Hey guys it's Nate, aka the Foot Accountant. Welcome back to the channel and let me be one of the first people to welcome you to January 6th. Team of the Year in FIFA Ultimate Team is coming out today and we need to talk about what could be coming today and especially on our usual Monday market outlook video talk about this stuff that we're seeing on this screen with the loading screen. Talk about all the content that we are going to be seeing for this next week of FIFA Ultimate Team because today is January 6th. Team of the Year starts today. The day that we have all been waiting for so I'm super excited to see what happens with the game today. Super excited to see with what content comes out. We're going to talk about the market in this video. We're going to talk about player prices all across the board from low stuff to really to really high priced cards. SBC fodder, SBCs again that are going to be coming out and of course the team of the year cards themselves. But the first thing I want to talk about because a lot of you a lot of you guys seem to be wanting to buy teams and again I want to put this in your mind and make you before we start going into the video I just want to put this point out there for you guys and I'm talking about this a lot right. Make sure that you have a plan going into the year right. So I'm looking at my squad right now and I'm not going to be buying a team during this team of the year period. I want to be doing the upgrade packs and I want to be making coins during this period and maybe trying out a team of the year card. That's kind of my goal and my aim for this team of the year season and this week of team of the year. If I want you guys to make a goal and I want you guys to have a plan because when you plan ahead and when you think about what you're going to be doing right you're planning ahead you know am I going to build a team during this promo. Have I bought players already is that my main goal. When you sort those things out when you have a plan going into promotion knowing what you want to do with your coins knowing what you want to do with your club then you can plan accordingly and make moves accordingly and maybe save yourself some time and some coins on this game. So again plan what you want to do. I want to do upgrade packs I want to do a ton of them. I'm going to be buying some rare gold cards tonight very soon to get them in my club so that I can possibly do those SBCs for cheaper later this week when the upgrade packs do come out. But I do want to talk about this market. If you're somebody who wants to buy a team during this promotion especially some of the meta golds I talked about this in my videos earlier this week that I thought Thursday was going to not Thursday excuse me Saturday yesterday Saturday was going to be the lowest day on the market for some of the most OP meta cards on this game that a ton of people want to buy for their teams because that's what we saw on Black Friday and I think this promotion is going to resemble Black Friday in itself and again yesterday for a couple for a decent amount of cards today we're going to look at a lot of cards that went up from Saturday into Sunday because most people think that Sunday is the buy day now because they think okay week only is ending and I should go buy my team that I want to use next week I should buy it on Sunday which is you know last year that was true but this year the absolute cheapest day is usually Saturdays and especially Saturday nights on open bids and stuff you can see that today Messi from Saturday night went from 870k all the way up to 915,000 coins so yesterday I think we nailed that one right on Messi was the cheapest yesterday actually 865 at one point same thing for Neymar as well Neymar as well was really cheap yesterday I think he was like 645 Saturday he was 630 636,000 coins according to foot today on Sunday he's risen up to about 676k so he's up a decent amount and that also happened with some really really high tier cards that I'm going to show you guys right now on my transfer target so it looks like some people went out and they bought some of their cards today for their teams just because they had been panicked sold so much now I do think that a card like this Mbappe might have another chance to be panicked sold again it's going to be interesting I'm going to talk about that a little bit later when we mentioned team of the year cards but this card right here you can see right now is 1.73 mil 1.71 and he's been like 1.7 plus all day last night there were cards of this caliber selling around 1.6 and that's kind of a buy price that I put for a lot of you guys on this card in the streams when we were talking about him so if you bought him around 1.6 I think you're doing good he's he was up to 1.75 already today this Vieira card last night this one sold for 1.76 I saw multiple Vieiras around 1.7 flat were under and he's like 1.8 mil today there's a couple undercuts showing up right now but he was 1.8 mil for most of the day today this Firmino went from 690 to like 730 at 1 point today this card's up like 20k from last night as well so it did look like some people bought their teams back now some stuff has kind of stayed the same some stuff on like the cheaper levels if we're looking at some more um you know maybe cheaper level cards like let's say Levin Dossky's team of the group stage some of these cards dropped a little bit more today than they weren't then what they were on um then what they were on saturday night but you know what most of the drops happened on saturday and overall most of the cards today either stayed the same or uh went up a little bit because people actually went out and they maybe bought some cards for their teams because they don't expect them to get any cheaper during the team of the year promo which I think especially for some of these guys that are not getting in the team of the year like Levin Dossky I do think it's possible this card could make a nice rebound as it's the best version of his card uh into that so that's what I wanted to start out this video by talking about that and some of those market movements that we saw I also want to talk about some other things in this market art you know we talked about um some cheap cards in yesterday's video I mentioned that some of those cards are definitely going to be a buy before the team of the year is absolutely released right so this is the guy that I used in the video yesterday as an example deli ali's inform I mean he's been over 50k multiple times this year um and this is the type of card that I don't feel like is going to be hurt or affected by team of the year right like this card really shouldn't be affected that much by team of the year he was 52 000 coins just this weekend his highest so far this year has been about 52k he's kind of just bounced back and forth between 42 and 50k but uh you know deli ali's playing really well in real life right now and this is the type of card that somebody who maybe just started the game uh in maybe just started the game they got it for christmas this is like the perfect type of card for them that they would maybe want to put in their squad so maybe this guy if you're on playstation sorry excuse me um some informs like this that are you know that are still riding really like this guy at 40k two of them get listed there at 42 I'm honestly kind of tempted to snag either one of those uh because that's kind of the cheapest that he's been all day today if we go back to the footpeg graph and take a quick peak 44 45 42k is like the cheapest that he has been he was even 50k yesterday so you know what I'm going to go ahead I'm going to pick these two deli ali's up and I'm going to throw them on my transfer list because I do think there is an opportunity for some of these cards uh to rise after team of the year uh enduring team of the year because these cards shouldn't really be affected too much by the team of the year promotion just because somebody who wants to buy a team of the year card or do upgrade sbcs they're probably still going to be able to afford this card after all that stuff happens so on the informs that are under like 200k even even a guy like I was looking at nine goaling like this is a card that a lot of people like to use in their teams right this inform nine goaling card the center mid one uh 86 this is a card that definitely wouldn't be impacted too much by team of the year I don't think so a lot of these guys are can I have continually stayed low 131k for this card is not bad it was like 150 this weekend so if I saw that drop maybe another five to 10k I might have to pick up on some of that extra panic and possibly flip that card next week into team of the year because I do see I did this last year boys I can honestly we're going to look back in a little bit at some of my videos from last year during this time frame I bought cards last year during this promotion and I flipped them into the team the year because they were panic sold before and I bought them and of course people who who people who can't afford a team of the year make up a large portion of the player base of this game and they just basically maybe did some upgrade packs and they ended up buying their teams back over the course of the week right and you saw some of those informs under the 200,000 coin range rise back up kind of like last year I bought Fabinho's informs because he was a meta popular use inform so that's something if you're wanting to invest in something right now get over here on the market look through the team of the weeks you know go through the team of the week page and basically just look at a bunch of graphs and this is also going to be very good for you because you're going to be learning the prices of cards in this game that are very usable and that of people for trading wise as well like this for me knows 137k this Suarez is 100,000 coins flat right now at the moment which I think might be cheaper than he was this is even cheaper than what Suarez was during team of the week that looks like it to be a like a 10k undercut there as well so look look through some of these team of the weeks look through what you see and with some of these cards again if it's an under 200,000 coin card and maybe for somebody like this Royce I'm kind of interested in Royce because he's been usually around 300k and he's dipped down to 250 that would be a card I don't feel like would be hurt too much by team of the year but again those are the types of cards that you might be able to put some coins into right now again just make sure that it's an out of pack special card that it is something that people use in their teams and that it is under that 250k ish range maybe 200k or under you know kind of suiting to that player base which would not really be looking into buying something for team of the year but they might have sold their team off because you know a lot of people have been instructing sell your teams for this promo and then you know buy it back because its price are going to go down right team of the year so I've like I played weekendly this weekend I'm sure you guys have as well I got a couple more games to play I saw so many gold teams this weekend of people that are just like stripping down their clubs to an untradable gold team or rockin a cheap 100k team as they head into team of the year and you're going to start seeing some of those cards slide back up this week after we exit and as we go through team of the year speaking of team of the year cards though I want to talk about those cards that are coming out this week and some of the content that we can come out this week as well because after all this is a market outlook video for this weekend fee for ultimate team the mark the videos that we drop every Monday you guys know this so the first thing I want to look at is team of the year cards and we're going to look at attackers first because I think EA is smart the last two years they have dropped attackers first I believe I think I think that is correct they have dropped attackers in packs the first day so the first day they do attackers second day they did midfielders in the third day they did the defense and the goalkeeper last year and in FIFA 18 I believe and then FIFA 17 they did defenders first but I think they should do attackers first again because people who have been saving up packs feel like they can't wait can't wait any longer and they go out and they actually open up packs sometimes for the attackers because you know there's team leader cards and packs and they just can't wait any longer and that does actually happen so I do think it would be smart on EA's part to drop these cards and packs first last year in Bapu I want to look at this first day in Bapu it was around 1.1 or 4.1 on PS and he actually rose to 4-3 on Wednesday before they got put back into packs on Thursday the entire team and then he went down on Friday at 3.7 and then rose back to 4-1 on PlayStation but I want to look this okay again we're looking at I need to show you guys Xbox prices but these these cards are going to fluctuate from day one to day two to day three if the attackers do come out first this is a screenshot and look of me last year on Xbox so this in Bapu was around three six or three hundred three point seven mil on that first night when he was in packs so the first day he was in packs that first night they the team of the year cards got decently cheap this is from my Monday video last year and this is my Tuesday video looking at his price he went from three seven which was where he was around here three seven to three eight and he went all the way up to just over four million coins right about four point one on Xbox before he got back into packs and the Xbox graph doesn't really show it on the daily basis but he did he definitely did make that fluctuation of around like three hundred thousand coins and the fluctuation was even bigger on PlayStation right so if you're somebody on PlayStation that could be a trend right away that if you want to try a team of the year card maybe watch those prices right away as they get packed and as they get put onto the game that would be something that you could be watching out for but then through the rest of this week for the rest of this week what you see is like the attackers are really rare their impacts they're really rare for one day and then they all get released into packs right the entire team of the year gets released into packs and everything just just goes down last year this and Bobby went from four point three to Friday he he hit his lowest at three seven people starting buying the cards for weekend league they started investing in these cards because they just knew these guys are going out of packs and then boom by Sunday he's back to four mil and then he stayed at four mil until next Friday which is when future stars came and all these cards got panic sold which was ridiculous in itself but we'll we'll talk about that kind of stuff after this week of team of the year is over or is there going to be a crash for future stars again so if you're looking to buy a team of the year card this week this is what I would tell you I think again that probably sometime in that Friday or Saturday timeframe might be your best you could either take a chance of being Saturday night weekend league sell-off as a time to buy some of those team of the year cards although I don't feel like the team of the years would be that much panic sold right I don't think they people would panic sell a team of the year card that much maybe a little bit of a drop on a Saturday night or Sunday of this next week as the team of the years are going out of packs but I think that that Thursday or Friday would be a great time to look at these cards because it's technically still in the earlier part of weekend league they'll run the lightning rounds on Friday they're going to be like lightning rounds every single day of this week we'll talk about those in a second as well but yeah for most of the team of the year cards we'll look at Kante as well most of these cards if we look at their graphs from last year we'll go generations FIFA 19 and Golov Kante the 96 rated this card is probably going to come out again it's going to be insane it's going to be insane boom look at this first day 3.5 mil for Kante drops down on Friday to like 2.2 which was his lowest according to foot bin and then by Monday he was 2.5 so there's on all the cards that I have looked at so far at least most of them there was a rebound from Friday into the end of the weekend when they went out of packs and then of course we had the panic for future stars but if you're looking to buy one of these team of the year cards like to play within this upcoming weekend league or to get them at a good price of course we have to look at the pack weight first day we'll be looking at that kind of stuff and seeing how many of them actually do come onto the market but that Friday looks to be kind of like the quote-unquote buy day or time where you might be able to find these cards for a good price when they are impacts on this game speaking again of all the packs that are going to be open let's go back to foot bin here for a second and look at the cheapest players by rating I want to talk about the SBC fodder stuff this weekend FIFA ultimate team as well and I'm honestly perplexed right now because I think a lot of people are investing in high-rated we're undoubtedly gonna be getting SBCs right the loading screen if we head back to FIFA the loading screen that is on foot right now shows us that we are getting player moments and that we are getting flashback SBCs and supposedly we're getting seven of them based on what the loading screen says which four flashbacks and three player moments and those SBCs you think okay boom there's seven SBCs that means seven days of team of the year one SBC per day it would make sense right we will see what actually happens if that if they do one SBC per day a lot of times on the first day the promo they don't drop an SBC but regardless I know a lot of you a lot of people maybe not you guys watching this video but a lot of people want to know when do I invest for some of these SBCs and people have been buying high-rated golds to do the icon SBCs because there's nothing to do in this game right now until we get to team of the year and they've also been buying them to um invest for the SBCs coming during team of the year but I'm still not somebody who wants to buy these cards and I would not recommend buying these cards at inflated prices because they are inflated right now 86s are up like three to four k uh 85s are up like two k even the 84s are up like a thousand coins 87s are up two three k 88s are up three four k 89s are up like four to five k so I would honestly just wait for the lightning rounds to hit on Monday and then reevaluate what happens because if no SBC comes on Monday and these cards are going to get packed and packed and packed for lightning rounds uh they were going to see basically every single day of this week then they're going to drop right you're going to have people that invested in these they're going to be panic selling on top of that and it could be a fantastic time to get in on these cards and regardless if we do have an SBC that comes out I don't feel like the demand for that SBC if it's overpriced like half of the SBCs we get on this game are the first flashback or player moments that we get maybe today on Monday is there going to be that much demand that these card prices stay high I really don't know it all just depends on the SBC so definitely keep your eye on some of these especially the 85s 86s and 87s early on in the promotion due to just the pack supply and the amount of stuff that is going to be entered out in this game via uh via packs so that's one thing I wanted to mention as well and then talking about more about lightning rounds just really quick to end the video last year I did a bunch of trading during lightning rounds last year I'm going to talk about the first day especially because it worked out the best on the first day I was doing basic icon lipping now this might not apply to all you guys you might not have a ton of coins but if you're somebody who's got a bunch of coins this is what I would do basically set yourself a filter to get to the 59th minute on icons now right now there's not going to be that many so I'm going to get there with a pretty broad range filter right I got to the icon 59th minute in three pages which is pretty crazy but it's not going to be this way today when all the lightning rounds happen so you're going to have to kind of tighten your budget a little bit so what I would say is if you're somebody who's got like two million coins set yourself a a basic chem style filter for the icons where basically what you're trying to get is panic selling and people packing these new icons from opening lightning round packs you're trying to get super low undercuts on the 59th minute that are going to rebound later in the day and last year I saw a perfect example of this with baby george best because he was a right wing card obviously they released the messy right wing and then bop a right wing uh team of the year cards and those two team of the years were um came out of the game and people maybe panic sold their icons to go and buy a team of the year card I think you're going to see a decent amount of that tomorrow but what happened was that george best I bought one last year in that first lighting round for 900 000 coins flat and I sold it literally the next listing later as a fresh card uh for 1.03 mil or something like that so I made like 40 000 coins after tax or something like that if my math is correctly maybe 50 ish 1000 coins after tax um because somebody panic sold that card and then they basically wanted to go buy it back because they realized how expensive those team years are or somebody just wanted to go buy a fresh george best so there's going to be a lot of panic on the icon 59 for the basic chem styles tomorrow especially in the attacker range because if they do release the attacker uh today I keep I keep saying today and tomorrow Monday Monday Monday Monday if they do put the attackers out first then there's going to be some panic on the attackers uh icons that are released in this game because people are going to see those and they're going to say I'm going to go panic sell my icon and try to pick up one of these cards uh that is new team of the year item so basically that's what you're going to see with those a lot of icon panic probably some panic on other areas of the market as well and also probably some panic buying now I want to mention this really quick too at the end if you don't see a guy like Neymar get into team of the year tomorrow because or today they will release the team of the year today they will tweet it out and they'll show the entire team of the year so we're going to know who's in it this center attacking me Neymar is on so many people's buy lists if he does not get a team of the year or it doesn't look likely that he's going to get a team of the year via the team of your vote if there's going to be like a 12th man or if they're going to put in a couple extra guys that didn't get into the starting 11 for team of the year you might see this card make a quick surge upwards in price as people run to go buy it and put it in their teams because they want this card so bad for their squads but he's not going to get in team of the year so this would be the best version of his card you kind of understand what I'm saying and one more I'm coming with that I mean people you might see some panic buying on cards especially like that Roberto Firmino screen version that I think a lot of people are actually investing in already because it links to so many guys that are supposed to be in team of the year like if Sadio Mane if Trent if Robertson Van Dyke and Allison all get into team of the year that's a lot of Liverpool players that people will try to link to and I think a lot of people have invested in this card already it's up like 20k today because they think it's going to be a good link investment to team of the year players so you might see this take another spike up in price when people see the team of the year you also might see maybe cards like this get panic sold tomorrow when team of the year is announced because some people who still have this card might see Mbappe getting into team of the year and then might go and panic sell this version so that's something to look out for as well there's probably going to be some mixtures in the market of some things being panic sold right away today on Monday and also some things are going to be bought up as people realize what is happening and what cards are coming out for a team of the year especially let me look at one other example of that for you guys this Conte card right so Conte should be getting into team of the year if he is announced tomorrow you might see a quick drop in the road to the finals price you might see a quick drop in this card's price just because it's around like 720 I think right now but if people see that he's getting into team of the year they might go and panic sell this card really quick because just people like to panic sell in this game so you might see a dip in the card like this Conte if he gets in a team of the year you also might see a dip on that Vieira card like 88 Vieira that we just looked at last year there was a ton of panic selling on Vieira's prime card there was a direct panic sell on this card last year based off of team of the year Conte's price and I would expect that again to happen this year especially because this card is just super duper rare at this point but again I have a slight hunch that the team of the year cards might be just stupid expensive and if they're stupid stupid expensive day one with the attackers being released if that's how they decide to do it and then you might see some of the other areas of the market pick back up because people would just realize I don't literally have enough coins to buy these players even if I want to try them out basically so yeah that's kind of the market outlook for this week guys there's going to be tons of market movements I do think though the general movement of the market is you're going to see things go up especially in terms of out of packs informs out of pack special cards and stuff like that that people are going to be buying their teams with again a lot of this promo is putting coins onto the market getting coins in people's clubs saved up packs help with that lightning rounds help with that there's going to be plenty of those types of packs this week there's also going to be plenty of chances to make coins through flipping through investing and stuff like that so we'll be covering all that stuff this week and of course we will be smashing upgrade packs when those are released as well it's gonna be a fun week all right all right boys hopefully you are ready hopefully you got your plan set on what you want to do during this team the year week and hopefully you guys enjoyed this video as well if you did smash a thumbs up on a comment down below if you have any questions and subscribe to the channel if you are new that's the market outlook for this week it's been Nate the foot accountant catch you guys later peace out


總統先生,今天在座的差不多所有的人都程度不同的參加過中美之間的經濟貿易科學技術和學術交流活動其中很多人到過美國,或在中國,即來過美國的同行他們從所有中國人民一樣願意看到中美關係得到穩定持久的發展期待兩國的經濟貿易科學技術和學術文化交流發展到一個新的水平現在我們請總統先生講話各位女士們,我是很 honoured to come before you todayThe first American president ever to address your nation from the great hall of the peopleMy wife Nancy and I have looked forward to visiting the people and treasures of your great and historic landOne of the world's oldest civilizationsWe have marveled at Beijing's sweeping vistas and we have felt the warmth of your hospitality touch our heartsWe only regret that our visit will be so briefI'm afraid it will be as a Tang dynasty poet once wroteLooking at the flowers while riding horsebackBut you have another saying from the book of HanWhich describes how Nancy and I feelTo see a thing once is better than hearing about it a hundred times12 years ago, former president NixonArrived in BeijingStep down from Air Force OneAnd shook hands with former Premier Zhou EnlaiPremier Zhou would later tell himYour handshake came over the vastest ocean in the world25 years of no communicationWith one handshake, America and ChinaEach turned a new page in their historiesI believe that history beckons againWe have begun to write a new chapter for peace and progress in our historiesWith America and China going forwardHand in handShe has show Bing JinWe must always be realistic about our relationshipFrankly, acknowledging the fundamental differences in ideology and institutionsBetween our two societiesYes, let us acknowledge those differencesLet us never minimize themBut let us not be dominated by themI have not come to China to hold forth on what divides usBut to build on what binds usI have not come to dwell on a closed door pastBut to urge that Americans and Chinese look to the futureBecause together we can and willMake tomorrow a better dayWhen Premier Zhou was in the United StatesHe told us China has opened its doorAnd will never close it againWe'll permit me to assure you todayAmerica's door is open to youAnd when you walk throughWe'll welcome you as our neighbors and our friendsWe may live at nearly opposite ends of the worldWe may be distinctly differentIn language, customs, and political beliefsBut on many vital questions of our timeThere is little differenceBetween the American and Chinese peopleIndeed, I believe if we were to ask citizensAll over this worldWhat they desire most for their childrenAnd for their children's childrenTheir answer in English, Chinese, or any languageWould likely be the sameWe want peace, we want freedomWe want a better lifeTheir dreams so simply statedRepresent mankind's deepest aspirationsFor security and personal fulfillmentAnd helping them make their dreams come trueIs what our jobs are all aboutWe can work together as equalsIn a spirit of mutual respectAnd mutual benefitI believe in Chinese, you sayWu Jing, Hu WeiWell, America and ChinaAre both great nationsAnd we have a special responsibilityTo preserve world peaceTo help fulfill that responsibilityThe United States is rebuilding its defensesWhich had been neglected for more than a decadeOur people realize this effort is crucialIf we are to deter aggression against AmericaOur allies and other friendsBut we threaten no nationAmerica's troops are not massedOn China's bordersAnd we occupy no landsThe only foreign land we occupyAnywhere in the worldIs beneath gravesitesWhere America shed their bloodFor peace and freedomNor do we commit wanton actsSuch as shooting 269 innocent peopleOut of the skyFor the so-called cause of sacred airspaceAmerica and China both condemnMilitary expansionismThe brutal occupation of AfghanistanThe crushing of CampuchiaAnd we share a stakeIn preserving peaceOn the Korean PeninsulaI think our two peoples agreeThere can be only one sane policyTo preserve our precious civilizationIn this modern nuclear ageA nuclear war cannot be wonAnd must never be foughtAnd that's why we've proposedTo the Soviet UnionMeaningful negotiations that go beyond rhetoricTo actual arms reductionsAnd why we must all work for the dayWhen nuclear weapons will beBanished from the face of the earthAmerica's interest in ChinaOur friendship for your peopleAnd our respect for China'sMany contributions to theProgress of civilizationDate back to the beginningOf our own historyYou might be interested to knowThat personal dinner settingsUsed by our first three presidentsGeorge Washington, John AdamsAnd Thomas JeffersonWere of Chinese originEvidence of our founding fathers'Attraction for your country'sHigh artistic standardsBack in 1784When the first American trading shipThe Empress of ChinaEntered your watersMy country was unknown to youWe were a new republicEager to win a placeIn international commerceA slightly homesick AmericanSailor recorded that first dayIn a letter homeMy dear father he wroteIf ever you receive this letterIt will acquaint youThat after a passageOf six months and seven daysWe came to anchor at WampooThe Chinese had never heard of usBut we introduced ourselvesAs a new nationGave them our historyWith the description of our countryThe importance and necessityOf a trade here to theAdvantage of bothWhich they appear perfectlyTo understand and wishWell, since those early daysOur countries have both profitedFrom the exchange of peopleGoods and ideasChinese settlers helpedTame our continentDuring the 19th centuryToday their families descendantsJoin other AmericansIn cooperating with youTo build a new prosperityIn ChinaHow did AmericaWhich began as anImpoverished countryAnd a melting potAttracting immigrantsFrom every corner of the globePull together and becomeThe leading economic nationIn the worldHow did we go in so short a timeFrom living by candlelightTo exploring the frontiersOf the universe by satelliteFrom each farmerLabering with horse and hoeFor an entire yearJust to feed four peopleTo running his farmWith the most modern machineryAnd producing enough to feed75 peopleBeing AmericaThe breadbasket of the worldWell, we're peopleWho've always believedThe heritage of our pastIs the seed that brings forthThe harvest of our futureAnd from our rootsWe have drawn tremendous powerFrom two great forcesFaith and freedomAmerica was founded by peopleWho sought freedom to worship GodAnd to trust in himTo guide them in their daily livesWith wisdom, strength, goodnessAnd compassionOur passion for freedomLead to the American RevolutionThe first great uprisingFor human rights and independenceAgainst colonial ruleWe knew each of us could notEnjoy liberty for ourselvesUnless we were willing to share itWith everyone elseAnd we knew our freedomCould not truly be safeUnless all of us were protectedBy a body of lawsWe treated as equallyGeorge Washington told usWe would be bound togetherIn a sacred brotherhood of free menAbraham Lincoln definedThe heart of American democracyWhen he said, no man is goodEnough to govern another manWithout that other's consentThese great principlesHave nourished the soul of AmericaAnd they have been enrichedBy values such as the dignityOf work, the friendshipThe workers and the warmth of familyLike China, our people see the futureIn the eyes of our childrenAnd like China, we revere our eldersTo be as good as our fathers and mothersWe must be betterTrust the peopleThese three words are not only the heart and soulOf American historyBut the most powerful force for human progressIn the world todayThose who ignore this vital truthWill condemn their countriesTo fall farther and farther behindIn the world's competitionFor economic leadershipIn the 1980s and beyondBecause look around usThe societies that have madeThe most spectacular progressIn the shortest period of timeAre not the most rigidly organizedNor even the richest in natural resourcesNo, it's where people have beenAllowed to create, compete, and buildWhere they've been permitted to thinkFor themselves, make economic decisionsAnd benefit from their own risksThat societies have becomeThe most prosperous, progressive, dynamic, and freeNothing could be more basicTo the spirit of progressFor a farmer, laborer, or merchant Than economic rewardFor legitimate risk and honest toilA little over a century agoUlysses S. GrantWho was then a former president Visited your countryAnd saw China's great potentialI see dawning, Grant wroteThe beginning of a changeWhen it does comeChina will rapidly becomeA powerful and rich nationThe population isIndustrious, frugal, intelligent, and quick to learnWell, today, China's economyKrackles with the dynamics of changeExpansion of individual incentivesFor farmers in your new responsibility systemNew bonuses for workersAnd more disciplined managementIn terms of profits and lossesImproved methods of market distributionOpen your economy to the worldThrough China's membershipIn the international monetary fundThe world bankAnd through your invitation to tradeAnd invest, especially in yourFour special economic zonesAnd your commitment to attractCapital and scientific knowledgeTo create a high technology baseFor the futureAll this reflects China's new roleIn the international economic communityAnd your determination to modernizeYour economy and raise the standardOf living of your peopleUnlike some governments which fearChange and fear the futureChina is beginning to reach outToward new horizonsAnd we salute your courageProgress, Premier Zhao has told usProgress, Premier Zhao has told usLies in our efforts to emancipateOur thinking in a bold wayTo carry out reform with determinationTo make new inventions with courageAnd to break with the economicMoles and conventions of all descriptionsWhich fetter the developmentOf the productive forceWell, we Americans have alwaysConsidered ourselves pioneersSo we appreciate such vitalityAnd optimismToday I bring you a messageFrom my countrymenAs China moves forward in this new pathAmerica welcomes the opportunityTo walk by your sideIncidentally, I know Premier ZhaoHas demonstrated mastery of his subjectWhen he was directing agriculturalPolices in SichuanPolitions went from food shortagesAnd forced imports to bumper harvestsAnd rising exportsIn fact, I'm told that becauseOf the work he did, it is saidIn Sichuan province, if you want riceGo see ZhaoWell, China's growth is in China's handsYou will choose your own pathTo developmentBut we're not surprised to seeThe fresh breezes of incentivesAnd innovation sweeping positiveChanges across ChinaAnd behind the statistics of economic growthAre reports of personal success storiesPointing to a new spirit of progressChairman Deng has a sayingSeek truth from factsWell, today in ChinaThe reality of more small enterprisesDoing a thriving businessMore families profiting from their ownHard work and the bigger harvestsThey produce and more investmentIn science and technologyPoints to more opportunityFor allPresident John Kennedy often usedA metaphor to describe such progressA rising tide lifts all boatsIn the United States, as I mentionedEarlier, we've always believed deeplyThat incentives are keyAnd that free people build free marketsThat ignite dynamic development for everyoneFor a time, America's governmentHad drifted away from this key principleAnd our economic growth sufferedWhen we took office in January 1981We said to the peopleLet us make a new beginningFrom now on, if you work harderAnd earn more than beforeYour reward will be greater than it wasWe're putting America's future in your handsYou can spark the spirit of enterpriseYou can get America moving againAnd they haveIn three short yearsThe American people have revivedA dynamic growth economyBolstered by incentives of lower tax ratesStable prices, reduced interest ratesA rebirth of productivityAnd restored currencyOr confidence in our currencyHope is highConfidence is strongAmerica's future looks bright againWith a strong technological basePioneering sunrise industriesAnd modernizing older onesThe United States is beginningAn economic renaissanceAnd helping pull other nationsToward worldwide recoveryI see America and our Pacific neighborsGoing forward in a mighty enterpriseTo build strong economies and a safer worldThe United States and ChinaHave an historic opportunityWe can expand our economicAnd scientific cooperationStrengthen the ties between our peoplesAnd take an important stepAnd a better lifeAnd there is much we can shareWe think progress in four areasIs particularly promisingTrade, technology, investmentAnd exchanges of scientificAnd managerial expertiseIn a few short yearsTwo-way trade has risen sharplyThe United States is now China'sThird largest trading partnerOur bilateral trade showsGreat promise for the futureIn areas such as machineryTechnology, oilEquipment, petroleumAgricultural and manufacturing productsLast June, I instructed our governmentTo liberalize controls over the export to ChinaOf high technology productsSuch as computers and laboratory instrumentsOur policies on technology transferWill continue to evolveAlong with our overall relationshipAnd the development of broader cooperationBetween usMay I emphasize to the members of theScientific community here todayThe relaxing of export controlsReflects my determinationThat China be treated as aFriendly, non-allied nationAnd that the United States be fullyPrepared to cooperate in yourModernization. DuringPremiere Zhao's visit to our countryWe took another step forwardSigning the United States ChinaIndustrial and technologicalOur joint commission on commerceAnd trade will discussImplementation of the accordDuring their next meetingIn Washington in MayWe will focus our efforts onThe sectors to which ChinaHas attached greatest priorityOur trade and developmentProgram will facilitate ourProgress. ExpandingCooperative ventures isAnother area of promising growth.American firms have investedAlmost $700 million in jointOffshore oil exploration inChina, making the UnitedStates your largest foreignInvestor. We welcomeYour determination to improveConditions for foreign businessIn China, streamliningBurocratic procedures, establishingA more predictable system forInvestment through domesticLegislation and internationalAgreements, reforming pricesTo make them internationallyCompetitive and providingWith the offices, housingAnd schools they and theirFamilies need to workEffectively will stimulateMore American investment.For your part, some 50Chinese firms have establishedOffices or branches inThe United States and ChinaHas invested in severalJoint ventures in ourCountry. We intend toStrengthen these trends.When Treasury SecretaryReagan was here last monthAgreement, Monday our twoCountries will sign thisAgreement, which I amPlease to report will increaseIncentives for even closerCooperation betweenAmerican and Chinese firms.And we are continuing toWork toward conclusion ofBy latter agreements onGreater investment protectionAnd many other areas ofCooperation. I amParticularly proud that theAgreement on cooperation andThe peaceful uses of atomicEnergy. As many of youKnow the negotiations betweenOur two countries go backAlmost to the beginning ofMy administration. We haveHeld a total of six sessionsIn Washington and Beijing. WeMade great progress duringPremiere Zhao's visit. AndOur negotiations have justNow concluded successfully.The result and agreementFor cooperation in peacefulEnergy with several of thePeople here made majorContributions to this effortWhich meets the requirementsOf both sides. OnceApproval is complete, itWill open broad opportunitiesFor joint work and developmentOf the energy base, whichChina needs for herModernization. Scientists,Engineers, business leadersAnd officials of bothCountries interested inPeaceful nuclear energyWill welcome thisAgreement. China hasOne of the world's mostAmbitious programs forPower generation. And IBelieve that America'sEnergy technology, notJust in nuclear energy, butAcross the board isSecond to none, and perhapsMost suitable for China'sVaried needs. OurAgreement is founded onImportant nonproliferationStandards. We haveNoticed recent statement ofChina's nonproliferationPolices, particularly thoseBy Premier Zhao inWashington and BeijingOver the past severalMonths. Premier Zhao and蔡英文I can tell you that ourCountry share the sameBasic principles ofPreserving world peaceand preventing theDestabilizing spread ofNuclear explosives. Neitherof us will encourageProliferation nor assistAny other country toAquire or develop anyNuclear explosiveDevice. OurCooperation and the peacefulUses of nuclear energyWill be based on sharedPrinciples of和不可抗拒我們也有很大能量的能量的活動去增加能量和學術的學術經濟我知道很多人從中國的文化中聽過了九個月的學術學術學術學術學術學術學術學術學術學術學術學術學術學術我是想要提出我們已發明不錯的新學科給予全三年學術學術學術學術學術學術學術學術學術學術學術學術學術學術學術學術學術學術學術學術We have a very productive agreementwith exchange programs in 21 specific areas.We are sharing the benefits of research in medicine,energy, and other technical fields.Our scientists are learning a great dealfrom each other in public health,agricultural sciences, and many other areas.Men and women of vision already see thatworking as the zero gravity environment of spaceoffers dazzling opportunitiesto improve life on Earth.學習中有做過的效果是在我們空間的發展中展示了生命的醫療能夠在空間採取4次的優惠同樣的醫療在地上它們可以採取超過400次的優惠所以1個月的產生在空間採取30年的產生在地上我們也看得出能夠採取超過400次的發展的優惠這是對於共產品的產生中的超過8000變化產生中的超過5000變化產生中的超過5000變化產生中的超過5000變化產生中的超過5000變化和相對 escapeis an social sciences hundreds of American Chinese scholars have visited eachother's countries to teach and study subjects ranging from lawand economics to poetry and history,for our part we welcome this New Pacific tidelet it roll peacefully on carrying a two way flow of people有更多的想法可以削弱障礙和毕竟而建立了互聚和互聚的关系未来是我们要建造的伸展的危险和恐惧有些 may be difficult但我认为挑战是比较值得的比较大互联人们必须是很棒的当他们描述他们的精神一起继续继续达到更高的好处那么多时候我们看到每天每日的生命和爱情的行动让我们有信心相信我们和希望有更好的未来1981年一位年轻美国学生John Zeitman来这里学习中国学习和找新朋友他是一位很棒的人在北京街上和学习和学习他在爱上和你的国家他在20月的生日随时死亡但他的旅游拒绝了新生命John的家庭和朋友发生了中国学习的计划在华盛顿企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院企业学院A tree that now blossoms with the beauty of friendship and cooperation.If our people could go forward in this same spirit,planting not one tree, but millions,and then tending each, so it may grow sturdy and tall.Then the dream of a single youth might grow into the golden dreams of all mankind.Thank you very much.總統先生


So welcome back everyone. Now we're at the last talk of this session. We'll be hearing a talk by Mireille Kibbuti on homological methods in rewriting. She is currently a PhD student at the MIT, advised by Adam Klippuschlid-Powler. I hope I didn't mispronounce it. And while she will be speaking about work she did completely independently, which was awarded by the best paper award for junior researchers at the FSCD conference last year, which is the conference on formal structures for computation and deduction. So we're really curious to hear about your work here and please go ahead. Okay, so hello. My name is Mireille and this talk, I'm going to talk about homological methods in rewriting. This talk is about real equation theories or term rewriting systems, TRSs for short. So I begin by clarifying terminology around them. Both equation theories and TRSs consist of a set of variables, a signature, and a set of rules. A signature is a set of constant or function symbols like this. And if we fix a set of variables and a signature, we can construct terms looking like this. Then our rule is a pair of two terms. For equation theories, we use an equality sign to represent a pair. And for TRSs, we use an arrow. The only difference between equation theories and TRSs is that if we don't care about the order of two terms, then it's an equation theory. If we do care about the order, then it's a TRS. So let's move on to the main question in this talk. We are given an equation theory or TRS. Then can we tell if there is any smaller theory or TRS to equivalence to the given one? Smaller here means smaller number of rules. So in other words, can we know how many rules are actually needed? This talk will give a lower bound of that number using algebra. Later of this talk, I will explain a brief introduction and history of the algebra we are going to use. Here's an example, the theory of groups. The theory of groups is presented by these five axioms. But it's known that these three axioms are enough. They can derive the other two axioms on the right. And even smaller presentations are known. The theory of groups is equivalent to these two axioms. And if you use a symbol for division instead of multiplication, then there's just a single axiom, which is known to be equivalent to the theory of groups. So this is the smallest. And in terms of the number of axioms, but do we have a single axiom over the same signature, like this one? It has multiplication, inverse, and identity. So we don't count the last rule as the smallest one. Since it's over the different signature, it doesn't have any information about identity E. So our question is this. Is this still possible to give a single axiom equivalent to the theory of groups without changing the signature? Tarski, Neumann, and Kuhnen shows the answer is no. Then what about other equation theories or TRSs, not just the theory of groups? If we are given a theory of TRS, can we tell in a generic way how many rules are actually needed? At FACD 2016, Melvos and Mimran gave a lower bound of the number of rules. Their theorem tells that if we have a complete TRS, complete means terminating and confluent, then you can compute a number Mm, and any equivalent TRS has at least Mm rules. The number sign here means the cardinality of the set, but not many TRSs are known to have Mm greater than 1. So for many examples, the number Mm just tells you at least 0 or 1 rule is needed, which is of course trivial. And for equivalent TRS here, the two signatures can be defined. So for those reasons, we want another lower bound. Here is my theorem at FACD 2019. We fix a signature sigma and let's say R is a complete TRS over sigma. We use a new notion degree of R to describe the precondition here, but I'll explain it later. The theorem is if the degree of R is 0 or prime, then we can compute a non-negative energy E of R such that any equivalent TRS over the same signature has at least sharp R minus E of R rules. Equivalence here is the equivalence between two TRSs over the same signature, and this lower bound is greater than or equal to Melvos-Mimran's lower bound Mm. Using this theorem, we can prove Tarski's theorem earlier in a very simple way. Let's say R is a complete TRS of the theory of groups. Computing the degree and E of R, I got the degree is 2, which means our theorem is applicable, and sharp R minus E of R is 2. So we can conclude any TRS equivalent to the theory of groups over the same signature consisting of multiplication, inverse, and identity as always two rules. And this is the outline of this talk. We are going to see definitions of degree and E of R first, then example, then a quick overview of my proof. So far, this is the same with my FESD 2019 talk, but since I have more time today, I'm going to share you about homology, which is the most important thing in the proof. So we begin by the definitions. We define the degree of a TRS first. We assume every variable is written like x sub a natural number for simplicity. Sharp sub i of the term means the number of occurrences of variable x i in the term. The degree of a TRS is defined by doing this. For each rule and for each variable, count the variable in the left term and in the right term, take the difference of them, then take the dcd of those numbers. For example, let's think about this TRS. For first rewrite rule, x1 occurs once in the both sides, so we get the difference 0. And for x2, it occurs twice in the left side, but doesn't occur in the right side, so we get 2. For the second rule, in the same manner, we get 3. Taking the dcd of them, we see the degree is 1. As a special case, having the degree 0 is easily described. The degree of a TRS is 0 if and only if the rewrite relation of the TRS preserves the mod set of variables like this example, x1 to x1, x2 to x2, x3 to x2, like that. And to define e of r, we introduce a matrix associated to the TRS, which plays a very important role in connecting rewriting systems and algebra. Let's say r is a complete TRS that has n rules and m critical pairs. If you don't know, that's why a critical pair is a pair of two terms related from a single term in two different ways that satisfies some conditions. I don't tell the condition now, but you can find the definition in rewriting text books, and it's not something I defined. Then we fix a rewriting strategy and d of r, which is I defined, is n times m matrix whose ij's entry is computed using the i3 write rule and the j's critical pair. Here's the j's critical pair, and we normalize them, and we count the number of i3 write rule, li to ri, in each normalizing pad, and then take the difference of them. That is the ij's entry. Here's an example. This TRS preserves variables, so its degree is zero, and it has four critical pairs, so d of r is a four times four matrix. For the first critical pair, the first rewrite rule a1 appears once in the both sides, so we put 1 minus 1, 0 here, and a2s appear twice in the left side, so we put 2, and the other rewrite rule don't appear, so we put 0s. Doing this for the rest, we get the whole matrix. Then we can give e of r, which defines our lower bound. First, we consider the matrix d of r over the integers modulo the degree d. That is, we are going to consider matrix operations over the ring z over d z. Remember that our main theorem has a precondition, d is zero prime. If d equals zero, z over d z is isomorphic to just z, and if d is prime, z over d z forms a finite field of order d. If d is prime, e of r is defined as the rank of d of r, and if d equals zero is something similar to the rank, but a little more complicated, e of r is the number of ones or negative ones in the Smith-Normal form of d of r. Smith-Normal form is a normal form of a matrix over z, looking like this, obtained by elementary row and column operations. The only diagonal elements can be nonzero, and each nonzero elements divides the element on its lower right. Then let's look up some examples. We already computed the matrix for this example. By row and column operations, we can reduce it into this form. e of r is the number of ones or negative ones, so in this case, it's one. Applying the main theorem, we see any TRS equivalent to r has at least three rules. In other words, there's no equivalent TRS with two or less rules, and we have an equivalent TRS with three rules, a one, a two, and a three. For the theory of groups, a complete TRS has 10 rewrite rules and 48 critical views, so we don't compute the matrix here, of course, but I implemented a program to compute the matrix and e of r. I run the program and e of r, that was eight, and which implies any equivalent TRS has at least two rules, so the theory of groups cannot be presented by just one rule over the same signature. My program computes Malbos and Mimram's lower bound also, and e was zero. The last example is a TRS for average and successes of natural numbers. It has one critical pair, and d of r is the five times one zero matrix, since the left path and the right path have the same number of each rule, so e of r is zero, and the same e of r is zero, means the original TRS r is already one of the smallest. This can be generalized like any TRS whose critical pairs are of this type, then the TRS does not have any smaller TRS, smaller equivalent TRSes, and this type here means the left path and the right path have the same subset of rewrite rules. I think it's pretty interesting. I don't know if it's shown by another way, but let me know if you know. Now let's see the proof. From now on we assume the degree is prime. Assuming the d is prime makes the proof simpler because we can use linear algebra. The over d z forms a field, not just a ring, and its n-th Cartesian power forms an n-dimensional vector space. We also use Melbus Niemann's result. Even though their lower bound didn't give many interesting examples, their theory behind that is actually important. So we first look into their theory briefly. To define their lower bound, they introduced two linear maps between z over d z vector spaces whose dimensions are sharp... I hear echo. Is that okay? This sounds okay. I think it's okay. I think that was my fault. Sorry. Is it okay now? I think so. Okay. So to define their lower bound, they introduced two linear maps and the dimensions are the number of rules, the number of signatures, and critical affairs. And yeah, they are named one tilde and two tilde. And in fact, we are already familiar with the second map. The matrix T of r we've seen is the matrix presentation of the second map if the TRS is complete. Melbus Niemann's lower bound mm of sigma r is defined by the dimension of the quotient space of the kernel of the first map by the image of the second map. Here the kernel of a linear map is a subspace of the vectors that map to zero and the image is the subspace of the images of all vectors by the linear map. Then mm of sigma r is less than or equal to the cardinality of r. This is shown by abstract linear algebra like the dimension of the quotient space is less than or equal to the dimension of the space we are taking the quotient. So we have kernel and since the kernel is a subspace of z over dz to sharp r, so we have this inequality and it's sharp r. And the core theorem is that their lower bound is invariant under equivalence. Equivalence here is the version where the signatures can be defined. It's shown using homological algebra and the quotient vector space kernel over image is called the second homology. By showing mm is invariant under equivalence we can change r into any equivalent TRS. So it means any equivalent TRS has at least mm rewrite rules. This is a very rapid introduction to Morbos Mimlan's paper. So let's move on to our proof. Our lower bound sharp r minus e of r is equal to the dimension of this quotient vector space and we name it V. We can show it from some basic facts from linear algebra as well like the dimension of the quotient space is the difference between the dimensions and the first term is just sharp r and the second term is the rank of the major exp of r. And by definition it's just e of r and also by some theorems from linear algebra the dimension of V is equal to the sum of the dimension of these two vector spaces and it's less than or equal to the cardinite view r. And kernel over image here is the second homology so it's invariant under equivalence. For the dimension of the image of del 1 tilde it does depend on the signature but it's invariant under equivalence between two TRSes over the same signature. In summary we get these equalities and sharp r minus e of r is invariant under equivalence over the same signature so you can replace r with any r prime equivalent to r and then we get the desired inequality. Here remember the statement of the main theorem. We have a precondition saying the degree of r is zero r prime and we've seen the case the degree is prime. The proof can be extended to the case d is zero but not to the case d is neither zero r prime. Basically it's because the ring d over dz is more complicated in that case especially it has zero devices like if d is four two times two is zero module of four and many useful theorems don't work for rings with zero devices. For such rings for instance Smith's normal form used to define the lower bound is not well defined for matrices. Okay so we saw something called homology in the proof. For now I want to talk more about homology and its relation to rewriting. Even before my and Malbos and Mimons research homology groups appeared in rewriting in 1980s so but for string rewriting so let's talk about string rewriting. String rewriting systems SRSes for short is the same with TRSes but it's about strings instead of terms. We have an alphabet that is a set of characters and a set of rules and for instance we have this kind of SRS here and ipson is the empty string and if we have the string abab then we can rewrite ba here to ab so we get this and abb is written to ipson the empty string so we get a. For the first step let's see how string rewriting relates to algebra. There's a relationship between an SRS and a monoid. If we have an SRS we can define a monoid that is the quotient of the strings over the alphabet by reflexive symmetric transitive closure of the real-life relation and the multiplication is the string concatenation. In this situation the SRS is called a presentation of that monoid. For example let's say our alphabet has just a single character a and a single rule aa to ipson then the monoid has just two distinct equivalence classes. We write brackets for equivalence classes. The class of aa is equal to that of ipson because aa is written to ipson. For the next example we have two characters ab and rule ba to ab. Then the monoid looks like this since the class of ba equals that of ab by the rewrite rule and not just we can define a monoid from an SRS any equivalent SRS is present isomorphic monoys and also we are given a monoid. If we are given a monoid we can always find an SRS that presents the monoid if we are allowed to have infinite alphabets and rules. So this is the relationship between monois and SRS. So let's move on to homology groups. What are homology groups? Homology groups appear in many contexts. The most famous one is the homology of a topological space and this sounds somewhat confusing but there's something called homology groups of a group and not just groups. Chilean fields medallists noticed that we can define homology groups of a general algebraic system and for any of these one common thing is that homology groups are avian groups and they extract some information from the object we are given. Here examples for topological spaces we have a sequence of homology groups for each topological space and for any two topologically isomorphic spaces like a sphere and a cuboid and then the the sequences are the same and for a sphere it's known that the zeroth homology is the first one but we count its zeros. The zeroth homology is z and the first homology is the trivial group that consists of just the identity and the second is z and the latest are all trivial. And for a torus the first homology is z times z the product of z and the others are the same with the sphere. For surfaces like these the first homology tells the number of holes for a sphere since it has no holes then the first homology is just just has zero and for a torus we have a hole here and we can think of the space inside the torus as a hole so we have two holes it corresponds to the things that the first homology has to z and for homology groups of groups I don't give the concrete examples because it's going to be complicated but basically there's a sequence of homology groups for each group and any two isomorphic groups gives the same sequence and the homology groups of a group also called group homologies are important to think about homology of a theory or TRS. Similar to monoids there's something called group presentation. A monoid presentation was a pair of an alphabet sigma and a set of rules r but for a group presentation r is a set of strings and strings can have a formal inverse of a character in the alphabet. Then to present a group from the sigma and r first we create a monoid presentation from them. The new alphabet is sigma with its inverse and the rules are w to epsilon where w is a string r or xx inverse or x inverse x for each character x. Rules xx inverse to epsilon and x inverse x to epsilon means the identity rules so this equips the presented to monoid with a group structure and any group can be presented in this way as well as monoids and the reason why I'm talking about groups is that group theories has worked on small presentations of groups for a long time. Since we are interested in small presentations of equational theories or TRSs that sounds similar. A result by epsilon is that if group g is presented by finite sigma and r then the difference between the number of rules and the number of characters is bounded below by something using homology groups of g. s and s is the number of generators and rank is the torsion free rank. If you don't know about them both s and rank are something similar to the dimension of vector space but even though homology groups here are not vector spaces and if we move the minus sharp sigma to the right hand side we get a lower bound of the number of rules when we fix the alphabet. So that sounds more similar to our result for TRSs and indeed the result for TRSs is the consequence of generalizing this inequality here to TRSs. This is how I reached to the result but for generalization we need homology groups for monoids or TRSs so let's see things about that. For monoids since monoids are pretty similar to groups we can construct homology groups for monoids or SRSs in the same way with groups. Also in this case any two equivalent SRSs or isomorphic monoids gives the same homology groups but not many people had worked on this homology and no application to rewriting were known until 1987. In 1987 Squire a group theorist published an amazing paper he solves an open problem at the time he the problem asks if there's exist monoid with a solvable world problem that cannot be presented by any finite complete SRS. A monoid with a solvable world problem means that checking the equality of elements in the monoid is decidable and notice that if a finite complete SRS present a monoid the world problem of the monoid must be solvable because if you want to check if two elements are equal then you can rewrite them by the SRS until they cannot be written anymore then you can just check the resulting terms are equal so the problem is asking if the converse is true or not. What Squire discovered is that if the third homology group of the third homology group of a complete TRS is not finitely generated then the SRS must be infinite. His theorem is actually stronger than that but and remember that if two SRSs are equivalent then their homology groups are the same so if we have a complete SRS whose third homology group is not finite regenerated then not just that SRS is infinite but all SRSs equivalent to that SRS are infinite that means we cannot have finiteness and completeness at the same time. This is how he solved the problem and this was the very first time that homological algebra was applied to rewriting and I have to mention something about this. I found out about this topic when I was reading the French Wikipedia article on rewriting the English version doesn't have this so thank you the French speaker who wrote this article. I published a paper received on ours and was invited here it's all because of you so thank you so much the French speaker merci beaucoup and remember that we saw this diagram here SRSs are closely related to monoliths now let's see what about TRSs instead of SRSs for strings we can define multiplication as concatenation then what about terms instead of trying to define multiplication of two terms we define multiplication of two tuples of terms by substitution for instance we have these terms and then their multiplication is substituting the first element of the tuple for x1 in the left term and the second element for x2 then we get this term and we can multiply these two tuples we have a tuple on even on the left of dot in the same way the first element for x1 in the left the second element for x2 in the left the third element for x3 so if we multiply an m tuple with k kinds of variables and k tuples with m kinds of variable variables then we get an n tuples with k kinds of variables multiplication and yeah so to multiply them the k's here must coincide so yeah and yeah so multiplication now is defined but it's not always for it's not for any two terms we have something like type something like pipe like these two must coincide so tuples of terms form something like a monoid but the multiplication is typed what is that algebraic structure called it's called a category more precisely let's see how the category of terms is defined first objects are natural numbers so morphisms are defined between natural numbers natural numbers morphisms from k to n are n tuples with variables x1 x2 to xk and composition is substitution or multiplication here as the usual composition we can see the type matches well since k on the left is the kinds of variables and k on the right means we have k tuple and finally the identity morphism from n to n is the tuple of x1 to xn this category is the term version of the free monoid or the queen style of an alphabet with multiplication by concatenation we can generalize that category to something called lovier theories a lovier theory is a category whose objects are natural numbers and each object n equals the n's categorical power of one the later part means that any morphism from n to k is a n tuple of k to 1 to k and the relationship between t r s is and lovier theory is pretty much the same with the relationship between s r s is and morphoids any lovier theory is presented by a t r s in the and in the presented lovier theory any term t is identified with a term s if and only if t can give s by a sequence of rewriting in either direction this is how we relate a t r s to an algebraic structure and i can't give the definition of homology concretely it's complicated but for the timeline homological azure on lovier theories is investigated first by zivo aze and pierce sivili they defined cohomology groups of lovier theories and cohomology groups are something due out to homology groups and melbus and heman defined the homology and figured out the second homology is computable when the given t r s is complete and showed that the number of rules is bounded below by using the homology and i gave a better lower bound as i talked in conclusion um we get a lower bound of the number of sorry let's see sorry about that yeah uh in conclusion we have a lower bound of the number of uh rewrite tools to present a t r s of a fixed unit and we saw a relationship between rewriting and abstract algebra we have new algebraic tools and i'm hoping this work gives more research directions of t r s s so that's all thank you for listening thank you very much very much it was really exciting work and just a nice application of interdisciplinary results it well it shows us that how important op max is in science so see in wikipedia and that we all should learn french so are there any open questions i would have a question whoops okay continue yeah after you uh just about the is there a constructive way to get the smaller uh t r s in your method or is there an hope to to build it i say i didn't hear you well um is it constructive constructive the way you get a smaller uh system of holes uh yeah um uh let's see for topological spaces you mean from is is your question whether from the lower bound can you extract a constructively a t r s that achieves that lower bound yeah yeah uh yeah we can constructively compute the number the number but not the new the new system of holes from the old one the smaller one yeah on your implementation the does it your software does this i'm sorry your your your code you you spoke about your your code to compute the number but does your code also compute the new system the smallest system oh sorry um we can't compute the system so it's not you know um it can find a lower bound but it won't construct the system okay thank you yeah uh rick's statement here uh have you thought what i'd like to do is instead of using simply the number of occurrences of the variables i'd like to replace that by an unspecified polynomial depending on that and then replace the degree with uh instead of being primed with an irreducible polynomial seems to me everything after that would work just as well and i'm wondering whether that might get you extend the lower bounds to other possibly powers of primes i know it's a very vague question but the thing is that the algebra i think would work equally well on an irreducible polynomial is it wouldn't on a prime number yeah yeah that's interesting um yeah i'm not sure what's what will happen right now but yeah thank you for the suggestion yeah i would have a question related to bringing that a little bit closer to lambda calculus we know for instance that there is the s and k combinator calculus and then there is also a single axiomatization of the same with rosser's x combinator i don't know if your lower bound can tell us if we could search for even simpler forms in some of those situations the other thing is that in intuitionistic propositional logic which is a bit more complicated than the bullion one minimal axiomatizations are known but i don't think there is a proof that there is one that's minimal so maybe applying the techniques that you have for some of those could tell us about that it's more more of a possible suggestion of uh to reuse the same formalism yeah yeah that's interesting yeah so this one applies only to like the first or the term rewriting so it's not now it's not applicable to you know lambda calculus higher order things yeah but that would be interesting have you considered extending this to higher order rewriting systems i thought about that but it's not it's not an easy thing so i'm not sure how to extend i was wondering one thing just so the simpler case of linear term rewriting systems which is a special case of what you're doing do you get something um i mean something nice in that in that case from your your bound so in your case yeah yeah that might be much simpler i'm not sure right now yeah yeah okay um yeah i mean i had lots of questions it was a very nice talk one one thing i wanted to ask about uh the looking at the example of monoids the axiomatization of monoids i mean like the example that you gave her groups but can you can you extract a lower bound for the axiomatization of uh of monoids so like an associative and unital operation yeah i yeah i tried that but i didn't have any non-travial inequality so yeah okay is that known what is the minimal axiomatization like do you need three axioms or i don't know um yeah i don't know about that okay okay uh but yeah thanks thanks again it was a very nice talk no welcome perfect are there any other questions perfect so thank you very much it was a really nice talk so thank you again really this ends the first part of this after