a downward dog.  Keep the fingers spread nice and wide and grip the ground and then let's lift the hips. Keep the legs bent to begin with. Just let the body open up gradually. So from here, starting from the fingertips, I've spread the fingers. Now I'm gonna wrap the shoulders like I did in my cat stretch. So draw them across the back. Imagine you're sort of squeezing your armpits together to help feel that tension in the shoulders. And the back is nice and flat. So if I feel like I'm really rounding through here and too much weights in the shoulders, just bend your legs a bit more. And now what I'm gonna do is start by walking the feet. So maybe I extend one leg, switch side to side. I don't want you to worry at all about getting your heels down to the mat. So just focus on the sensation of the body, walk through the feet. So waking up the whole back of the body here and also strengthening the shoulders and the wrists. And then let's come up onto the tiptoes, lower the knees to the mat, back into that tabletop position. This time we'll take the hands up nice and wide onto the fingertips. If you need more and you've got two yoga blocks, you can raise the hands a bit higher. Stick the bum up in the air, take the chest down towards the mat and just stretching into the shoulders. Bring the body back up, press out into a cat stretch. Now let's hook the toes, lift the hips and go back into a downward dog. Wiggle about when you're downward dog. See what feels good, move around. Make it a bit more dynamic. And then what we're gonna do from here is take our right foot up high into a lunge position, just lower the back knee down and straight away we're gonna kneel up nice and tall. In this shape here, what I'm gonna focus on is opening this front hip. So, tuck your bum under, pin the rib cage down.  And then while I'm tucking my bum, I'm pressing this hip forward so it's giving me a nice opening into the front of the thigh and into the hips. And then from there, we're gonna come all the way down. So you can use blocks or books for this bit. Maybe you need to elevate the hands just to get into the hips a bit more. Obviously, if you don't need that, you can take that away or perhaps come down even lower into more of a lizard deep lunge here. You can raise the back knee or keep it down. So if you're finding in this position, your elbow that's closest to your foot is raised up off the ground and you're almost forcing it down but then really rounding the back to try and get there. Just elevate the forearms, don't try and force that position. Well, a nice flat back. We're going into this flexion in the front hip, so a steep bend in this hip here, which is definitely gonna help with my setup for my frog stand variations. So once I'm here, if I am on the forearms, try and keep them sort of roughly in line with that front foot. Wiggle about into the hips, move about into them.  Again, you can extend and lift that back leg if that feels good. Maybe we open that arm in further back and then bring that arm back down from here. Back onto the hands, step back to your downward dog, walk out through the feet, wiggle about if it feels good. And we'll go to the other side to take your left foot up. Doesn't matter if you've done opposite sides, it's all good. Come up nice and high into that lunge position and then we're pressing that hip open again. So tuck your bum, press your hip forward, keep the ribcage locked down. The front heel's pulling back as the other hip is pushing forwards. Breathe in and out of the nose and just feel that opening into the front of the thigh. And now let's come down onto the ground. So both hands are inside that front foot. You can keep the back knee down. Obviously, if you need your block again, place that inside the foot. Hands can elevate on it. All forearms can. Getting that elbow that's closest to the front foot down if you are going all the way down, maybe extending the back leg. This front knee is driving out and open a little bit without the foot rolling. So keep that foot grounded, wiggle about into the hips. And then let's add the twist. So you can still on the forearms or go on the hands. I'm gonna raise the arm up that's closest to that front foot. Breathe into there, maybe lower the back knee if it's feeling too much. Opening that top arm if that feels okay. Just opening the chest as we turn. And then coming back. And we'll step back to down the dock. Walk through the feet again. So from here, bring your thumbs to touch at the top of the mat or in the space. We're gonna hop or step the feet outside the hands. So we are now in a deep squat position. If while you're here, this is feeling too intense, you can come up onto the balls a little bit more or perhaps use cushions or a block under your bum just for a bit of support if you need that. And then while we're here, we're just checking in with the ankles, move side to side. And let's open into the hips. So for these hip openers, you can kneel down on one side and make it a little less intense if it's feeling too much, okay? So we're gonna take hold of one ankle. Don't worry about which one. Press the other knee open with the elbow. If that feels okay, you can give it a little bit more. See if you can press it with the hand, but if not, just do the elbow. And then we come into the center and add a twist. So we'll do a few on this side then switch. On this twist, keep the elbow up rather than the hand if this is too much. And then I can kneel down to make this a little bit less intense. So kneeling on one side, press, keeping this foot grounded, maybe press with the hand and then twist or open the arm. Bring it back in. Okay, so that's the other variation. Press the knee with the elbow, push with the hand and twist and open that chest. You feel this all the way through the side body and into the outer hip as well, it's a really nice twist. This is gonna help prep us for some of those poses while we're coming into a side frog today. So press, press and twist. And one more, push open, press with the arm and twist. And let's switch. So again, kneel down again if you need to. Push with the elbow, press with the hand if that's accessible and then twist to the elbow. And then obviously your twist can be with your elbow if the arm's too much. Opening the chest, couple more, push, push, twist. Last one, push and twist. Okay, so opening into the hips a little bit more we can rock forwards and back. So this isn't our frog stand balance but we're gonna actually put the weight into the backs of the arms on this one and this is just stretching only into the hips. So pressing the knees open with the backs of the arms, letting the weight sink into the arms a bit more, basically the opposite of what we're gonna do when we're in our frog stand. And then from there, just come back down, maybe sit onto your bum for a little bit, give the legs a shake out if it's feeling too much. So we're gonna go from here into our first frog stand variation and to begin with, we want to set the base up. So I'm gonna have my hand shoulder width onto the ground, spread my fingers and grip them up. And by spreading my fingers, that's creating a more stable base for me to control my body weight. From here, I want to lift my bum up nice and high. I'm going to collect my legs with my arms. So I need to bend the elbows here, collect them like a shelf. And now it's all about letting the shoulders tip forwards. I've got to let that weight transition forwards. So now you can see that my elbow is above my wrist. From there, this might be as far as you go for now. Maybe you can lift one foot, maybe the other, maybe both starts to come up. Find a position that you can move into nice and controlled. What we don't want to do is really try and throw ourselves into it fast. It's a lot harder for you to then, one, stop face flooding the floor. We don't want to do that.  Hands are shoulder width, lift the bum, collect the legs with the backs of the arms. Now my knees are really high here. You may find that you're more down there and that's fine as long as we're above the elbow. The higher we get the knees, the higher the hips are going to be, now I can start to pull my feet up towards my bum. I'm pressing down hard into the ground and my eyes are forwards. I'm not lifting my head up like this, but I'm looking just between my hands, okay? So have a little play with that one. I'm going to go through some variations for those of you that might need a little bit of help getting into that position. So if you are getting worried about face flooding the floor, good little tool you can play with is using a block under the head. So I'm going to set up, my head is on top of the block. I've got that base of my hand shoulder width. My head is on the block here, so it's resting down a little bit. I'm still pushing through the hands. And now I lift the feet and it's my goal now to try and lift my eyes up and look at the block. So I'm lifting my head up and off looking at the block and now I'm holding it. So if we can get to that point, we're now starting to really feel what that position is supposed to feel like and how to press into the ground with less fear about collapsing forwards. Soon as you can obviously take that block away. The other variation is if you're struggling with the height of the legs or getting the knees high enough in the arms and I'm going to place a block under my feet. Obviously if you can use two blocks, just make sure that they're not wobbling about.  So the hands come down from here shoulder width. My knees, you can now see they're a lot higher already. So I'm going to lift my bum and now they're right up into the armpit. From there I can start to bring my feet up off the block.  So while we're in our frog stand, we need to think about pressing into the ground and not letting our weight sink down into the arms. A lot of the time I find people complain about pain into the triceps or bruises. To be fair, when I started practicing these balances, I also got a lot of that. But part of that was also down to not understanding how to feel my weight on the backs of my arms. So when we're in that balance, I want you to think about rather than being down here, you want to press up here, imagining like I'm just going to float up into a handstand which I'm not going to do that right now. I'm not ready for that one yet.   And that is going to make those hips come up nice and high. And you'll really feel less weight sinking down. So if I let my legs collapse into my arms, that already now I can feel that discomfort into the triceps. Okay, so give that one a go, thinking about growing tall. So from here, I've got a couple of variations I want to play with. The first one is maybe you feel ready to go into your straight arm frog stand. So if I'm going to do that, this is going to be really demanding on the wrist. The extension of the wrist is going to feel pretty intense. So I need to really focus on that gripping and bending those fingers and of course in this position. I'll start in my normal frog stand and from there I'm going to push up. So I'm in my bent arm position, eyes are forward, grip the mat, press and round out those shoulders like I did for my cat stretch and press up to extended arms. Okay, so that is our straight arm variation which will feel like a really heavy push. Okay, so have a little play with that one as well. The transition don't want to have a little bit of fun with is transitioning from our frog stand into a handstand. So what I like about this is with my hand shoulder width, as I tip my head down gently to the ground, I'm going to be set up in that nice tripod position, hopefully naturally.  And now I'm in that nice tripod position where my heels of my hands are down, I can still grip the ground. This might be as far as you go and that's fine. Otherwise we'll see if we can lift the hips and the knees come off, maybe then extend the legs. The legs are extended, we squeeze the bone, we press hard through the hands, I don't want all of that weight into my head. And now I'm going to draw the knees back in, find the tops of the arms, lift the eyes, push through the heel of the hands and then transition back down to the ground. So how I play with that one, go once more, hands shoulder width, find your frog, tuck the chin, lift the hips, keep breathing and then we come back in, press and now I'm going to push up high again. So I come back into that frog stand and back down. That's if you're trying to then find that frog stand on the way out as well. So a lot of the time when you come out of your headstand into your frog stand, like the first one I did, my hips were super low, the second one I pushed and that lifted me back up again so I could find that balance. So that's a fun variation you can have a little play with. So the next ones we are going to move on to are our side frogs. If the ones that we've done so far are enough and you want to continue the rest of this class doing those then by all means focus on them. This next one requires quite a lot of rotation of the trunk of the body. So we're going to start by sitting either in a squat with the heels up or kneel down on one side. Okay, so I'm going to add a rotation and take my elbow outside my knee or my thigh. The other hand can just hang back here. We can bring your hands together like this. And from there, I'm adding in that nice twist into the trunk of the body, which is going to help prep me really nicely to come down into this side frog. As I'm twisting one, I'm working hard to balance. I'm going to try and pin the knees together so this back knee doesn't start to pop forward. There, press the knees together. Press the hands together if you've got them together like this and look back. Keep breathing and then from there, let's jump around to the other side. So again, you can lower one shin down to the ground, hug the outer thigh, start to rotate open the chest. Maybe the hands come together and you push them together. This will really help the chest open and lean back. That back knee wants to creep forward so I'm going to squeeze them together. If this is too much here, just hold on to the outer thigh and then release. So our side crow. We have a basic side crow and then we can, side frog, I'm going to do that a lot. And we have some leg extension variations that we can have a play with. So take it as far as you like. We start into this squat. We add our twist. I'm going to take both hands to the ground into that same frog stand set up so my fingers are spread. And now the elbows are collecting the outer thigh or the hip and lifting. I can now lift the feet up. I'm pushing through the hands and I've created a shelf. So there's that side. Breathe into it. There's another way of going into that which you might, you may or may not prefer. So from a crouching position here or there's a cross leg variation where I sit with one leg over the top of the other, add my twist, bring the hands down to the ground and collect this hip. So we've got to rock the weight across. Okay, and this is going to really help me then go into a leg extension as well but you might just go here, that's fine. Okay, so we'll do the same on other side. Feet together while you start in that kneeling position. Add a twist, bring the hands down shoulder width. We're going to collect the lower body, lift the hips. Bringing the feet up, pushing down with the hands and back down. You'll probably find you'll have one side that feels more comfortable than the other. I always feel less comfortable on that side. So just try your best to always balance it out both sides. Generally speaking, if you start on your worst side first each time, it's probably the best way to go. I've done it backward, but just make sure. Okay, this next one is with some leg variations where I'm going to open the legs up. As we're pushing and we extend the legs, we're not only active in the upper body, we need to engage the leg muscles. Don't let the legs hang because if you just kind of let the legs go floppy, it will just pull you straight back down to the ground. So I've added my twist, collected my hip, my lower body. And now I'm going to extend the bottom leg and lift the top leg. But my legs active, they're not hanging down like this. That's making me feel really heavy. Oh, I'm eating my hair. And then from there, come back in. So you see the difference between if I'm just letting my legs hang. I'm less stable and my upper body's actually having to work harder. So engage the leg muscles. Onto the other side. We twist, collect the lower body, extend the bottom leg, lift the top leg, engage those leg muscles, push down hard and back in. So notice how the shoulders feel as well. If my shoulders are tipping down, I'm going to end up with my face pretty much on the ground like this.  And so I'm nice and active in those shoulders, not sinking them down. You may also prefer a slightly different grip with your hands. So I tend to do this one more, but that doesn't necessarily mean that's better. So I'm going to rotate. This front arm comes outside the thigh. The other arm comes out to the side here. Now I'm going to tip and lift the body up. I can play with other leg variations. So give both of those grips a go too. I know some people prefer one to the other. So see which one works best for your body. Okay, keep going just for another minute or so with those leg variations. I'll quickly recap you with the one kneeling down. So one leg has come over the top of the other thigh. And then from here, this setup is going to take me into the leg extensions kind of straight away. So I twist, come onto the side. And now all I've got to do from there is extend both legs. And I'm already in that setup again. Okay, so hopefully maybe one of those variations works well for you. And let's move on to another one. So obviously stick with that. If you don't want to go any further, I might be moving a bit too quick through these. So sitting down, this next one is called eight angle pose in yoga. I'm still using that same kind of frogs down setup with the shoulders. However, this is a bit more demanding onto the hips and the hamstrings. So if this one feels too much, please don't force it. One leg is out. I'm gonna try and get this knee as high on the shoulder as I can or above the elbow. So I'm gonna bring the arm inside that knee, lift this leg up, bring it as high over that side as possible. Don't worry if it's not like right over the top of your shoulder. Just go to where you can. Place the hand down straight away. So I've put the leg over, hand goes down. I'm gonna grip the ground. Now this top leg squeezes around this arm. My other hand comes down. And now I'm gonna cross the feet. Still squeezing that leg around the arm. I'm now gonna rock my body forwards and extend the legs out. So you have to let the body rock forward almost like you're having a headbutting football. Extend the legs out, squeeze the legs around the arm and press hard. If you wanna go further, we can press the arms up straight, maybe then come back in. Maybe I'll just show enough now. Give that one a go. And then we'll come to the other side. Give legs out, one leg in. Take the arm inside that bent knee. Get this leg as high over the shoulder as you can. As soon as you are there, place the hand down, bring the other hand down, cross the ankles. From there, we're gonna rock. Extend the legs out, press hard. I'm not sinking the chest down like this. My bum doesn't wanna be on the mat, so lift the hips and press hard. Maybe extend, back in, and there you go. That is your eight angle pose. So have a little play with that one. Hopefully you're still with me. So there is one more I'd like to go over. If you've had enough at this point, then you are welcome to just stick with the ones we've done. This last one is flying pigeon. So it's going to be quite demanding on the rotation, external rotation into my hip. So I want you to just be really mindful and listen to your body. You should have absolutely no discomfort or niggling going on in the knee. So just listen in. If you do feel that, then please stop. So for this one, what I'm gonna do is take one leg on top of the other thigh and I'm flexing this foot. The bottom leg is gonna bend and I can just stay here. This might be as far as you go. You're gonna know pretty quickly if you've got anything going on or the hip or the knee doesn't like it. So this will feel really nice into the outer hip and glute. You can stay here, practice and work your balance. Maybe bring the hands together. But this is gonna be my entry point into my flying pigeon. So this foot, my foot that's waving at you, is gonna be doing quite a lot in terms of the pressure I wanna feel in the back of my arm. So I wanna resist my foot against my arm and resist my arm against my foot. And I'll explain that a bit more now. So I've got the foot on top of my thigh. I'm going to bring my body down. My foot is hooking around the back of this tricep. So the hands come down into my frog stand, set up the foot, hooks. I'm gonna press into this shin and into this foot, into the back of the arms. And now I can lift the top leg up. But I'm actively pressing away. So what I don't wanna do is be all relaxed into this shin and let it hang into the arms. If I do that, I'll show you. So if I'm not resisting the arm and leg against each other and I balance, when I extend this leg, it's gonna be more down here. So I wanna press, wee, and lift high, okay? You may also find, like I was then, your foot sliding down back of your arm. Maybe I put too much moisturizer on, I don't know. But you really wanna hook the foot and grip as best you can. So again, if you're wearing socks or anything like that, it's probably not gonna help. So that is your flying pigeon. I listened to your body and hopefully you've had some fun with some of these balances. And hopefully you're as out of breath as I am. Stubbornly to breathe now. Great, so thanks so much for joining me today, guys. I hope that you've enjoyed those frog stand variations. If you have taken any photos or videos throughout the class today, then we would love to see them. So as always, tag away, send them away. If you've got any questions, give us a shout. We'd love to help. And apart from that, hope you'll keep them well and managing through this time in isolation, hopefully taking lots of advantage of all the online content. So apart from that, guys, look after yourselves, take care and class dismissed.


Live from Las Vegas, it's theCUBE, covering ServiceNow Knowledge 2018. Brought to you by ServiceNow. Welcome back everyone to theCUBE's live coverage of ServiceNow Knowledge 18. I'm your host, Rebecca Knight, along with my co-host, Dave Vellante. We're joined by Chris Beatty. He is the CIO of ServiceNow. Thanks so much for coming on the show, Chris. Thanks for having me. So we're hearing so much about improving employee experience and this is the goal, your goal, and also the collective goal of CIO. So can you tell us a little bit about why this, and how do you see your role in this? Yeah, for sure. I mean, if I rewind three or four years, I don't think experience was really on anybody's agenda or not high on the list. I think what we've come to realize or I've come to realize is that experience is critical to actually getting the right behavioral and economic outcomes. It is not optional anymore because with the amount of transformation that we're driving through technology, it's changing processes, changing the way customers interact with us, suppliers interact with us, and that change needs to be easy. And not just easy for easy sake, but otherwise we don't get the business outcomes we're looking for. So for me, it's very purpose-driven to say that for us to get those economic outcomes, we have to focus on experience. I feel like the CIO role is evolving and we've talked about this before. I'd love your thoughts on it. You know, it kind of used to be, all right, we're going to keep the lights on. Granted, that's still part of the role, but it's table stakes. It doesn't go away. It's still part of the role.  Okay, that's shifting. You know, with the digital economy, machine intelligence, the economy booming, this war on talent, especially in Silicon Valley, things are changing. How do you see the role changing and where do you see it evolving to? Well, I think the CIO role is changing. It's driven really by what's going on in every industry. If you think about it, everything, how fast your company operates, how efficient your processes are, how engaged your employees are via employee experiences, the mode in which you're able to interact with your customers, how digital your supply chain is, everything is powered by technology platforms. And CIOs are the ones governing and managing and delivering those technology platforms to deliver those outcomes. And I think it's only going to increase where technology has a bigger and bigger impact. And I think that is really driving a shift in the CIO role where CIOs need to be front and center. There is no more, here's the business strategy, here's the technology strategy. They are one in the same thing. And I think in our consumer lives, we talk about the digital divide, sort of the haves and have nots. I think the same thing is going to play out in enterprises where those enterprises that can figure out how to harness these newer technologies to drive meaningful business outcomes are going to start to separate themselves from the competition. And that separation is only going to get bigger with time. So I think there's a tremendous amount of urgency on this topic as well. I was reading a recent article that's talked about CEO's priorities for IT and saying favoring speed over cost. And I don't think that's because all of a sudden we're going to become frivolous with our spending. But I think, again, it just speaks to the urgency and the need for businesses to transform. And it's now. So it's not just harnessing the technologies, it's also harnessing the employee behaviors that need to change in order to create these cultural shifts that you're talking about, right? Yeah, for sure. And I would say, and we had our CIO decisions yesterday, one of the key topics was, you know, driving cultural transformation. And I find that's a lot of what I'm doing. And that involves a lot of selling, quite frankly. I mean, I don't have sales in my title, but by the very definition of it, we're saying this technology has the promise to unlock a new business model, unlock a new process, get to that next level of efficiency or productivity. But you're selling a vision, right? And that means change. And people don't like change. As long as someone else is changing, they're fine with it. It wants it's themselves. So we have to focus a lot and really double down on transformation efforts and play a key role in that. And to link it back to your first question, that transformation gets so much easier if we can deliver compelling experiences, right? So it's all kind of tied together. Four years ago at K-15, Frank Slutman sort of threw down the gauntlet to CIOs in the audience and said, you must become business leaders. If you don't become business leaders, you'll be a dinosaur. How are you a business leader and how are you becoming a business leader? I think it's really shaping IT's agenda based upon what's important to the organization. And that's going to be different for different organization, but largely it's going to be things tied to customers, how productive and engaged are the employees? What can we do to drive margin, which is top and bottom line improvement in the economic model and making sure that IT's goals and objectives are one and the same with the business goals and objectives. So for example, we do at service now in IT, we have a shared contract with every function, marketing, sales, you know, professional services, that here's the business outcomes. So on my dashboard, you'll seldom see a whole bunch of IT metrics. It's all about, did we get to the business metric or not? So if you're not measuring that, then I'm not sure what you're measuring. Okay, so you, and I'm sure you have a lot of IT metrics too, but you're able to then tie those IT metrics to business metrics and show how a change in one flows through the value to affect another. I mean, where the role was, that doesn't go away. And it's a critical part of the role and I don't want to undermine it, which is all the invisible things that just happen in corporations, you know, the utilities of, is the networking and phones and all that, that has to be rock solid, that's table stakes. But yeah, for the next part of that, it's really driving those transformational business outcomes.  How do you see machine learning transforming the modern work experience, the modern workplace and the employee experience of the modern workplace? I think at a very high level, it's around speed and effectiveness of decision-making. And machine learning I think has the promise or the opportunity for all of us to unlock that next wave of productivity. Just like in the late 90s, we had ERPs and they drove a lot of automation and supply chain and finance organizations around the world got better. They got faster, more efficient. I think machine learning can do that for the entire enterprise by leveraging platforms to help people make faster and better decisions. I know there's a lot written about replacing humans and things like that. I don't buy into that. I think it's just helping us be better. And I think there's use cases all over the enterprise. The biggest barriers to machine learning in my mind typically come with talent, how do you do it? And the good news is here, I mean, what we embedded with machine learning and the ServiceNow platform, you don't need an army of data scientists that are super hard to find, almost democratizing the ability to leverage machine learning. Second biggest one that when I talk to CIOs, it's lack of the right data. And they don't have the right data, perhaps because they haven't yet digitized their processes. So that's a critical precursor. You got to digitize your processes to generate the right data to then feed the algorithms to get the outcome. But yeah, machine learning I think is going to materially transform how we operate dramatically over the next three to five years. I mean, IT systems continue to get more complex. They, in many cases, becoming more of a black box. I wonder if I could get your thoughts on this. I mean, I remember reading Michael Lewis's book, Flash Boys, and he talked a lot about the flash crash. And nobody could explain it. They chalk it up to a computer glitch. And his premise was a computer glitch is computers are so complex, we can't explain them anymore. AI, machine learning, machine intelligence, going to make that even more complex and more of a black box. Is that a problem for us mortals? I think it's a problem for us mortals. But I think it's a problem and I'll tie it back to the transformation in human behavior. We're, I'll call it, prototyping and rolling out and leveraging machine learning in our own enterprise. And one of the things we've observed is that us humans, us mortals as you call us, we need to know why. So if a machine is making an algorithm based recommendation or a decision, we need to know why. And our employees at a hard time accepting the ML based recommendation without knowing the why. So we had to go back and rework that and say, how do we surface the why in the context of the recommendation? And that got people over the hump. So I think it is a super important point where as these algorithms get more and more sophisticated, our human brains, the way we interpret it is we still need the why. Yeah, so you're trying to white box that is what you're saying, which again is not easy. I often use the example of a computer can tell me if I'm looking at a dog, or I joke Silicon Valley, if you watch Silicon Valley, hot dog or not hot dog, but try to explain how you know it's a dog. It's hard. It is challenging, especially if you think about data scientists, they are incredibly cerebral and way smarter than me and they often have a hard time simplifying it enough where it's consumable, if you will. So it is a challenge and I think it's something that'll evolve as we start to use more of it because we'll just have to figure it out as an industry. I want to ask you about one of the things that we're hearing so much about this conference is the neat things that you're doing around eradicating employee pain points and taking care of all those onerous, annoying, tedious tasks that we have to do, the filling out of paperwork and all that sort of thing. What are sort of the next things you're thinking about? The other parts of the workday that are annoying for all of us when you sort of think ahead to the product lineup? So I think one of the things we do is figure out where you are and digital transformation is great, but it has so many different meanings depending on your company or your industry. So what we did internally is we actually gave definition and an answer to the question of how digital are you? So we take every process and a collection of processes to a department and bubble it up and so on forth and we rate every process on how fast it is, how intelligent it is, which is a measure of machine learning and what's the experience we're delivering. And taking those three measures we're able to come up with a score and more than anything it gave us a common language around the enterprise to say how do we move this from a score of 50 to 70? How do we move this from a 60 to a 90 and which processes are most important to move first, second and third, right? And without that it gets really hard because digital transformation can just feel like this abstract concept and as business leaders we do better when we have measurement. And once we have a number and a target and a goal it's easier to get people aligned to that. So that's been helpful for us as well on a change management aspect. So true. Coach K, you guys always have great outside guests come in and speak at your CIO decisions conference. I mean Robert Gates is one that, you know, I mean as much as you've accomplished in your life you haven't accomplished nearly as much as that guy. Very humbling. Coach K was one of your guests this week. You host that event. I do. Share with us some of the learnings from Coach K. We have Coach K, Duke's basketball coach. I would argue best coach, best basketball coach. I'm a Tar Heel. Sorry, Tar Heel here. Exactly. A couple in the audience. He said he's no Dean Smith the other day. Well. And I am a college hoops junkie, so for me it was a massive treat. I just wanted to talk to him about so many games and things like that, but he really gave a great talk about just how to be a better leader, how to constantly be learning and applying yourself. I mean he's 71 years old and how he needs, he talked about how he had to reinvent himself at least 10 times. He's been coaching for 42 years to meet the players where they are and changing himself. And every season, the day after the season ends, having a meeting with his manager saying what do we need to change? And it could be they just won the national championship. So never resting on his laurels, constantly learning. And he had really interesting anecdotes about when he coached the US Olympic team. And the difference of 18 year olds right out of high school versus these are the superstars of the NBA, massive egos. And one of the interesting things, he said so many interesting things, I could keep going on, but he said don't leave your ego at the door, bring your ego, because that's what makes you great. I need you to have that ego, Kobe, when you're taking that last second shot, because that's what makes you you. But also what he spent a lot of time is getting them aligned on values. Here's the core values at which we're going to operate as a team that are going to allow us to be successful. And I think that leadership lesson applies to any team. He applied it in a very difficult environment where millions of people are watching, but he talked about how he took that collection of individuals and made them a unit. And that was super powerful. Yeah, he coached the first dream team, which was magic. Yeah, I think he coached like four or five and... I think Bert, I think Bert might have heard, but he played and Jordan. I mean, that's trying to bring that eclectic mix together. And then to hear, have someone be so, you know, have done all these things and then be articulate enough to be able to say, and this is what I did. This is how I brought out the best in people and... Super humble and just, again, constant learning, right? I mean, John, our CEO talks about be a learning animal. I think Coach K embodied that in spades. West Point grad too, right? With a lot of discipline in his background. For sure. And then he talked about that. That's where he learned a lot of his leadership lessons. Really, yeah. Yeah, at West Point. Well, Chris, it's been so fun talking to you. Maybe we should get Coach K on with you. A little... That would be a treat for me. You and me could talk about Duke Tarheels. Yeah, well, okay, all right. We could bring John Wooden into the greatest coaches ever, conversation and fairness. We could. To the Wizard of Westwood, I mean... Cool, well, thank you. Thanks again for coming on. I'm Rebecca Knight for Dave Vellante. We will have more from theCUBE's live coverage of ServiceNow Knowledge 18 coming up just after this.


We will start this all off with a Happy New Year to everybody and thank you for joining the average golfer for what is going to be a full on week of tailor made testing. And we are going to start with the M5 driver. We will stick a shaft on it don't worry but there's been lots of hype already around this club all we've managed to see so far is a little brief snippet of that club face and these two red screws. We are going to find out exactly what they are supposed to do in helping this perform well in the hands of people like you and I. So it is all about speed, it is all about ball speed across the whole of that club face maximising that and getting it as close to as they possibly can to the legal limits in terms of COR.  We are then looking at this underneath the club you will see the T bar track system that has been introduced so there is plenty of adjustability. Two 10g weights allow you like I said plenty of movement within this club and optimise the ability to custom fit for you as an individual. Now once again we see twist face technology is going to continue into the M5 and M6 drivers and into quite a few more products of tailor made as well. So they certainly believe that this has been performing well, this concept they introduced last year about reducing spin in areas of the club face so low heel, high toe and again plenty of videos out there from last year explaining exactly what twist face technology does. But it basically allows us in theory to get better dispersion with off centre hits. We are then looking at this introduction of the resin into the face, this speed injection as they are calling it and surely we are going to produce a pretty impressive driver. In terms of looks it is some images thrown up on screen now. I want to stay out of the looks opinion to be quite honest with you. I think that is very much down to the individual so I will leave that one to you. Maybe we will talk about this a little bit more when I sit it behind the ball very very shortly which is what we are going to do right now.  But first of all let us get over to four golf chester and start hitting some golf balls with this M5 driver. Right so you have heard what Taylor made, I have got to say you have seen all the glossy pictures, you know what this thing looks like now, it is into four golf chester.  Let us talk about what I am seeing immediately from the performance of this golf club. Let us talk about address first of all, I love this carbon crown, I love it in this matte muted finish as well. They have also gone back to thinning this top line, you had a white and silver band in the M1 through to M4 series of drivers which is sort of three quarters of an inch thick. I like that to be fair, they have a problem with it, however they have said that, they have thinned it down considerably, it is a silver band that is still there and it looks good. I really like the finish on this, it has definitely got this sort of motorsport formula one theme running through this whole look of this. One thing I really do like is in the grooves here they have put a circular colouring, white colouring, a little cut away in each of the grooves which highlights exactly where the centre of this club face is and it is really handy at address. I really like that touch that they have added, this is set up at nine degrees, I have got the weight pushed quite a bit far back in this and we will look at this T bar in closer. A closer look at it in a later video but for the time being that is how it is set up and I am using this Tensai Red Shaft in stiff or Orange Shaft I think it is actually.  That is not the feedback I was looking for I was going to be a little bit more subdued but that is a bit of a ball to start proceedings off. First impressions, let's hit a few more balls before we draw any conclusions because that one is absolutely rocketed. Now that is a ball that comes right out of the toe. It will be interesting to see if I would love to know if there is a way of measuring the impact of twist face technology because if there is ever a shot that requires some assistance from twist face it was that because We have not got this club marked up, taped up as such to see where that came out of but I can tell you from feel, it has come high toe and it did okay to be fair. What would it have done without twist face?  That is a golf ball, absolutely flown. Right, let's stop it there.  Would have made it like rough I think but it certainly wouldn't have found the fair way, the other two balls, first balls, an absolute bullet. First things to note are, difference in sound and feel in this M5 driver from previous models that Taylor made. It is a more muted sound, I think it's a softer sound, I again prefer the feel. Feel doesn't mean say you're going to get better performance but it certainly might persuade you into why you might buy one driver to another. And for me, the sound that's coming off that which resonates in feel is a good one. Looking at the first, the two out those three drives, it's gone, it's flown. We're expecting it to but ultimately drive ball data will tell us exactly how it is doing in performance and I will do that a bit later on. But for now, that's my opinions in here.  So that's exactly where we're going now. So finishing up some feedback straight off the course from the M5 driver. Once again, it's really nice behind the ball, plenty of confidence to address, I love the way it frames the ball. For me, I could probably tweak it a little bit in terms of a lot of the shots that I'm hitting are slightly down the right hand side. Which is perhaps a good thing for me because I'm struggling to move that ball to the left at all. But like I said, that's a pretty much positive based on the way I swing the club. But yeah, maybe that adjustability, I can just make a little change in the weight down there. But overall, the ball seems to be coming off fast. I'm hitting it pretty much out the middle so I'm not exactly testing twist face right now. Which is unfortunate. But reporting on what I've seen out there on the course. This ball travels, great feel, great sound. Once again, very difficult to be critical with this club out on the golf course. Any errors? Well, they're down to the swing, I'm afraid. Right, OK, so that's pretty much all we can do as far as the testing this far in anyway. And I'm going to say this far in is because I'm going to spend a lot more time with this driver and others in the weeks ahead. To spend more time looking at data in particular, trying to analyse the impacts of twist face if we can ever do that. But it is a very small amount of data that I've collected so far with this club. But I took it out on the course, we've had it down at Fogolfchester and here is my evaluation of it. I'll give my opinion on the looks. I love it from the top side, I love the crown, I love that matte finish. Great looks. From the underneath, a very, very personal thing. I'm not a fan of adjustable drivers in the sense of all the nuts and bolts that are on the bottom. They don't ever look right on a driver to me. It's far too mechanical looking. So that's something that's very, these are never going to appeal to me adjustable drivers. And once again, I think it's pretty, pretty ugly from the bottom end to be fair. And not something from sitting on the shelf I would be gravitated towards. That is again, nature of the beast. If you want that adjustability then you've got out these nuts and bolts and these elements of screws to be able to move this weighting system around. That'd be an interesting point for me is how much difference could be made by moving these weights around. There are so many different permutations and in the timescale that I've had, I've probably not been able to optimise this club in terms of settings for me. And I think that's down to the limitations like I said in time we've had since I've had the club and the embargo date. I will be looking to make further videos on this, trying to have a closer look at adjustability and what impacts it has on this club. Because if you look at the dry ball data I obtained, it's a small sample shot of data, carry distance 243 on average, launching very, very high for the amount of loft that I had on this club. Up at what is it 16, 16.7, average in terms of pretty constant, very high ball flight. Incredibly low spinning considering the high launch. Combination not far off being good if you get probably, I would have thought, when I'll lower that launch down a little bit, bit higher spin and I think in the adjustability elements of this club you'd probably be able to do that. It's off the feeling, definitely there's no doubt about it. It's a different sound, a sound that I prefer. In terms of performance, how do we measure it? Well, that's a difficult one for me. How do you gauge on the small sample testing that I've been able to do? And don't forget this on the course and dry ball data.  How do you look and see if the resin that's been impacted on this face has it produced faster ball speeds across the face? I think we need a lot more testing in the weeks ahead to try and find out if we can identify how TaylorMade managed to achieve that. I can't back up that from the small testing that I've done. All I can say, it's a very, very good driver. The M3 and M4 in last year's top five for me were well up there. I think I had them at number two and three. They were very, very good drivers. So for me, it's still a very, very good driver. How much has it come on from M3 and M4 for me at this stage? Hard to tell. The only thing I can say is in terms of sound and feel there is a difference in terms of overall performance. I can't really see a great deal of difference at this stage, but I will promise you I'll do some more testing on this and we'll try and establish more of those facts. I think by spending more time hitting a larger quantity of shots. I'll get these out to the rest of the taggers team as well and we'll get a few different handicaps hitting this club in the weeks ahead. So for now, that is my first look at least at the Taylor M5 driver. I've got plenty of these to come, so stay tuned. Thanks for watching this one. Comments down below and I'll speak to you soon.


So, we are proving a Myhill-Nerode theorem and the theorem says that the following three statements are equivalent and we propose to prove this theorem by first showing 1 implies 2, then 2 implies 3 and then 3 implies 1. Last time we completed the proof of 1 implies 2 and just to remind ourselves what the proof was? We start with the assumption that L is regular, L is regular means there is a D F A M which accepts the language L and then we considered R M for that machine M. We defined the equivalence relation R M last time and we found that time the R M, the equivalence relation R M for M accepting L, we found that of course it was an equivalence relation and R M that equivalence relation was right invariant we proved that and it had finitely many equivalence classes. So, therefore it is a finite index and of course the language is the union of some of its equivalence classes. So, 1 implies 2 we can say proved by R M just considering the equivalence relation R M. Now, we would like to prove 2 implies the statement 3 here we start with the assumption that L is such a language satisfying the conditions here and then we would like to show that the equivalence relation R L for this language is of finite index. We defined R L last time and it was like this again R L is a equivalence relation on sigma star. So, it relates to finite strings you know in sigma star this relation is pairs of such strings and we said that x R L y if and only for all z in sigma star may be I will write it here. So, when will x R L y hold it will hold only when for all z to take any string z in sigma star then you take the 2 strings x z and y z by concatenating z first to x and then to y you get these 2 strings. Now, you see what are the possibilities x z in L and y z in L x z in L, but y z is not in L x z is not in L and y z in L and both x z and y z both of them are not in L right these are the only possibilities that can happen when you get these 2 strings either both in the language or exactly one of them in the language or none of them in the language. Now, for x R L y to hold these 2 cases must not be there. So, you see for 2 strings x and y if it is the case that for every z when you form these 2 strings x z and y z either both of them will be in the language or both of them will not be in the language only in such a case will say that x is related to y. How do we check whether x is related to y of course, here is the definition, but that is not the point right now the point is to understand the definition. First of all let us quickly see that R L is an equivalence relation see reflexivity and symmetry are quite obvious. In case of reflexivity you need to check the condition for x z and x z of course, you have only one string x z and. So, therefore, the conditions will be true this things that we have written symmetry is again you see this the way the definition is either both of them x z and y z in the language or none of them is in the language. So, this definition the way we are saying is symmetric. So, this is true and finally, transitivity let us check suppose that x R L y and y R L z and what I need to show that I would like to show that if x R L y and y R L z then x R L z. Now the reason for that is you take any string w now because. So, x R L z will be true if both the strings x w and z w satisfy this condition. Now let us say x w is in the language right then from this fact what do you get since x is related to y. So, suppose x w is in the language implies y w is also in the language and now from the fact that y is related to z I get z w is also in the language right and actually this is an if and only if statement x w in the language because of this condition I can say if and only if y w is in the language and y w is in the language and because of y being related to z if and only if z w is in the language. So, what I have is x w is in the language if and only if z w is in the language. So, x R L z. So, it is clear that R L is an equivalence relation and just to fix our ideas let me just consider an old language that we had considered before. So, let us say L is the set of finite binary strings such that x has even number of zeros and even number of. So, now of course, sigma is the binary alphabet 0 1. So, now consider x to be some string let us say 0 1 1 0 1 and y to be 1 0 1 0 will they be in the language or rather of course, we know that both of them 0 1 1 0 1 this is not in the language because this has even number of zeros, but odd number of ones and this has odd number of zeros and even number of ones. So, both are not in the language. So, that is ok, but we would like to know if x R L z. So, you see just consider the string sorry x R L y these two strings x and y we are considering and we would like to know if x R L y holds or not for this language L. Now actually x R L y is not true does not hold why because just consider this string z to be 1. So, when you get x z that will be the string. So, write x z here 0 1 1 0 1 and then 1 x z is in the language because it has even number of ones and even number of zeros this is in the language and now take y z y z is what 1 0 0 1 0 1 0 0 and then z again 1 this has odd number of ones and odd number of zeros. So, this is not in the language. So, that shows that these two strings 0 1 1 0 1 and 1 0 1 0 0 these two strings are not related by the relation R L for the language. On the other hand if you take another example for the same language L that is the context supposing I say 0 1 0 and 1 1 1 it is you need to argue, but you should be able to see for example, any string that you take see the point is this has even number of zeros and odd number of ones this string also has even number of zeros and odd number of ones. Now for this string to become a string in the language after augmenting some z that z must have even zeros and odd ones. So, that together they have even number of zeros and even number of ones. So, now if you add the same string here what will happen this already had odd number of ones you added through the string some more ones. So, two odd number of ones when you add them their number becomes even. So, the total number of ones will become even here and since here it is even of no zeros and only zeros are here. So, again the together this entire string will have even number of zeros and even number of ones. So, whenever this string is going to be in the language by adding some w that same w also will make this string when added this w when added to this string to the right that string also will be in the language. And similarly we should be able to argue that any string if it you know if you take any w if that w does not make the entire string in the language that case is will be the case if and only same w when added will not make the string in the language. So, you see what I am trying to say for this language l the two strings 0 1 0 is related by this relation r l. So, you are more or less clear about the relation r l and you should be able to see that this r l is defined for any language l whatsoever because you know we just follow the definition. So, and this is an equivalence relation therefore, it will partition sigma star in some equivalence classes. Now, what the statement three is saying that this r l is of finite index. So, let us prove this that if two is true that implies three we now prove that the statement two implies statement three. So, you see statement two is saying that we have a language l in sigma star such that there is so this is sigma star. What this language l is l is the union of some of the equivalence classes of an equivalence relation on sigma star satisfying these two conditions that that equivalence relation is right invariant and it is of finite index. So, let that equivalence relation be r dash. Now, first of all it says that one of the things it says that r dash is of finite index and r dash is an equivalence relation. So, therefore, it partitions the set sigma star in some manner right. So, these are the equivalence classes induced on sigma star by the relation r dash. Two things we know already one that the very fact that it is it is inducing this partition of course means r dash is an equivalence relation we have made use of that two other things is r dash is right invariant and which means the number of equivalence classes for r dash r dash is finite. So, this number of such equivalence classes is finite and now we are saying that l is the union of some of the equivalence classes. So, the language is solely consists of we take some equivalence classes entirely may be this one also this is just a picture of course. So, the statement two is saying that my language consists of the union of these strings and these strings and these strings and now from here I would like to show that that language l which is the union of some of the equivalence classes of r dash where r dash is an equivalence relation which is right invariant and it is of finite index that language l for that language l if I consider r l r l is of finite index this I need to now prove and which will show the implication two to three well the proof is by showing that any such r dash is a refinement we defined refinement last time and to prove that r dash is a refinement of r l what I need to show that is if x r dash y that means whenever two strings are related by r dash that means take any two strings in any one of these equivalence classes that would imply these two strings are also related by this is what I need to prove. So, of course, I know from the definition so, let us that this will be true if I can show remember it is just the definition x r l y holds if and only if for all z x z and y z these two strings either both in n or neither is. So, now consider the two strings x z and y z because r dash is right invariant I immediately have x r dash y if I take any z it is the case because the relation r dash is right invariant what I have is x z is also related by r dash to y z that follows that is something which follows directly from the definition of r z. Now, suppose x z is in the language that means what that x z is either here or it is here or it is here because remember the language l is the union of some of the equivalence classes of r dash to suppose x z is here. Now, what is r dash saying r dash is saying that x x z and y z of course, since they are related that means y z also is in the same equivalence class. So, if x z falls here being we have assumed x z is in the language so, it is in one of these dashed equivalence classes and now since it is all related by r dash y z is also in the same equivalence class which means if you assume x z is in l that implies y z is in l it cannot be the case that x z is in l and y z not in l and of course, this is true the other way also that if you assume y z in l in the same argument shows that x z will also be in l therefore, if x z is not in l y z also cannot be in l so, that shows what that whenever x is related to y by this r dash I can find no z such that exactly one of x z and y z will be in the language the other not in the language. Therefore, clearly x is related to y by r l as well and that proves that r dash is a refinement of r l now remember our notion of refinement. If something is a refinement of some other equivalence relation what does it mean we discussed that last time it just means that r dash at most breaks up some of the equivalence classes of r l into new equivalence classes. In other words the equivalence classes of r l are made up of by made up by combining some of the equivalence classes entirely of r dash now what does that mean that number of equivalence classes of r l is less than equal to the number of equivalence classes of r dash because r dash equivalence classes are made up by breaking apart some of the equivalence classes of r l. So, now if since r dash is of finite index that is what 2 says r l also has to be of finite index and that proves this that is indeed the statement 3 that r l is of finite index.  To complete the proof of my hill Nerode theorem we now need to show that 3 implies 1 the statement 3 implies. So, what is statement 3 saying that I have a language l and the relation r l is of finite index from here I would like to show to show r l is the language l is regular. So, from the assumption that r l is of finite index we need to show the language l is regular and the way we will show the language l is regular by actually constructing or defining a DFA for l using this relation r l. So, here is a definition of m a DFA using the equivalence relation r l and we will show that this DFA m which we are going to define now will accept the language. So, first of all my DFA m is going to be as usual q sigma delta q 0 and f I mean I need to tell what this q delta q 0 and f r sigma is of course, I know that sigma is the language sigma is the alphabet l is defined over. Now, we define q to be the set of equivalence classes the equivalence relation r l. Now, let me use this part of the book remember that we our assumption is r l is of finite index. Now, r l is an equivalence relation on sigma star the set of all strings over sigma and what more r l is of finite index. So, the number of equivalence classes r l induces on sigma star that number is finite. Now, what we are saying is that I will have an equivalence class standing for a state of my DFA you might like to think of this way that for every equivalence class this is r l induced partition sigma star and these are the equivalence classes an example of course, just a picture and. So, I would like to say that these I have a state of the DFA for this for this for this for this. So, in this case here there are one two in this picture there are one two three four five six seven equivalence classes. So, in such a case q will consist of seven states q will have seven elements. Now, I think you know to what each of these equivalence classes consist of they consist of strings. Now, how does one denote an equivalence class one way of denoting is that take any element here let us say x which is a string and you might remember that x denote. So, let me in fact write this. So, for any x in sigma star this is the standard notation that this denotes which is this stands for the equivalence class to which x belongs. My definition for q therefore, is of this machine m can write q to be like this in this manner although it looks that for every x in sigma star I am doing something, but the number of these equivalence classes is finite because that is what I have my statement three said that r l is of finite index therefore, there are only finitely many equivalence classes. So, this set the right hand side set is finite and therefore, we can take that set to be the set of states. So, after that how do I define delta now remember the way we define the transition function is what it was delta for any d f a delta was a mapping from q cross sigma to q. In other words q if it if delta will look like this that it will take a state it will take a symbol and it will tell me what does the d f a which state the d f a goes to from state q on symbol a will define delta. Now, as I said the delta takes two arguments one is a state and one is a symbol. So, our states are of this kind equivalence classes of r l. So, let us see I have this state and the symbol is a then we define it to be that state to which the string x a belong. Now, this definition looks all right however we need to ensure one thing remember that equivalence class x consists of many string. So, suppose y is also here. So, in other words the equivalence class x is same as the equivalence class of y. So, the way we are doing it of course, that this equivalence class we are representing by the string x you know well we write like this equivalence class we are representing by an element of the equivalence class. And then we said that this is the equivalence class and the state corresponding to that is the state to which my machine will go if it is in this state and the symbol a comes. The natural question therefore is that since the state is same whether you represented it as this or this that is the equality is the same state because this is the same equivalence class. Then this for this definition to be what we call this definition to be well defined it must be the case that if I had taken some other representative for the class let us say y in this case that delta y and then a of course by this definition I will get y a d must be same x a and y a must be same only then this definition is meaningful otherwise it is meaningless as it turns out that will be the case why remember what do we know of x and y that they belong to the same equivalence class of r a. So, in other words let me now write it here I am trying to prove my definition this is the definition of delta and that definition is done properly it does not depend on which representative of the equivalence class I take and that is why I am saying in order to make this particular definition you know to convince you that this is a proper definition I need to show this that given that there is another string y in the same equivalence class. So, that is why we are writing the equivalence class of x is same as equivalence class of y this I should show that the equivalence class of x a is same as the equivalence class of y given that x and y belong to the same equivalence class. Now x and y belong to the same equivalence class means this that is the definition they are related by R L and this will be the case if x a also I can show to be related to y a. Now to show this that x a is related by the same relation y a related by the same relation to y a I need to show that for any z x a z and y a z whenever x a z is in the language this means if and only if y a z is also in the language. Now remember x a z x a z in the language now consider this as the entire string a z which you are appending to which you are concatenating to x. Now since x and y are related x with this string appended if that is in L since x is related to y then y a z also will be in the language and vice versa. So therefore, it is indeed the case that x R L y implies x a R L y a and therefore, this definition is well I mean it is well defined. So, we have done that so my definition for delta is complete and in order to show what the machine is I need to provide the definitions for q 0 and f. Now what is q 0 q 0 is the state initial state where the machine begins. Now it should not be too difficult to see intuitively that if I take q 0 to be that state or that equivalence class to which the string epsilon belongs. Remember epsilon is the empty string and epsilon will be in one of these class one of these equivalence classes that equivalence class is the initial state of the machine. See we are you know we have considered that equivalence classes of R L they constitute the states of the machine m. So, please keep that in mind so at one level we are talking of equivalence classes which are strings but at the other level we are thinking in terms of this entire equivalence class is a state and takes a little while may be to appreciate this, but really there is no difficulty think of the way we have done it here at least q you can see there will be finitely many elements delta this is the way we have defined makes sense. So, this also makes sense that q 0 is something q 0 is a state of the machine m and. So, therefore, it has to be an equivalence class of R L because that machine m the states of the machine m are the equivalence classes of R L and we are defining the initial state to be that equivalence class to which the empty string belongs. Now, will show of course that this makes sense but before that let me complete the definition and finally, the set f f is a subset of q remember this is the final states. So, this I put it this way all those equivalence classes such that you see we are what we are saying again x is in the language of course language is infinite that is fine, but you think of you these strings which are in the language the there are only finitely many equivalence classes. So, they will distribute to some of these equivalence classes those are the classes they when thought of as states of the machine m are my final states of the machine. So, this is a definition, but does it make sense first of all does this machine as we have as we have defined does it accept the language L that is what I need to show we need to show that 3 implies 1 and in that starting from the definition of RL I defined this machine m, but ultimately what I need to show that this machine m the language accepted by this machine m I need to show that this language is nothing but the language L well look at it this way. So, first of all let me show that if I take a string in the language then that string will be accepted by this machine m let see how to suppose x is in the language and the string x consists of a 1 a 2 a n these are the symbols the string is of length n and this string x is consists of this symbols a 1 a 2 up to a n. Now, how does the machine behave on this string initially the machine is in this state epsilon then this symbol a 1 comes. Now, look at the definition of delta if it is in that state epsilon then a came what is the next state it will be in this state the state to which the string a 1 belongs then the machine m will be in this state. Now, a 2 came the machine will be in this state by definition by definition of our delta. So, now finally the machine after scanning all these symbols these n symbols it will be in this state this string a 1 through a n is accepted provided the state is a finals but of course, that is the case because since this is an element the string is in the is an element in l is a is one of the strings in the language. So, you replace a 1 a 2 a n here that is in the language. So, therefore, this state this state will be a finals. So, this proof is very simple we just followed the definition and what we see finally we are in this state and that state is of course, a final state. So, this is accepted now what what else to what actually I have shown is that this language l is a subset of l m because that is what I just showed that you take any string in the language that string will be accepted by definition. Now, what about the other way I need to show that l m is a subset of the language l one way of proving that is it if I take a string which is not in the language then I should show this machine the DFA that I have defined will go to a state which is not a final state. So, this is like this I am trying to show this part. So, what I need to show that starting from x in l I mean starting from x in l m I should show that the string x is in the language and now use the contra positive of the statement that this I can show one way of showing this is saying that if x is not in the language I should be able to show that x is not in l m. In other words if I take a string which is not in the language then I if I manage to show that string does not take the machine to a final state then I am done. So, again the thing is very simple supposing that string now let us say a 1 a 2 a n is not in the language l. So, how would the machine behave on this string the same way that we had shown at the end of this string when this string is scanned by my machine m that will be in this state, but this is a string which is not in the language and. So, therefore, this is not one of the final state if this equivalence class to which this string belongs is not one of the final states of the machine because by definition the final states of the machine is all those equivalence classes which are consist which consist of strings which are in the language. Now you might wonder see this is again it looks kind of we are just you know waving hands and proving things, but this proof is rigorous. You might wonder what if I have a string I have a equivalence class of R L like can it happen that x is in the language and y is not in the language can never happen because consider this just the string epsilon x epsilon will be in the language, but y epsilon will not be in the language and that would mean that x and y could not be in the same equivalence class. So, really one of the ways R L one of the implications of the definition of R L is the language L consist entirely of the union of that is you know some of these equivalence classes we take their union and that gives me the language L. So, it cannot be that some strings from here and some strings from here will be in the language L that we can prove and therefore, all that make sense what we said and that completes the proof that three the statement three of my hill narrowed theorem implies one just to just to wrap up we started with the assumption for a language L to have R L which is of finite index that means it has finitely many equivalence classes from there I define a DFA and then I showed that DFA accepts that same language L. So, therefore, three statement three was R L is of finite index and statement one was L is regular and that is proved because my DFA that I define this M that I define this is the definition of the M that indeed accepts the language L and you see the nice thing about one of the very important things this theorem tells me is what is that the notion of minimization of states let us see why so my hill narrowed theorem tells me what is the minimized automata what will be the minimized automata minimize DFA for a regular language L is it supposing L is regular then that means that I have a DFA M such that L M the language accepted by that machine is the language L by definition and then if you consider R M what do I know that R M is a refinement of R L that was the statement of the basically one implies two that is what we said now then that means what whatever machine that you may take that machine the states number of states cannot be more than sorry cannot be less than equal to the number of states of number of states of the machine of the DFA that we define through R L once more you recall that from the definition of R L gives me a DFA for M dash for L and if you if you if you just go back to the statements of the my hill narrowed theorem what you know what you can see immediately that any machine M accepting L the number of states in that machine has to be equal to or more than the number of states defined through R L DFA remember we just used a DFA definition from R L now minimization of states will be that coming to the DFA which corresponds to R L from R L that definition of the DFA that we got that is indeed the best in terms of number of states and this is what we will follow up in our next lecture.