 So I'm going to talk about induction today. And I want this to be quite an informal talk. I might give some proof sketches and some brief instructions. But mainly, I wonder, I'll give you guys an overview of several different subjects that we can talk about, and how some pieces of math can actually tie this word together better than you had imagined just from hearing the philosophers talk about something very different called the induction. And so the general theme today is going to be induction. And the thing that will connect all these different notions together, I think, is something like getting global facts from local factors. And so what are facts? I don't know if you're familiar with the latest meme of alternative facts, alternative truth. Well, there will be nothing up back here today. But we'll be talking about several different kinds of things we can do. And it's also not really clear what we need by global and local here. Like, give all that analysis so you know the kind of, like, human and local definitions in terms of episodes and developments. I'm afraid to disappoint you. There will not be a single exulant of that today. But there will still be some kind of notion of locality. If global facts, it's like something is true for all, for all statements, by one statement. And so at the very basics, we have, like, these proofs. Perhaps you've created students who have written these proofs, and they go something like this. We conclude that some statement holds for all n. For all n. And so here we have two kinds of local facts. We have the local facts when n equals 0. And we have the local facts of how to get from that something works for n to the fact that it works for n plus 1. You've perhaps, like you've almost definitely also seen this strong induction, where instead of just using, like, this local fact n gives us n plus 1, we use that it works for all values smaller than n to show that something works for n. This can be generalized to induction on the ordinals. And so induction on the ordinals looks something like this. Here we have some numbering. We have some alpha is an ordinal. For those of you who have to take a set theory, an ordinal is basically one of these guys, or something that comes after a set of these guys. So we have the same kind of operations. We start with 0. We can take plus 1 to any ordinal. And whenever we have any set of them, then we get an ordinal that's, like, bigger than this whole set. So it's kind of like least upper bound property of the The reals don't actually have the least upper bound property because the set of all the leaves doesn't have the least upper bound. It has no bound. But the ordinals, you always get to add a new one in. And so because of this, there is no set of all ordinals. So it's written with a capital letter into the note. It's a proper class. If we ever had a set of them, we'd get to one after the rest. Like the infinity of the primes. We have not the infinity of ordinals. We already have infinitely many natural numbers. We get proper comms, I think, of the ordinals. And so when you prove things on ordinals, well, you do this in some theory, model theory, when you have some kind of process that continues past infinity, and then past the next infinity, but the next one, and next one, and next one. And so you, like, pass all of these infinities. And so you need some kind of way of, like, making this into something manageable. And so the proof is very similar. And so you say for alpha equals zero, it's the same thing. Suppose it works for theta, and you have this case that's different from the natural numbers. And you'd say, suppose we usually use lambda to not limit. Now it just means that it's not the successor of anything. And so in particular, this means that lambda is equal to really the union lambda if the ordinal is identified with the set of elements that are smaller than it. So for any ordinal alpha, you say that alpha as a set is equal to beta. And so you can prove something is true for all ordinals, even though there isn't really a set of them. So far it's pretty standard, at least if you're doing the logic. What comes next is something that sometimes calls structure of the function. It's an ordinary note here that the calorie property we use here, it's not really, it's a finite step that we're taking. Because the limiting process could be infinite. It could be of any internet size. But it's a set size that we're taking. We're only taking set-sized steps. And we're getting something that's true for something that's too large to be a set. So it's still some kind of notionable palette that's given us something more global. Structural induction is like in terms of sizes of things, it's not much worse than integers. It's about finite objects, usually. But instead of just taking them after each other, we can take them in different combinations. So I think someone once said, is that your logician, if you treat variables and formulae as if they were mathematical objects that you do proofs about. And so I'm a logician, so I will have to apologize to you that I'm about to treat formulae and variables and things that will reason and do proofs about it. And so we have, let's say, you want to prove you can do a simple thing. You can say, but fine, basic. That's just a formula that contains some, that's just some propositional variable or it's perhaps some relation between objects or perhaps it's some equality between terms or some other property between terms. And you say, let, fine, be, sign. And then you check that. Then you say, and you do the same thing with, or, and, for all, x, simply. Some people talk about this as being an induction on the length of the formula. And you can think about it that way, but that's not really what's going on. You're not actually doing induction on length. But you can think about it as strong induction on length. But what's actually going on is that you have this notion of predecessor, where fine has, as its predecessor, here it's just a sign. It's the negation of a sign. But if it's the conjunct of sign and chi, then it has both sign and chi as its predecessors. And so you're using both of those as your base cases. There are similar things for trees, but really, formulae and trees are the same things. You guys see these things more often. So you can use the different immediate sub-trees of a tree to prove a thing about the tree. But for trees, it seems more natural to use a strong induction, which you don't get as an induction monster. This can be generalized in general to what's called inductive times or some kind of things defined by recursion, basically. You can think about type theory, get various forms of induction there. I don't want to go into that, but it's all basically variations of this thing where you combine one or more things to point to the next successive thing. And you have to check them all the ways you combine them. Instead, let's talk about the mathematical thing that all things so far are in common, and it's that it's based on some kind of well-founded relation. So basically, what it means for a relation to be well-founded is that you can use it to do the induction. Here, the relation we have is less than. We're using the fact that any set of these or any set of those has the smallest element. Here, if the relation is really sub-formula, you can also think about it being the predecessor over there, but it doesn't really matter which one you use. It's the difference between the weak and the strong induction. And so we can write this down formally. So it means that if for all x and our domain of discourse, the domain in their relation r is defined. We have that for if for all y. We have some fact that is true of y. If this thing implies should it be those ys so that y are x? Oh, yes. Thank you. Otherwise, there would be no r in here. And so we can't really say that this is r in anyway. Thank you. So if for any x, we can use the fact that whatever property we're trying to investigate holds on all its predecessors, if that collection of facts, this for all statement, implies that we get the statement at x, then we can just straight up conclude that for all things we have x. And this is true even for the base cases. Actually, base cases are a lie because base cases are the things that have no predecessors. And so then this statement will be quantifying over an empty set. And so it will be actually so true. And so you have to still check the base cases. But like, base cases are actually not special. They're just like the empty set. They don't have any predecessors. So it's helpful to think about them separately. But structurally, they aren't really different. And is there no unlock to the sort of infinite predecessors of the limit-ordinal? Like, you can talk about having something on a limit sequence. But the thing is a limit sequence is like, there are many choices of them. And things can be different depending on which limit sequence you pick because the limit sequence has an order in it. So for example, an ordinal like omega squared, you can pick the limit sequence, which is just all the ordinals in all of it. But that will have an order type of omega squared. But you can also pick the limit sequence of omega to the m. And that will also be a limit sequence for the same thing. But it will have an order type of omega. So you can talk about it being defined that way using some object that encapsulates all of the ordinals that are smaller. But it's kind of like not really convenient to do so. So I want to step away from all of these things that they usually call mathematical reductions. And quite familiar in this context, in a department like this, and talk about. So he was one of these philosophers in the 18th century. So this was in the 700s. He kind of wanted to know how we could know things at all. And I believe the example with the sun is due to him. You've seen the sun rise yesterday and the day before that and the day before that. But how are you actually really sure that it will rise tomorrow? How do you know that the things you've taken for granted, various natural laws, different laws you think are valid, will be remained valid in the future? Usually we kind of just want to be, like, pragmatic and say, like, come on, stop it. Like, we know the sun will rise. But it's kind of troubling that we can't actually give any good answer for why we know this. I don't really have anything to write down immediately. It's like this puzzled people for a long while. But then people started, like, actually giving science some decent underpinnings. At first, the thing I don't really want to talk about, there were these logical positivists that kind of thought we could, like, make some observations of the world and, like, then do inference on them and, like, this way get, like, really good knowledge. And then Popper came up philosophically. And Popper said things like, yeah, perhaps we can't actually, like, know things. Like, the sun will rise tomorrow. But we can at least refute them. We can at least make observations that are inconsistent with some theories of the world. So, for example, if we have some theory that says that there would be a graduate student seminar every Tuesday, and last week there was no graduate students because we couldn't get any speaker, then, like, we have refuted that theory. Like, it's no longer something we can believe in because we all remember last Tuesday we didn't get any there. But, like, this is still, like, really unsatisfying. Well, 20th century, what, you know, lots of exciting things happen. Some people actually invented probability theory. And this is cool stuff. Specifically, I want to talk about the Asian probability theory. And, like, this is kind of like a release of the equation to write in a mathematics department because, like, it's just so simple. It's the, but you have some theory. And you look at some things in the world, like, evidence. This is just the probability of having both the theory by the probability. This is from the fact of, like, just how we define conditional probabilities to just make sense. This kind of, like, allows us to actually get a start of, like, tackling how sure are we actually that the sun will rise tomorrow? Perhaps there is some kind of, like, gamma ray burst coming in that we can't see because it travels at the speed of light that, like, completely knocks out this planet and the sun or, I don't know, something. That means that the sun will not rise tomorrow. And it would be very sad because, like, if the sun doesn't rise, it's likely because we'll die. And, like, we don't want that. We want more graduate students and ours with cases like this. And so, but, like, there is some probability. And so we have to account for that. And we have seen lots of evidence that the sun will rise. And so we have to find some way of accounting for that. And so here comes the problem, like, yes, sure. You can write down this silly formula and you can plug some numbers into it, but, like, where are you actually getting your numbers from? This is the thing that's called the problem of priors. Like, where do you have this notion of, like, how probably do we think that the theory is as a thing from the start? How probably do we think that, like, evidence is from the start? Like, this equation might be easier to think about if we write it not this way, but as the probability of seeing the evidence, even the theory, as a probability. And the probability of evidence is, like, and so, okay, this is perhaps slightly easier to get some numbers here, but, like, what else could be a possible theory? We should, like, allow for anything, really. Like, quantum mechanics was a really weird theory before people actually discovered it. Like, it's not something you would have let on. And, like, how do we give the theories different probabilities? Perhaps these numbers are things we can calculate because, like, yeah, the theories have some equations, we can do some kind of stochastic simulation. We'll get some numbers out of that. But the other ones, this one, this one, yeah, it's hard to say that you can get numbers for anything interesting. And so, people were arguing about this, people had this notion of, yeah, well, reasonable numbers are numbers that you would bet on. So, for example, like, if you have a bunch of money in the bank, if you think that the sun will not rise tomorrow because the whole world will end, then it's probably a good idea to go out drinking tonight and spend that money. So, like, wait, wait, we can have some fun. Or, like, spend it to call your family or something. And, like, the fact that people don't do that is a kind of a sign of them assigning enough probability to the sun rising tomorrow to feel like this thing will not be to be worth it. You can spend your money better tomorrow or next week or next month or something. And, but, like, this is still a little satisfying because people are really weird about how they spend their money. And so, what comes to rescue is this thing called Solomon of induction. It goes by many names, it's also called, and so, this is from the 1970s. What this is about is you take one of the widest classes of theories conceivable, which is the theories of environments that are computably described. So things that can be described by some kind of Turing machine, some kind of program that you write up and spins out the environment. Before I continue about that, I just wanna touch base again about this notion of global and global facts to connect with this. Here we had this local forming through these local connectives, complicated things, giving us global statements about all formulae. Here we have really, like, local observations. You've seen the sun yesterday, you've seen the sun last week. We know about this particular graduate student seminar to make statements about what we think happened for, like, theory. What do we think holds for all time? And so, if it's something you want to, like, calculate with, it's kind of something you should be able to get some numbers out. And you get some numbers out by doing some kind of computation, some kind of calculation. And we have a universal language for describing that, like, the language of Turing machines. And so, what this really is about is that you don't really get a probability, but you get a semi-measure. And so, a semi-measure is, and I usually don't define you, like, well, it's no more than one, but perhaps it doesn't quite add up to one. And we'll see why in a moment. So, as X, we take really all bit strings because we deal with something computable. We can encode it as a sequence of zeroes and ones. Like, we encode all pictures as zeroes and ones, all, like, letters, everything, or different particles are moving as just one, zeroes and ones, position, momentum, all of it. Well, we get that the universal semi-measure of some set A, yeah, it's a sum, even though we're doing something over this uncountable set, and, like, it's gonna be okay because we're only gonna deal with the computable we get in here, and there are only 10 to the many of those. So, it's sum over all, which is from machine. And so, if M holds, we write two to the minus. And so, you might see that these things might not quite, might add up to something really large. Here, you can have, really, two to the N, here in my case, of, like, N, like, every subnation of zeroes and ones. So, we're basically adding, at worst, one for every possible length. So, this sum should be in fact. Well, it's not if we do a particular convention. And the convention we use here, that is important is we have a prefix for the, what this means is that if M2, in one, being an initial segment of M2, nor M2 being an initial segment of M1. So, that's how computing is anchored, just like, when you feed your computer a string, once, you can't have machine accepting two strings, one is the initial segment of the other. Exactly. So, when you hit return, it really returns. Yeah, so, a way this is kind of, you can accomplish this on a computer, is that there is a, like, you have every eight bits being ASCII character. There's a special ASCII character called M2 file. And so, when you give that to them, the C compiler knows that the file has actually entered. But we don't have to care about the details of how this is actually implemented, but the thing is we can do this, and then I think it's carbs inequality that says that this sum is one of the one. Like, you can also basically see it while I'm drawing on the tree, since we're not double counting anything, it's really not possible. So, since we're getting this thing, it means that we can't get that thing, like, you can imagine each branching gives half the measure to each side, and, like, you can stop branching and then, like, you keep your measure and give it to the Turing machine, or you can, like, continue and, like, give the measure to each other. But the problem is also that this doesn't actually add up to one, because there are lots of Turing machines that don't try it. So, they don't talk about anything. They just continue going. Why don't you just normalize it? Just some of the whole Turing machine and stuff? You can do that, but that makes this thing very incompatible, because you can't tell the probability that that Turing machine will hold this. So, what people do is they actually use this signature, and it's kind of neat, because it's still nothing, not something you can compute, but it's something you can approximate from the low level. So, it's lower semi-computable. Like, you can go through all Turing machines, and when they output, and you see that it's in the center, then you can add that measure you get from that Turing machine to your full of measure. And, like, eventually you will add everything you have to adding, but you can never tell when you're finished. In fact, in most encodings, you will just, like, keep adding things. We can also do this with, like, minimum message, like, that would be, like, yeah, we say, we only take the smaller one, smallest Turing machine, short institute description that produces this output, and take two to minus that. And, like, yeah, I think this just looks major as I did measure. But you get basically the same thing. So what's going on here is that you have smaller values of, like, the things that have shorter description, things that are less complicated, that don't need as much computational power to specify it. That means that you think simpler theories of the world become more probable. This is exactly what people kind of, like, usually take as the, like, constant thing, like, what's more likely, like, you might have heard about Occam's Racer that's many centuries ago. And this basically gives a formal notion of what it means to be simple. So what it means to be more likely as a description of reality. And of simplicity is that we can do, what Shane Leggan, our researcher, did, and give a definition of intelligence. This is the... Instead of just giving outputs from the environment just the way the world behaves, you can have a Turing machine having several tapes where you're getting both the output from the environment but you can also interact with the environment and give it some inputs. And there is a choice to make a particular, a choice of a certain part of the output you get from the Turing machine as your reward function, like, how much, how good this outcome is. And so what you have then is that, like, you're having input. And you say that the intelligence of an algorithm, A. So just the sum over the different machines that could be its environment to the virus. And then you have the total reward of the algorithm, A, in the environment, M. So just what you expect its total reward will be. And, like, this is really about intelligence being not a measure of, like, how well you perform on one particular problem, but a measure of, like, how well you generalize across all possible problems. And so this kind of tries to measure that. And so, you can't do well literally everywhere because there are no free lunch theorems. But if you have this notion of simplicity, then you can actually get some kind of an optimality. And if you have access to an halting oracle, then you can use this very notion of optimality to write down an algorithm that is uncomputable, but, like, computable with a halting oracle. That's called IXE. That's the optimal thing for this notion of intelligence. Yeah? How do you formalize the environment in terms of M? Is M still kind of treated as a binary string? So M is some kind of Turing machine. And an alternative formalism, instead of just having a Turing machine with one tape, you can have a Turing machine with three tapes where one tape is input, one tape is output, and one tape is reward. And you have the, like, agent, the algorithm writing to the input to the environment, and, like, reading from the output and reward. Is there any reward that's contributed? Yeah, yeah. Like, this is just, this is all under assumption of supercomputability, like, the fact that you're getting a reward from the environment in this way is a very unrealistic assumption. Like, it works perhaps if you're trying to write an agent to play some kind of, like, game of pong or something, but, like, it's not something you actually want to, like, have doing real things. But it's, you can actually get stuff from, like, this very abstract notion if you then add, as well, the penalty term for running time. That's enough to play some very simple arcade game. And one of the people behind this definition of intelligence, Shane Legh, was one of the founders of Google Declined, which, like, does AI that's, like, very useful for various real things, like playing Go and some all kinds of stuff. Using very different techniques from these, but, like, ideas from here have actually, like, leaked out into reward that people are doing. They're kind of cool and abstract in themselves. So, let's talk a bit about problems with this. Because you mentioned the computability aspect. What is, like, yeah, what if you reward, like, a good something that's not computable? What if, like, the whole environment is not computable? Well, like, yeah, then you can't really do anything because, like, you won't be able to see reality there. It won't be in your space of possible theories. But it's, like, even worse, the things we can write down that use these things have to have access to a halting or a goal. So they can't believe that they are part of the world that they're modeling. Yeah, like, it's very Cartesian. Like, it's very kind of, you know, the car, I think, therefore I am. And he didn't even stop to consider that perhaps there is some problem with his thinking, so that, like, he just thinks that's his thinking. Like, it doesn't actually exist. Like, these kinds of problems of, like, yeah, perhaps the way I reason is not just, like, some kind of, like, abstract thing out there that we can take for granted, and, like, will be, perspically, interfacing with the world. They're kind of new. You have to ask yourselves, like, this guy's not a verdict, he can promise or anything, like, you can take some drugs, and, like, your judgment is impaired, or, like, you can drink what's at alcohol and your judgment is impaired. There's, like, a bunch of people over at the psychology department studying, in which various ways humans don't really approximate these kinds of, like, ideals of reasoning. And that's kind of, like, troubling. If you can't have this notion of intelligence be anything that's, like, actually real, like, what is good for. Another problem that we, like, don't really see here, we can't really take, we have to, like, take explicitly into account this formalism, that I think is kind of similar, is observation selection effects. So, you might think that, well, humans did evolve and, like, were smart. And so, this is evidence that evolution is quite likely to produce intelligent things. Well, that might not necessarily be the case. You might have evolution going on for, like, a long, long time on, like, millions of planets, and then there's only one of them in which it produces intelligence, just by chance. And it happened to be, like, that's why we happened to see this thing. And so, the fact that we're making an observation is sometimes actually evidence that we have to take into account. The fact that we're asking ourselves a question is sometimes evidence we have to think about how it actually interfaces. Now, with the question of intelligence we have, like, other evidence. Dolphins evolved quite independently from us. They're, like, almost human intelligence, like, do they have tool used or do they have small groups of dolphins that, like, have worked together in this system? So, like, from mammal to intelligent mammal doesn't seem like it's actually that hard in, because it happened several times on Earth, and things like that. But, like, there might be other things. Perhaps forming DNA is, like, really hard. We were just, like, do we have to be on one planet, or, like, whether that did happen? And so, like, that's kind of similar to, like, taking into account the fact that you're thinking and having that be part of your models. Another thing that's kind of like troublesome with this is that, like, we expect this to produce good predictions about the world because we expect the world to be kind of simple, so that there are short descriptions describing the world. But there might be short descriptions describing other worlds where intelligence also evolves. And there might be also short descriptions of things, of, like, things that those intelligences in those worlds can influence. And so, it might be that if we're using this, not to reason about the whole world, but, like, reason about particular things, it might be that most of our probability maps is assigned not to things that model are, like, real-world for, like, the good causal reasons we observe out there, but actually model what messages or influences these alien civilizations living in other possible worlds are trying to, like, send us an influence. Like, you can make this specific by actually writing out how many bits you think describes something. And, yeah. Wait, so just to make sure I'm understanding this point you're making, you're saying that perhaps our world has a very complicated description. The simplest description of our world might be very complicated. So, like, it doesn't think... It might be the case that there's a planet that evolved similarly and has a similar looking. We might observe similar things about it, but that planet had a simple description. And so, if we're trying to think about what string described us, we would accidentally say, oh, we were probably described by the simple string. So, that's not quite it. Okay. So, the thing is, like, say, maybe described by something like a 150-bit string. So, it's not, like, really long. There's... Alligents, perhaps, they have classical Newtonian physics rather than quantum physics, and quantum physics takes a bunch of bits to describe. That's described by 15, this... So, then we add up, like, how hard it is to find this location and how hard it is to find this world. And we get that totally it's 100 bits of hardness to find this particular location in this particular world. And even though that might look nothing like ours, if it contains intelligence that cares about other worlds, other hypothetical worlds, then they might want to influence those locations in ways to manipulate us towards certain decisions. Yeah, it's kind of risky to, like, go with our current understanding of these things. If we actually want to use this to kind of, like, create some kind of, like, world government to, like, rule by perfect probability and logic, it's just what I'm saying. It's like, yeah, there is still stuff to understand. Like, this is not, like, the perfect ideal of reason. If these things actually contain life, then, like, we get these kinds of weird influence. Maybe, maybe not, maybe it gets dropped out by all of the other stuff, but it's just worth noting that this is the kind of problem that I'm finding. And so, let me talk for the last 20 minutes or so about some more recent stuff. Don't really remember if this was from the 90s or the early 90s, but around that time, I think. So, I have a good question. Can you explain again the connection between this stuff on this semi-major intelligence and the sort of local, global behavior of induction? Right, so here, you're basically doing local observations in your environment. In a piece of theory machine. Yeah, well, like, you're not really observing the whole theory machine. You're observing a little piece of its behavior. Like, you're observing that this theory machine is giving me only once for the first 10 bits, for the first 10 outputs. Well, perhaps it will continue to get once. Like, how much do I know? Perhaps it gives me the first 100 more times. Well, will it give me the 100 in the first plan or not? So, like, we want to know the global behavior of, like, the whole world or whole environment, this whole theory machine. But we've only observed small parts of the output. So, this is also interesting for, like, just reasoning about theory machines. You have a perfect description of, like, there is a thing called RISES theory, I mean, computability theory. So, if you have two descriptions of two theory machines, like a quantum general, telepath, which of them belongs to a certain extensional class of input output behavior, just from this effective description of, like, the code. Because they might encode something like, oh, search for an inconsistency in ZFC and output one, if that happens, otherwise keep outputing zeros. And, like, there is no computable way of determining that you're only with infinite time just by running all of them. And so, I want to talk about, this is a more recent idea from last year, by Scott Gerrobrand and some others. And it's called Gerbrand induction or logical induction. And I think it actually managed to unite some of these different pieces of reasoning this way about algorithmic probability of, like, some environment you don't know how it's behaving with just reasoning about mathematical facts. The place these ideas came from is that they were trying to figure out how do you know whether certain mathematical statements are true or not? Like, you can try writing down proofs more and more of them and say, you've proven it or you've not proven it, but that's not all we mathematicians know about things. We know some things are, yeah, probably true. We can find some proof if we sit down for an hour or 10 minutes or something. And we know that some things are, yeah, it's probably true, but we don't know quite how to prove them. And some things, like, we actually know that, yeah, like, no idea. Like, if I ask you for the millionth digit of pi, like, you'd better be putting even odds on all the 10 different digits if you haven't memorized them, I think we're wrong. Right? And if someone comes to say, and tells you, no, I'm 70% certain that it's actually seven, then, yeah, perhaps he has calculated it, but like, if he hasn't, then just speaking up is us. And so, this tried to, this was coming out of where they're trying to find a way to consistently assign probabilities to logical statements. Actually, all statements in the language of arithmetic in a way that it's, provides some kind of probability distributions on models of kind of arithmetic. But like, of course, that's not computable. You can only do that with some analysis, fix point tricks, but like, what if you want something computable? What's like this kind of coherence properties you can hope for? It turns out that you can get some really neat things by actually considering simulating this property of a market. This stuff we talked about earlier, the way people argued about, yeah, probabilities of things that you can bet on. Well, it turns out that if you actually formalize this and formalize this for mathematical statements, you get some critical behavior in the limit. And so, what we'll do is, it's like the things that are either zero, one, or true or false. And this will capture this kind of like, how the environment looks, the input-output behavior. You can do various tricks to get like, input-output behavior, rather than just describing environment to like, whatever you want. And so, we're looking at statements, statements are formally with no free variables. So, if you have any x, you have to like, say that for all x, so there exists x, not just x. The original paper, it's not quite probability assignments, they just converged probability assignments. You can do this in a way that you actually output probability assignments at every stage. And the construction going on, you also simultaneously have a productive process going on that outputs a bunch of proofs of different statements. And so, the different stocks that these trailers trade between each other are each stock corresponds to a statement. They are given like the prices for all these stocks. They can decide which stock to buy as long or so, as long as they do it in polynomial time. And so, at every stage, you're only considering a finite set of these traders. But you can do it in such a way that you asymptotically get all polynomial time traders under consideration. And you can also set this up in a way that traders only trade among themselves. So that the market has really a bounded rule. Like the market maker, the person who organizes the market, has a bounded loss facing towards them. So the market maker gives all the traders the prices, and they decide what to buy or sell, and their wealth accumulates, or it goes down to zero. And the market maker only loses a finite amount of money. So I'm a little confused about the setup. So the trader here is someone who's going to make a decision about what proof to buy, or what statement to buy, or... Yeah, an option corresponding to a statement. So what can you give us an example? Let's say the statement is a remand hypothesis. A winning base should be sold in 10,000 years or something. So, like, you can decide at what price twice you're willing to buy the remand hypothesis. So if you know that you've given $1,000 if the remand hypothesis is sold, like, for that ability to be given $1,000 when the remand hypothesis is sold, how much would you pay? Not $1,000, right? Because when you're, like, just sending money to future you, you can just keep the money. But perhaps you're willing to bet $500, because it will be sold in time that's, like, longer than $500 to grow to $1,000 if you put it in there. Perhaps you think that, oh, my God. It's very likely. We did just all the things we could sell. Yeah, yeah, perhaps, like, yeah, I don't actually care about this. Like, I think it's more likely than... But, like, even if you think that it's equally likely that it will be sold or not, it might be worth, like, I think $0.50. So then the poly-time constraint means that the traders have to make their decision in a polynomial amount of computation. So the traders can do a little bit of scratch work and say, ah, I think pre-modern hypothesis is good. There's $400. This is polynomial in the stage and we're at. Oh, okay. So, like, they trade today, and then out come a bunch of probabilities, and then they trade tomorrow and out come a bunch of new probabilities. And it's the number, like, the particular day that they have to be polynomial. So they get to spend more time thinking each day, and it's limited. Like, the fact that today are phenomenally time limited, that's not actually, like, really important. The important thing is you have a limit. Like, with this... Yeah, so with this limit, like, the traders are polynomial, they have computable, like, this whole algorithm, and so being doubly exponential, I think. So it's not really practical or anything. But you end up getting, like, something. And it's neat because all of this is computable in finite. So it can be part of the work and it can have, as part of its hypothesis, the fact that itself exists in the world, which is kind of cool. So on your assuming you have a market with, for each possible trading algorithm, you have trade algorithms. Yeah, like, you just... Yeah, you just, like, consider the set of all possible trading algorithms that run in polynomial time. You can set it up so you have... And you generate only the ones that run in polynomial time. Yeah, you basically calculate some kind of fixed point thing so they only trade with each other. Like, you can do, like, the paper where this is in is, like, 100 pages long. So there's, like, a lot of details I haven't actually, like, read up on. But I'm just, like, a bit... Michael, did you have a question? And so what happens when the time goes to infinity is that for any phi get that Pm. And so similarly for anything that is so that if Pa disproves it, then we get that the probability test is error. Except won't this happen quite suddenly, because at some stage, one of the traders will actually be able to exhibit a proof? Yeah. I think there's a way of, like, seeing how this happens tends to happen before that. Okay. Like, I don't really understand this detail, but in a way, this go... this thing tends to truth, not at the speed at which you can prove truth, but at the speed at which you can recognize truth. How interesting. So in particular, there is, like, if you have a sequence of things that are true, if there is a way of recognizing the sequence in polynomial time of light, just generating the sequence, then the probability will also attach to one. So that's, like, kind of a probabilistic short cut to prove. Yeah. So if for any sequence phi n exists... What you're saying is basically that some agents will find the proof before it's and, well, basically, the way we see this is that we can write down an agent that just buys this thing if its price goes below some point. So that agent will push this price to which one. Because we can show that, like, in that little suspect. Oh, and since we're not going to disprove it, we can always write down someone who's willing to buy it. Well, like... Sorry. Because all of them will be provable, the agent will always get back its money that it puts into it so it can make new and bigger bets. On the other hand, if a statement is neither provable nor disprovable, then we can do the same thing for both the statement and its negation, and so we get the probabilities bounded away from both 0 and 1 with something that, like, measures the statement. Because, like, these traders can't get equally any amount of money to trade them. But, like, more complicated traders will have to figure out who that's money so that the money adds up to something final. Um... So, is that, you know, increasing? It's not necessarily monotonically increasing where the price can go up, but then it goes down, and go up, but down, and down, and up, and, like, it's also not about truth, this is about provability. So, like, there are two statements, like, pan arithmetic is consistent, that it's not provable in pan arithmetic, so the probability will be bounded away from 1, but, like, it's still true. So it won't be disposed, so it's, like, at least bounded away from 0 as well. And so, this gives a cut. It takes us back to the realm of mathematics including things about the integers. And so, here we, instead of, like, actually doing this explicit induction reasoning, we're just letting all of these traders do our inductive reasoning for us. And so, together we get some notion of, like, whether a thing is, like, probably true or false. And not only do we get that, but by adding this string of, uh, booleans, we also get that, um, if we have, uh, P of N, like, if we look at P infinity of some string of these booleans, well, it will, in fact, um, dominate, or perhaps this is some kind of asymptotic domination, I don't quite remember, uh, well, it will dominate with some kind of constant complexity factor, uh, this universal semi-measure, which can be thought of, like, you think of all the things that begin this way, with this finite string. And so, we get something that reasons both about mathematical truth, allowing us to get, like, global mathematical truth from just local observations and local computations. But it also reasons about these general computable environments, okay, in the same way. And so, this fact that it's, like, within some constant factor of this, uh, universal semi-measure, uh, this is kind of tells us that even if we didn't add this fact that we have this deductive process that makes sure that the things that are provable in PA office groups to one, it will probably recognize something like an arithmetic, because it's quite simple. Objects like natural numbers are very simple. And so, it will be able to kind of, uh, deduce what the probability correct behavior of pi is, even if it doesn't have access to any mathematical theory that describes pi. Uh, but, like, both here and the original thing with, like, Solver of Induction, I've kind of brushed on the design here that there is this choice of encoding. Like, we made this collection, but it's a prefix free encoding. But there's still a choice of, like, which prefix free encoding do you think. Lots of different ways of encoding this. And you can find, like, particular tricking codings to give you, um, arbitrarily high probability of any particular thing you can think of. As long as it's below one. And so, like, there is a lot of dependence on language both here and here. And so, it's kind of, like, not completely universal. But it's universal within, like, this kind of constant factor. And so, within the constant factor, everything is really nice. But, like, there's still, uh, cool stuff that's left about. Yeah, how exactly do you think about something that's, like, part of reality and trying to do optimal reasoning and reasoning about, like, your own inoptimality and, like, this comes a bit closer but, like, it's still not there. I think I will... Going back to the Bayesian case, um, what you're talking about, they said I mentioned you and so on. Does that actually allow for a, um, I won't even say efficient but computable method to actually do Bayesian reasoning? Uh, it's lower semi-computable. So, you can, um... So, the difference between these things is that, like, this is kind of, like, simply uncomputable, but, like, you can approximate this thing from below. So, like, you get computable approximations of different stages, and you know that different summation is always smaller than the thing you're approximating. And you know that when time reaches infinity, you will get your perfect approximation, which is the thing you're trying to compute. But here, the behavior can go, like, both up and down. So, in that case, even though the Bayesian, um, inference is itself an imperfect inference based on price, we still can't get the imperfect inference in finite time using this formulation. Is that the case? So, like, this thing doesn't guarantee you anything in finite time. It only guarantees I mean the, um, the Bayesian. This semi-measure, or which part of the Bayesian thing? Well, basically, is it possible to do Bayesian inference using the semi-measure formula? Like, you can do approximate Bayesian inference. There is an algorithm called AIXITL, which has also, um, time, not only length of the Turing machine taken into account. And that does approximate Bayesian inference well enough to actually be used on, like, some simple games. Thank you. Any other questions? What advice do you want to buy? What would you like to pay? Well, positive. Yeah, I think it would be an interesting experiment to run this. I actually run a market here at the department. It'd be fun. Oh, yeah. Yeah, no, like, I don't know, it's interesting that, like, you get to this really neat limiting behavior. Perhaps, like, this is the way mathematicians should finance themselves. Like, you, you, you run these prediction markets and, like, whether things will be proven or not. And, like, I don't know, like, some practical mathematics that perhaps will get me to actually, like, check all the really complicated proofs and make sure that they're correct. You know? All right, guys, let's spend that. And if you want to buy a proof... Yeah. If the price is right, you know? Yeah, yeah, it's right. Yes, and I usually do that. Yeah, and David, I forgot to introduce Hene, that's also our first year guy, and then me, that's also my friend, and they're all in logic. Okay, so...