 So I Did to thank the organizers for for inviting me and Start with the talk so so this is a joint work with me Kauai and and I wanted to talk about a single use restriction Which is and like something like an approach that we believe should be used When doing transducers that work on infinite alphabets, so that the known Models all but some of them some of this will be still like is still working progress I will tell you exactly what what results have their proofs and they are written down and what are still like in progress so the The classical model defined by coming skin Francis for for those infinite alphabets is just that we have a set That's countably infinite and the elements can only be compared for equality. So this means that This one over here is just as good as this two over there, but they are just not equal So so we cannot say that a letter is even For example, but we can say that first and second letter are equal or the first and the last last letters are are equal So this is a model by coming skin Francis from 94 and here is like a very Typical language over this alphabet so the first letter appears again, and this is an equality property So we're fine with that and automaton like this has a state which comes from a finite set of states and a set of registers So in this particular case of this language the automaton will have only one register and at the beginning it's empty but then we go through the world and we can just store that The letters in that in that register. So now when we want to recognize this language, we just go through the The word and just look for the letter. So for example here We see that one is not equal to two so we don't do anything just go forward But at some point we meet a letter that's equal to the letter that we have in our Register so we go to a special state that says found we've already seen the first letter And then we just continue with the state to the end of the world and then at the end If we are in the state after the last letter then we know that we have found that Everything is good So now the modification that that we propose is to consider automata like this But every read access destroys the content of this red register. So if we try to repeat this construction We just save the letter in the in the register, but now at this point We want to compare whether this free is equal to this one and when we do this we lose the free because we've Compared it. So so now we have no idea what the first letter was and Therefore we'll have no no idea at the end whether these words should be accepted or or not and Like it can be proven that there is no single use register automata and that will recognize this Language the first letter appears again So so it's a weaker model But it's it's not that bad because we can for example do a language There are at most three distinct letters and and the first idea for this language also seems to fail because we can store the first letter in the first Register and then like once we know it's different then we lose it and we have no idea of knowing What was there in the in the first place? So we do a slightly more involves construction that will still be working with the single use Restriction effort and for this we need six registers And now like the the convention is that everything in one column will store the same value just Multiple copies because now it's important to count the value. So we might need to store multiple copies of some of some values and So at this point We just check if this one is equal to this two and it's not so we lose one copy of this one But then we move the ones to those like two registers and we can fill those free registers with the value two and Then we proceed similarly with free. We just like check this one. This one. They are both Unequal so we move them forward and then we fill this row with the free because like we have access To this free so the single use restriction doesn't Apply to the input letters the input letters. We can take in as many copies as we as we want So now we See a one We compare it with the first three then with with the two and then with the one so now We moved those two registers here and since we know that this one was this one We can fill this one here So the invariant is that we have one value in three copies one value in two copies and one value in in one copy And we can continue like this Till the area and the variant of this word and now here when we lose the last two the last Copy of the two. We already know that the word is not in the language So we don't have to remember anything more We just remember that that we're in a failing state and we propagate the state to the variant of of the word okay, so this this class of languages recognized by by the By this model already has two different Characterizations and it's orbit finite semi groups which was introduced by Miko in 2013 and Ridley guarded MSO with that of our comparison, which was produced by Colcon Bay Gabriel and And play yeah, yeah, and and yes, so so Then we can also prove that both one way and two way Single-use register automata are equivalent to this like we can give it translation and it's actually in a paper So so this is not work in progress. We can show that The standard construction that would translate a two-way automaton to orbit finite semi groups It's basically the good way of translating and then we can construct for every orbit finite semi group a one-way automaton That will that will recognize the same the same language and Yes, so so this is something we would like to point out that that that in this world of Languages with infinite alphabets this this class of languages seems to be very robust because it admits so Remarkable like it's not very common for a class of languages to be to be robust and for example like if we if we take away the single-use restriction or the Rigidly guarded restriction for the MSO with tilde then all those classes of languages will recognize a different So will be different classes of languages and now I would like to talk about like how to apply this to make transducers with atoms Which is the work in progress part of this of this talk? so for example like a typical equality only Transduction would be to remove the repetitions from the input so We just want to get rid of this tool those freeze because they They are repeating them themselves and just to have the same word, but with no repetitions And here I will also show an example of a single-use register transducer that will do that It's a one-way transducer, so it starts it will have two registers and It just stores the previous value in both of them Registers and then When we are in this state would we just compare whether this letter is equal to this letter And if it is then we don't do nothing, but if it's not Then we still have one more copy to output it because we will also lose the copy when we output the the copy So like this we go through the world Outputting only when there is a change We also have to remember to output the the last letter after the last After the end of the world and this is basically that the transaction that we wanted to to achieve so a nice property of this type of transducer is that it Admits a chroner outs the composition And I would like to talk about this now. So basically that there is five prime five functions and every Function that's recognized by a single use register automaton one way Can be expressed as the composition of those of those functions and I would like to go through every table function so Basically, we can have a homomorphism between a letter to a word So every function from a letter to a word will extend to a To a function from word to word and this homomorphism has to be a quivarian So in a sense looking only on the equality also so so it cannot be for example a if the atom like if the atom odd or even just has to only look at the equality and example of The function that is a homomorphism is just doubling the letters So we just translate one atom like one to one one two two two and then this is how we can double all the letters So another prime function is the delay function and It can be used for example to do this example that we did Before so removing all the repeating atoms. So so what it does is just moves the word forward and Like makes the end of word and like the beginning of word and end of word word marks At the beginning and end and then with the homomorphism We can just filter out all the equal pairs and then we just keep the top row And that's the function So now finite group on on prefixes This is basically we have any kind of Alphabet that will just stay unchanged but on the other Coordinate we have a group and we want to for every prefix. We want to just Change this so the value to the value of the group product on a on a prefix So for example, we might use it to use to do this function remove letters from odd Positions and we'll do this like this That we just add the one which is a member of that two group in this particular Group and then we just apply this function So it would change us change like one zero one zero one zero and now we can just filter out all the zeros And we get our Our input and then the last thing In the classical chrono steep theorem is the flip flop semi group or mono it on Prefixes and this this is like sending one bit of information to the future So so basically we have a device and we can either set it to one set it to zero or leave it as it as it is So it has three three elements and One will mean like just leave it because it's a monoid one a means set it to one b means set it to zero Slightly confusing because of the one that That doesn't change and the a and b so so that's why and here Here that that's written all the operation. So every time like that. It's basically the last The last letter unless of course it's it's one so So we only care about the last operation that was changing the the thing and we can for example use this function To compute remove everything after the first repetition. So we start by the delay function And now We just map every non repeating part. So non equal part one And then equal equal part two to a and then once we apply the flip flop Monoid we get like after the first repetition We we only get get a so like in here. We didn't even have to use the the b But maybe we wanted to do something else like in the future. This might be possible and now you can just filter out all the a values And get the output And then that's it. So like for the original crown rods t theorem. It says that A function f is recognized by a one-way transducer The classical model will find it Alphabet if and only if it's composition of those prime functions So for example remove repetitions is just this homomorphism that will filter out the the equal pairs and so it's An example like this And then when we want to have the infinite alphabet case, we need one more Prime function and this is the letter propagation so now above like Basically like operations on one register so we can have You can write it down So okay, we have And we sometimes will want to send a letter To the future and and for this we might want to store it at some place and then output it in some other place and we just Want so this will move this too In here and this is also subject to a single use Restriction so when we have seven eight and for example here We also do output and it won't have any effect because this two is Already used and we haven't stored anything here. So this one won't have any effect and This is useful to for example do this function which changes the last letter to the first letter So again, we start with the delay monoid and this is only because to get this end of word mark and we just Add a letter like a save Command here and a read come Command here you you you you think you're finitely supported So equivalent homomorphism And then we just propagate this letter to the end of the word and apply another homomorphism to get To get what we wanted so if we add This one letter propagation to to chronological theorem we get exactly so The class of single use one-way transducers And then we might want to add two more prime functions, which I think Were mentioned by miko wine Earlier to today and they also Make sense in the classical so finite alphabet word And one is just iterated reverse which Reverse every block of numbers And the other one is iterated duplicate, which will just duplicate every block of Numbers and then if we add those two Then we get a theorem which says that When when we look at the two-way automata and the transducers would work basically the same as one way But can also go back then we get exactly the the class of function That's done by the two-way Prime functions so those are Those functions and those functions plus the letter propagation function, which is specific to the to the infinite alphabet's word And now we have two coloraries, which Are not immediate from so maybe are immediate but So that one-way single use Registered automata are closed under compositions And this can be also Proven in another way that's kind of simpler But this two-way single use register automata are closed under compositions for Follows immediately from this crown roads the composition and and We think it's it's the easiest way to to do it And then we have one more colorary, which is non immediate that like Most of those will require some of Some more work, but only like a little more work. So it's it's not that hard Once you have the crown roads decomposition for two-way automata And it basically says that all of the following recognize the Same class of transductions so this is The first model that recognizes all the All the then so those are the models mentioned here Okay So this one was By shepherd's son, I think and I don't remember which year But I think it will appear on the on the later slide Then we have this singles like string streaming single use register Transducer and this is this is nice because we have like the previous copy less restriction that was introduced to make the string streaming transducers Equivalent with the two-way transducers and it somehow seems to To play very well with this single use restriction that That just ensures that That the automata recognized the same class of languages as the as the monoids And then the regular fleece least functions introduced by by miko I Krishna and and Loreta So oh, yes, this is shepherd's son model Yes, yes, yes, so so sorry. So those so Yeah, I mean like so What I meant by original model is that the atom free version of the of the model and here I just like A single single Use extensions of of it. So like register. So maybe not single use but Atom register of of those functions. So that will extend those to infinite alphabets And then I would like to finish with the general picture of the situation so we have one class of languages Of sorry of transactions, which are all those transactions from the Previous slide we have this one-way transducer which seem to coin which coincide with With the prime functions and letter propagation This is sequential functions. This is This is the equivalent of rational functions. And this is the equivalent of regular functions. And this is What also might be very interesting to To look is how does like weather Read it msso transactions with with this tilde to compare data data values and lake This is where we're not sure yet, but this looks promising So thank you very much for