I'm really excited for today. It is Friday. Julie and I always like to have some fun. We like to bring some levity and we are so glad to Fundraising Academy for bringing to all of you nationally the Ask and Answer episode. So thank you so much to Fundraising Academy and thank you of course to our other presenting sponsors. We are so grateful to have each and every one of you, all of you equally hold a special place in our heart and of course, you know, continue these conversations. Now 347 national episodes, so 347, that means we hit 350 next week and we are super excited, not a single stress line, frown line, any like age marks on us, I'm sure. So it's been a lot, it's been a lot of fun but thank you of course to our presenting sponsors and thank you to Julia Patrick for coming up with this wonderful idea.  We're only gonna do it for two weeks. And again, this has been the longest two weeks of my life but I really do love it. I'm Jarrett Ransom, also known as the non-profit nerd, really excited to continue this journey and ride this roller coaster right along with you Julia. So I'm CEO of the Raven Group, my own consulting firm but each and every day, Julie and I come together, we join forces and you can go back to the questions. We join forces to really provide that levity, which really comes on Friday because Monday through Thursday, we are talking to so many amazing experts across the nation and really, really grateful to have their voices, their time, their expertise. And Fridays are reserved to provide you some answers, some solutions as we all continue to navigate this journey. So we've got a set of questions prepared for today. There are always four ways, more ways, more ways than four ways to get in touch with us. I know I feel like I just wrote my own Dr. Seuss book. That's actually really good. Right, there are always more than four ways to get back to this. So yeah, let's do get us started. Of course I say that right as you take a sip of coffee. It's really good. So I have a couple of things I wanna talk about. First, I was unable to attend at the last minute an AFP conference in my community. I reached out to one of the organizers for that day who happens to be Moves Management, Sasha Lewis, one of our sponsors. And I said, hey, something's come up. I can't get downtown. Please, if you can give my ticket to somebody else. And so I got this hilarious email from a viewer whom I've never met and she was clutching her pearls and she sent me a picture and she's clutching pearls and she's like, I got to meet Julia Patrick today. Clutching, I clutch your pearls. I got to meet Julia Patrick today. So anyway, I just wanna say thank you very much. You know, we're big supporters of AFP globally. They do amazing things. And I was really looking forward to being a part of a larger discussion, but thrilled that one of our viewers could actually attend and do well. We had a really interesting viewer, our viewer guest on yesterday, Jared. Yes, Mitch Stein, CEO of Pond. Okay, now give us some background about him because he had a really interesting thing that he was doing for all of his clients. Yeah, so he's a tech startup. We met on LinkedIn. I loved his authenticity and just his true transparency to his true self, but we met through our guest that's coming up next week, Jessica Campbell. And again, that is a, Jessica's doing a virtual conference. So Mitch is one of her presenters. I'm one of the presenters of the three-day conference as well. And he jumped on board to a comment thread, but essentially created Pond. Pond is for a lack of a better term and he used this. It is like a dating site for nonprofits. So it is a nonprofit in need of a service, a consultant or service provider that can provide content and product solutions essentially to that organization. And so it's a match made online, match made in heaven, match made online for organizations and providers to assist one another in this nonprofit space or in Pond as he refers to it. And so Mitch was on, he was doing this, is doing currently, he's in New York right now doing a tour of Pond meeting with constituency bases. And I joked around and I said, oh, this wasn't on the show, but on LinkedIn. And I was like, I wanna buy that tour t-shirt. Like when can I get that tour t-shirt? Cause that sounds like a really fun tour, right? Like a band concert. So he shared with us yesterday that literally last minute, you know, he received, I don't know, a call stating that his event had to really shift in shape in a whole new form. So he got back to you. He got back to us and he, this was all about the resurgence of COVID. He had a very large venue space. It was an art space for a very large group of people. Everybody was excited. He had all these RSVPs. The mandate came down, but they had to shift the number of people that could be assembled at any one point in time. Very creatively, instead of just canceling their event or hoping that not everybody who RSVP would show up or denying guests at the door until other people had cycled out. They came up with a three-tier system. They broke their event down into three different periods. They emailed everybody, got on the horn and said, hey, we've got to readjust. Can you pick one of these three times and we'll cycle you through? And what he found, what he was hoping in this email he sent to us, he said, yeah, it reduced the number of people in the venue space, but it allowed our team to have better quality time with those people that we were entertaining. So it ended up actually being something very positive for everybody. The guests had more room. His team had more room. It wasn't just that mad cocktail party kind of rush. So I really appreciated what he had to say yesterday, Jared, because he was like so many nonprofits to work really hard to get on an event going, really put a lot of financial hope into that event and then it changes on a dime within 24 hours, less than 24 hours. Right, all completely out of his control, all completely out of his realm, right? But what I loved was he had that mindset of, there's a solution, right? You may not know the solution, but there's always alternatives and there's always solutions. And he applied that to the tech industry and to Pond, which he's the CEO of. But yes, you and I were talking about how we are consistently navigating now the Delta variant, navigating this inner space. And you aren't alone, right? But we did have our guest yesterday, literally like live in this transition in the moment. And he still took time out to honor his commitment to us, which as you said, you were like, I would have been a little more frantic. I was like, dude, I would have like said, I can't do this, go away. You crazy ladies that are asking me all these questions, no. It's part of the process, right? Yeah, and he was lovely and he was, I felt like he would have given us all the time in the world. He was calm. I was like, wow, I was super impressed. Okay, all right. So two other things before we get into some questions. I've had some people that have emailed us and said, we do not believe that you actually have an executive producer, Kevin Pace. And so I'm going to ask Kevin Pace, you don't have to come on camera. Oh! I just want you to say hello. Is it you? Apparently we don't. Kevin Pace. He's still here. Oh! Also called The Voice of God. The VOG. Okay, VOG, thank you for making sure that we know you're a real person. I do realize that you haven't shown yourself on camera because you are actually editing on the fly. As we every day for 350 shows, our executive producer has been there without fail. Jared and I get to take days off once in a while, do other things. Kevin Pace does not. So I just wanted to say yes. It's a real human being. Okay, Jared. Yes ma'am. This question came in, I don't have a slide for it because it just came in yesterday. And it says, so I have a question for the Friday show and I'm not certain it is appropriate to ask. What is the best? Maybe the best. This is like, you know this is going to be good when that's how they start. Of course. What is the best way to address and thus be able to work with an unvaccinated colleague who refuses to wear a mask in meetings or when gathering with donors? This person infers she is vaccinated by what she does not say and by not wearing a mask. Are others experiencing this dilemma or am I too wedded to messaging from Dr. Fauci and the CDC? Wow. I can imagine that this is not the only situation happening. There's a lot going on in our world. There is a lot. But I also know there's a lot of hate going on and that really does not sit well with me. It really unsettles me. So I don't know the best answer to this, but I would think that we, this is an HR policy issue, a personality issue, right? Yeah, I did reach out to Sky Mercer. Oh, good. Who is an HR consultant in the middle part of the country who deals with a lot of nonprofits and she have a bunch of different things to say, but the boil down is what I took from her response was understanding what the expectation is and developing a policy. And that's a typical HR response. I mean, HR people are really good, I think at distilling what a course of action should be, whether everyone agrees with it or not, the concept is that if we all understand what it is, then we can measure ourselves against it, right? We can determine if we're gonna follow it, if we're not, you know, that whole thing. And so this is a new problem, if you will. We're talking a lot about it across the nation in regards to schools and all of this. But I'm wondering, Jared, what your thoughts are about this because you've been such a good leader, thought leader in how we communicate to our teams and how we bring them together. What are some of your thoughts on this? Well, so I always like to go to, you know, the system, like what's the policy, what policies do we have in place so that it's not a people issue, it's a system issue, right? So like what do we need to do as an organization to create a better system policy procedure standard? And I think it needs to remain standard. I often use the analogy of, you know, if you allow someone to do something even though that's outside of policy or outside of bylaws, it is equivalent to letting a dog jump up, right?  And so I'm not calling your board or your staff dogs, but I am used to a very similar training style and consistency. So it's all about consistency, right?  Or your front door. So to me, it's consistency, right? And again, I use that dog behavior because I think many of us, you know, we can relate this and boil this down to what happens in our home. I have another story, very similar, but it's like, you know, if you allow your toddler to walk around with grape juice, that time you go visit your great aunt, I don't know, Pearl, right? And now the toddler's sitting on the white couch with this grape juice, like scary. So I have a couple of those stories that I like to enter mine. So for me, I think it's that consistency. This is our policy, this is our standard, this is what we need to stick up for. Now I also know there's a lot going on now, even restaurants requiring vaccination cards, right? So having that proof of vaccination. And so again, that goes to a policy. That goes to, you know, the restaurant itself is stating a policy for its dining patrons. And so if you're meeting with a donor, you know, I really think it comes down to what is our internal policy? And how are we keeping that policy consistent at all times and enforcing it at all levels? Right, I agree with you. And you know, I think it's bigger than the virus. I think it manifests itself in how maybe a dress code or whether you are, you know, it's appropriate. I don't like to use the word loud, but whether you have a policy for drinking while you're entertaining donors or travel expenses. Work schedule. Yeah, I mean, showing up on time for meetings, whatever it is, my sense of it is that it is a policy issue. Now, having said that, this is going to be a heavy lift. You know, getting to the point where you have a policy, that's where it's gonna get to be a sticky wicket. And that's okay. But I think that without some sort of policy and you could set a policy that will be for the next two weeks or until further notice or until the first of the month or it gets reviewed every Friday, Friday. I don't know what, but yeah, the vaccine Friday. I mean, I think that that's probably the best step to take so that you can defer this hurt feelings and anger and upset that is really escalating in our country. Yeah, and it's really, you know, again, I've said over the course of the last two years, we've been navigating pandemics plural. And oftentimes now the mask mandates, the mask policies have become political. And so it adds that other layer of the pandemic, right? So we continue, I think we all continue to watch the CDC guidelines as that is the platform and the expertise to provide these policies. And I know here in our community, right? Like our children are going back to school, our city and official offices are now mandating policies of the mask wearing again. So there's a lot changing. And I know so many of us are looking forward to fall events. We've put them on our calendar. We've planned to have an in-person event because we are all craving connection and desire to see one another in the flesh. And so now this is adding that layer of anxiety, added layer of stress. But if you joined us on the Chitty Chat Chat, the thing I did say is, and I have to look down because I wrote it, is change is mainstay. Change is not going away. That is the one consistent is change. And I like that you said that because I think that allows you to get some perspective on this unsettled times because these unsettled times are gonna be with us. And then the repercussions of going through all of these pandemics are gonna be with us for a while. So, you know, understanding that we need to look at the ecosystem of change and how we react and how everyone else around us reacts. And no one likes change. Like there's, I shouldn't say no one, the majority of people don't like change. Some of us actually embrace it, but it's still, and I would say I'm an embracer of change. I actually welcome change, but it doesn't mean that it sits well with me, right? It still makes me unsettled. I'm a very type A driver personality, go figure. So in my home, I know, shocker, I really like to have things in place. I like there to be consistency. I thrive on a level of normalcy and a level of having that routine, but we are living in a time where our routine is out of our control. So there has to be a bit of willingness, right? Willingness to continue to navigate this change. And again, as we said 18 months ago, Julia, provide grace, right? Ease and grace, give it to yourself, give it to others, give it to the situation. So I do feel like we're going back to 18 months ago. It's like groundhog day and the same alarms going off.  I think it's really, really, really concerning. And I think it's going to test the metal of our sector. I think that we are in this period of the great resignation for many reasons which we've talked about and we will continue to talk about, but the reality is the nonprofit sector is even more in a perilous situation because we have emotional and physical fatigue and we have economic fatigue too for a lot of our team players and staff workers. So yeah, this is a really going to be a really intense time. Well, let's get into a couple of the questions that we have today because these come in all the time. As Jared mentioned, we have four ways that you can get to us. And Josie from Dallas, Texas writes, we have a board member who wants us to start a retired board member board. Have you heard of this? We talked about this a lot, the emeritus board. What's emeritus Latin for? Old soldier. Yes. I'll be ding-danged. Good job, Jared. Yeah, emeritus Latin for 400, please. That's it. That's the only Latin I know. Oh, yeah. Josie, you know, emeritus board, we have courses on the American nonprofit about it, nonprofit Academy website. We've done episodes about this. You've got to investigate this. It will be really a great tool, I think, for your organization. Yeah, absolutely. Yeah. Check out the AmericanNonprofitAcademy.com. I know, thank you. Okay, name withheld, Scott Still, Arizona, you know how I love my name withheld. Can or should a staff member also serve as a board member for the same organization? I personally think this is a conflict of interest. I personally agree with you and I professionally agree with you. That is a conflict of interest, right? And I think you should have your staff attend board meetings. I don't know if it's policy on attendance. I have attended many board meetings of, sorry, as a staff and I don't have a voting rights. Certainly don't have a voting right. And I, in fact, walk into that feeling as if I don't even have a voice right, which sounds weird saying that out loud, but literally I'm going to observe, I'm going to kind of like listen and watch what's happening. I tend to take a lot of notes. If on occasion, you know, I am asked to provide my opinion or feedback on a certain thing, I certainly do that. I do think once you get to know the culture of the board and how they're run, if there's something that you feel you could add to the conversation for the greater good. I have on occasion said, you know, if appropriate, I would love to add some comments here, you know, if that's of interest to you and the board. But I ask that permission because as a staff member, that's not, you know, that's not the place to be present and vocal. No, so now this is where our 20 years of age is going to come into play. I know clutch the pearls, clutch the silver collar. Hey, first and foremost, I believe as you staff should be at the meetings, not at the table literally, but they should be there as source experts in case there are questions or data points or concerns that only a staff member can ask. But this is the gig. If you, pardon me. Only a staff member could answer perhaps is what you. Yeah, what did I say? Ask or answer? Yeah, I'm sorry, answer. Okay, see, there is a 20 year age difference. We are source experts, that's what you're saying.  You should be prepared so that you can accurately answer a question. And of course, if you can't, then you have to say, I can't answer that, but I'll research it and get back to you. Bottom line, that's what they're there for. This is where the rubber meets the road. Your board meeting should be run by Robert's rules of orders because this is a fiduciary commitment. It is a legal component of how your organization is operated and reported back out to governmental agencies, contractors, donors, all of that, funders. In Robert's rules of orders, when you are discussing an action that's gonna be voted on, there is a point in time where you can ask for, you open the floor for questions or comments. And so that is a structured point where staff could then engage. Now, I'm not saying you bring, you know, 45 people. You bring your C-suite, your executives, you know, the people that are in charge of the things being discussed for the most part, but there is a process for that. And that, I think, is how you protect the entire discussion. And Jared's like, oh. I know, I'm smiling because all of this consistency is process. Our whole episode has been about staying the course with the process. There are procedures in place and it is to, you know, build value, to build culture, to build consistency. I do like Robert's rules of order. And there are some individuals that are really sticklers on it and they know it so very well. And that's wonderful, right?   So yes, but what I'm hearing and kind of like smugly, you know, smiling about is the process. Absolutely. Everything has and should, I just should it on you. Everything should have a process. Well, you know, and I think the other thing too is it breaks my heart to think that you as an educated, intelligent, engaged woman of an organization who has a lot of insight. And frankly, a lot of times this is what's missing with board leadership. They don't have boots on the ground insight. Yet they're making decisions that impact the performance and the management of the organization. So not to have that voice is really a missing link for many, many organizations. You know, and we talked yesterday with Mitch sign of Pond that there is an evolution happening right now. So I had mentioned the innovative disruption but Mitch really took it further to say, you know, it's an evolution. And I agree our sector is in an evolutionary time. And so that means even some of these, you know, board operations, the policies, the procedures of how the board meetings are run is also in an evolutionary time. Absolutely. And if you think about just the movement to virtual board meetings, you know, like our friends are boardable. Okay, we got time for one super quick question. Okay, thank you again. I know it's really, it's good and short. Sherry from Cincinnati, Ohio writes, who should pay for professional membership specifically looking at AFP? Oh, how interesting, we brought that up earlier. We have some differing opinions in the office. What say you? So typically I am aware of like professional memberships, personal development, professional development budget lines. And from my experience, most departments have this. So the organization has figured out kind of what that amount is. And then it trickles into various departments. Now, Sherry, one of the previous ask and answers, I was a huge proponent for asking your staff what they want. What do they want? What do they need? What are the cost of that to help you build this budget? So that's kind of taking your question to a whole nother level, but really assessing with your team what it is that they are seeking and building that opportunity. So I do think it comes down to the departmental budget, but that should be included with building your overall organizational budget. Right, I think that's important. But the organization, by far, the organization I think should pay the membership. I do too, because I think it's gonna be who the organization, it's a lot of times these groups don't have membership packages that work for under the umbrella of an organization. So you'll have to kind of explore that. I'm also gonna just, because I know time is short, I would also Sherry recommend that you, your team, whomever look into sponsorships or look into a reduced rate because those do exist, especially in the Jedi space when it comes to organizations looking to diversify their membership base. So I would look and see if there's any way you can cut corners and dollars by receiving some of these sponsorship dollars. Love it, that's very wise.  Hey, hey, I'm Holly. I was gonna say I'm here all day, but really we're not, because we're signing off. Peace. We're signing off. Hey, I'm Julia Patrick, CEO of the American Nonprofit Academy. I've been joined today by the nonprofit nerd herself as evidenced by her eyewear. Jarrett Ransom, CEO of the Raven Group, really one of the great luminaries of our sector and we are thrilled that she's with us. As Jarrett mentioned, we have four ways to get a question in. Kevin Pace, our executive producer is actually the man who runs our Twitter feed. And so- And you heard his voice earlier. He heard his voice today.  Hey, again, our thanks to Fundraising Academy who stepped up to be our premier sponsor of Friday Ask and Answer. It's a lot of fun and they are an amazing partner with us. Again, all of our presenting sponsors, we could not do this without you. So we wanna say thank you. And Jarrett, a great week. Next week is gonna be really interesting for many, many reasons. And we want to remind everyone to stay well, so you can do well. We'll see you back here. Have a restful weekend. You too. Thanks everyone. Thanks.


Right, welcome back to the Sportsmax zone. For the second consecutive year, President of the World Anti-Doping Agency, Beto LeBankar is in Jamaica for Widers Forum for Caribbean Sports Ministers. Approximately 18 ministers of sport will be in attendance for the two-day forum set to be held on Tuesday, February 16 through to Wednesday, the 17th. In reference to the forum, Jamaica's Minister of Sport Olivia Granger said, We will report on the progress we have made as countries and as a region in implementing the Anti-Doping Action Plan that we discussed at last January's forum, right here in Kingston. Additionally, we will discuss a number of clean sport issues and initiatives, including athlete education, strengthening national anti-doping organizations in the region and the critical process of updating the water code. Well the occasion marks the third year that water will be meeting with sports ministers in the region to discuss ways of strengthening anti-doping mechanisms. The Sportsmax zone is very pleased to welcome President of the World Anti-Doping Agency Beto LeBankar. Welcome. It's a pleasure to be with you. Yes, so you know you've been in Jamaica twice already. The first thing I thought as a trainee is you of course enjoy coming to Jamaica and these conferences are happening here often. Definitely, I enjoy Jamaica. May you are very lucky as Jamaicans. You live in a really beautiful country, a great sporting nation, so it's a pleasure to be here again. Right, really happy to have you. So tomorrow, big conference again. The second one within a couple of months. What's on the agenda, President? Well, it's a very important meeting. We're going to discuss how to strengthen the anti-doping system in the region, together with the ministers responsible for sport from the Caribbean region. This is their responsibility to take care of anti-doping policy in their respective countries. We as a water, we regulate the system, we oversee the system and our duty is to make sure that athletes compete in a level playing field. So this is the goal. We're going to discuss many important things like compliance procedures, how to follow water rules and processes, and you mentioned education. This is the key. We want to educate better athletes. We want to make sure that athletes are well informed about the procedures, how the anti-doping system looks like. So it's going to be tomorrow in the agenda and in the discussion with the ministers. Yeah, how impressed have you been with the first meeting and the turnout by the ministers? Because, you know, of course, it's such an important education forum. How impressed are you? Yeah, we see the progress. Of course, you know, there is always room for improvement. And we see when we look at the Caribbean region that there are some, you know, countries and places where the anti-doping system doesn't look perfect. Let me put it in that way. And that's why we are grateful that our friends from Jamaica took the leadership as a country which has quite strong anti-doping system supported by the government. That's why Jamaica is hosting the event. And so education this time is a key topic. Last year, we discussed the compliance procedures. We encouraged the ministers to do more. We tried to convince them that the strong anti-doping system is a matter of credibility. Right? If you have a good athlete, if you're an athlete, the athletes from Jamaica deserve to compete with the athletes. From other countries in the same conditions. So this is the responsibility of the ministers of sport to make sure that the anti-doping policy in their countries is working well. How far off is the Caribbean to where you would like it to be? Well, I mean, we want more testing activities from the Caribbean side. I mean, from the countries. We want more education actions too. That's why tomorrow I'm going to discuss with the ministers and appeal to them to do more, actually, to strengthen the anti-doping system. You know, we really believe that sport has this power to unite people, to build something stronger. You know, sport is the best teacher for young athletes, but only sport without corruption, without doping this pure clean, beautiful sport. And this is the responsibility of the governments to build this credibility, to protect these values. This is actually our role as a water. President, how often, globally, would you have an exercise like this? Well, in all continents, actually, we have meetings, we have gatherings like this, assemblies in Asia, in Africa, in Europe. This is one of my duties as a water president to collaborate with the governments because our structure is unique. 50% of our representatives and contributions come from the sports movement and 50% regular contributions from the governments. What I was established in 1999 after the big doping scandal where there were no harmonization anti-doping system based on this unique structure, governments and the sports movement. So we have to collaborate with the governments without working well with the public authorities who are not able to achieve our mission. We are anti-doping global regulator. But, you know, the anti-doping policy in the countries, this is the responsibility of the governments. So, I mean, you know, the anti-doping system is very unique because I always, I love to use these arguments that, you know, we harmonize, we have, we oversee almost 200 countries. We have 700, more or less, 700 code signatories. I mean, those who signed the code word anti-doping code, the most important document in anti-doping. So can you imagine, you know, I know, criminal or administrative law coordinated or harmonized for two or three countries politically is impossible. And we managed to do it in anti-doping. Starting many years ago that we oversee, you know, almost all disciplines, all countries. And this is something very unique. And but of course system requires development, changes. And from our side, from my perspective, I have to say that we are in a really better and good place than in the past now. Having said that, a few decades ago, the general feeling was that drug cheats were maybe five or six steps ahead of water. Do you think drug cheats are still ahead of water? And if so, are there fewer steps ahead? Well, let me put it in that way. We will, it's not possible to eliminate doping from sport. You know, you will always find someone who will be able to cheat the system. You know, this is like with a with a crime in every dayness. You will not eliminate it definitely. But when I look at our tools, which we have now, we are in a much better position. Look, the anti-doping, it's not about testing now. Only testing. We have intelligence investigations. We work with Europol and Interpol with law enforcement. We conduct investigations currently only in Europe together with law enforcement. We have 55 operations. Thanks to these operations, we're able to detect, you know, hidden laboratories to confiscate dozens, tons of steroids. So what is not only about regulating the system, but working with law enforcement. We have tools like athlete biological passport to monitor, you know, behavior of athletes. I mean, how the body exactly reacts. We have, we are able to retest the samples after many years. We have compliance procedures. We have tools now even to sanction, you know, countries which are not following the rules.  If not, if your national law is not regulated, adjusted to the code, you can be non-compliant with the code. As a country, what does it mean? It means that one of the consequences is that your flag will not be you will not be in the world championships exactly. So you cannot be the host of major sporting events. So those consequences are really strong. So it shows our position currently in the sporting ecosystem, how strong we are, what we can do, what we can impose when someone is not following our rules. Yeah, and you just referenced drug cheats being phoned out years after testing. Initially, there was an eight-year period, but it's now up to 10 years that an athlete can be tested positive, which gives water that span of time to, you know, having probably upgraded and improved its testing that athletes can be phoned guilty even 10 years after performing. Exactly. Today I had a very interesting meeting with the athletes, Jamaican athletes, and I have to say that I received a lot of tough questions from the side, very nice fair discussion from them. But and this issue was raised by them, you know, and I've told me, no, guys, I was an athlete. I was running 400 meters. So I understand that sometimes this feeling that after many years, you know, when you are an athlete, you want to win this medal exactly in these championships, in these Olympic games. And after many years, it's like, you know, but it shows where we were actually and where we are now with the testing that methods, you know, now we have devices more sensitive to detect those, you know, violations which were committed by the cheats many years ago. So it's a very successful project only from the London Olympic Games. We are able to test positive more than 100 athletes after many years. Retrospective testing is the label it's given. Yeah. Yeah. Interesting you speak about being an athlete. I do remember you competing at the 2007 World Championships in the four by four for Poland running the second leg in the heats. Did you return for the final? Yeah, no, I didn't. I was not in the final because of the injury actually. So it was a difficult moment for me in Osaka because I was in a really good shape. But after the qualifications, when I ran with my colleagues, I got the injury. So I of course got the medal because my colleagues won the bronze, but I was only in qualifications. And then next year was another because I wanted to be the Olympian. I was in a good shape, but another injury. So yeah. And then you called it quits as an athlete after London? Yes. And you know, when I look at my career, I'm happy that it went not so maybe how I expected because things that I retired early, I am where I am because I was, you know, in my previous capacity, I was the minister of sport and tourism in Poland. And then I have a honor to be what a president. And then, yeah. And just 39 years old as well, you've achieved so much. If you had continued running, then we wouldn't be having this discussion with you probably. It was exactly my argument when I spoke with my wife, when I told her that I want to finish the running. She was like, no, no, you have to continue. I said, no, no. Yeah. You have a plan. You didn't know the plan. It was a good decision. It was a good decision. How much does it help your time as an athlete, especially identifying now with other athletes and the issues they face in this whole anti-doping process? And you speak about the conversation that you had with the Jamaican athletes earlier today. Were you able to identify more with a lot of their questions given your own experiences? Yeah. I was, yeah, it was a very good discussion because I received fair questions about the strict liability. Why we have to be so responsible? Why you cannot, for instance, certify the supplements in the world? So I explained them, it's not possible. We will not give the stamp of credibility to all the companies in the world which produce the supplements for that you want to be more confident. You have to be responsible for what is on your body, what you want to eat, what supplements you want to take. And, of course, we discuss about the whereabouts, the procedures, add-ons that athletes have to deliver the information where they are to be able to be ready to be tested. So it was the discussion, fair discussion, how we can strengthen the system, but how we can maybe facilitate the processes, the system for the athletes. And again, coming back to education, I encourage them to be in contact with Jatka, with the anti-doping commission from Jamaica to receive the information, what they can do, what they can take, what they cannot. So this is the responsibility, and I've passed the message to them, you know, guys, if you want to be an international level athlete, you have to accept the rules, first of all, sorry, but it is what it is. And then you have to be responsible for your body, for your career. One aspect of the rules, the therapeutic use exemption, that over time, well, definitely in this region, I personally feel there is a lack of understanding of how TUE's work and under what circumstances an athlete can apply for a TUE, and how long does it usually take to get it and so on? Yeah, I mean, this is exemption, this is the process which allows you to take, you know, prohibited substance if there is a real need to do it.  Asthmatic is one of the... Yes, but yeah, you mentioned asthmatic. A lot of misunderstanding is around, you know, the asthmatic, the accusations on some nation that the athletes from this country is taking too many TUE's or medical meds, but to be fair, not too many, I would say, asthmatics, these medicines are on the, on our prohibited list at the moment, exactly. So this is not like, maybe it's a matter of communication, we have to communicate better the whole process, how it looks like, but it's quite clear, you have to, first of all, to ask for this TUE, your respective commission agency in your country, then it's reviewed by WADA, by our experts, and finally, if there is a real need to take this concrete substance, because it's dangerous for your love, there is need to do it, you have a permission to do it. Yeah, that explains it perfectly as far as I'm concerned, but just to be absolutely clear for those who are watching and may still not understand, so let us say an athlete turns up at a championship and they get the flu, they can't just go, oh well, I'm not feeling too well, I'm gonna apply for a TUE. No, absolutely not, it's a serious thing. For instance, there is a strong injury and I know some dangerous situation in your life, and then there is no alternative that you have to take, use this substance, do this method to recover or to be healthy, then doctors are reviewing it, so this is not like this great example that, oh, I feel bad, I will take the protein substance and I said, oh, sorry, I didn't know. Oh, I wanted TUE retrospectively, no, it doesn't look like this. Yeah, definitely, it does not work like that. You spoke earlier about nations that can be considered non-compliant. Is there any nation, as far as you know, in this region that is even close to being categorized as non-compliant? We have different tiers, right? So, the countries most developed when it comes to sport, economy, the different factors are taking account are in tier one, so some countries which are not so developed economically, you know, when it comes to sport and so on are in different tiers, so we have, I mean, we require less from them, right, to be compliant. The process is quite, but as far as I know now, it's not, there were some countries which were close to which were on our watch list, which were close to be non-compliant, but I don't want to crucify anyone now here, I mean, until the country is non-compliant, which is actually, obviously, it's a public information, I don't want to. What we are doing, first of all, this is the last tool which we use. It's not that we are happy to make someone non-compliant, right? So, this is actually set, and this is the last resource. It's not like we assist, we help, we push, and then if there is no alternative, we have a process in place that our CRC independent committee recommends some sanctions, let me put it that way, and then executive committee is taking the final decision. What type of work do you do with these federations, specifically when you realize that they are trending down the wrong path and there is the possibility they could become non-compliant? Yeah, we have department responsible for it. So, as I mentioned, we assist them, we monitor them, we have special procedures to monitor, some forms, we require documents, we review their legislation, how the system works, and then our experts are telling them what to improve, what to adjust, what to change. We are giving them timeline and then the assistance. So, it's not like that we are telling them, oh, it's wrong and do whatever you want, but you have to do it before March, right? In three months, otherwise you were gonna be non-compliant. No, no, we assist them, we help them, and then sometimes it depends. When you look at the politicians where they are ignoring us for months and then when it comes to the sanctions, they wake up and this is sometimes too late. Yeah, yeah, doesn't sound good at all. You want to cooperate as much as possible. We're almost out of time, but I have to get this one in. Vital Banker, 39 years old, a former 400-meter runner, is being asked to take on a retired Usain Bolt at 400 today. Who would win? Oh, my God, Bolt, it's, you know, I would not be...  I would... He would still beat you? No, you don't think you're in better shape? Absolutely, it was 50 kilos ago when I was, you know, when I was strintered. So, no, I think it's... It would be an academy of strange moves from my side, you know? Still, I'm running from time to time, but it's jogging, right? So, you know, I don't want to... No, no.  Yeah, okay. You are very well prepared. Not to show what's happening with them, no Champions League titles since 2007.  This generation, actually. You have a different team now? Now, as a well-known president, I cannot, you know, say it publicly, but, you know, I'm from Poland, so you can... Figure it out. You can figure it out at, you know, Lewandowski. I know him very well, so... The goal-scoring machine. Well, the goals haven't been coming, but yeah. I think, still, he will come back in a very good shape. I keep my fingers crossed for him. Yeah. We all do. Bitol Manca. It's been an absolute pleasure having you on the sports map. Thank you very much. It's been a pleasure having you back in Jamaica, and we wish you all the best with the forum over the next two days. Thank you very much. It was a pleasure for me. Thanks. Yeah, man, great. Yeah. All right.


We started today. Last week of the semester. Are we excited? No. How are you guys excited? Because I'm not going to have you as a teacher. Well, depending on how you do in this class. Generally, not you particularly. Just generally. I'm too busy being terrified to be excited. Too busy being terrified. Two more weeks it's all going to be over, right? So you did a workshop Thursday? I had probability class. Is there anything else? Yeah, actually, you recorded it. I'll put it online soon. I just haven't done it yet. I'll put it on YouTube. I recorded it just like this. Some of the stuff on there, you don't actually have access to the server because I have taken it down. Yeah, giving random students access to servers in my lab is not like a great long term strategy, but yeah, it'll be up there and all the content will be up there. I think I'll probably put the slides on there too. That should be pretty easy. Yeah, if you're interested in that kind of stuff just let me know. Cool. Alright, project five do Friday, right? Is it Friday? It's on the website. I don't know, you can't force mind control me to give you a different day. Alright, let's see midterm and just midterm threes. So now we're back to our old friend Landicatulus. So, why are we talking about Landicatulus? I know it's kind of early to ask this question. Do you want to know when Nathan is going to be crazy? I don't know. I'm not going to promise anything until it'll be before your final grades are released. I can guarantee you that. That's a fact. What if I tell you, well, you guys should just assume that you should study really hard for the final, right? That's what I want out of that. Just tell us we all failed it. It's good. And we're all going to have to work hard just to... Ah, it's fine. I don't think you guys failed it. Actually, my money is on you did better than on midterm two and midterm one is my guess. We'll see. I will give you one more homework on Landicatulus because Landicatulus will be on there. But pretty much everything that I have tested on in the past has been on either a homework or a practice midterm. So I feel like you have all the study material you need basically up to the final. So it's basically going to be through Landicatulus, beta reduction, and the final will be comprehensive. Anything that we've talked about in class up to this point is fair game for the final. Okay, so why are we talking about Landicatulus? Since it's been, it gets on the curriculum and we have to part of it. But if we're just doing that, I would just go, oh, Landicatulus is a thing and then we put it on, right? Well, it's like a formalized way to describe how programming languages work. Yeah. So it gives us a formal definition to think about programming languages, right? And it gives us we've already studied Turing machines, right? And so we can relate and actually see that, wow, this completely different paradigm of a machine or computation actually have the same expressive power as a Turing machine. So that's why I think it's really cool. So we've been talking about Turing machines. Landicatulus, I hope that you've gotten familiar with the syntax and disambiguation, right? That was on the midterm and on the midterm practice. So that would be good to study. So now we can get into really the semantics. So we talked about combinators. So what's a combinator? Something, an expression, yeah, that has no free variables, exactly. So any expression that has no free variables is considered a combinator. And this is a definition we're going to use now that we're going to talk about in the future. So bound variable. So I think this is also a review, right? So we went over this on Monday, right? So basically, you can think of it in two ways. A, it's the opposite of a free variable, right? So it's a free variable. Yeah, it's okay. There you go. Yes. Right? So the variable's not free, it's bound. And if it's bound, then it's free. Or it's not bound, then it's free, right? But the question we always want to ask is specifically the bound variables, right? With free variables, free variables have no enclosing abstraction. For bound variables, we want to know not just are they bound or are they not bound, but what's the scope? So what is, which abstraction bounds that meta variable? Okay, so we need to go into the formalisms to describe exactly what bound variables means. Okay, so if an occurrence of x is free in some expression e, then it is bound by lambda x in lambda x dot e, right? So if we look inside the body of an abstraction, we say just looking in here, it's a free variable. That means that if we put a lambda x around it, put some abstraction around it, then that means that that variable x is bound by that specific abstraction. If an occurrence of x is bound by a particular lambda x in expression, then x is bound by the same lambda x in lambda z dot expression. So the idea here is that by, so basically these are defining our scoping rules. So this says, hey, if it's bound already in e, then just adding more abstractions on the end is not going to change what it's being bound as, right? You can't change it by adding more abstractions higher. You can only change it by adding more abstractions kind of lower down. And that would be the same if that lambda z dot expression was lambda x dot x. Exactly, yes. That z could be anything. Yeah, so it could be a lambda x. Right? Even if z is equal to x. So lambda x dot lambda x dot x, right? So which lambda expression, I'm going to give you the numbers, one or two binds x. Two. Two. That makes sense. We look at the inner body, right? And we see, okay, x is x free here in this inner body in the very inner body inside here. Just look at this. Is x free? Yes. And so we know by this first rule, okay, lambda x dot where x is free, that means x is bound to this abstraction. That's what this first rule tells us. And the second rule says it doesn't matter how many lambda x's we put on in the outer outside, right? It's going to be bound to this lambda x, this abstraction that it's originally bound to. This kind of does our scoping rules as like the closest, tightest bound possible. And we have to go through kind of all the cases, right? So if the occurrence of x is bound by a particular lambda x in expression one, then that occurrence in expression one is tied by the same abstraction lambda x in expression one, expression two, and expression two, expression one. This just says you can do, application doesn't change the binding, right? It doesn't matter what happens in any of the applications in the left side or the right side of an application. Bound things are still bound. Questions? So some examples? So now here, what binds... What is this x bound to? The first lambda x. What about this x? The same lambda x. Yeah, exactly. What about this x? Also the same. Also the same. What about this x? It's free. It's free variable, yes. This y, bound here. Yeah, the first y, the z. Free. Free. This y. Bound this y? Free. Exactly, right? So these bind all here. The italicized y is all bind. And x and y are both free here. Cool. So here, okay, this x is bound where? The first lambda x. The y is bound also to the lambda x. This x is free. And z is bound here. Let's see. What about this? So what is this x bound to? The first x. The first x. What about this x? No, the second x. So this inner x more tightly binds this x. Think about scoping, right? This is like an outer declaration of some variable x, like maybe a global. It's not really a global declaration here. This is some inner local variable x, which shadows that outer x. Okay, cool. So why don't you think about this question. What does it mean for two functions to be equivalent? What does that mean? How could you do it? How would you... There are inner expressions. More generally, like I said. The bound variables and the free variables are the same. Like the ones that we were italicizing and folding and underlining, those were the same for two different expressions. Okay. Why wouldn't we just say they have to be... So are two functions that are textually identical that have the exact same string? Are those going to be functionally equivalent? If both of their variables are the same. Yeah, the exact same text. How do you know the operations? What operations? How do you name operations with that context? They have to be textually the same and their free variables have to be the same. So they're textually the same, right? So like the exact same string. So if we have a function foo, that's some body, I don't know, x plus y. Just generally not thinking about Landon calculus, right? x plus y. Then we have a function bar that also does x plus y. Are those equivalent functions? Yeah. Sorry, I wasn't going to answer that. Okay. I was thinking about my question. Once you look at it just like what we were doing before is structurally equivalent, would you look at it like that? Yeah, you could look at it like that, right? So if they're name equivalent, name equivalent and functions, right? Yes, functions of the same name are the same, right? But what if you have two functions, foo and bar, foo returns x plus y, bar returns x plus y? They're functionally equivalent, but they're not name equivalent. Are they textually body equivalent, maybe? They have the same body, x plus y? Body, yeah. They have to be in the same scope. In the same scope? Yeah, why? Well, because x can represent different things in different scopes. Right, I think that goes to the point of here, right? Is it depends if this x and y, if there's some global variables or maybe there are other, you know, what do these x and y refer to, right? So let's take parameters, if they're both parameters, x and y, and they return x plus y, are they the same function? So we have a function foo, takes in parameters x and y, and returns x plus y. We have function bar that takes in, I'm not going to give them types, but are these functionally equivalent? Equivalent functions? Yes. Are they untyped? Yes. What if they're, okay, what if they're tight in the same type? Yes. Yeah? What about this? I get rid of this, and I say bar, is that, are they the same? Are they textual? I mean, if you look at the text of the body of the function, are they the same? No, but they still do the same function. Why? I think they're the same. Because of the plus operator? I think they're the same, because variables, they're just two variables, they're not constrained to a certain type. So I think they're still the same. Even if they were, so let's give them all ints, say they're all ints, x, w, y, and z are all ints. Even if they're either untyped or all their types are the same, then both, those both are the same functions, because you use the same operations to evaluate the function. Is that right? The function is evaluated exactly the same. Yeah? Well, if you were setting the value of the return for that into another variable, the output, if the output for them is the same regardless, then they're equivalent. There's no way to define that, just based on generic information. I don't think you need more information depending on what kind of influence you're trying to define. Yeah, so well, let's think of it in a couple of ways. Could I use, everywhere I was using foo? Could I substitute it for a bar? Is it going to change my program? No. Why? Because it does the same thing. It simply combines two values. Yeah, the evaluation is exactly the same. Yeah. It doesn't, it depends on if you're doing like pass by value or that sort of stuff. Does it? You tell me. I mean, yeah, they're just copies of what you're passing into it. Well, if it's pass by reference or pass by name, does it matter? What are their names? It doesn't matter what you put in them. They're just there for understanding what's going on. It won't matter what the names are. Because you're putting in different names in them anyway. What about this one? It is also pass. That would be different. It's just the same because of how, you know, addition to work. Yes. Right? You look at this and you know, you have the equality right? Well, a plus b is equal to b plus a, right? You know this to be true. But if we just look at the functions, right? Assuming no knowledge of plus, are these the same? No, right? Because they're doing different things. Even if I change this to be x and y, this is going to be different than both of these. But these would, I would argue are functionally equivalent, right? You could use one for the other in here. And it would work just fine. So, was this, are these two expressions functionally equivalent? Yes. Yes. Because they're just declaring. It's just self-referencing. Right. So it's a lambda expression that, well. That returns. So why? So talking about lambda expressions, right? What do we know about this expression? How could we try to actually determine using what we know that these are the same? Well, the variables are both bound in the same, not the same. Like they're bound in the same way. The inputs and the outputs are the same. The operation is the same because we don't know what any of the variables are. But they're both expressions that simply bind a variable to a meta-variable. Yeah, so let's talk about this. Could we completely, so does it matter that we call this meta-variable x and this meta-variable y? No. Just like we write a function, does it actually matter what names we give to our parameters? No. No, right? And so the idea is, well, if we can rewrite our function such that we change all of the parameter names, or here the meta-variables, if we change those and we get another function we'll say that those are functionally equivalent, right? What about something like this? It's not equivalent. This is the same? Yeah, it's defined a variable to a meta-variable while containing a free variable. Doesn't it matter if they're in different order? It doesn't matter that they're in different order. I don't know. Why? No, because the bound variable is the first one. No, they're still equivalent. Very equivalent. Yeah, because it's given an input apply. We're going to return the same output? No, no, no. It's just a function which, given an input, applies another function to the input. Let's pick what we talked about with the addition, right? Could you use one in relation to the other? And you'll get exactly the same output given the same input. Okay, so the free variable, because the free variable is not found by this function, it can be different anytime you call it. Is it the same free variable in each function? Not necessarily. So does that maybe change possibly the semantics of what this function does? Normal kind of example, right? If we had... What's the scope of this variable X? Yeah, we don't know, right? Some global variable, right? What's the scope of this variable A? So are these functionally equivalent? Could you use one or use another? No. Why? Because we don't know what the scope of A is. In regards to bar. This could be a function inside of... So even if we had this... We still couldn't because in A and X might not be the same thing, so passing in a value wouldn't actually return the same thing. Right, so if I give it the same input, right? Because I'm not going to get the same output for each of these programs, each of these functions. No. Why? Because A and X are not the same, so... Right, the global A and X, right? They're each referring to global variables, or variables outside the scope free variables, right? And so it depends on the context of when we execute this function, what those global variables are. But if you write exactly like this, though, they are functionally equivalent, right?  But we're going to get the same output for the same input given the state of the system. What if I have later on I have X is 10 and A is negative 10? Then I call, I don't know, foo5 and bar5, right? Are these going to give me the same output? Because it depends on what these global... what the global state is, right? We can think of it in Lampti this term, X and A are both free variables here, right? We don't know what they mean. They could be anything, right? But now, what if I change this of these functionally equivalent? The same free variable, right? Yes. It's not the fact that they're relying on global variables. It's the fact that they're actually relying on completely different global variables, right? Where as here, yes, no matter what I pass in A or B, right, this will each of these invocations will return me the exact same result. So they're functionally equivalent here. It is the exact same thing. Okay, so we're going to formally define what we mean by equivalence and we're going to call it alpha equivalence. Well, somebody else called it alpha equivalence and we're going to use that name. The idea is alpha equivalence is when two functions vary only by the names of the bound variables, right? Which kind of matches our intuition. If we have functions, we change the bound variable names, right? We can change it to anything we want and almost anything we want and we'll get new behavior. When is the case that we could not... Yeah, exactly. So, for instance, could I write, so I'm going to claim that I can just write all of these new functions, right? Baz, C, whatever, right? Changing this bound variable. But can I write foobar? Can I write foobar x? No, because that overrides the global scoped variable. Right, what was a free variable is now bound to x, right? So, exactly. So this is...you can't just rewrite every bound variable, right? You have to have some rules on how to do that. Okay, so how do we do this? So we set it to expression. Expression one is alpha equivalent to expression two, but we need some way to remain variables in an expression, right? We need some way to actually...some operator to do that. And actually what we're going to build up, this operator is going to...we're going to use this to do function applications. So this will help us in learning about that. Can we do a simple find and replace? For instance, here can we rename, like, x to foo? Yeah, we could rename y to bar could we rename this y to x? Because it's changing the scope of the inner... Yeah, it's changing the scope of the inner x, right? Is this x free? No, but is it free in the body of this abstraction? Yes, so that's y, right? That is why we can't rename this variable y to x because there is a free x inside the body of that abstraction. Okay, and then we can't just rename x to z, y. We have a free z, exactly. That's going to mess things up. Okay, so we're a renaming operator. We're going to use this syntax. We're going to say e curly brace yx, which means we're going to replace the y's with the x's. So this is pretty simple. So we have to...we have our cases, right? So we say, okay, for an expression to evaluate this operator for an expression, we need all the different cases. So here we have x of y replaced with x. I guess I just got that backwards. We're replacing x with y. So we're renaming x to be y. So if the expression is just x, the result is y, right? Seems pretty straightforward. Applying this operator to another id, let's say z, where z is not the same as x. Then it just returns z. So the left side is replaced by the right side of the... So the first one is backwards. So the first one is backwards. The way they're reading this is replace all y's with x's. It's the other way around. It's replace all x's with y's. And so the idea here is this is an operator. We're applying this operator to an expression. So we have our cases. We know an expression can be either an id, an application, or an abstraction, like an abstraction and parentheses. So this is what we're defining here. So here, does this case take care of all id's? Every id is going to be in either one of these two cases. Expression can be an id. Is that correct? In our parsing rule, expression can be an id. If it's an id, it's either going to be equal to x, in which case this rule applies. Or if it's not equal to x, this case will apply. Does that take care of every possible id? No, we also need to account for a name that belongs to a free variable. So we're actually building up an even more basic building block first. So we're first going to build just a formal operator to do renaming. And then we're going to use another substitution operator to actually do the proper substitution. So we first need this, but we're just defining an operator. So we're saying replace all x's with y's. Great. If I see an x, I know I should replace it with a y. I know I have something that's not an x, so I leave it as is. I don't change it. And that covers all of my id cases. Something is either equal to x or it's not equal to x. Those are the only cases. Then we have the case of application. And it's pretty easy. Basically this just means nothing happening. We apply our operator to the left side and the right side. There's nothing special. Nothing special happens. We just say, okay, if we have an application, that means we apply this operator to the left side and apply this operator to the right side. What's the other case? Yeah, lambda. And so we have two cases here. If the meta variable is the variable we're interested in. So here we're replacing the meta variable x with y. And then we call, we pass the operator, we apply the operator into the body of the abstraction.  So the other case here is if the meta variable is not equal to x, then we don't change that, but we do apply the operator inside. Any questions? Very simple, very stupid, just renaming operation. So if we do something like this and we say we're applying the renaming operator, we want to rename x to foo. So the first step of this operator is going to, that last rule is going to, well the second to last rule will apply. We will replace the meta variable x with foo. And then we will apply this operator to the body. The body here is x. We see x, we're going to replace it with foo. Pretty good. We got more complicated examples. We're going to replace all x's with bars here. So this outer is what? What type of expression is this, the outer one? Application. Application, right? We have, actually it's an ambiguous one, right? We have this guy here, this whole thing here, and then an x and then a y. So we can actually just, we know the rule, we can actually just distribute this across the x. Across the left one this thing, the x and the y. Right? So we just say, okay the only thing we do is we apply that to every part of the application, right? Nothing special there. Then we can just choose which order we evaluate these things in, right? There's no, nothing that says we have to do them in any specific order. So we can evaluate the right one and say this is just y, renaming x to bar of y is just y. We can do this x here, renaming x from bar to y is bar. Now we're here, now we're at the lambda, right? So now we need to replace the lambda x, the outer lambda x with bar and then apply this operator to the inside, to the body. And then once again we're going to do this again. We're just going to replace this bar with this x with bar and then this is going to happen again on the inside and then we're going to find and replace that x in the middle with bar. Super simple. That's the facility to find and replace. Okay, so for alpha equivalence we're actually still having to find the alpha equivalence, what we actually want to have happen here. So the idea is basically we're going to find what it means for things to be alpha equivalent. So the idea is for all expressions, any expressions you want and any variable that does not occur in that expression, this is defining lambda x dot e is alpha equivalent to lambda y dot e replacing y with x with y. So what does this actually mean? So what does it for all expressions e mean? Yeah, just any lambda expression. Here you can think about it, we're actually interested in talking about if two functions are equivalent. So you can think about it as the body. So if you're interested in a particular function talking about the body, but any expression e, so give me a random any kind of lambda expression you want. And all the variables that do not occur in e, so what does this mean? That do not occur in e. Actually it's even more broad. Any variable, all the y's are anything that is not free and not bound that doesn't even appear in e. And then we're saying that if you have some lambda x dot e, then you can replace in your original expression, you can replace every instance of x with some new variable that does not appear in e. The new thing y, and it's going to be alpha equivalent. So why can we do this operator here? Why can we do this e x replaced with y here? Yes, but why? You're replacing the bound variable in the initial expression with another bound variable and that's what alpha equivalent is by definition. Exactly. But the important thing that we're doing here is that we're putting conditions on what we can change things to. We're not saying we can change why can't be any arbitrary variable because we saw that that actually can cause problems. What does y have to be? Something that's not in e in the body exactly. That's why you can do this renaming operation on e because you know why this does not appear. There are no bound variables called y, there's no free variables called y. This is why that you can do this operation and not worry about things that are bound or not bound or renaming. Let's think about it like this. Are these alpha equivalent? I heard some yeses. We actually think we already said yes, they are alpha equivalent. Can we actually use this to show that they're alpha equivalent? If you please. So what would be the e in this case? x. It could be either one you want to do it. If we're keeping exactly with this way, we could do just y. e is the inside of this. We can take y and does x occur inside the expression that's just y? No. This means then we can put a lambda y dot y on the outside and then we can have over here a lambda x dot and then replace all y's with x. Now we know that those are both alpha equivalent. What about something like this? No, why not? There are free variables that are not the same. This one? Yeah, w and z. Exactly. We can't change those free variables. This is what they're saying. The only things we can possibly change are the bound variables. The key problem is this w and the z here. These are free variables that are getting changed. These are not alpha equivalent. Would you make something that is alpha equivalent to this? Absolutely. Renaming operator was incredibly simple. It's just replacing one variable name with another. Remember, we tried to get some intuition for function applications here. When we did something like this, we had lambda x dot plus x1. We reduced that when we were doing our reductions, just like the examples we replaced that with plus one two. What did that actually do? It substituted x for two. Application means take the meta variable, replace all instances of that meta variable, but remember when we talked about the bound rules, do we want to replace every single possible x in there with two? No. We only want to replace those x's that are actually bound to this meta variable. Anything inside we shouldn't be touching. It's essentially parameter passing. We're passing in two for x. We need to replace everything that refers to this x with two in this case.  Simple. This is why we're going to use another operator which we're going to call substitution that's going to be a little bit more intelligent. It's going to actually be able to change only the bound variables inside here. What we're going to do, we're going to use this syntax. Our substitution operator is e square brackets this time. x goes to n where e and n are lambda expressions and x is a name. This is the operator that we're going to build and we're going to use this to show how this actually just does function computations and function calls. Questions on why we need a substitution operator? The substitution operator just replaces anything. This n is an arbitrary lambda expression. That's another difference whereas the renaming operator was we could rename an ID for an ID. Here we need to substitute x for some lambda expression n. In our function application here, this could be expression space expression. This on the right can be an arbitrary lambda expression. We want to be able to substitute that in for x. Some arbitrary expression. The renaming was bad because we could only use 98. Yeah, that's part of it. Exactly. Part of it we can only rename to another ID. We can't say this ID is actually lambda x dot x. The second reason is because of the bound variables. The renaming operation just renames everything and doesn't, with this substitution operator, we only want to replace or substitute for those x's that are bound to this x. We're going to define this actually seems pretty easy. I've already solved this. We want to replace some x with 2 in this plus x2. This should be what? Yeah, plus 2, 1. What about something like this? A, first question, should we replace this x with a 2? It can be any arbitrary expression. This 2 could be lambda x dot x. A, we can't have this expression here as a meta variable. B, what is this x bound to? This x. What is this x bound to? Nothing, it's free. Then should we replace this x with 2? What about this? Again, remember we don't have a plus operator. Don't worry, we'll get there. If I had this, I applied 2 to it. What is this return? lambda x dot plus x, yeah, plus x1. Here I'm replacing every instance of this lambda x with the expression 2. How many instances of this meta variable x are there in here? What does this x refer to when it's bound? The inner lambda x. When we're applying this for the outer lambda we only want to replace all instances that refer to this x with 2. Just like if we were just to do lambda x dot plus x1, apply that to 2. We need to replace every instance of this meta variable x inside the body which is here with a 2. We're going to replace this x so this is how we get plus 2, 1. If we look at it from a substitution operator, essentially what we're doing is we're taking the body the body here is plus x1 and we're substituting x for some expression 2. This should return plus 2, 1. Here when we do this application, essentially what we're saying is lambda x dot plus x1 sorry the spaces, not plus x1 and all the terrible limitations of real paper. With a substitution operator, apply and replace x with 2. What should this return? What was this? Lambda x dot plus 2, 1? What happened? Why did we replace this? This x, but here when we tried to do this application just by looking at this application what do we want to get? Lambda x dot plus x1. This is a function that takes in an x that returns a function that takes in an x that does plus x1. Eventually we want to try to use the substitution operator to do to formally define exactly what we mean when we do this application process to make it more formal so we know exactly what happens. Informally what this means is we want to replace this x and we want to replace all instances of this x with 2. Inside of its body we want to replace all x's with 2. But this x does not refer to the same outer x. It refers to this inner x here. We do the substitution operator here and another way to think about it is we're kind of saying we want to replace all free x's with 2. If we want to replace all free x's with 2 in here then what would the result be? Plus x1 which then is exactly what we wanted to get from doing this application. And the same thing if we use that same logic here replace the free x in here with 2 do we get the same thing? Plus 2, 1. That only implies the free variables. Like we said earlier, it only implies the things that we're not using the expression for. Yes, because let's think about it like this. So we have some lambda expression, lambda x dot x, lambda x dot like this. So all the x's inside here, the ones that refer to this x when I remove this outer lambda x the outer x is free and why is that outer x free now? Because it was bound to that other x and we got rid of it, right? So this x is bound to the outer lambda x and this x is bound to the inner lambda x. What happens when I get rid of this binding abstraction? The outer x is free and why is that outer x free now? Because it was bound to that other x and we got rid of it, right? So this is the idea is basically you kind of think about it if I get rid of a function, right? If you throw away a function and you just look in the body of that function every free variable in that function that refers to that same meta variable it's now free because you got rid of that lambda, that abstraction. What was a free variable is now not a free variable. So this x is free but once I put an abstraction of lambda x around it then I know that this x has to be bound to that abstraction. So if I want to replace every x with let's say lambda y dot y I'm doing this function application, right? I want to replace every x with whatever this is, lambda y dot y. So what I can first do is remove the outer abstraction. So I'm left with lambda x, lambda x dot x and I'm saying inside here replace now every free x with lambda y dot y, right? So will this replace this inner x? No because it's not free exactly so it will be lambda y dot y. Yes, so this will be another application and then if I do this again then what happens? What is this application doing again? So here I applied this to this so now if I apply this to this what happens? Oh, so it'll take away the lambda y which will leave you with y and then it'll replace y with lambda. It just didn't function application. Now there is, I just want to go over really quickly there is one tricky case that's going to come into play when I have lambda x, I have lambda y, x, y and this applied to so I can have any arbitrary expression here. So doing what we just did we want to replace we want to get rid of the outer so we want to replace inside lambda y dot x, y we want to replace every free what? Every free x with y, z but what happens when I replace this y, z inside here? There's a new bound to y. Was y bound before here? No. So now I have this free y think about a global variable y here but I'm substituting it in for x here if I just did this substitution directly I'll have lambda y got y, z, y Is this y free? Is this z free? Was the y free before? And now it's bound which is a huge problem we've changed the semantics of what we're substituting in there Well, you wouldn't do that because of number one So going back to functional alpha equivalents what if I change all the bound y's here to w's Can I do that? It's still alpha equivalent Can I replace this x with y, z? Does anything happen? No, now I don't have a collision so I return lambda w dot y, z, w and now this y and z are still free like they were before and does it matter to functional equivalents that I change the name of this bound variable? No, not at all, right? Questions on this? I think this is a good before going into the formal definition of substitution I think this was a good way to go All right, cool, let's say