So, regardless of when I get here, we start at 6.32, which is not incentive for me to get here at 6.27 again. Starting the Montpellier-Roxbury Board of School Directors meeting at 6.32 on December 20th. It's our final meeting for the break. Thank you everyone for coming. Also, I just want to thank the Board for really putting in a lot of work over the last month and a half.  So really appreciate the extra work and, you know, the dedication and thoughtfulness that this has involved. With that, I do want to say we've got a couple of people who appear to be under the weather and online. And I know we're all rushing to go to the holidays, and this is really kind of a check back from the last meeting. So just because it says it's going to go for two hours, we are under absolutely no requirement for it to go the full two hours. I want ample discussion to happen. But I think everyone would appreciate it if we kept this meeting as efficient as possible. And with that, let's open it up to public comment. There'll be two sessions of public comment tonight, one at the beginning, one at the end of the budget presentation and board discussion on the budget presentation, which will be followed by more board discussion on the budget when we get to process some of the public comments from that. You should just be aware that if you can speak at both times, if you don't speak at the initial public comment, you are certainly welcome. If you do speak at the initial public comment, you are certainly welcome to speak again. And again, public comment is a listening time for the board. We take it very seriously. It really helps our decision-making process, and it also oftentimes will bring things to our attention that need addressing. We do not engage in a back-and-forth, and we don't respond in real time. So our sitting and listening does not mean that we are not hearing you and does not mean that we are not going to take action. That's the way it is structured. And also, please do speak a couple of comments. We really appreciate it. We know it can sometimes be intimidating or difficult to come in from of a public body in a public space, but we really appreciate what it takes to do that. And also, if you would prefer a less public forum, there is a school board at MPSVT.org. We'll go to all of the members, and that is another way to weigh in with your thoughts, where you can do both. So with that, we have several members in the room. If there's anyone who wishes to speak in the room, could you raise your hand so I can get a sense of how many we have for the initial public comment? It's like, anyone online? Either do the raise hand function, which is in reactions at the bottom. If you hover on it, a little raise hand will appear. Otherwise, if you want to go off camera and just wave, feel free to do that. Okay. So it looks like no public comment at the initial section. Again, we'll have one after the budget presentation. We have a very short consent agenda. The consent agenda is where we put items that do not require generally consideration by the board in terms of discussion and are kind of pro-formal, like approval of minutes, et cetera. So do I have a motion to approve the consent agenda? Do I have a second? Any discussion or comments? All those in favor? Are we opposed? The consent agenda passes. So now on to the budget presentation and board discussion, the second draft of the FY25 budget. Do you want to just say that Jason just texted me to say that the live feed wasn't working, so I think we should continue the meetings recorded and all that kind of great stuff. Okay. Okay. And just to clarify, can the board members here? Yeah, yeah, the Zoom works. It's just the live feed. So they don't work on it? Yeah. Okay. Yeah. All right. So this budget presentation, the second draft is always a very quick draft. It's really just to show board members any updated numbers that we have gotten. Usually there's more updated numbers than this presentation shows, because it's usually two weeks instead of the one week. So really only one number has changed so far. So Kristina's just going to walk us through that and then open it up for discussion. You'll be sick of seeing me by the end of January. So yeah, this is a much shorter presentation because like Olivia said, only one number changed and it was our long-term weighted average daily membership. So on the second page of the presentation, the budget at a glance, this draft of the budget is still an increase, budget to budget 11.51%. And as you can see, we gained, from the last version that I showed you, we gained pupils in our multilingual weighted section. So what this changes as far as the budget is concerned is the equalized residential tax rate, our actual, if we were not capped. That's the number that's going to be changed by that. So we are still seeing the cap at $1.33. And the next slide we kept in just so you can still reference back to see what the tax bill will be increased by. And the non-residential rate has stayed the same. We haven't heard anything change on that. And then we also included the residential tax rate history for your review. That's it. There is a big one change on this slide right here. Christina and I are both traveling. We had March 4th for the budget informational meeting in town, meeting before town meeting day. Christina and I were traveling on that day. So we moved it to them. We're going to move it to the morning of March 5th. We have to have it 10 days before the vote. Before the vote. So we can do it. The city does theirs the morning of town meeting. And so we're going to do ours the morning of town meeting. So that's just another change, too. Thank you. So we have to have it 10 days before the vote? No, you have 10 days before the vote to do it. Oh, it's somewhere in that window. Yeah. OK. Got it. It always lines up with vacation weeks. Right. So it's tough to schedule that. Yeah. And have good attendance. And with that, we can open it up for discussion. Yeah. Question, comments, discussion. Jill and Jill's got our hands up. Jill, go first. Oh, I was just wondering, thank you, why our multilingual student number changed? The agency of education was using the wrong number in the first version that we saw. Yeah, you might want to step in. Yeah. So the first version, we didn't realize it at the time, but they needed the ELL students by grade level. So we contacted our data person, and they resubmitted the data, our student counts. And so they increased our numbers. And I did just get an email about two hours ago that version three will be out in its way soon. The version three of this number? Of the student alone. Long term weighted average daily membership is going to be a new number the next time we see this. There'll be a third version. A third number, OK. Which is not odd. That happens every budget season. So this is not out of the ordinary. And I'm not sure we'll see much change, because we've finalized our data. So you might see a small change, but I don't expect anything significant. In this version, I think it was about 2020 that we gained from the last version. Yes. But it's not likely to go change it that much the next time. We were expecting our number to go up, because we knew the data that they needed by grade level they didn't have. So once that was resubmitted, it didn't correct us. Yeah, that's a related question, because last week, there was a little arrow pointing to that number saying estimate. And so my question was going to be, is this one also an estimate? And I think you answered that question. Oh, yes. Technically, yes. Probably times two. But it's gone up, which means the ed spending has gone down per pupil, right? Yeah. Which is good, right? Because that's the. That's what we want to happen. So where are we in terms of year over year? In terms of the equalized. Yeah, like somewhere around 9%. We're just under 9%. Just under 9%. And I can't remember what it was. But last meeting, there was some news that you had just gotten that you were like, they'll hear that next week. That was the L2W number, yeah. Gotcha. I've been on pins and needles for a week. I bet you were waiting. I just have a question, process and schedule. Can you remind me what format does the budget informational meeting take? Is that a virtual, is that in person only? Because Roxbury has an in person kind of full morning sometimes to the afternoon town meeting. It's virtual and in person. And in my six years, I've had four people attend. Because it's the same budget presentation that we've given the board across budget season. There's nothing different about it. And there's no change. There's no ability to change the budget at that point. So it's very similar to what the board will see in the last, in the 17th. The 17th, January 17th. So what time do you know like the time that that, because I just wondering in terms of coordinating with Tammy. I mean, if people wanted to attend and they hadn't been paying attention, I understand that the budget wouldn't change at that point. But it is like it's an information access point for the communities. And since they'll be voting on that budget that particular day, I get it, we don't have high participation rates. But I'm just wondering how that will sync with the schedule of town meeting day for Roxbury. I don't know. I don't know when the town meeting happens in Roxbury. And Christine and I just realized that both of us were traveling on the fourth like yesterday. So we changed it very recently. So we haven't decided on a time yet. Okay. I would just say that. Typically you run. Typically Roxbury town meeting, I think starts right at nine o'clock. It goes from nine to 12, sometimes to two or three. It really kind of depends on the year and what's up for discussion. So is the fifth actual town meeting there? Yeah. Okay. The informational day is on town meeting day. It will be this year. Please, I'm held the day before. And what time do we do at 6.30? Like 6 p.m. Yeah, something like that. And are we, is that, are we allowed to do that? Mm-hmm. Okay. It's my statute. Okay. And it's just like Roxbury town meeting. I guess it's a similar thing. So it has to be done on the day or? 10 days. Yeah. 10 days. 10 calendar days. Yeah, four. Uh-huh. Annual report will be to you in February. Go to printer January 23rd or so. So as soon as that's printed, it's out. First sort of out of context, I just wanted to take a moment to publicly recognize and celebrate that our student, one of our student board members, senior Alar Kohn got accepted over the weekend to her first choice of college, Smith College. Oh, congratulations. So congratulations. Congratulations. Thank you. Man, that's gotta feel good. Not as good as being here. Good answer. Good answer, Alar. That was quick too. And a question I had was, so there was a slide in the last presentation about staffing changes, and I was just wondering if anything has shifted since the last presentation from that slide. No. Nothing has shifted except for the long-term weighted average. Okay. Other questions or comments? Regis. Regis. I say back in November, we were looking at a worksheet from the agency that calculated our current equalized pupils. They made it look like equalized pupils for FY 25. And so that's what we were comparing. And with the next version, we started using the long-term weighted average daily membership instead of that calculation on equalized pupils. As far as losing revenues, I'm not sure I understand that question because we're gaining revenues. We're not losing revenues, right? We have to increase our tax capacity. So what's the percent increase in revenues? You said, I think you said it was 18% last year. What is it now? Or last week? And what? Education spending. It's 11.51. 11.51. 11.51 is our increase in education spending from FY 24 to FY 25. So we have an increase in education spending, even though we have technically more weighted pupils. So that means that we have to, but that is capped, right? That's capped. So it's not going to be, yeah. So it's not going to be 11% increase. Make sure I understand. Thank you. Yeah, no, definitely. It has definitely moved in a more manageable, still difficult, but more manageable direction. Mia. Since our Ed spending per LTWADM is now under 9% higher than than our current year, is there anything you would, should we put anything, make any changes to what our original plan was? That's the board's purview. My advice would be no, because this is a five-year challenge. Yeah. And we're going, remember that 11% as Jim said, is a percentage that we have never dealt with. And if the cap weren't protecting the board, that would be significant. Yeah. Right? Absolutely. So we need to be thinking, not just for this year, but we need to be thinking for five years. Well, one of the things I was thinking about was we were planning on drawing more from our fund balance, and maybe it would make sense to put, to draw less than we were originally planning from our fund balance, since we don't necessarily need it to stay under the 10% and it's the kind of thing that we may need to use later in a few years, next year or in a few years. That would be the board's purview. I would encourage you to wait until the next, the third draft to have that discussion. Oh, okay. You don't need to make that decision now. Okay. So you have one more draft look. Okay. In the beginning of January, before we show you one that needs to be voted on. And what? Gotcha. What, are we gonna have more or more stable numbers? Most of the, on the third draft, we have mostly stable numbers-ish. Yeah. We're looking forward to doing the CLA. That's the one number that I haven't seen. In January one is the deadline for that, right? That's what Jill told us last week. I think she's gonna- And not a moment's center this year. And in terms of, I know you want some wiggle room, right now we're a little under nine percent. Like where are you comfortable in terms of wiggle room? Look at Christina for that answer. Are we comfortable? Is now kind of the sweet spot, could it creep up a little? Like at what point do you start to get nervous that we don't have enough in case there is, yeah, the yield changes? Yeah, the yield won't be known for a while. No, it affects your 10% that you're trying to stay under. Yeah. It's only the pupil count that will affect that. And we're pretty solid on expenses and revenues right now. So like, let me say we're a little, we're under nine percent. So it depends on the board's comfort level of the expenses. And also the more we put in, the tougher it's going to be. Yeah, so if we were not capped, we'd have a 20.4 or 5% increase in our equalized residential tax rate. So we have about 15% that the Ed fund is going to have to make up for us. Okay. So we're capped. That's this year, right? Yes. Yes. So you are not in the 5% club this first year. You can't get there. So that's just something to consider. So as the time goes on, more information is coming in. The level of uncertainty is shrinking. The amount of buffer you would need is also shrinking. I said this last week and I'll reiterate. I would be more inclined to increase, like we're leaving money on the table by keeping our increase lower than nine when we can go as high as 9.99 and not pay any difference in taxes. I mean, the problem is like to get to the point in 2030 where we're not going to hit a cliff. The more we like rely on that 5%, the bigger that cliff is going to be. Certainly if you're making changes to the budget this year that would impact future budgets. But if we're deferring some of the maintenance on the building this year and know that we're going to need to do that in subsequent years, it seems to me that we should be doing some of that stuff now when our budget is well below the 10% increase threshold that we want to stay below. The facilities budget that we adjusted after we met with you first in November was furniture and classroom kind of updates. So Andrew still has available his capital projects fund that's still in here, the transfer. And he has a fund balance in there so he's still working on the maintenance. But if we don't replace furniture this year. We have insurance money.  No, no, no. We're going to get that. It was still in the budget without thinking about it, without the insurance piece to it. But when Andrew realized that, he was like, oh, we'll take that out of the budget and still get the furniture. And still get the furniture but have somebody else pay for it because of the flood. Yeah, absolutely. Didn't you also mention that, I mean, this is related, sorry to just building on this, didn't you also mention at the last meeting that there were some changes that would come after budget season that would? Budget to the printer in January. What changes after that is the dollar yield because that's set by the legislature and the governor. And so that can't happen until May. And that's true every year. This happens to us every year. And typically in my 17 years of experience it's gone up a couple hundred dollars. In the last couple of years it went up a couple thousand dollars one year. But now we're, I don't know, Jake might be able to speak better to this. Now they have a new calculation for the dollar yield or a new assumption for the dollar yield which is a decrease. But is that one of the reasons why we would maybe want to build in a buffer and not be right up against the 10%? The dollar yield doesn't affect the 10% cap on your ed spending per long term weighted average daily membership. That's a factor after. The yield is after that calculation. Are there questions or should we open it up for public comment? I would like to better understand, I mean I still don't totally understand how the five years out and then the cliff and how not getting up to the 10% would sort of help us in 2030 tax wise. So if at some point if someone could summarize that it doesn't have to happen tonight but sort of project out like, and I know we can't get good numbers but I've asked for it before just some rough sense of like if we were able to be at 8% versus 10% over the next five years this is how it would help the cliff. I'm not quite understanding that. I mean we don't have to cut later. It's the most basic way to say it. So we don't want, if we're presenting a budget that says yeah we think this is good, we don't need to add anything back in then we're not adding dollars back in. Eventually we're gonna have to have conversations of which dollars we're taking out and that would be one less thing to worry about. I think that's the most basic way to say it. Yeah, so basically we've got like the 5% cap and we've got what we'd actually be taxed at. So the closer we bring this to this at the end of five years, I mean that's the cliff. So if we spend like this for five years, that's our cliff. If we spend like this for five years, that's our cliff. Okay, and then this year we had four and a half positions that had been covered with grant money that we kind of added to the budget and we had to subtract some of that money from the budget in order to like keep it under 10%. But next year, hypothetically, we don't have those four and a half positions to add. So wouldn't we be in a better position next year? Or?  Four and a half. So what the challenge may be next year is to increase our budget to get to the 5% club, the cap. We may come in, we generally, in a typical year-to-year scenario, our budget doesn't raise by 5%. So this is a unique year in that way, right? I look back in other budgets and we're at 3% or 4% each year. Do you mean the tax rate increase? The tax rate increase, yeah. And so we may not be in the 5% cap next year based on just that, right? That we're not going to be adding so many positions to our budget. We may be, I can't necessarily predict that, but I can see a scenario where we decrease and we continue to slowly decrease our budget, right? Because we know we have to. And we are not increasing so much, so we don't get the cap anymore because we're coming in under it. We're not increasing order to qualify for the cap. I think we're gonna be hitting the cap for a while. I think we're gonna be hitting the cap for a while. It could be. Because the cap is 5% and we're at like 20 or something percent over the yield. So that's the gap currently. And we're gonna keep on going up by increments of 5% and still till we converge. And our salaries and benefits are gonna keep increasing. That's the piece that you can't anticipate is how much that's gonna, and we don't know how much benefits it's gonna influence our budget. I mean, the health insurance thing I don't think is an operation. I think you could expect that. Going forward. But so there's a 10% per pupil spending cap. If we don't, are you saying that if we don't increase that at all, that we won't qualify? Well, there's a point where there's a potential that we don't qualify for the 5% cap because our budget isn't increasing by that amount. Because our- If we fall under 5%- Right. If our tax rate increase- Then we don't qualify. Naturally is under 5%. But would we lose out of the 5% club if we fall under 5%? Yeah, that's bananas. I didn't realize that. Yeah, sorry. Okay. But I don't think it's gonna happen at least in the next few years. It's interesting because when Brad James was talking to the agent, to the business managers and superintendents a couple of weeks ago, he was pretty, he was like, I think most of you won't reach the 5% cap in the next few years. It was really interesting. He was pretty confident about it. So it's interesting to say, here Jake says something else and Brad says that the agency says something. You know, like, but that's the story of our lives with the agency of education. So I don't really know what's gonna happen but there is that potential. Well, he's looking at all district budgets. I'm mostly looking at ours. Yeah. So I don't really know what the whole landscape is. Does the podcast discuss the next five years and the cliff? Can we direct listeners to the podcast to learn more about the trajectory of this? I talked about it, but it was really fast and maybe boring also. I think it was, I think it was a problem. The podcast can actually slow down. Oh yeah. And you can pause and rewind. It's really helpful. I can't wait to listen. At some point in the future, I think it would be nice to understand the trajectory of the next four to five years a little bit better as a board.  Well, I mean, we're getting to the point where we're at the point where this budget is moving forward but I would like to better understand sort of the long-term planning. I think part of my, what I'm curious about is it appears as if we have people understanding that. So what is it that you're not understanding so that we can better support you? If everybody, well, I mean, I just said it. I just said that I'm not totally understanding why if we, you know, over the next five years, if we, I'm not understanding exactly what the cliff looks like. If we're not adding four and a half positions each year that like in my mind, that seems like we would eventually get to a pretty good place where there wasn't a big cliff. It seems like this year seems like an aberration in terms of if we're gonna compare the first year to the other four years, we wouldn't have four and a half added positions. I mean, I can't imagine that that's in our plan every year for the next four years. So if that's not in the plan every year for the next four years, then it feels like we should be in a better place to be going like this. And so I just think it would be good for our community to understand that. I'm sure I'm not the only person in the community at large that like isn't quite grasping what the next five years looks like and what the cliff might have in store for us. So there's a possibility of like a convergence like you were just doing with your hands and there's also the possibility of Jim's cliff with his hands. The Ed Fund, it's difficult to know. There's a lot of districts who are in our position and if they keep on maximizing their spending, it's gonna put a lot of strain on the Ed Fund, which will make the yield go down and will make kind of the possibility of a cliff more likely. So it's difficult to know. So the responsible thing to do is to like, gradually get yourself to a place where you're not looking at a potential cliff. Yeah, I think the difficulty of the next five years though is we don't know what's gonna happen to the Ed Fund. We know there's the possibility and perhaps the likelihood of a lot of strain on it. We've already seen the dollar yield go down significantly from last year. That could stabilize, that could continue to go down. The legislature could step in and fix things, but there's likely a lot of strain on the Ed Fund that's going to in some ways make it hard to really know what the next five years is gonna look like. The other piece, as I see Jill have her hand up so we'll get to Jill in a second, but Jill jogs. My piece too is that the CLA's done, the reappraisal is done every 10 years. So we will be right in the middle of a cycle for appraisal, particularly here in Montpelier so that 100% is gonna continue to decrease. If the CLA decreases, your taxes increase. So there's another part of a perfect storm kind of brewing with numbers outside of our control as well that we can't. We don't know what the CLA will be, but we can predict that it's gonna be lower. Mia and then Jill. I was just gonna say, I think what I hear you saying, Emma, is that you understand the cliff in the abstract, but it would be helpful to have numbers that would help it be more real and not abstract. Yeah, I mean, I've asked for this before, like even if they were very hypothetical and I understand the hesitation of like putting any numbers forward because of all of the unknowns, but if we could like name a fictitious district and just run some like pretend numbers just to sort of get a sense. And even if they were off by many percentage points, I think it would help people understand the math better. I think one of the reasons that it's confusing is that the five, the cap really helps us for the next four, this year and the next four years afterwards. But once it goes away, it's like it never existed. And so it's like, we care about staying under 10% for this year and for the next four years after that, under 10% increase per pupil spending for the 5% cap, but we also have to think about it more long-term than that and realize that any money increase even under that 10% means tax rate increases. But I know that didn't actually answer your question, but I think one of the reasons for the confusion is that the 5% cap really matters right now, but then once the five years are up, it's almost like it never existed, which is why we have to also think about, almost think about it like it doesn't exist when we think about budgets year to year. Yeah, I guess what I'm recommending is that after maybe even after town meeting day, we make some time for a community listening session slash education session around Act 2027 and the potential impact over the next five years. And I think we have a lot of really brilliant minds when it comes to Act 127 and the impact. And so I think we have a lot of people that can explain it pretty well. And I don't know, my experience as an educator is like, if one of my students in the room had a question, it doesn't, you know, and they're the only person asking the question, it doesn't usually indicate that everybody else understands totally. That's not what I'm asking. I'm trying to better understand where your confusion is so that we can give you the information we need. I wouldn't even say it's confusion. I think it's just like there's a lot of unknowns which keeps being said. There's all these numbers that are unknown. And I think the closer we can get to like projecting out and sort of clarifying what this could potentially look like because talking about a cliff in 2030 is still very abstract, I think, for most constituents. So we showed this in the very first budget presentation in November. It assumes where the tax rate is right now. The orange is FY24 as we see it. If we had a 5% cap, that's where it would have been at $1.27. This assumes that we add about $2 million to our budget every year between now and FY30, which is a typical amount that we add that I've gone back four or five years and 2 million is a pretty good number to add. That's about what we've added, right? So this is pre-CLA. And if you look to FY30, that means that this year in FY24 our taxpayers would be paying $1.35. And if we just continued spending just our typical amount then they would be in FY30, they would be asked to be spending $1.84. So that's without the 5% cap, right? So the cap would have been at $1. If we stayed just at the cap the whole time, that's the blue line. And if we just kept, and if we didn't have the cap but added our typical amount each year, that's the orange line. So it's a difference of 14 cents on this with tons of hypotheticals in these numbers because we don't know what the dollar yield's gonna do. We don't know anything like that. Let's say this could change. Let's say the dollar yield drops precipitously, continues to do some because the ad fund is so stressed then that orange line is gonna go up higher. But the blue line is gonna stay the same. But then isn't it the case that if you jump from, when you jump from FY29, assuming that cap at $1.62, we'd be going to $1. We'd go $1.62 to $1.84 instead of staying at $1.70, right, yes. So that, yeah, like that last section of the blue line disappears and it's like the blue line has to jump to the orange, is the eclipse. Let's also be honest, like 5% for five years in a row is higher than we've been. Let's not forget about Jill. I know, I know, I know. And not insignificant to the fact. I mean, it's, it's, it's outspacing inflation. Jill, you've been super patient. It's okay, we're captive audience here. We don't have a choice. No, thank you. I'll try to be quick. I am, I'm surprised to hear that we wouldn't want to revisit the budget numbers of specific cuts that we were talking about. Now that we've gotten a little bit of good news. Emma, I'm, I'm with you. You're not alone in confusion. And I kind of live in this world and I'm not saying I'm confused or needed more explanation, but for me, it's really hard to be so strategic and so sure about the cuts we're talking about making now. If there is so much unknown, as Scott said, and we've gotten a couple things of good news. So I'm a little less comfortable with cutting an existing staff position by 0.20. That seems very specific and very draconian and very like a drop in the bucket of this overall, this overall budget. So if we're so unsure about what the next four years are going to be, how can we be sure that we don't have to make any changes to the budget that we've been working on? Like Mia said, maybe we set aside more of the, the fund balance. I just, it surprises me because we spent a lot of time on the most specific expenses. Now we're just gonna, we've got some good news and we're not gonna take advantage of that buffer. I think it is strategic to make cuts to the union elementary school staff ratio because there's numbers that are telling us that. That is strategic and that is a long-term plan that we have based on enrollment. So just a reminder, you know, point to all of the existing educator who is currently employed is very different than like retirement incentives for not filling vacancies. And I think it's a drop in the bucket of this particular budget, but it's life-changing for an individual who's a beloved current staff member who works directly with students. That's all. Thank you. Right, that's the same note. Correct. But Jake also had his hand up for me. Might as super quick. All right. So these numbers bounce around a lot, but the gap between the blue and the orange was 14 cents, I think in the graph that Libby showed. And now it's 20 cents because that's the difference between our cap rate at $1.34 or roughly, and what it would be if there was no cap 1.534. So that gap, which is like the potential cliff is now a little bit bigger. And obviously the numbers are always evolving, but that's what it is at the moment. That's a good point. Not now. Sorry. Also, Shannon said that the technology position at RVS is a little bit redundant. So as I remember it, either the 0.4 of those two positions that are currently actually occupied would save $45,000 or 0.2 saved $45,000, which was that, which was right there. So the science position at MHS is half of that. And I personally, when I have some, you know, a student participant on the board mentioning how that would affect them directly, that seemed like the most directly affecting cut. And I had my fingers crossed and hoped that maybe that wasn't still there. And I know that, you know, there is a cliff, but it's $22,000 in whatever it is that needs to be 735,000 or 800,000, whatever it is that we needed to get to, it's 25,000 or less. So that was my opinion, but just my two cents. I got an overarching comment about some of these cuts. I think it's really easy to compare the big number to the small numbers, and you get to the big number with small numbers. And I know all of these cuts are hard and many of them affect people we know, but if we go around and say, well, we have to cut 80,000, what's 25,000 and 800,000? Let's take one that one out. Well, here's another 25,000 over here, that's just 25,000 to 800,000. And here's 40,000 to 800,000. Doesn't sound like much, let's pull that out too. Suddenly, you've lost your cuts. And you've lost the number and you've lost the ability to get to where you're going. And I, you know, all the cuts are hard and I don't think we would prefer to make any of them. I also think the administration does a really good job of looking at the numbers and coming up with the numbers they think make sense. And I also feel we're asking questions about are these cuts, and I think board level questions that are appropriate, are these cuts undermining the values and the mission that we've put together and that we've put together through a process? And the answer we've been getting thus far is no, that these, you know, cuts while, you know, certainly there's value to what we've been spending the money on and certainly it's hard when it affects people. But, you know, the answers we've been getting from the administration are these are not things that are undermining the kind of core mission that the board has set out for the district or things that, you know, it can be made up in other ways with resources that are there. So I know each one of these cuts is hard, but I also think that there's, you know, the type of questions we should be asking are the type of questions around, you know, we've set, you know, three, we've set our priorities, we've set our values of how are these cuts impacting those? And if we're hearing that they're not and we're at a place where we have to make some tough choices, these seem to be, you know, the choices we've made and they're also kind of within, I think, the general parameters we gave the administration in terms of where we'd like to see the balance of cuts coming, coming from. So I just want to put that out there kind of in terms of what our role is and also just the, you know, the real difficulty, like all of these when you burrow down into them, someone's getting impacted and that's awful. But, you know, I think we have to step back a little and see what our role is and also trust that the administration's really, you know, combed over these, you know, because, you know, Libby's the one who works with her people who has to have all the tough conversations around these who has to deal with the disappointment. So, you know, I just want to kind of remind folks of where I think we should be. Yeah, I really appreciate you saying that and I do, I feel uncomfortable when as a board we're talking about like specific recommendations around decisions that are not our purview and so I appreciate that you reminded us. I also, you know, the priorities that we've set and the guidance that we gave was based on information that we got like a month ago which has changed and I certainly, and I've voiced it and I'll say it again, like my feelings have changed as the information has changed and it sounds like, yeah, it just, it sounds like maybe, yeah, we have a different perspective now as a board than we did a month ago and I would like to see the budget reflect our perspectives now, not a month ago. Yeah, and I think it's okay to kind of re-ask the core questions and see if things have changed in that direction and also give, you know, some guidance but let's go to the public. And anyone in the room who wants to speak just not a huge crowd so feel free to come up. I have an empathy with Emma. I'm a visual person a lot and I think Nathan Suter, resident of Montpelier as always thank you. As I understand, I'm trying to understand the cliff idea as well and so I drew a picture of sort of flows coming into our school the cost of educating our students if we had no change in Equalized Pupils, et cetera, 80% of this year, this next year's budget would be covered, we'd have no tax increase. To stay level, in fact, we have to add 20% compared with last year. If we were just paying the full tax is what I just heard Christina say, right? It is 20. Would be 20% without the 5%. So our tax payers will shoulder 5% and the ed fund will make up the 15% and then we get to educate our students, right? And then whenever that 15% supplement goes away now or in five years assuming that VI keeps going at 16% a year assuming that right costs go up. Whenever we take off that supplement our community makes that jump, that's the cliff, right? My other question is, I think it's fascinating that if we, did somebody say if we do not, if we do not get protected in the 5% group this year we do not have access to that lien in the following year? That's fascinating. But part of what puts us into that class every year is by staying below 10% per people spending against the previous year, right? So to Scott's point, to my mind, one piece of game theory says get as close to 10% this year because then our potential growth over that next year is lower, right? If we govern our per pupil increase this year to say 7% and then I actually wish we'd gone up higher because then next year it's 13% and we're in trouble. I don't envy you Libby or Christina. Anyway, I don't know if that helped me think about it. Thanks. Yeah, and I just want to echo, I mean, it is confusing. There is a lot of uncertainty. I don't think anyone has the next five years figured out. I mean, I think we know some of the trends, I think we know some of the predicaments, but as Jake was just referring to, the strain on the Ed fund is going to be huge or the CLA is going to be big in terms of the next five years, what the legislature does or doesn't do also may influence it. I think what we do know is there is going to be a squeeze of some sort. At these meetings and looking at the presentations and listening to the podcasts and going to the webinars and Nathan's coming to meetings and sitting here for all of this and there's a lot of voters out there who haven't really seen any of this and so that's where I'm coming from. It's just like a place of sort of like, if people feel they need more information can we provide an opportunity for that even after, I'm not even talking about this budget cycle, like. Oh, we're going to be dealing with this a lot over the next several months. Yeah, and just to be mindful to like put some, I mean, there's been a lot of great resources created on our website, which is amazing, but like can we sort of break that down, maybe provide some visual stuff. Some people learn better by having a discussion and so I just think like creating some sort of forum at some point to educate people if they're interested feels pertinent at this point. Hello, Joe Carroll, MREA president. I think I just want to underscore the sort of reconsideration of the rifts for this year and I guess I would say that I think the people whom the rifts would affect are aware that that's a possibility. So if there's any chance that those funds could be cut in creatively different ways that wouldn't affect human resources, I think I would just want to add my way to that tonight. I think you all know my position on this. I've come to all these meetings, but I think hearing the momentum for that the news is less bad, I think I would just add my support to reconsidering the five and a half positions that are gonna be rough. Thank you. I'll leave it there. Thank you, Joe. Anyone else in the room? Alison? Alison Waring, science teacher, Montcalier resident taxpayer. So when I look at these numbers, I see like a 12% increase in overall budget and if I'm not wrong, I think that almost directly parallels the increase in salary of teachers under the new contract. And so while I am very appreciative of that, I'm imagining it's not going to increase in our next contract cycle by quite that much because we've now been brought up to like surrounding school district comparative salaries there. So I'm imagining next budget year, we're not going to see quite as much increase in staff salaries. And when I look at the uncapped tax rate, I see it at $1.53, is that right? And when I look at the tax history of Montcalier, I see 1.53 is kind of on par of what we've seen the last five years, right? So I see 1.7, 1.7, 1.6, 1.1, 1.3. So even at like an uncapped tax rate, we're kind of on par with what Montcalier voters have approved the last five years. And so when I see these numbers, I'm not really seeing any huge like, oh my gosh, our numbers are out of control crazy, right? I'm seeing we're having a large increase in salary due to a new contract. And that even our uncapped tax rate is on par with what we've seen the last five years. Am I interpreting this wrongly or? You aren't interpreting it wrong. The last five years keep in mind was under a different educational finance formula. But it still has the same impact on taxes, the dollar value. Yeah, we just don't know what the new formula is gonna impact in the future. Okay, there, go ahead. The big dip in that graph is from a re-prasal that happened in Montcalier. So all the property values increased by 50% on average. So the missing piece is the tax amount. So everything kind of got rebased. So even if we were at $1.54, it's not like people's taxes would be like $1.54 right from a few years ago. It would be 50% higher. Because the value of the homes increased. So the thing that's not here is what you're multiplying $1.33 or $1.34 times. Yeah, but again, I'm not seeing anything that's like hugely out of what we've seen the last five years, I guess is my point. Morgan? Morgan Lloyd, fourth grade teacher at Union Elementary School, Montcalier resident taxpayer and parent of two kids at the high school. Thank you all. And with respect, I think I wanna just ask the board to please ask our administration and business office whether there are different cuts that could impact students and mission less. Basically positions that are not student facing. Thank you. Anyone else in the room? Anyone online? One, let's start with James's iPhone. And if anyone else online wants to speak up, go ahead and raise your hand and give him a cue. Hey folks, Jim, Mike and Mary, sorry I couldn't be there in person tonight. Thanks again for all that you do. I would feel like a sustainability position would really fit in line with a lot of the core values that we think are central. And so I would encourage you all to consider if we've got some leeway in the budget to make some room for that, if possible. And I encourage again, as we're trying to think about what these next five years are gonna bring, just full of uncertainty. And let's see if there's anything that we actually feel like we have even a modicum of control over and see if we can run with that. Cause the Ed fund and all these other things, nobody knows, we could guess now and it can be wildly off four or five years from now. But if we have an ability to push on enrollment like Libby had mentioned with getting folks that are paying to come to our school district either from other districts or as exchange students, can we run with that? Cause that seems to me that it's worth investing in some money now, even if we had to put more into the budget to make that happen, if that's gonna pay off and help us not have the cliff in the future. So whatever kind of fits, and I think Rhett had thought of it as like a revenue stream and each student is kind of getting you revenue. I think that's a great idea. I think we need to find those things that we might be able to have some control over and to push really hard these next four or five years so that when we come to some perhaps inevitable cliff, that cliff is not as catastrophic as it could be. So again, thanks for all you do and appreciate it. Great, thank you, Jim. Anyone else? David. Hi, and thanks Echo Jim's comments. Thanks to all of you for kind of slogging through all this and going through the struggle. I really appreciated Emma's comments earlier and found it. I'm just struggling to kind of understand. I don't follow the broader context of school funding. I think not nearly at the level you folks do, nor the specific issues facing the school. And I saw the upcoming schedule and I guess I was just wondering if there was a time for the public, like one of those meetings who wanna kind of understand that and when numbers are firm enough to really have meaningful comment, like what is the thing in the upcoming timeframe running up to town meeting that is like the best time for us to learn but also maybe have an opportunity to comment. So thank you, I'll go off now but I really appreciate the opportunity to listen and all the work you guys are doing.  So Tim, I don't think you could hear Christina from all the way over in the corner there but January 3rd is the meeting where we are doing another presentation of the budget and it is the official public forum for budget season as well. So that is a great time. Yeah and we can start that off with some sort of primer to try to educate people. Well maybe folks could listen to the podcast. As much as we know, yes. I think listen to the podcast episode 19. Yes. Sorry, Jake Feldman. You can find it on all places, you can find, I don't know, just kidding. You can, actually. I just stopped being an advertisement. Yes, no, that's a great reminder. We did Jake and Libby to do a podcast on Christina. Well and our budget, the budget page of our website. And the budget page of our website. MRPSVT.org backslash budget has a lot of information and anytime somebody is going there and doesn't see what they're looking for, they should definitely email us and say, hey, I have this question that isn't answered on your FAQ or isn't, you know, I still don't understand X. Please email us and we will try and get the answer. Yeah and we also got an email about the difficulty of us having meetings when the city council does. We usually do not. We are having extra meetings this month because of the complexity of the situation and kind of the gravity of the decision we're making. So we usually do not have meetings on the Wednesdays that the city council does. It's, we usually alternate. So, you know, apologies for that. As I know, the city council is also discussing some pretty weighty topics. So, and I know it's hard to keep up. Anyone else online? Oh, someone in the room? Yeah, go ahead. So my name is Carlton Anderson. I'm a renter. I'd like to buy and don't want to pay a lot of taxes. Eventually, I have social anxiety. I have friends who even have more social anxiety, so I'm asking the question on behalf of them. I hope I'm not stepping in it, all right? Because I'm not too sure I have no children, but I recognize we need an educated population. So with the flooding and the damage to Montpelier's schools, as well as the lower admissions, when would it make sense to review economics and technical viability of merging with U32 and would require vote from the voters? So, yeah, it's a very good question. And so for the two districts to merge, it would require both districts to want to do it and to approve it. I believe it's something that has to be approved by the voters of both districts. And the process would be, and just to give some history to explain this earlier in the year, kind of with this coming up and also with the flooding that we had and a process of examining this building and also this quite honestly has been a topic that's been around, I think as long as I have. Libby and I met with Megan Roy, who's the superintendent of the Washington Central District and Floridia Smith, who's the chair of their board. And we talked about talking about a merger. And they are going through a kind of facilities review process where they're answering some questions inter-district. And we're also going, kind of before we knew the gravity of the budget situation, also going through a facilities review question to answer some questions about this building and the rest of our buildings in light of the flooding and just generally. So what we agreed to do is after those processes are over, which is in the spring, that we would touch base again and see what further conversations made sense about how our two districts future, they or may not coalesce together. I think what we're going through this budget season, they're also, I think, going through a pretty tough budget season is going to, I think, add to that conversation. So we're going to come together in the spring and kind of revisit that. And... Informally. Just to touch base. Informally. Not really thinking off the process. Yeah, informally. And see where we're at. And if there's an appetite to have more serious talks, we'll move in that direction. And if one or both of us are feeling that that is not the best path to take, then we won't take it. So that's kind of where we're at. So it's on the radar. It's discussions we're having informally. But for it to happen, both districts would have to come together, agree that they wanted it to happen. It would probably involve some sort of community study committee that would come back to the boards with a proposal. The boards would probably have to approve that proposal, both of them independently. And then it would go to the voters. So it would be a long process. But it's, you know, I think it's on the minds of both districts, but there's a lot of steps between now and it happens. Can I ask another question? With that, when we talk about the cliffs that we're avoiding, trying to avoid, is there an idea of what their cliff is and our cliff so that we can possibly say the urgency is to merge so that we can avoid each other's cliffs together? That, I think, is some of the questions we have to answer. They are under a different set of, it's a very good question, and they're under a different set of pressures than we are. They actually are a district that has been minorly advantaged by Act 127. However, they are facing a student decline. A decline in enrollment projections that's really precipitous and that's putting a lot of strain on their budget. And then you throw in, when we're all dealing with the declining yield. So I think I would like to see some of those numbers, but it's possible that there could be savings between the two districts. It's also possible that that may not be the case when we actually look at things. And one of the things we hope the facilities report is gonna do is at least give us some more information to ask some of those questions about building expenses. And they've got some other issues too. Like they have PCBs at U32 that they're gonna have to go to their taxpayers and ask for probably a fair amount of money to redo a couple sections of that building. So a lot of things to discuss, but it's certainly something that we will be discussing I think in the spring, at least informally, it's on our minds. And we're thinking about it, but it's many steps from I think being a serious reality. A lot of things. Yeah, I think it's a lot of things. But thank you. It is a great, great issue to bring up. Lisa Burns online. Hi, good evening. I just wanted to follow up on the question of the last gentleman that I appreciate it because I've been talking to people in the community, more and more people ask that same question when you look at the school districts around Montpelier is the donut hole in U32. And so many of those issues relating to Act 127, which of course was initiated many years ago because the cost of education in Vermont has been increasing wildly while our results have been going down, academic results, that is. So Act 127 was an effort, we'll see how efficient to make education spending more efficient in our state. The PCBs, we haven't been tested at Montpelier High School but I think our high school was built two years before U32, so we're likely to find the two that are flooding all that. So my question would be, you said that you will go to the soup and the people there at U32 discuss issues, but is it just your decision or do voters and taxpayers have any ability to influence what you choose to do? That's my question. Thank you and Merry Christmas, Happy Holidays. I mean, I will say if U32 is interested in having discussions, I think this board is likely interested in having discussions as well. I think we'd have to talk about it, but it certainly is a board level decision. And U32 just went through a difficult merger where I think there was a lot of, it was not an easy merger. And I think they're at a place where going through that process again is something that they're a little wary of. This community cannot force Washington Central to have talks with us they did not want to have. And Jim, didn't you just say that ultimately the decision is up to the voters? It is ultimately up to the voters. Our decision may be to enter a process, but ultimately the decision to merge is up to voters. Ultimately the decision is up to voters and it's not just up to modular voters. It's up to both district voters. So if a district is just not interested in going on a date in the first place, you're not gonna get a marriage. So that's the way it works. Oh, oh, sorry, yeah. You say that they have to agree with each other. So the citizens of, for U32 would have to agree just as up. Is there a way we could package a tax saving deal for all if we merge and promote it that way? And so that we're essentially stating that two districts merged into one will cost taxpayers less money. I think that's an assumption that people make that may or may not be true without studying it with the two districts. We don't know if that's accurate or not. Do you feel common sense to it? Not necessarily, because you need the same spaces. And so you need to- It's less janitors, it's less cafeteria. No, because you need the same amount of buildings to house the kids. So you need all the same amount of support staffs and that kind of thing. So we'd have to study it though. Like we can't just make the assessment that it's tax savings because we really don't know at this point. In your experience, would you know a hypothetical upswing for the citizens if it did occur? Do you know any positives that would occur if two merged, as opposed to not doing anything because the negative is two different cliffs? Yeah, I mean, I think to really figure that out you need to study it. And it takes information from both districts. So it would really take, I mean, it was kind of like a business venture. Boy, we could sell a bunch of widgets together and here's why. And if you say, I don't wanna sell widgets with you, like I'm fine. I really can't do anything to make you wanna sell widgets with me. But if you're kind of like, okay, let's sit down and see what this business venture would look like. And that's questioned some of these assumptions. Then we can get through the process at the end where we can look at it and say, wow, this would really work well, let's do it. Or like, boy, some of our assumptions didn't work.  And yeah, and then they've said, you know, we're going through a process, give us some time, let's come back, let's see how we feel. In spring, if both districts are saying, you know, the spring are saying, boy, this might be a great idea, let's study it. Then I think we can ask and answer a lot of those questions. If one of those two districts is kind of like, we're feeling fine, you know, thanks, we'll be nice neighbors, but we're okay over here. There's not much the other district can do to, you know, to move that forward. Mia. Well, I was just going to say to try to answer your question about hypothetical positives tonight, worries me as an individual board member about over-promising something for our community. Yeah, I understand. And there are a lot of factors to consider when thinking about merging or just deciding to merge with another community. And taxes absolutely are one of those factors, but there are also a lot of other factors, which is why it's just an impossible question to answer tonight.  You know, you put them out in the back 40 and businesses could start saying, wait a second, we're losing our lunch customer. So there's just a lot, a lot to discuss. And I understand my questions are not, they're not my views. Oh, I know. I'm trying to grasp all this in its entirety so I can make sense of it. I'm also just understanding now what you're saying is, because we all live in this community, we all know the feel of each other. And so the idea of it happening, we already know whether or not it's viable based off of the reaction.  Well, if it looks at me with a smug eye, I know the answer already. So it's like, is there any type of unspoken type of vibe to even considering the merger informally? I think that's what Jim said in the beginning. So we've met with the board chair and the soup over there and had that initial conversation. And at that time they said, not right now, we've got too much on our plate. So come visit again in the late spring. Yeah, they said that right now, but not never. Like we want to figure some things out. The article about this question, I think it was in the Times Artists or maybe the bridge, where they kind of gave some historical background context to this merger question that has been going on since you 32, the building was built. And when it was built at that time, as I understand it from the article, it was discussed, should we merge these two schools? And I've heard, I mean, I was raised here in this town and I've heard the conversation, come and go and come and go. And I think my gut feeling is that because it has come and gone so many times, there's an interest, but there's never been enough momentum to like make it happen. And I agree totally with what Libby said about like, there's no guarantees on the other end of this. And we've seen this with the mergers all that happened. There's some savings, but it's not like the answer to everybody's tax problems are gonna be merging the two districts. And we have about 400 students in this school. You 32 would not be able to take on, we can't just like move everybody there. We can't take on all of their students here. We just don't have the size for it. So there's just a lot of questions, but I think the reason why we're not in a process right now to explore that as a possibility is because we know that they are in a process right now of exploring it as part of a feasibility study. Yeah, well, that's a Washington Central. Yeah, where they're just looking at their own. They're building usage. Yeah, they're building usage and... In holding. And one of the questions in that process will be potential merger with no. No, I think the information they get out of that will be useful information if we want to move forward with that. And I think it'll give them a sense of some of their numbers, but yeah, and also there's one of the things where like the savings could be realized, but there's other considerations that could come with the savings, including the future of certain schools and what that means for certain communities. It could mean things like redistricting towns. So right now each town has its own elementary school. That might not be the most efficient way to go forward and people might have big feelings about those changes. So it's a really complicated discussion. Yeah. Then they are having right now. That they are having. But if you layer it in, our district on it too, you'd amplify some of those and complicate some of those questions. And really that all of the communities have been having since the 70s. Yes. Like it's been a cyclical topic of discussion in these towns. I hope that you get around to studying it personally. And I'm a tax guy, but for me taxes are probably not the main issue. Like I'm really concerned about flooding. So I think that's the thing that's maybe different than the past 30 years is that climate change is real. The high school could, another eight inches and Andrew said another eight inches. The kids would have been going to school at the Berlin mall. So I think it's real. I think we got to talk about it. And I hope that we do do that work eventually. Yeah. I similarly hope that we're able to come together and at least have a really comprehensive study of what the options are and what it looks like. But yeah, I know they're wrestling with their things over there. And hopefully they're welcome to that conversation. And we're certainly gonna see if that's something they're interested in exploring. And we do have a facilities process happening right now which could also help us. Yield some more information. Especially about the flooding and the impact of the buildings. Yeah. Thank you very much. I appreciate you. Yeah, no, thank you for bringing it up. I know that. Yeah, yeah. Thanks. Thank you. All right, I think that. I just wanted to ask a clarification of something that was said during public comment. Yes. It was said that Act 127 was meant to make spending more efficient in the state, but I'm not sure that's actually true. I think it was meant to make it more equitable. And those are two different things. Very different things. Efficient and equitable. I just wanted to just make sure I was not confused about that. You were not confused about that. Thank you. Yeah, no, it's about student equity and making sure that the. Scholars from the Education Fund go to the students that have the most. Yeah, and the way that the students are calculated, like one student does not equal one student. Different students have different educational needs and those different educational needs require different resources. So the purpose of the Act 127 is to better match the educational resources that are going to students to their educational needs. No, it is not about efficiency. So with that further further discussion are we at a good place till the third? And I think on your suggestion, Emma, if we could do a maybe start the third with a little primer on where we are with Act 127, or Act 127, that might be helpful just to, you know, just some of the considerations and kind of what we know. I think it can be real high level 101. Yeah, I'll connect with you, Emma, to just dig in a little bit more, so I understand. Cause I want to present it in a way that's understandable. So I just want to make sure that.  Jake and Jill both work for the state in ed finance. So like, they have a better understanding than probably most board members across the state of Vermont. But for the rest of us, I would put that invitation out to everybody. Like if there's questions that people still have about this, I think it would be wise to send those questions to Libby and try to get them answered at the next meeting. Yeah, absolutely. The board at MPSBT.org. Yes. It's the Libby and the rest of us. Any questions you have? So policy monitoring, we have B3 alcohol and drug free workplace. And B7 tobacco prohibition. I move to approve the policy monitoring reports for B3 and B7. Yes. Would you have a second? Scott. Scott seconded. Any discussion or questions? All those in favor? Aye. Any opposed? Motion to adjourn. I think Rhett came off mic to move to adjourn. I also wanna just thank everyone's participation tonight and the questions. And I hope that the everyone that's out there keeps asking questions because we can't answer them unless you ask for them. So I appreciate everyone's participation. Yeah, absolutely. And I move to adjourn. Do I have a second? Second. Well, thank you everyone. Thanks for the tough work. Thanks everyone in the audience. Continue to stay engaged. All in favor? This is my little speech. Happy holiday. Oh, I'm sorry. Okay. Everyone has a great holiday. Get some rest. I was gonna upload. You're gonna vote no? Aye. No, I wanna acknowledge that Nathan said my name in the same sentence as game theory at like Nobel Prize winning, you know, discipline. And that went like this to me. I've been glowing ever since. I almost gave up. Quite honestly.


Hi, I'm Ekansh Deep Gupta. I'm from Chennai Mathematical Institute. And today, I'll be talking about the well-structured problem for Pressburger counter machines. And this is joint work with Olaf Inkel. The title has two things, well-structured problem and Pressburger counter machines. Let's define the first one. So for that, I'll define well-structured transition systems. So an ordered transition system has a set of configurations. It has a transition relation and an ordering on the configurations. And it is well-structured if it satisfies two conditions. The first one is that the ordering needs to be a well-caused order. And the second one is called the monotony condition, which is that if there's a transition from S1 to S1 prime, and if I start from some configuration which is bigger than S1, then I should be able to reach some configuration which is bigger than S1 prime in the reachability relation. That is the monotony condition. And there's a natural stronger condition which is called strong monotony, which imposes that it should be reachable in one step. Earlier, it just needed to be reachable. Now it should be reachable in one step. So these are respectively called the well-structured and the strong well-structured transition systems. So this is quite a robust class of transition systems. Like all of these are examples, PetriNites, vector addition systems. All of these can be seen as WSTS under appropriate orderings. And so the reason this is studied is because the coverability problem is decidable on WSTS. The coverability problem is given two configurations. Can you go from one to some configuration which is bigger than the other? So it turns out this is decidable. And there's a backwards algorithm which came much earlier in 1996. And there's a much more recent forward algorithm on downward close sets, which came in 2017. So now we come to the motivation for the problem that we studied. So suppose we're given a class of transition systems, which is not decidable in general. The problem is not decidable. But we're given a specific instance of the model. And we want to check if the problem can be solved on this model. So if the model is nice enough, if it satisfies monotony, then we can use the results on well-structured transition systems for this model. So for example, this Minsky machine, if you look, it satisfies strong monotony. So we can solve the coverability problem, et cetera, on this. So we wanted to come up with these sorts of things. So the question we asked was, given a transition system, can we check whether it is well-structured or not? So that is the well-structured problem. And correspondingly, the strong well-structured problem, whether it is strongly monotonous. So these are the problems. Now let's introduce the model. So Pressburger arithmetic, everyone knows, I'm sure. It's the first-order fragment over natural numbers with additions. So these are just some formulae you can write. You can check divisibility, multiplication by constant, and so on. And yeah, turns out it's quite a nice logic. Firstly, because it is decidable, that is given a formula in Pressburger, you can check whether it's valid or not. And this is shown by quantifier elimination on the formulae, whereby you add modulus operators and yeah. And the second thing is, this is quite expressive in the for-counter systems. So the formal model is that you have decounters, some number of counters. You have some states. And your transitions are labeled by formulae, which have twice the number of free variables. The first copy is for the value before the transition. The second copy is for the ones after. So this is an example of a Pressburger counter machine. The formulae x prime equals 19 minus x, and so on. So for instance, this is a sample run. You can start with the counter 0, and then take the loop, and go to 19, and so on. So these are quite simple formulae. You can express quite complicated things as well in Pressburger. But so this is like a recurring example. We'll see this later on as well. Right? There are no tests in our history. No, no. This is quite a simple model. But yeah, you can have zero tests. You can check divisibility, and so on. You can have quite complicated things.  Yeah, yeah. You can express quite a lot of. You can express guards and everything in Pressburger. Right. So we look at Pressburger counter machines, and we also look at a lot of subclasses of those. So one is functional counter machines, where each formula, each transition formula, is a partial function. Each input has at most one output. Then there's Minsky machines. These can also be expressed as Pressburger counter machines, where you have translations or zero tests. Then Affine-Vass, this is quite a well-studied model, where all the transitions are of the form, x prime equals ax plus b, where a and b are matrices. Note that we're allowing negative values in the matrices as well. And since we can multiply by constants and so on in Pressburger arithmetic, this is also like Pressburger expressible. And then we look at some restrictions of Affine-Vass, which is positive Avass. And here we only allow positive values in the matrix, whereas earlier we allowed negative values as well. And then we have totally positive Avass, which is another restriction, where we say that b can also only be positive. So these are all the classes of models that we look at. And then these are the results that we have. So we look at the well-structured problem and the strong well-structured problem for all of these classes. So it turns out well-structured problem is quite hard. It is undecidable for most of the classes that we look at. But if you do have decidability for this model, which is one Avass, we have one counter and affine transitions. And the results highlighted are the ones I'll be discussing in the stock. For the other results, you can check out the paper. And okay, so I'll get the strong well-structured problem out of the way first. So it turns out a strong well-structured problem is decidable for all classes because strong well-structuredness is sort of a local property. If there's a one-shape transition, then we want another one-shape transition. And that is Pressburger Expressible. So we can check validity of that formula and you can decide it for all classes. The well-structured problem on the other hand is quite hard. So okay, so first we look at this result that given one PCM, we'll restrict it to one counter with Pressburger transitions. We'll show the undecidability of the problem. Okay. So the undecidability, the reduction goes from Minsky machine reachability, right? So given a Minsky machine, we'll construct a one-counter PCM such that it is well-structured if and only if all states are reachable in the Minsky machine, right? And so here we have just one counter in our counter machine, in our Pressburger machine, whereas we have two counters. So basically we encode it using the Godel encoding, whereby we store the values of the counters in the exponents for two and three, right? So note that this allows us to multiply by arbitrary constants, arbitrary large constants, and that is sort of key for getting the reduction, right? So since the other factors don't matter, we only care about the exponent of two and three, we can define sort of an equivalence relation on the configurations of the one PCM, where two values of the counter are equal if the exponent of two and three are the same, right? So now I'll just show the reduction on an example. So suppose this is the Minsky machine that we have with two counters, C1, C2, and you have some, this thing. So the corresponding one PCM will sort of simulate the Minsky machine with the Pressburger machine. So you start with the same set of states, then you add the corresponding relations which simulate the counter operations. So for instance, incrementing the first counter corresponds to incrementing the exponent of two. So you multiply by two here. Similarly, decrementing corresponds to division. Now these are not Pressburger formula, but you can write equivalent Pressburger formula. So that's fine. And zero test corresponds to checking divisibility by two. So this is the guards that you can have guards like this, right? And so this is not sufficient. This is just a simulation of this thing. And then we add two more classes of transitions, yeah. So when we add the red transitions, so from every state to the initial state, we add a transition labeled by this formula phi. So what phi does is, it allows you to jump back to the initial state while resetting the exponent of two and three. So you can jump to any value which is not divisible by two and three. So this sort of corresponds to like a counter reset and since we're going back to the initial configuration and resetting a counter, this does not affect the reachability relation in the simulation that we have, right? But it does allow you to jump to arbitrarily large values and that is sort of key. So if you add these transitions, it gives you this property that if all states in M are reachable, then N will be a well-structured transition system. And the way to see that is, suppose I want to reach from Q1 to Q2 and I want to cover Q2, N, for instance, then what I can do is I can jump from Q1 to Q0 with an arbitrarily large value, which is greater than N. And then since Q2 will be reachable in Q0 by the hypothesis, then I can just follow that reachability path and I will get a larger value than N. So if all states in M are reachable, then N will be a well-structured transition system if I add these transitions. And the second class of transitions that we add is, it's simple, it only allows the transition if both initial and the final values are zero. Now, note that here our initial value will be one and no transition allows you to reach a zero, your counter value to be zero, right? Because you only have division, multiplication and checking for divisibility and so on, right? So the blue transitions, they can never actually be taken in the counter machine. However, they give you the property that if this is well-structured, then all states will be reachable. And that is also easy to see because suppose, because you have Q0 to Q10, right? And therefore, if N has to be a well-structured transition system, then if you start from Q0, 1, which is the initial configuration, then there should be a way to reach Q1. And same for all the other states. So upon adding the blue transitions, you get the property that if this is a well-structured transition system, then all the states have to be reachable. So this is the reduction that we have. And so what we've shown is we've taken a Minsky machine and converted it to a one PCM, such that this is a well-structured transition system if and only if all states here are reachable. So that gives you that the problem is undecidable for one PCM, right? And the same is true for all these other classes. The proofs get sort of complicated, but the idea is the same, so you can check it out, right? So the well-structured problem is also undecidable for functional one PCMs. Note that this reduction is not functional because the formula phi is not a function. It is independent of the input, right? But there's a way to do it such that you get undecidability for functional one PCMs as well. Then for two counter-Minsky machines then two hours also you get undecidability, right? So right now we just proved this result, the undecidability of the problem for one PCMs, right? And now we look at the one, the decidability result for one hours, okay? So I'll go over the model quickly once again. So the model is this, you have one counter and all your transitions look like x prime equals A x plus B, some linear thing. And A and B can obviously be not negative or whatever, so this is the model. Again, this is the example that we'll use. So note that this is not, this is stronger than a VAS because in vector addition systems you can't have transitions like this. You can't have 19 minus x. So these are not just translations, you can have negative transitions as well, right? And so one of us have the following properties that they have been studied before unsurprisingly. So reachability and coverability are decidable which was shown in a paper in 2013. But what we do is we come up with an algorithm which can take a configuration and compute the pre-star of that configuration. By pre-star I mean the set of configurations can reach this configuration. So we show that this set itself is also computable. So if you have this then obviously you can see that reachability and coverability are decidable because you can just check membership in the set and so on. And not only is it computable, it is also presberber-expressible this set. And we use this property that the pre-star set is computable to come up with the decidability result for the well-structured problem. We'll see how, right? So the pre-star algorithm, the one which computes the pre-star. So suppose you want to compute the pre-star of QF and NF, the configuration QF and F, then well you see some things that these are some of the key observations which allow you to get the algorithm to work. And the first thing is that the set will always be presberger-expressible. So if you try to compute it with a presberger formula, that is sufficient. Like you won't keep discovering new values and so on. The second is that given a cycle in a one-averse, you can compute sort of the back acceleration of it. So what I mean by that is you can compute all the sets, all the configurations, which can reach your configuration while using the cycle sum arbitrary number of times, right? And the third thing is that computing these back accelerations is sufficient. That is like if you're, so our algorithm is a backtracking algorithm. And if you keep backtracking like this, then you will eventually compute the entire reach a pre-star set. That is you cannot get into a state where you keep backtracking and you keep discovering more and more values. Like that won't happen, the computation will stop. And these are sort of the main things. And I mean, we prove them, they're not, they're sort of tricky to prove, but yeah. And once you have these things, then the algorithm is the simplest thing that you can think of. That is you store a presberger formula for each state and then you just backtrack. And from results like this, we know that that is sufficient. So we'll compute all the back accelerations of the cycle and given a transition, we'll just compute the backtrack from that. And that is sufficient. So proving that termination and so on is slightly wonky, but yeah. So okay, that is the main thing. So I'll just run the algorithm on an example to give you. Okay, so suppose I want to compute pre-star of Q119. So we start with formula phi one and phi two for states Q1 and Q2, right? And we initialize phi two as bottom and phi one is N equals 19 because you want to reach Q119. Then if we compute the back acceleration of this cycle, we'll get 19 and zero because zero can also take it. And if we compute the back from this, we'll get 19 and zero in this, in phi two. Because from Q2, you can start with 19 or zero and reach the configuration that you want, right? And then if we compute back accelerations of the cycle, then we will get this. That any value which is greater than 19 and which is one mod three, you can take this cycle arbitrarily many times and you can again reach this thing, right? And then we just keep going, we take this. So the values here get translated and you get them in phi one because of this transition. And then yeah, you just keep going. And then yeah, eventually you'll get to a state where backtracking from anywhere won't get you new values and that is when you stop, right? This is the algorithm which computes the pre-star. Okay, so now we just were at the well-structured problem. So the result is that the problem is decidable and the way we show that is, okay, so given a transition, if the transition is a translation, like if it's a positive transition, that is, okay. So by a positive transition, I mean one where A is greater than equal to zero, like X prime equals AX plus B.  And the result and the thing is positive transitions are already monotone. That is if you can go from some counter value and using that and you start with some bigger counter value, then you can already reach a bigger value. So that is fine, like that won't violate well-structuredness, right? But for negative transitions, we make the observations. So for instance, if this is a negative transition, then if you start with Q one zero, then you can reach Q two five, right? So for this thing to be well-structured, what that means is, if you start from Q one comma any number N, then you should be able to reach something greater than five, right? And the way we formulate that is that if we compute the pre-star of upward closure of Q two comma B, then Q one cross N should be a subset of that. That is from any Q one comma any number, you should be able to reach this set, Q two comma upward closure of B. And this is obviously necessary for it to be well-structured. Turns out this is also sufficient. So the decidability result is as follows. You look at all the negative transitions in your transition system, and then you check that all of them should satisfy this property. That if this is the negative transition, then this should be the case. And if that is the case, then the transition system will be well-structured.  So that is all I have. And so some further questions that need to be explored are. So the complexity analysis of the problem. Like we have a pre-star algorithm, but getting the complexity bounds in that is slightly hard, we're still looking at it. So that is one thing that can be done. And the other thing is, of course, extending the problem to other models. That you can look at the same problem, well-structured problem, for push-down machines, T4 automata, exchanges of petrinets, and so on. So these are things that we're working on right now. And that's all. For well-structured position system, there is a sort of variant for the future. Look at the somehow stronger, for instance. You can see the well-step between well-step. And I can see the way to the well-step. Look at the name of this. I mean the resolution of the star. That is a strong way. So for strong property, I think the vis-à-vis the structure, vis-à-vis the other. Yeah, so the stronger structured property is decidable for everything. And we will do that as we encode it as a formula. So if you look at the machine, somehow you extend the semantics of the automata to show a condition relation, given by a representative formula. But the relation of configuration, the way it is sent is component-wise, component-wise, or component-wise, or component-wise. And then it's possible to replay this relation by a presclerger to find another relation. The question is, does there exist a presclerger formula so that your system is from your resolution? So if you have a presclerger in the machine, you would like to know if it's a strong waste structure, but not for the usual order, because maybe it's a strong general, a strong space. But for another one, it's explicit in the presclerger logic. So actually, that is something that we started looking at. We don't have any results like this, but yeah, that is something that can be done. We don't have any results like this. That's the forward direction also, the post. So the post thing will not be presclerger-expressible because you can simply have some loop which is multiplied by 2, right? And that will be the power of 2 which is not presclerger-expressible. So the post star is not presclerger-expressible, so I don't know that. But pre-star certainly is. That's why we looked at pre-star instead of post star. Thank you.


Western states propose new sanctions on Russia. The European Commission proposed that the new 9th package of sanctions will be imposed on Russia, including a ban on transactions with three Russian banks and other measures. European Commission President Ursula von der Leyen said in a statement, We are stepping up the pressure on Russia with the 9th package of sanctions. As a new package of measures, the Commission proposed adding almost 200 other individuals and entities to the European Union sanctions list, including the Russian Armed Forces, as well as individual officers and defense industrial companies, members of the State Duma, and Federation Council ministers, governors, and political parties, among others. The European Commission also proposed sanctioning three other Russian banks, which would include a full transaction ban on the Russian Regional Development Bank and a ban on broadcasts of four Russian media channels. The proposal also included further economic measures against the Russian energy and mining sector, including a ban on new mining investments in Russia. Von der Leyen's proposal also includes a ban on exports of unmanned aerial vehicles to Russia. Additionally, the European Union is seeking to impose new export controls and restrictions, particularly for dual-use goods, including key chemicals, nerve agents, electronics, and IT components.


A boat conveying no fewer than 100 persons including a father and his four children has capsized with scores reportedly dead at a blue village in the Partigui local government area of Quara state. The Hsu Partigui Ibrahim Umar Bologhi II, who spoke with our correspondent confirmed that the boat was reportedly overpowered by river waves, leading to the boat capsizing and leaving many people dead while others sustained varying degrees of injuries. It was gathered that about 50 corpses have been recovered, throwing the entire local government area of the state into mourning. The victims were said to be returning from a wedding before the sad incident occurred. This incident happens to my people, but the brief incident I was told about was they had to, those communities are river and communities actually, they went for a ceremony when coming back that is when the ship, the boat capsized and a large number of people, according to report we have it correctly, about 150 people are already dead on the incident and that's a very large number. And His Excellency, the executive governor of Quara state Malabra and Abdur-Azzaq actually called me to come and serate with us and told me that I should engage with them, inshallah also try to engage our people so that we can condone them for and see how we get possibly to avoid such incidents in the next future. According to report we have it, it actually happened the day first, the day night around 3 a.m. in the morning, actually the report we have it, it was the engine had the fault, so it's, according to report we have it, it's about 300 people and the boats, yes. We look how people at the community are doing their best correctly to see how to get to recover the other crops that are remaining in the water, that's what it's about.