 Continue. Okay, so please, Keisuke, present us your open problem. Okay. How many minutes can I use for presentation? I think... How long? Up to a quarter, so it's absolutely fine. Yeah, you are a co-host, so you can share your screen. Can you see the screen? Yes, it's visible. Yeah, now it's even better. Okay. Okay, I'm Keisuke Nakano from Tohoku University. I like to explain the open problems about combinators in combinatorial logic. So it calls... It calls... I just... I call the low property of the B terms. The B term is a combinator from the only this B combinator. B combinator is defined by like this. This is a function composition. It takes a two function and argument, and it returns the result of the X of the Y of Z. This is just a function composition. It is a well-known combinator called B. And B term is a combinator built from just only B and application. And the... I call the low property of the combinator like this. This is just... First, I'd like to define the self-right application, the repetitive application. So for example, if you consider X subscript N, then it means that X, X, X. Here is N number of combinators of X. So we assume that application is associative. So we will have the parenthesis here and here and here. So this is just the right application. So this is just the X of N minus one times. So formally we can define X1 is just X. And XN plus one is defined by XN and apply to X in this way. Okay. And then I like to define the low property is just... If this combinator X... Okay, let me explain. First, X has a low property. If this set, the all of the XN, this set is finite up to the beta-eta equivalence. So maybe I need... It is better to show the examples of like this. If you consider B combinator, only just single B, then if you compute the equivalent, the lambda terms, then just this way. And then B, next, we consider BB. BB is like this. And if you compute a BBB, then you will get this one. And if you compute in this way, then you find that B10 is equivalent to the B6. So which means that B11 is equivalent to B7 and B12 equivalent to B8 and so on. So this is kind of the loop here. So in that sense, Bn is just finite. Just nine elements can be generated by the right applications. So I just said the B has a low property because the B10 equals the B6 in terms of the beta-eta equivalence. On the other hand, the BB52 equals to the BB32. So BB also has a low property. I checked by computer. Okay, so I like... Let me show the program, my programs. I implement the checker of the low property. So if you compute B, then you will get... You will find the B10 equals B6. And this is the size of the cycle, just 10 minus 6. And if you consider BB, then you will find BB52 equals BB32. In this way, it's actually... However, I found that BBB does not terminate anymore. So actually I have proved that this BBB does not have low property. But if you consider B of BB, then it has low property. BB294 equals BB... B of BB has a low property like this. And I conjectured in 2008 like this. B terms, that is only B, has low property, if and only if this X is beta-eta equivalent to the B and B. This is a charge-numeral. So N4 application of B, left application of B. So if N equals 0, just a B. And if the K plus 1 then defines that B of BKB and so on. I compute several examples instantiated by 4N. If you consider B N equals 0, we have the low property and N equals 1, and it has low property, N equals 2, and so on. And I checked up to 6. Then we have low property with big numbers and big cycles. I couldn't find B7B because of the time. Because my implementation still takes the... maybe almost one week to compute these numbers. But I couldn't compute B7B, the number of B7B, even if I compute one year. But maybe we need more efficient or parallel implementation, but it is hard to compute these numbers. But of course, B7B may not have low property, so I couldn't prove any more. On the other hand, the other part also is conjectured, but that means that if X is not equivalent to this form, then that X does not have low property. In my paper with Mirai Ikebuchi who gave the talk, invited talks yesterday, she helps me to prove that this part by showing if X has a special form of B terms, then it does not have low property. This is a conjecture. This is just my presentation. I omit almost a motivation, a motivating story, so maybe you think why I consider this kind of problem, but I have no appropriate answer. But please look at our paper, which includes the motivational story of these problems. Thank you. Thank you very much. That would be my first question, why you consider B terms. If I may ask, are over combinators also so interesting? Actually, I can show you some results. If you consider C-combinator, that is, the C-combinator is defined by this X, Y, Z, and it's X, Y, just swap the argument, then we compute low property. C also has a low property, C4 equals C3, and also maybe you can find some low property. But almost other combinators is like a trivial, just the size of cycle is one, or immediately it enters into the cycles. And also if you consider the S-combinator, then actually we cannot find any more, because it always expands the size of the terms, so S cannot have low property. This is why I do not consider these combinators. And also if you may, maybe you may be interested in K-combinator, but it is also trivial. But if you consider the B terms, then we have such non-trivial cycles, so this is why I consider in particular these combinators. Okay, thank you. I think we can move with discussing this problem to an appropriate channel. I think there can be many questions concerning this. Yes, please, Nam. Okay. Could I just ask a couple of questions? Sure, I think it happens. Yes, please. One question is if you've thought about this problem from the standpoint of typing of the combinators? Yes. Yeah. For any combinators, or the B terms? For B terms. Yeah, I'm sorry. Yeah, that is a good point, because B terms is actually isomorphic to the type of B, because of every B terms, every equivalent B terms has the same principle types. This is a non-result, because even if we have the C and I, then we have the same combinators should have the same principle types, so we can consider types instead of the terms to compute the law property, but I'm not sure we have, it will help to prove my conjecture using the types information, but I'm not sure, I have tried, but type requires some kind of unification, so it makes it hard to prove, at least for me, but it may help if you have any idea for such. I don't really have an idea, but there's one kind of, I'm not sure if it has a connection, but it just makes me think of this recent work, I don't know if you're familiar with, but of Nguyen and Bach, sorry, Nguyen and Pradec, which was at ICALP, implicit automata and typed on the calculator, and it was about characterizing, so connection between terms that you can define in typed, non-computative lambda calculus, so some kind of planar lambda calculus, and this notion of a periodic monoid, so I guess I wonder if there's some way of using their techniques to prove something about lack of the real property, at least for some class of terms. Okay, thank you for the information, can you put some pointer to the paper? Yeah, so I'll put the pointer here, and I can also put it on the Discord channel. I put it in chat, but I'll put it also on the Discord channel. I would like also to ask a question which is somewhat connected, maybe to know whether this problem which is set in combinatory logic has other interpretations, for instance on graphs or maybe also on lambda calculus to see any kind of other combinatory structure. Yeah, actually in our papers I use different characterization from the lambda term itself for the terms. I can show you. We developed different representation of the B terms like this. Any B terms can be equivalent to these forms where P1, P2, Pk is decreasing or non-increasing. So we can use this representation instead of B terms and then we can consider to consider observing to these forms instead of the lambda term itself. Actually we use this representation for my implementation. But I'm not sure there are many other good representation to find the property but we need to consider more the other form. Okay, thanks. It would be nice if you could add a link to this presentation or to the paper. I think you link the paper with Mirai if I'm correct. But it would be nice to have this as well. And I don't know whether there are any other problems to be presented. Is anybody interested in showing us some open problem of theirs or somebody else? I don't see any more volunteers. Can I? Yeah, sure. So to Japanese people for these open problems. Can I share a screen? One second, let me make you go host. Okay, so actually this is a somewhat supplemental talk of yesterday's my talk and I hope that most of all participants are familiar with what the asymptotic density is. It is a fraction, a limit of this fraction value. And recall that we call a language L now if its density is zero. And we call a language L now if the complement of it is null or equivalently density one. So my conjecture are three kinds. The first conjecture is that I saw the first one I have no much confidence about this, the first one problem, but I have some confidence. That means that for every non-null context free language contain non-primitive words. And if it is true, then the primitive of the conjecture is sort of affirmative. But maybe there is a counter example. But I tried to construct in such context free language, actually context free grammar, but maybe I cannot. So maybe the machine generation of some counter example may help us. And actually to solve the conjecture, actually we can consider more relaxed problem. The case exactly the language is called now, very large context free languages. And the third question is can we give alternative characterization of the class of such very small context free languages and very large context free languages respectively. For the regular case, we have many different both algebraic and automatic characterization of null and the conal languages, which I proved in my master's doctoral thesis. And but we have no hope to have a decidable characterization because of the recent result obtained by Nakamura. The first showed that it is undecidable whether or given CFG generalize null context free language or conal context free language respectively. And here, finally, I want to give us some remark about these conjectures. Yesterday I summarized some known approaches and I said that there is some analytic method for probing that given language is not ambiguous context free language. Because we have Chomsky's Zenbeja's theorem. Okay, we have good theory for the generating function of ambiguous context free language. But I think for general context free language, the generating function of the general context free languages could be very, very complex. So, I think this line of approach is very difficult. I have some one research which represents some complexity of the generating function of general context free grammar, which is given by WIC. Okay, so first we define for a given context free grammar G. So we can define its ambiguity function, how it is ambiguous. So the ambiguity function dc takes the natural number n, and it returns the maximum number of the number of pathways of our same world, which is the word with language smaller than n. Okay. So this is a non decreasing function on natural numbers. By definition. Okay. And the dc is constant and its value is one means exactly the produce our ambiguous context free language. Right. And then we proved that the, this is a somewhat surprising result that for any computable divergent and non decreasing function f. We can construct context free grammar. The foods or ambiguity function is smaller than f, but diverges to infinity, if n tends to infinity. Okay. So, I remember that yesterday, maybe Sergey asked, asked a question about, can we analyze the multiplicity of words, so that we can apply analytic method to do the general conjecture but the this theorem roughly states that the ambiguity, the multiplicity of words of context free grammars can be very complex. Okay. This is surprising result. But I have never read the proof details of this paper so I cannot believe this result but maybe it is true. Okay, so that's all my problem are three kinds. Thank you maybe there is a counter example. Okay, thanks a lot for recalling that this problems. Are there any comments concerning these problems or any additional questions. Okay, if there are any later, of course, we can discuss it on this court. Thank you. Thank you. Any other problems or suggestions what problems can be presented, we still have some time before the lunch break. Okay, I don't see any but anyway, there are some discussions still open on this court so I would invite everybody to go there and see what's what's new. Since the morning there were some new posts so I think it's interesting to go through. There are no more problems to be presented. We'll meet at 12 UTC so in roughly an hour and a half.