 And now we welcome Michael Mordkat to speak about the Tamari Order for D3 and the Ritability and Semi-Associative Lambic Grishen Calculus. Okay, so thanks for suggesting this conference to me. It's always nice to speak for a new community. This talk will be actually on a linguistics problem on the different degrees of Ritual of Freedom as you find them in natural languages. The way we are going to approach the problem is via formal language theory. So basically what we will be studying is two types of languages of higher dimensionality than usually considered and the primary object of our study will be extended to Dijk languages, k-dimensional Dijk languages, which consists of words over a k-letter alphabet and that's what we will call the dimension of the language. And the words are satisfying two constraints. One is a multiplicity question which requires everywhere to contain the k-letters with equal frequency. And the second is the prefix constraints which says that for every prefix of a word the number of first alphabet symbol is at least as much as that of the second alphabet symbol etc etc to the final alphabet symbol. So our familiar language of balanced brackets, usually studied the Dijk language would be the two-dimensional case or the two-letter case. So that's one family of languages and then the closest relative of these families of languages are so-called mixed languages which one could say are just preserving the resource sensitivity of the Dijk language in the sense that they respect the multiplicity constraint but that they drop the prefix constraint. So in these languages there is equal Dijk constraints on these languages. So actually I wrote a paper on these extended higher dimensional languages in the first year for Jim Lombick's ninth year birthday. Here is some numbers related to small cardinalities of the of the letter frequency and the dimensionality. Of course what one does then is you generate these patterns, you have nice sequences of integers, you go to the online integer sequence. It's Clopidia and here you find the you know the culprits for these sequences which as usual give you interesting ideas as to problems. Now the problem has a little history because I gave this as a talk and a paper for Jim's ninth year birthday and I gave it as a challenge to my logical language students. Students couldn't solve the challenge of the exact place of these extended Dijk languages in the Chomsky hierarchy but worse than that I couldn't solve it myself. So since what is it 2014 this problem is with me I promise myself not to think of it anymore but then there is always people who make this thing resurface like Noam's and Noam's suggestion to actually contribute to talk to this work. So here we are again with the extended Dijk language and the kind of patterns that you that you find with these extended Dijk languages give you what is known in linguistics so-called crossing dependencies. So take the pattern, the formal pattern A to the end, B to the end, C to the end. This is typically a pattern that requires expressivity beyond context free. You can apply the pumping number for context free languages to show that this goes beyond the context free boundary and it is a kind of pattern that you find in actually in natural languages at least if Dutch is a natural language. So the example sentence that I give here translated would be something like he hopes that John helps Mary to teach the kids how to ride a bicycle. But complex sentences are grouped together at the end of the phrase. All the compliments of the verb, the subjects, the objects, etc. etc. are preceding a cluster of verbs and the arcs in the little picture here are actually capturing the semantic notion of who does both. So the young, the John here is helping, the Mary is teaching and the kids are doing, are doing a cycle. So this is a crossing dependency pattern. It's one of the patterns of the three-letter and the three-letter Dijk language, for example. And since the 90s more or less, the feeling has been in formal linguistics that what we would need for these patterns beyond context free would not be a jump to context sensitive languages, which are going to show many languages. But there is actually that there's actually an interesting class of so-called mildly context sensitive languages, which was studied initially by Arvin Joshier at the Bangor University. And so he characterized this idea of what this family of mildly context sensitive languages is namely the mildly context sensitive languages will properly include the context free languages. They would allow a bounded degree of crossing dependencies of the type that I illustrate with the example here. And the feeling is that this bound would actually be a low bound. So that's the reason that for the rest of the talk, I will be concentrating on the three-letter extended languages. And the most important thing for computational linguistics is that this jump beyond context free would still allow you to maintain the polynomial. Which of course for practical etiquette is a benefit that you don't want to lose. There's some other properties of this mildly context sensitive, which are not directly relevant for the for the talk. So I just skipped it. Okay, so that's the background of the whole thing. Now the conjecture of let me show that the conjecture actually is conjecture by Makoto Kanazawa in personal conversation is that the k-dimensional dag languages would be recognized by a k-1 multiple context free grammar. Now, all this is the idea of a multiple context free grammar. Multiple context free grammars are generalizing the familiar context free grammars to higher dimension in the following sense. In a context free grammar, the non-terminalist of course as we know range over strings, so the semantics of the context free language it tells us that. Now in a k multiple context free language, the non-terminals are ranging not over strings but over string tuples with a maximum size of k. So if we say we have a two multiple context free grammar, we mean that there is certain rules in that grammar where non-terminals in this particular instance here, this particular example, it's a non-term a where they range over a pair of strings rather than just over strings. Right, so an ordinary context free grammar in this perspective of the multiple context free grammars is just a one-dimensional context free grammar and the conjecture for the for the three-letter dag languages would then be that a two-dimensional context free grammar would actually surprise to recognize it. Just a question, could I ask? So is the multiple context free grammars, are they also recognizable in polynomial time? Yeah, exactly, right. So they remain polynomial, but they become closer and closer and closer to context free as the dimensionality increase. Closer to what? As the dimensionalities, to context sensitive, sorry, but they stay with it as a sub-family of languages that retain the polynomial. So because we think these crossing dependencies only require a very low bound of crossing dependencies, concentrating for example on the three-letter dag languages is probably enough, but it's a conjecture, it hasn't been shown. And actually coming up with an actual two-dimensional context free grammar for the three-letter dag language has turned out to be very illusive there. There is a paper by Konstantinos Kokandidis and Orestes Melconia where they try to actually come up with a context multiple context free grammar for a three-letter dag and where they develop, say, a theory of increasing abstractness of formulating these grammars, but the result is still illusive. You can turn to that paper and you'll also find interesting code going with it if you want to experiment with this path. Can I have a question? Yes, of course. Okay, so do you know any partial result about this conjecture for the case K larger than 4? K larger than 4. No, so the conjecture is completely over. We know that we know that we need the mixed language with three letters, so that's just equal multiplicity of the letters but no ordering constraints. For that, the Kanazawa and Salvatthi have proven that indeed this must be a two-dimensional multiple context free language. But for the more restrictive case of the three-letter dag, this is an open conjecture. Of course we know that for two-letter alphabets, both the dag and the mixed case are context free. Yes, yes, yes. Okay. So this is, yeah, that's kind of tantalizing. My approach is not going to be via the rewriting grammar, so via multiple context free grammars, but it's going to be the logical approach via extended along the grammar section. Okay, so now let's look at some of these bijections that you find when you look at the online literature sequence. So one of the most I think appealing bijections is between k-dimensional dag languages and rectangular standard young tabloids. Now these young tabloids are mostly studied or used as an instrument in studying representation theory of the symmetric and general linear groups, so you could more or more have it to do with linguistics. I think this is a nice application of the tabloid. So remember that a young tabloid who starts from a partition of an integer n, so that would be a multi-set of positive integers which sum up to n. And what we do is we list the k parts of such a partition in the weekly decreasing order. Little warning here, this is the Anglo-Saxon way of doing it. The French, of course, have a different way and they do it in a weekly increasing order. We follow the Anglo-Saxon pattern. So a tablo, a young tabloid, is just an arrangement of k-boxes into a left-aligned row of partition bytes. And we say that the standard young tabloid is obtained if you place the integers one through n. In these boxes in such a way that the rows and the columns are both strictly increasing from left to right, from top to bottom. So here in the example, suppose that we partitioned the integer 9 into 4 parts, 3, 3, 2, 1. So then we have the empty diagram here on the left and we can fill it, for example, in the way indicated here. So 1, 2, 8, first row is strictly increasing, 1, 3, 4, 5. First column is strictly increasing, etc. Okay, let's see the young tablo. Now with a young tablo, you can associate the word, that's going to be the connection with the three letters, tag languages or k-letter tag languages, for that matter. The word that you may be showing is going to, again, be a word over integers, as many integers as there are parts of the partitioning of the integer that you are dealing with. And what the word is telling you is actually, so you have the first row here, and now it says that the first position of our word comes from the first row, the second position from the first row, and the first element of the second row. Okay, do we get it? Now in formal linguistics, rather than using integers to say which of the rows we will use A, B, C, etc., etc., so we will use our letter alpha. Now go back to the to the tag words, so for the the d-dimensional tag words, we will be dealing not with the tabloes in general, but with rectangular tabloes, where the fact that we are rectangular captures the equal letter multiplicity, there is as many elements in each of the range of the rows, and the increasing nature of the columns, the strictly increasing nature of the columns captures the prefix condition of these labels, so that there have to be at least as many a's, etc., b's, etc., c's in every prefix of a word that belongs to the language. Now we go to the numbers, you know, remember our display on the second slide. So for young tabloes, you have formula, same sublime formula, but if we have rectangular diagrams, that is actually much simplified to the formula I'm giving here. So for example, if you want to compute the number of elements in the three-letter tag language with letter multiplicity two, you take factorial of six, and then you take the, you divide by one times two, two times three, three times four, so you get five. Okay, so that's the numerology of the whole thing. Now we are going to try to move closer to the logical perspective on these things. So there was a recent paper actually by Nikolaus, a rectangular young tabloon, and what he calls a single input, single output product, co-product, pro-grass, with and product and hence also and co-product nodes. So I'm a linguist, I don't know whether this notion of a pro-gravity is something which you guys are familiar with, but for me it was just an interesting new perspective on the dyke language. So for the co-product nodes, we have a single input and two outputs. For the product nodes, we have two inputs and a single output. You can click these, these together as an example of the next slide. And then the correspondence with the dyke words is to go via depth left first, traversal of the graph that you obtain, the program that you obtained, with the condition that the output of the incoming edges have already been visible. And then the inputs of the co-products are giving you the entries of the top row of the tabloon. The left inputs of the products will be the entries of middle row, the inputs of the bottom row. We look for an example. So here's a nice picture. The word that we read off this graph, this program, would be A, A, A, B, B, A, C, A, B, C, C. I've indicated I hope that you can see it in the order of the depth first, left first, traversal. So we enter here and we read an A or an element of the first row. We go depth first, left first. So we come here. This is again an input of a co-product node to the second A. Now we go here and we have an incoming edge for the, which is going to be, we cannot exit here to continue the depth first traversal because we haven't found the second incoming edge for that product node. So we backtrack and we continue here. This is another B. This is another A. This is now first C coming in, another B, another C coming in, and the final C coming in. So that's our traversal of a Borey program corresponding to the two three by three rectangular tableau. And the other Yamanuchi word of that tableau to a word of our, of our three letters. Yeah. If you want to look at more pictures, by the way, so I have quite a notebook here, which is, upload a couple of samples if you want to see more. Okay. Now, now the Tamari order, the subject of the talk. So the Tamari order, actually for the two-dimensional case, would be a partial ordering on words that is induced by a semi-associative product operation. So rather than having an associativity isomorphism, there is an order here that says that you can re-bracket if you wish, a left-bracket product structure into a right-bracket product structure, but you cannot do it the other way around. Okay. So that's the basis of this, of this Tamari order for two-letter, two-letter alphabets. And now my claim is that, that for the three-letter alphabets, so for our three-dimensional type languages, we can extend the Tamari order during three semi-associative operations. One will be the product semi-associativity. Remember that these programs are built in in terms of product and co-product nodes that are clicked together. So there's the product semi-associativity on the one hand, and then by duality, there is going to be a co-product semi-associativity. It's one way, it's not going, the arrow goes one way, it's not a dual arrow here. And finally, there is going to be a mixed semi-associativity for the cases where the product and the co-product come in construction with each other. So if we have our co-product, which, sorry, our product which immediately leads into a co-product node, the semi-associativity here says that we can, if you wish, push up the product node over the co-product node. And maybe some of you will say, huh, this looks very much like directed versions, so one-way versions of what would be the equations of the Frobenius algebra, which would by definition have full associativity, both on the product and on the co-product, and which would have not just this semi-associativity for the mixed case, but would have the full mixed associativity. So let's see how this works out. So here on this slide, I repeat my earlier example of a Boree program. And you see that there is two opportunities, actually, of applying these rewritings. Co-product re-bragging, so the right branching structure here is turned into left branching structure for the co-product. But at the same time, you could have pattern matched on the product semi-associativity in blue here, which would allow you to turn the left branching, product structure into a right branching product structure. So the program on the left matches both the co-product semi-associativity and product semi-associativity. Okay, good. So now let's look at the simplest case of the order. So I take here letter multiplicity 2 for our three letter diagonals. And the idea would be that the bottom of the minimum element of the lattice, the tamari lattice that results, would be the regular words, a, b, c to the power n. So here a, b, c to the power 2. And the top of the of the lattice would be a to the n, b to the n, c to the, the example that I discussed right at the beginning. So how does this lattice arise with our three semi-associativities? If you look at the minimum element here, a, b, c to the power 2, it matches the mixed semi-associativity. We have a product node coming in construction with co-product node. And by applying that rewriting rule, we get, we get the words a, b, a, c, b, c. Now this word, I says, oops, is something happening for you? It's just for me. Is everything fine for you or not? It's, it's shrunk. It shrunk. Oh my god. Okay, I'm, I'm, I'm shrinking. This is Alice in Wonderland, I'm afraid. Are we back? Yeah, now it looks good. Yes, back to normal. Okay. So now we have a a, b, a, c, b, c, this point here. And we can just as in the previous example, actually, we can either apply the, the product associativity, semi-associativity, or the co-product semi-associativity, leading to a, b, a, b, c, c, or a, a, b, c, b, c. And then if we started with the product associativity, we can follow with the co-product associativity. And vice versa for starting with the co-product semi-associativity followed by the co-product semi-associativity. So to understand a little bit better what the combinatorics here is, it's good to think about the duality in these programs and on the words, Schützenberg's, Schützenberg's inclusion, which on the graphs is just taking the 100 degree rotation. And on the words, it's reading the word backwards with the, with the inverse output. So then what you see is that on the horizontal line here, the words that we have are self-dual. So let's try for this guy here on the word left, for example, we read the word backwards, and we replace the c by an a, the b stays what it was, and the a is replaced by a c. So read backwards, you get the same word back. So on the horizontal line, we have self-dual elements. And here, vertically, these are duals, but not the same. Another thing to remark here is that, of course, the minimal element of our lattice is regular, so it's just in general, it's abc to the power n. The maximum element is beyond context three, it's, what is it, two-dimensional multiple context three, namely a to the power n, b to the power n, c to the power n. And then the other elements in the middle here, sorry, these guys here are context three, right? So this is ab to the power n followed by c to the power n, or a to the power n followed by bc to the power. Okay, so that's how this lattice now originates, not from one semi-associativity, but for the natural three semi-associativities that you get when you start from these mixed product, co-product graphs. Here is an even nicer picture. The lattice for multiplicity, electric multiplicity three, which has 42 elements. Again, we start from abc to the power three, and we go to a, n, b, n, c, n. For n is three. The vertical axis here is the cell of dual elements. And if you mirror along the, along the vertical axis, you have the dual elements at the two sides of the vertical, the vertical divider. So there's really nicer combinatorics in this one. The next thing is that I just ask a question. So, so do the, do the, does the ordering have a, have a simple description on the words themselves rather than on the photographs? It has, but this is pretty elusive actually. So there is this book on combinatorics by, by Willow and, and even has this famous book from the 70 percent combinatoric algorithms. And actually, they have a way of enumerating the words of these languages in such a way that, well, but it's not, no, it's not really going to give you what you're asking for because of course there's not just going to be a linear order, which is what you get from the, from the Willow algorithm. But here there's a point of, there's points of re-entrancy that it's like, as we saw in the simple example, ways of getting to this point actually from two starting positions. So I wouldn't, I wouldn't be able to answer that question whether there is, well, it must be possible to directly formulate it on the word. Okay, but so was this, was this generated by, how did you generate this? Okay, so to generate these lattices is by simple breadth first unfolding of the, of the space on the search page. So you start with, with a minimum element abc to the power n, then you apply any of the possible ways of generating children from that, from that node using the three semi-associated. And associating, taking a word associating toward the corresponding program, then applying any possible rule and then associating back. So it's just centered breadth first unfolding of the, of the space, which for each point, for each node asks a question, what are the children of that node? And then you, so you generate, you have generations, generations, right, zero generation, first generation, second generation, second generation, going up to the, yeah. Okay. Yeah, so that's how you generate that. Then the tamari order for the, for the two-letter case is actually hidden. It's a sub-graph of our, of our three-letter case, which you, which you obtain by just restricting the words that have all the C's at the end, for example, or by duality that would have all the A's at the beginning. So here is the 14 element lattice of multiplicity for, for the, say for the well-balanced bracket language, if you just forget about the C's that are bulging at the end, right. So it's hidden in this, in this larger graph. Then the next, the next thing that I've learned all these things from, from Dom's papers on tamari intervals and, and Lombard grammars actually. So the idea of the tamari interval is to, to take the sets that, well, the sets, the sets of formulas every year, such that there is a C between these lower and higher points. There is a formula by Chapoton, which counts them for the two, for the two-letter case. And there is also a nice number in online as a group PDI here telling you what kind of animals that you find obeying that formula. Now the embarrassing thing here is that when we start counting the intervals for the three-letter case, I have a nice sequence here, my computer after the night of working stopped at this number here, but the sequence is not in the online as a group PDI. So this is an opportunity to either upload it or to think, you know, but be something really wrong about it. Okay, so, but it could be, of course, that, that because the, the Chapoton formula is for, for this rule to try and reconnect the, this pair that by looking a bit closer, it would be possible to also for the three-letter case to find the generating function for the, for the sequence, I don't know, and again open. So now my time is almost up, or maybe it is up, I don't know, which in a sense is good, because here I come to my, my own open, open problem session. So the thing that I found very intriguing in Noam's paper is that he was able to actually generate the two-letter stuff by order or to characterize it in terms of the rivalry between formulas A and B in a version of Lampic's 58 syntactic calculus that rather than having full associativity, which is what this system has, has a restricted form of associative. So Noam actually starts from the associative Lampic calculus and restricts it by having a version of the, the product left rule that only works on the first formula of sequence. Right? So the challenge that I gave to my students and that I couldn't solve myself is actually if you want to generalize this approach, so characterizing these orders in terms of logical derivability, we cannot do it with Lampic's calculus because it's of course well known that Lampic's system, both the associative and the non-associative one, that they are strictly context three in their recognizing power. So we have to move beyond in simple Lampic systems. And it so happens that what I've been working on the last 10 years is actually an extension of these Lampic systems which was proposed by Grishin in the 80s, 83, so called Lampic Grishin calculus, which you can, as Lampic has it with his own calculus, extend with either associativities or second year associatives. So the reference here if I want to know more about these systems would be a paper that I wrote with Richard Mote, where we give a display of sequence calculus for these Lampic Grishin systems, where we prove focusing, focusing property for that sequence calculus. We also have a proof net graphical calculus and a connection with the fragments of linear Lampic calculus going with it. So that's the wider picture of this whole thing. But just to give you an idea of what this Lampic Grishin calculus is all about, so Lampic actually the non-associative, so the most basic Lampic calculus starts from pure residual logic for a multiplication of product and its two residuals, the right slash and the left slash, expressing incompletely. And what Grishin did is he actually complemented Lampic's language with a co-product together with left and right difference residuals, difference of points. So the relation between the original Lampic product and its implications, it's the slashes. And the co-product and the difference operations is actually arrow reverse. So there is an arrow reversal duality that relates all the Lampic theorems to, if you wish, the co-Lampic theorems, which obtain for some and difference operations. So that's the basis, a non-associative basis with a product family and a co-product. And then one can extend that pure situation and co-residuation basis with structural postulates. So for the problem at hand, this would be semi-associativities of the same type, so either a product semi-associativity or co-product semi-associativity related by duality as we have it in the four re-graphs. And then the interesting case here is that there is also mixed semi-associativities which in the linear logic literature are known as linear distributivities. And they come in in Gleisch's terminology in different flavors. They are either the class one or the class four postulates. The class one postulates are probably what's most familiar to you people would be that you have a co-product which is dominated by a product. So which is under a product operation. And then the class one semi-associativity allows the co-product to actually win it over the product. It moves up. But in class four, in the class four case, it's just things are just going the opposite way. And I haven't been able to actually figure out which of the semi-associativities which would be applicable to what we want to do with the tamari order. So currently what I'm trying to do is first of all, of course, find an encoding of a three-letter daik word as a Lambert-Gerischen formula, which is what the tilebacker paper does for the two-letter. Then capture the inequality of the tamari order in terms of Lambert-Gerischen derivability. And the intuition here would be that if we have same type of associativities, sorry, if we have pure either product or co-product sub-formulas, they are subject to the same type of associativity. If we have mixed formulas, we would restrict to mixed formulas with equal numbers of products and co-product operations, as in the body graphs actually. And the linear distributivities would allow you to actually disentangle such a mixed formula so that all the product operations and all the co-product operations come together where the same type of associativities would be applicable. As we added more or less in our initial example here of the simplest lattice for the multiplicity 2k. So that's where I got and that's where the open problem session starts and where I hope to learn more from you these days and listening to the other talk. Okay, thank you. Thank you very much. We're actually running a bit late on time because then the next talk starts in five minutes, but we have time for a question or two. I mean, so of course I'm very interested in the problem that you posed at the end and the relation to Lambeck-Grishan calculus, but it's not clear how to go about it. I mean, so the relation between the co-product graphs to formulas is not clear to me. Exactly, it's not clear to me. The ingredients, so to speak, that we have in Lambeck-Grishan calculus are the same. So there will be a product operation and a co-product operation. And either of them can be subject to not associativity as you have it in the full system, but semi-associativity is fine. And now there's also formulas that mix actually the product and the co-product operations. And for this we have the linear distributivities to actually disentangled. As soon as they are disentangled, then the usual semi-associative distributivities for the same type of formulas are going to be applicable. For example, what you did for the two-letter tamaring order in terms of Lambeck-derivability for semi-associative Lambeck calculus is already here because that would be formulas that are entirely built up out of product operations. There's no co-product operations around. So that means that the only operation that is applicable there is actually the same type semi-associativity for the problem. So it's already there. Two-letter case is already there. And then the intuition would be that if it's a mixed formula with products and co-products, that that would actually be a way of reading such a formula as an encoding of a word. Right. As you have it in the Boree graphs that you say, well, you look at this object that is at these nice graphs, but then you actually read off the word that establishes the isomorphism with rectangular data blocks by means of this depth first, left first, the traverse of construction. So you could imagine that there is a similar way of actually going through a formula built out of these product and co-product operations that will spell out the word actually. Okay. Let me just point out in the chat, we have a comment from Orestes Milconian in case anyone is feeling adventurous and wants to explore Dicla languages. Feel free to use our tools to help you. Yeah. Okay. So, okay, I think we should we should stop there because the next talk is starting in just a couple of minutes. Okay. Thank you. Thank you very much again. Okay.