Pedro Domingos: Tensor Logic Unifies AI Paradigms

Tensor logic unifies not just symbolic AI and deep learning, it also unifies things like kernel machines and graphical models. I'm Pedro Domingos, I'm a professor of computer science at the University of Washington in Seattle and a long time machine learning researcher. And my dream from my PhD onwards has always been to unify all the different paradigms of AI into a single one.

My PhD unified two of them, my best known research unifies a couple others of them. I wrote this book that turned into a big bestseller surprisingly called the Master Algorithm that is precisely about this goal and where we are towards that goal.

separate problems? No. Very good.

So precisely the problem or one of the problems with GPT is that it it hallucinates even when you set the temperature to zero. What the hell? Right?

I want to have a mode. Yeah. Yeah.

Yeah. Right? Not I, but like every Fortune 500 company, if it's going to use AI, needs to have a mode where the logic of the business is just to bathe.

The security isn't violated, the customer doesn't get light, etc. We've got to have that orientated that they will not take off, right? And transformers can't do that.

Tensor logic can do that precisely because in this reasoning in embedding space mode that I just described, if you set the temperature to zero it does purely deductive reasoning. And by the way, the temperature can be different for each rule. Tensor logic is just based on this, to me, gobsmacking observation that an Einsam and the rule in knowledge programming are the same thing.

There is this thing called predicate invention, which is discovering new predicates, discovering new relations that are not in the data but that explain it better. I would say that you know in some sense discovering representation like that is the key problem in AI, is the holy grail. What was Turing's achievement that we now take for granted?

Turing's achievement was for which he is deservedly famous, right, is to postulate this notion of a universal machine. The amazing thing about computers is that the universal machine, which in his time was a was a completely counterintuitive notion. What do you mean a machine that can do everything?

The typewriter can type in like in like the sewing machine can sew, you're telling me there's a machine that can type with one hand and sew with the other, what are you talking about? So like this is the genius, right? So first step you want to have this property of having a machine that can do anything.

What we're missing to be able to do what the universe does and evolution does is universal induction. What is the Turing machine equivalent for induction for learning? That's what I'm after.

Just like I was doing deep learning before it was big. So very close analogy. So you should definitely watch Machine Learning Street Talk.

It's one of the best ways to not only learn about machine learning but find out about what's going on at a deeper level than you see everywhere and that is very important. So you should definitely subscribe to Machine Learning Street Talk.

What's the idea behind this paper? Well, tensor logic in many ways is the goal that I've been working towards my entire professional life, because I really do strongly believe that a field cannot take off until it has really found its language. And Tensor Logic, I believe, is the first language that really has all the key properties that you need in the language of AI.

For example, it has automated reasoning right out of the box. Like for example, Prolog has, right? The classic AI languages had a number of things that we just took for granted.

The transparent and reliable reasoning you didn't even have to worry about, it was just already available. At the same you don't have that in PyTorch at all, right? You have all these hacks to try and do reasoning on top of it.

At the same time, the Lisps and the prologues, they never had the auto differentiation, the ability to learn, right? One of the beauties of the current moment in many ways is that you barely have, you look at most papers, people barely talk about the learning because it's already implemented under the hood, so you want that as well. Right?

And you want the scalability on GPUs is the other thing that things like PyTorch and TensorFlow and whatnot give you. There was no language before that had all of these, and there's a number of others but these maybe are some of the key ones. So Tensor Logic is basically a language which as the name implies, is a marriage, a very deep unification, not just some superficial combination of the tensor algebra that deep networks are all built out of, and the logic programming that symbolic AI is built out of.

There's only one construct in tensor logic and it's the tensor equation. You can do everything with tensor equations.

There's just people who think it is. Right, everybody has their favorite language that they believe, you know, is the universal solvent, but it never really is. So we and we know also for fundamental reasons like, you know, going back to Shannon and whatnot, that there is no language that is the most, you know, pithy for anything you might want to say.

Right? Having said that, right, physics is a good example because calculus is so fundamental, you know. Feynman famously said that he thought in calculus.

Right? And the thing that I found with tensor logic is that, you know, I don't know how much of AI it's going to be, or how much it should be. But what I have found in many ways to my surprise is that in some ways Tensor Logic is more than just programming language.

It really I think captures the fundamentals of what you need in AI in a way that going in I didn't even think was possible. All of tensor algebra can be reduced to this operation which, you know, going back to physics, is called the Einstein summation. Right?

Einstein summation was something that was introduced by Einstein when he was working on relativity and got tired of writing summation signs. It was all about tensors, right? General relativity was all about tensors.

And he jokingly called it his great contribution to mathematics. But the bottom line, and you know there's this great paper by Tim Rock Taschel or you know, blog post saying Einsum is all you need. And truly you can do all of deep learning with just Einsum.

All of the matrix multiplications and tensor products, all of that are instances of Einsum on the one hand. On the other hand, in symbolic AI, it's all about rules. Right?

And tensor logic is just based on this, to me, gobsmacking observation that an EINSUM and the rule in logic programming are the same thing. They are actually the same thing, the only difference is that one is operating on real numbers, and the other one is operating on booleans, but that's just a different atomic data type. Right?

And then on top of that, so to summarize, at this point I think that it would be I look at all different things that I and others have done in AI and I think it would be crazy to not do these things with tensor logic.

No. Very good. It does several things.

So first of all, and this is gonna be an increasing order of importance, the syntax of EINSUM in this language, there's also this package called INOPs, is incredibly clunky. So at a very basic level, tensor logic is just a much pithier, more compact, easier to write and understand way to write EINSUM. And you know physics and mathematicians famously like to say that a good notation is half the battle.

So this might not seem like a big deal but my experience is that you can just think better and faster once you have this notation that like this funky procedure call with these indices and these arrows and these arguments. It's a nightmare and you know, the syntax of tensor logic is like, you write a ninth sum like you would write a rule. There's a tensor equation with a tensor on the left hand side this join of tensors on the hand side.

Right? So this is one aspect. Another very important aspect, and one that I think could prove decisive is that people don't use ions so much because it's not very efficient.

Under the hood, it's not as efficient as you tip you know, sometime I you know, this could be done so much better. Right? But I've done some of programming of this and I wound up even I wound up not using Einstein because it's so slow and clunky.

And all of that can be fixed. Once you have this one abstraction of the tensor equation, and you implement it on CUDA for example, you can optimize the heck out of it, and you'll just be able to, you know, EINSON will finally be able to reach its potential. Right?

But actually none of these things are actually the most important part. The most important part is that Einstein as we know it is only good for Tensor Algebra. Tensor logic is a language where the same construct does all the symbolic and all the numeric parts and any mix and variation between them including learning the symbolic part and whatnot.

These are all things that in world just didn't exist, right? You talk about the people in Einstein whether in AI or mathematics or physics and they just had no idea that any of this had anything to do with reasoning. You look at all the ways that people are trying to do reasoning today and just want to pull out your hair.

Let me ask a very concrete, you know, in some sense, I'm a simple simple man.

what I did in in, you know, maybe this is an important piece of context. So if you look at, so the simplest form of logic programming is data log, right, which is the foundation of databases, right? Know most SQL queries have variations on data log rules.

And data log rules are composed of two things, joints and projections. Alright, this is like databases 101. And what I have done is I have generalized join and projection to numeric values.

There's this thing which I defined called the tensor join and the tensor projection, which when the tensors are boolean becomes the regular symbolic database one. But now the numeric version has all these things as special cases, and by the way it's also more general than the EINSUM, right? So another benefit of this is that it actually goes beyond the EINSUM.

Now, an or, right, the way you get an or is is is by having more is is it just as in so how do you get an or in prologue or data log? Is by having, you know, multiple rules with the same head. And then those rules then they implicitly being disjoint.

Alright? So if I have AFBC and AFDE, then that means AFBC or DE. And the same thing happens here.

And you could also of course just put them all in the same equation because like you know it might be more convenient to say like well AB plus CD, right? So doing a NOR is a completely you know straightforward thing, but it's really not where the main action is, it's in the tensor joints and tensor projections.

understand your question. So again, it's more than that, it's a DNF. A DNF is exactly.

A junction of the normal And so what happens is that like the Einstein sum is, so in numeric land, think of a dot product, right, a dot product is just a sum of products, right, which in boolean land will be a disjunction of conjunctions. If more than one is true, get a number that is greater than one, which is you need to pass it through a step function, to reduce all the values greater than one back to one, yes.

constructs of the language are tensor logic. We can do everything with NAN, so why do we need high level programming languages at all? Right?

The point so there's two things that you want the language to be. First of all, want it to be universal, so you can, for some things you don't, but in general, right, for AI surely you want a universal language. You want something Turing complete and Turing logic and Tensor logic is that.

But then this is actually the most important and most difficult part. You want something that is at the right level of abstraction for the things that you want to do. And NAND definitely is not.

And I can show with a lot of examples I have in the paper for example, you can, you know, code the transformer in a dozen tensor equations. As opposed to a vast massive code. Right?

And then what happens when people have a language that suits their needs, is that then they just get used to that. They often wind up using it even for things that it wasn't the perfect thing for, but at that point it's what they're comfortable with. So my guess is at the end of the day people are just going to do everything in tensor logic, and they know, you know, in the back of their heads that yes, there are ores going on here, and you can think of them as ores, but they just think of them as joints and projects and tensor equations.

So in a way the master algorithm was laying out my agenda, right, was asking the question what is the master algorithm? I did say at the outset I'm not going to give you the master algorithm in this book, I'm just going to tell you where we are and why I think this is the central goal of AI. I would say that tensor logic is that answer.

Tensor logic, we haven't talked about that yet, but tensor logic unifies not just symbolic AI and deep learning, it also unifies things like kernel machines and graphical models. You know, the things that graphical models for example are built out of, and then you can compute probabilities with them, they are a direct, I didn't do this on purpose, but it just fell out. You know, factors that graphical models are made of, those are just tensors.

And then the marginalization and summation, sorry, the marginalization and the point wise products that are what probabilistic inference is made of, they are just tensor joints and projections on those tensors that represent potentials in the case of Bayesian network conditional distributions. So at this point we do have this very simple language where you can do the entire gamut of AI. Which honestly, I didn't think this was gonna be possible going in.

I thought the answer would be much more complicated. Now, is this the master algorithm? Tensor logic per se is not the master algorithm because it's just a language.

I would say that it's the scaffolding on top of which you can build the master algorithm. Now, tensor logic is not just the language, it's also the learning and reasoning facilities under the hood. So for example, one of the best things about tensor logic is that the autograph is incredibly simple.

Because there's just one construct, it's the tensor equation, and the gradient of a tensor logic program is just another tensor logic program. So this is all there. So the learning and the reasoning are all there.

Very good, so I actually go into that in the paper, briefly the paper is just an informal introduction to these ideas. Inductive logic programming is the field that deals with discovering rules from data. But it does this by things like greedy search or beam search and it's a very large search space and it's extremely inefficient.

Which is actually one of the things that killed it even though it could do all these things that people know in deep learning are just painfully rediscovering. In tensor logic, this is one of the best parts of it, the structural learning falls out of the gradient descent. The gradient descent actually does structural learning.

And then on top of that, this is actually the the best part as far as the learning is concerned, there is this thing called predicate invention, which is discovering new predicates, discovering new relations that are not in the data but that explain it better. I would say that in some sense discovering representation like that is the key problem in AI, is the holy grail. Everything that we know, you know, like when you look at the world, right, you don't see pixels, you don't see photons hitting your retina, right, you see objects.

The objects are invented predicates. All the way up to in science, right, the most like Newton's genius was to introduce a new quantity which is force, and energy, and entropy, and all these etc etc. Right?

So, in tensor logic that also just happens by gradient descent. Right? It's hard to believe but let me just give you a hint as to why this is the case.

There's this other thing that is folded into tensor logic which is tensor decompositions. Right? And tensor decompositions are generalizations of matrix decompositions.

And if you think about matrix decompositions to take that simple case, what a matrix decomposition does is it takes a matrix and decomposes into two new matrices that together are more compact but essentially reproduce the same data, right? And there's a generalization of that to tensors called the Tucker decomposition, there's others but the Tucker one is the most relevant one here. And so if you write in tensor, to answer your questions very directly, if you write in tensor logic a rule schema, you know, including a data tensor on the left hand side, and by the way your entire data can just be reduced to one tensor embedded as one tensor, we can touch on that later, but you write a rule expressing that as a function of a few other tensors, and the gradient descent, just as in matrix factorization, will discover the best values for those.

And then if you want to, for example, then discretize it, say like I'm going to threshold this and make it boolean again, you will see what is the concept that that learned, or you can leave it in numeric form. So the learning is actually extraordinarily powerful. I've always thought, and you know, I think, you know, a lot of people in deep learning really believe this, that, know gradient descent can do amazing things provided you give it the right architecture to operate on.

And in a way what all these million papers are about is about finding the right architecture for gradient descent to operate on, and of course transformers are a great leap forward, but I think transfer I think, you know, tensor logic is an even greater leap forward. How so?

find weights within that structure. No, but okay. I understand your question, but tensor logic does allow that.

You know, step one, you can you can encode a multilayer perceptron, the entire multilayer perceptron, and I do that in the paper with a single tensor equation. All the layers provided that they all use the same non linearity can be encoded in one equation. Okay, number one.

You can also have different equations for different layers or typically sets of layers you know, however way you please. But from the point of view of structure discovery, the thing to realize is that if you create, if you set up one of these very general equations that you can in tensor logic, then in some sense it can, it's a very broad classes of architecture. Then what the learning does is it discovers the architecture within that space.

Right? Which if you think about it at some level is what a neural network, when you compare an ordinary multilayer perceptron with a set of rules, right, a multilayer perceptron is can take in fact there was a system called K band in the early days that did this, very clever. It initialized multilayer perceptron with a set of rules because each neuron is a rule, Right?

But it's also more flexible because now you can have weights, right? But but but you know, a neuron, a single neuron can represent a conjunction and therefore a layer can represent a junction and so forth. So when you're learning weights in an ordinary neural network, you can actually see it as learning the structure of a set of rules.

What tensor logic is doing, this is at a more powerful level, that was just propositional, and now this is at the full level of generality of first order logic. But you can learn the structure, and then of course then there's more than one way to do that, and you can also decide how black and white you want the structure to be, what you want to leave as weights, what you want to discretize, but the structure itself can be learned by taking tensor equation. A tensor equation is a very general thing, right?

So to take another example, you can also do an entire ConvNet in just one tensor equation. And the quintessential example of like, complete connections don't work is a multilayer perceptron for vision, right? Which you replace with the convnet that actually has the local structure.

That is also a tensor logic equation. Now you're saying, how do you choose between the convnet and then MLP, right? Very good question, and now there's a range of things you can do.

You can actually these days start out with a very general structure, because GPUs and large server farms are an amazing amount of power for something like this, right? So you can almost, I would say, brute force that search, provided you have the data. I'm not actually recommending you to do that, right?

You can also however, and more interestingly, you can, and this is actually one of the key benefits of tensor logic, is that you can write down what you believe are properties of the say, right now what happens in when you for example program a network in Python is like, you have to commit, you say like here's the structure, and now the of the only thing that happens is the learning of the weights. Intensive logic you don't have to do that. You can you can set up one of these very general structures and then you say, let me give you a bunch of equations that are things that I believe to be true about the structure but do not completely determine it.

And those just work like priors, and indeed like soft priors, right? Those can, and again you can turn up the temperature on this or down, and say like, you gotta obey this equation and that one, you know, I'm sure you can overwrite. And then this, in my experience, this is actually what is important is that the gradient descent, instead of starting from a tabula rasa, have this kind of soft knowledge.

And then most importantly, you, the developer, you the AI researcher, you get to this is really the essence, is that it's not every deep learning researcher or data scientist knows this. It's like, you don't do the same priority and then push the button and hope for the best, right? There's an iterative loop of you set up the structure and then you learn, you get the results, and then you refine the structure.

And what this does is it makes that more efficient much more because you just have to, in your interpreter, you write one more equation or you modify an existing equation, and also the entire stack is of what you learn is much more interpretable than it was before. It's actually in some ways one of the most important properties of tensor logic is that you can understand what's going on much better than you could in two ways. One is that the code is much more transparent than the whole pile of things that you have sitting under a bunch of you know, PyTorch procedure calls, but also the result of learning, at least if you do it in in certain ways that I discuss in the paper, the result of learning is transparent in the way that a transformer just, you know, can't hope to be.

So we've covered some interesting topics on MLST before. Mean, of course, there's geometric deep learning, which is this idea that symmetries are are fundamental.

meaningless? You know, how do you know that you've got a good one? You know, great question.

And you've touched on several things there. Let me start with the geometric deep learning, right? I'm a big fan of this.

In fact, I gave a keynote at the second iClear on something that I called symmetry based learning, which is in some ways an ancestor of geometric deep learning. I really do think that the universe possesses these fundamental symmetry. Actually I don't think that.

This is known, right? In physics, right, the standard model is basically a bunch of symmetries. And this is extraordinarily powerful, right?

That such simple things could be such universal regulators that you then basically can build everything else out of. And if you think about it, in machine learning the problem is like, what is the learning bias that you should start from? Right?

Should you pull in a lot of knowledge? Should you have a very, you know, very vague architecture? The thing about machine and there's the no free lunch theorem, right, that says, you know, if you don't assume anything you can't ever learn anything.

The thing that's amazing about machine learning is that with very weak biases you can get very far, right? And I would submit that those weak biases fundamentally at the end of the day the most important ones are these symmetries. And tensor logic is precisely, you know, I think the perfect language for expressing those symmetries as the physicists will tell you, right?

It's what they use, in like, not the logical version, but the numeric version. Right? So I think we can discover those regularities, I have some suspicions as to what they might be, but I think, you know, we're not quite there yet.

But I think once we have those regulators in some sense, they will play in AI the role that the Standard Model plays in physics. Now of course as you say, there are people who say like, oh forget that, right? You know, going back to Marvin Minsky, right?

There's like, there is no small set of AI laws or anything, it's just one damn thing after another, blah blah blah blah, right? Like you're dreaming. Right?

And I respect that point of view, right? And you know, we will find out empirically, but what I if I had to guess how this is going to play out, at the end of the it's going to be like this. This stuff that I'm talking about gives you, you know, the eightytwenty.

You know, it gets you 80% of the way. And then the other 20% of way, have to do a lot of these things, have to do a lot of hacks, etcetera etcetera. But something like Tensor Logic still makes it much easier and faster to do those hacks than if you didn't have it.

So it actually gives you benefit both in the 80% part and in the 20% part.

right? Yes. And of course the whole notion that we now understand very well that systems are, you know, many systems are chaotic and therefore inherently unpredictable.

Right? And you know, complex systems and all of that. So where does and you know, the whole notion that more is different, right?

Exactly, which I'm a very strong believer in. So doesn't that contradict what I just said? Actually no, right?

I would say the following, from physics all the way to AI with biology in the middle. The universe is basically composed of two things: Symmetries and spontaneous symmetry breakings. Right?

God made the symmetries. The symmetries are the laws. As far as we can tell, none of these systems at any level violate the laws.

Right? Those symmetries are there. I mean you can go into that, there's a lot to be said there, but essentially, you know, most people, the great majority of people may be accepting, you know, some there are some exceptions, But they believe that the laws of physics apply to everything.

Like my brain abides the laws of physics, society abides the laws of physics. The problem is that the laws of physics are useless at some point in understanding, you know, even biology, let alone psychology or sociology or AI. Why are they useless?

Because we have inherited from the beginning of the universe a series of spontaneous symmetry breakings, right? And my brain is doing spontaneous symmetry breakings one after another continuously. And those spontaneously, like, those then, some of them die out, right, or become irrelevant, stay the same.

But others balloon into very big things. And that's actually what evolution is, is one of these things after another. And once you have that, so the computation with disability problem is that, at some level it is true that although in principle this is all predictable and reducible, in practice it isn't.

Right? But now here's the point, how do we handle that? Our brains know how to handle this in a way that AI doesn't.

And the way they handle this is like, you predict you computationally reduce everything you can to begin with. And I'm actually I've talked with Steve, you know, at some length about this. And I'm actually more much more optimistic about how much is reducible than he is.

And the things are like, your overall universe is not reducible, but it's full of these reducible pieces. And in a way, evolution is a our brain is an accumulation of these reducible pieces. So you do that, you want the machine learning to discover it, you want the inference to exploit it.

But then, after that you actually have no choice but to just keep gathering data and using that to inform your predictions. Right? In a way the physics goal of like, I give you the initial conditions and then I just predict like the, you know, Laplace is demon dream.

It is a dream, but I think that the problem that, you know, some of the complex systems people have not realized is that we don't have to do that. Ask any, you know, engineer, any aerospace engineer using a Kalman filter. What you do is you predict just what's gonna you know, or reinforcement learning, right?

It's like, you wanna have a sense of where you're going, but at at every step of time, you you you you recalibrate your predictions with the new data that comes in. So you actually only need to predict things well enough to control them, to make them predictable. Right?

We humans are always controlling the world to make it more predictable, and this is what robots need to do as well. And this is sort of like what I'm trying to, you know, support with a language like lightencer logic.

glad you asked that question. I am absolutely arguing that tensor logic is the language to use in all these layers, and let me give you some evidence towards that. Express relativity in tensor logic.

It's tensors and you know differentials of tensors and whatnot. That's, you know, tensor logic does that out of the box. Do the same thing with quantum mechanics.

Do the same thing with all these others. With all of the different pieces of AI that I know. And why is that possible and why does tensor logic do that?

Again, think this gets at a very deep fact about the universe, which you know, complex systems people and physicists have suspected as well, which is that the universe has this amazing property without which it would not be comprehensible, that you can have a lot of complexity at one level that then organizes itself into a new level at which now a different set of laws applies. Right? In a way what we do with computers is do that by design.

But here's the key: what you want is a language in which to express this process. The whole process by which multiple levels get created, by which multiple representations get created, including different representations at the same level. For example, going back to Herb Simon, many people at least and I have believed that the essence of human intelligence is your ability to switch between representations as the problem dictates.

And as long as you pick one representation you've stuck yourself in the box. But at that level, tensor logic is a meta representation. It's the way to construct representations.

And yeah, you know, a large language model, you know, to take a very salient example, what has a transformer learned, right, when it looks at all that text? Precisely I would say where a lot of its parts come from is that it has looked, you know, it's like, you know, Seb Bubeck says, it's like it has learned this soup of algorithms. There's all these different pieces and different ways of doing things that it has gathered from different places, and it doesn't choose between them.

It's the prompting and the fine tuning and what that then pull out the parts that are better for one thing or another. So we absolutely have to do this in AI. I think it also reflects a deeper truth about the universe.

I think there are going to be laws of this. You know, we're not then describing laws of the universe, and I think, Tensor Logic at least is my best attempt at having a language in order to do both this AI and this type of scientific discovery. I also believe, and I discussed that briefly in the paper, that tensor logic is not going to be just a good language for AI, it's going to be a good language for science in general, for several reasons.

One of them is this, but the other one is that if you look at the difference between the equations on the page and the resulting program from implementing them, often there's a lot of complication. In tensor logic it's almost, you know, the tensor equation is an almost symbol for symbol translation of the equation on the page. So now you can just do, you know, science, you know, on a different level.

Also the logic, if you look at scientific computing, right, it's usually these tensor operations with some logic wrapped around it. Tensor logic does the tensor operations and the logic in one language, but more importantly, the logic now becomes learnable.

star t because you needed that in order to express RNNs, right? No, no, no, look, great question. So there's two very important things to distinguish here.

One is, which start t is not, but let me mention that first, the RNNs also illustrate that, is syntactic sugar. Right? You always have syntactic sugar because for example, in an RNN you want to express x(t plus one).

Right? And I could, know, tensor logic is too incomplete, but I don't have the t plus one. I can this is a very simple syntactic piece of syntactic sugar to add.

Why wouldn't I do that? Right? Again, there's an eighty twenty rule of like which of these contracts you wanna have.

But the start t is actually completely different thing. The start t is there for computational efficiency purposes. Start start t is a hint about how to implement that tensor that saves a ton of memory.

Right? And you know this notion of a leaky abstraction, all abstractions are leaky famously in computer science. Tensor logic is no exception.

For the most part when you write tensor logic you don't have to worry about what goes on under the hood, but sometimes you want to. And this is precisely one of those things. The ideal of the is that, it's like, we don't have four loops anymore, right, which is great, forget all of that, but sometimes I don't want to be computing a new tensor or even just a new vector for every new thing that I do because that would be a waste of memory.

This Tarte is just saying, you know, you have one vector and you reuse it at every iteration, so you have the initial x zero and then x one is over it overwrites that, right? So this is a piece of the language, right? You can do everything without it, but it would be silly to not use it.

Alright. So let me let me push back on something because you mentioned it twice now, is like the Turing completeness.

Well Do you realize this problem? No, no, no. So actually there is no problem there.

Let me tell you exactly why, right? And let's just do this in three steps. First of all, Turing completeness doesn't matter at all whatsoever.

Because the only difference between a Turing machine and a finite state machine is the infinite tape, and in the real world there is no infinite tape. If you can implement the If doesn't finite matter, why do you keep mentioning it? That's part two.

That is part two, right? This is actually a very interesting set of questions, so let's get into parts of Turing completeness doesn't matter. What matters is that you want to be able to express any computation that you might want.

That's what matters, Right? You might choose a specific language for specific purposes, for something like tensorology, you want that generality. You have that generality irrespective of Turing completeness.

So this is part one, right? We can debate, but let's set this aside for just a second now. But, you know, the way I don't get to change the way computer science is, and Turing completeness is a shorthand for universality.

I just want to show people that tensor logic is universal. And now, I have a proof that tensor logic is computationally universal that does not rely on Siegelman construct. Right?

I chose to not publish in this paper because it would take too long. The beauty of that is that in one paragraph I can just say look, equation in the Siegelman paper, you can implement it here and we're done, right? I can also, you know, there's so many ways to prove that things are too incomplete, I completely agree with you and her that that construct is ridiculous, right?

It's silly, right? It has no practical significance, but the reason I use it is like, it's just my way of telling people in one sentence that and why, you know, tensor logic is too incomplete, right?

actually, there's even more than one other type of proof that is possible? Let me tell you what that one is and what so, here's two, you know, not just one, so three ways, there's the Siegelman way, right? Another one is, you have a finite control with access to an infinite external tape.

Right? That is a much more reasonable thing in my view, right? You have a memory, the memory is infinite, but all that you have to do in the tensor logic is know how to access that memory.

So like, it gets back to remember, a tuning machine is a finite control and an infinite tape. Right? If the if the tensor logic can realize the finite control, which obviously it can, and you give it an infinite tape, then we're done.

Right? And then on that note you can even just do it the following way, right? Which for example you know, Dale Schumann says a great paper about this is, know, people have come up with various very simple ways to set up a Turing universal, you know, computer, and one of them is there's a set of rules, right, that, you know, sets up that machine, right, without going into details.

And that set of rules, you know, you can just write intensive logic without even, you know, having to wake up from your sleep, so there you go. I totally agree with you.

course, but that is why tensor logic is more than tensor in math tensors in mathematics. Right? The tensors that people in mathematics just don't do this.

Right? But tensor logic does because of the of the logic programming side. Right?

If tensor logic can do logic programming, then it can do everything that a computer can. Have you specified fully like all the operators and tensor logic somewhere like on a website or something? There's only there's only two tensor project or three, right?

There's tensor projection, right, there's tensor join and there's univariate linearities. And the linearities are crucial, right? Tensor algebra is multilinear.

Algebra is linear, tensor is multilinear, right?

Oh no, I mean like it's not even projections or joins, right? I mean think of a trivial projection where you're not summing things, you only have one, right? That's what a write is, right?

Actually, let's not even worry about tensor joins and projects, let's just think about, you know, propositional rules which of course are what you if you want to implement propositional rules in tensor logic, all that you need is tensors with no indices, with zero indices, right? So all you're dealing with is scalars. And the right, right, is just a rule that says the target of the writing is on the left hand side and when the right is on the right hand side.

Now, to get very conclusive to the issue of an infinite memory, What is an infinite memory? An infinite memory is just an infinite vector, right? Indexed by the memory address.

That's all it is. Right? And so how do you write this infinite memory tensor logic?

You just have the memory as your tensor on the left hand side. It's kind of so, you know, it it it's so simple, it almost there's nothing to think about. I'll have to I'll have to, work through some tensor.

Sorry, just to finish how that do you advance the tape? Well, you just increment the index. And how do you, you know, move it left?

You decrement the index. Right? It's like it's done.

No, very good. So to clarify, is not a function. Is a notation about an index.

So for example, if I have a vector, you know, like, you know, x of I, right? Better example, a matrix, M. This occupies, if I and j are reach 100, this occupies 10,000 positions in memory, right?

But if what I do is Mij star, right, on the left hand side of my tensor equation, then this is just, you know, instead of being whatever, a 100 by a 100, it's just a 100. Because what this is saying is like, if I put the star and the j, what I'm saying is like, run through the I, right, and for every j you overwrite the result. You can do this in either dimension, but you know, so pick one, whichever one.

It just says like, keep overwriting the results, right? So you lose your old one so I think so let me put this way, is actually a vector. Is a vector where the only dimension is I.

J is actually just a iterator for a for loop. You see what I'm saying? And concretely, for example, in an RNN, this is what you want because x, right, x I is your vector and the j, let's call it t, right, x I t, at every new step in time when the state evolves, you don't wanna I mean, could, but in general, you just wanna overwrite the old state with a new one as in any state transition system.

Okay? Now you're you know, does this make sense?

how would I compute all the digits of pi? In infinite Turing machine land, right? I have a vector of the digits of pi that has a start but not an end.

Right? And what the computation in tensor logic does, it computes every successive so like, we didn't talk about this but there's, you know, how is inference done in tensor logic? Forward chaining or backward chaining.

Forward chaining is a general, they are both generalizations of the corresponding operations in symbolic AI. If you applied forward chaining to a set of rules that computes the digits of pi, actually just one rule because it's very simple, what it will do is in each iteration it will fill in the next digit of pi. Right?

Now, if your vector is infinite, this will go on forever as it should. If your vector is finite, well, at some point you run out of memory and you're satisfied with the number of digits, which is what we do with any real computer in the real world. I don't wanna, I always get us bogged down into Turing issues.

this is actually extraordinarily simple and it's to me incredibly frustrating that there's so much confusion about it starting with computer science and theoretical computer science and now playing out in AI and transformer land. And and and and if, you know, it it just boils down to this, right? You said earlier and I violently agree that, and correct me if I misinterpret it, but you said like, Turing completeness is not important but that shouldn't cause us to underrate Turing's achievement.

Absolutely. What was Turing's achievement that we now take for granted? Turing's achievement was, for which he is deservedly famous, right, is to postulate this notion of a universal machine.

The amazing thing about computers is that the universal machine, which in his time was a was a completely counterintuitive notion. What do you mean a machine that can do everything? The typewriter can type, you know, like the sewing machine can sew, you're telling me there's a machine that can type with one hand and sew with the other.

What are you talking about? So like, this is the genius, right? So first step, you want to have this property of having a machine that can do anything.

This is the foundation of computer sciences, of computers, as a revolutionary technology, right? So point one. But point two, and getting to the transformer part, right?

I don't know, unfortunately these confusions then build on each other and never get it's one of those symmetry We breakings, went down this road of defining things a certain way and worrying about infinity and now we're stuck there, right? NP completeness is another example, but ignoring that. The problem with transformers, so like, the real problem is the following: It's people like, oh, but if you only have this many blocks then you can only do so many computations.

The thing for example that inductive logic programming has and we want is that you can learn things from very small examples. Like children do in elementary school. You learn to do addition on tiny examples, but then if needed you can do addition on numbers of any length.

Of course your life is finite, you will never add infinite numbers, but that's not the point. Infinity is just a shorthand for something that's so large it doesn't matter how large it is. And what I want in machine learning is to precisely be able to learn to do to handle problems, graphs, structures, knowledge bases, inference problems, whatever, of any size from very small ones.

That's the limitation that a lot of these Transformers have. And that's the one that you Yeah. Want to fix and can't fix, and Tensor Logic helps you do that.

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very and I mean, and to be fair, Turing wasn't the only one doing things like this. And precisely now we know that all these things are equivalent and then extensions don't add But here's actually a really important point, right? The question that has been on my mind for decades is this.

A Turing machine is a model of deduction. It's universal deduction. What we're missing to be able to do what the universe does and evolution does is universal induction.

What is the Turing machine equivalent for induction for learning? That's what I'm after. Right?

That's what the master algorithm is. And I know it exists and again just as you can have a million different versions of Turing machines that are all equivalent, you can have a million different versions of the master algorithm that are all equivalent and that's okay.

No I'm just kidding. No if you're Bayesian it is Bayesian tensor logic.

and the type of reasoning it's doing isn't actually semantically meaningful at all? I would actually say that like of all things in the paper, this is the most exciting and important one. Is that you can do sound and transparent reasoning in embedding space with sensor logic and how come?

Right? Why is that possible? And to just sort of like give the gist of it, here's the key, right?

Is let's go to kernel machines for just a second. And like the gram matrix, right? The similarity matrix, what is it, right?

You're in feature space and it's for every pair of objects I, j, the dot product of their feature representations, right? And now, if you embed all your objects, we already know who do that, like, know, there's a matrix with the embedding vector for FJ object, whether it's a word or a token or anything else. Right?

Now I can do the dot product of the embeddings of two objects. And let's suppose that all unit vectors to keep things simple, right? And now what happens is that if I For the moment, let's say you're not even learning the embeddings yet, right?

Let's say you just have random vectors, Your embeddings are random. That's actually already useful for a lot of things, but of course it's not where the action is. And now there's the following very interesting property which is the dot product of a vector with itself is one.

But the dot product of two random vectors in a high dimensional space is approximately zero. So your gram matrix, your similarity matrix will be approximately the identity matrix. Okay?

And now what happens is like, if I have a tensor logic that operates in this way and then it has something like non linearity, right, then what's going to happen is that it's going to clean up that noise and it turns into the identity matrix. Right? And now I have all these rules that are just operating in a purely logical mode.

Right? It's boolean tenses going in, meaning relations, right? And it's boolean tenses going out.

Right? So that way you can do pure deduction in embedding space with these random embedding vectors. Right?

That's why I did something interesting. But now, let's say you learn the embeddings, which of course is the whole point, right? When you learn the embeddings what's going to happen by trying to minimize the loss function is that the embedding vectors of objects about which you tend to make the same inferences will get closer.

Right? Because if I'm saying something about one object and that one is similar, this like, you know, the gradient descent to minimize the loss is going to make them, you know, is going to increase their dot product. So you're going to wind up with a similarity matrix that has high values for objects that are quite similar, right, in the limit one in the diagonal, and has low values for objects that are quite dissimilar.

And now, if you turn the temperature parameter, meaning the stiffness of the sigmoid, right, at one extreme at zero temperature you have a step function. And and and and and the similarity make is is this criticized back to zero one. So so at the zero temperature extreme you have pure deduction.

more than one thing you can do with Tensor Logic, one of them is you can just reimplement existing things like MLPs and transformers and whatnot. And if all that you did was reimplement them, it will have all their pros and cons, right? It's the same thing, just implemented much more elegantly, blah blah, right?

What I'm talking about here, and talk about in that section of that paper, is doing something different. It's not an MLP, it's not a transformer. It's actually doing these things of like, you embed objects, you embed relations in a certain way that falls from the object, you embed the rules, you embed the reasoning, right?

So this is a different process. What this different process allows you to do is that when you raise the temperature, you get to do analogical reasoning. You know, Douglas Alstadter came up before.

Douglas Alstadter, I think, would like this because it's an analogical He has this whole 500 page book arguing that all of cognition is just analogy. Right? And again, this is one of the schools of thought, like this is one of the tribes in the master algorithm is reasoning by analogy.

You do reasoning by analogy because what happens is you generalize from one object to an object that has a high dot product with it. So now I get to borrow inferences from similar objects. And the higher the temperature the looser the inferences, the more analogical inference can be.

But for example, and again Douglas goes into this in some of his book, and any mathematician like, know, Terrence Tard the other day, I just heard him say this, right, is like, mathematicians reason by analogy. They notice similarities between things. But at the end of the day you need to have a proof.

Intensive logic in this scheme, in this particular scheme of embedding, know, reasoning in embedding space, this is just simulated annealing. You start out with a high temperature being very analogical and then you lower it, at the end of the day you have a proof. It's a deductive proof that is guaranteed to be correct.

But you couldn't have gotten to it because the search base is so large without the analogical part. Right?

Oh, it's not so again I skipped over some steps here. It isn't. Right?

So analogy so the the most powerful type of analogy so kernel machines in some sense are the least powerful type of analogy. It's just, oh, here's a similarity or nearest neighbor, right? I have a distance function.

That's not really where the action is. The action is in what is called structure mapping. Right?

Structure mapping was this thing proposed by Dederich Gendner, where you solve a problem by mapping its structure to the structure of problems that you know, right? And the canonical example is Niels Bohr's model of the atom which he came up with by an analogy between an atom and the solar system. The nucleus is the sun, planets are the atoms.

Turns out to be a bad analogy but it was crucial in the development of physics, right? And there's also this whole subfield of AI called case based reasoning, where I'm a help desk, you come up with a problem, and I don't try to solve it from scratch because I don't need to, that would be wasteful. I go to my database of similar cases and I find one, and then I tweak it.

So structure mapping is an extraordinarily powerful thing, but it's this combination of similarity and compositionality which kernel machines per se don't have. But tensor logic does. The point in tensor logic is that you do have all the parts of the kernel machines, but all the compositionality of the symbolic AI, So again, the the structure mapping just, you know, just comes out of the box.

You don't need to do anything more to add structure mapping, and all the power of analogical reasoning comes with it. Yeah. Can I suggest, a good analogy is to ad lib?

Do you think that's fair? It's like you've got the general structure there and you can plug in parts into the blank spaces and you get, you know, a solution. Right?

That's one mode in which things can function. Right? You can also Okay.

So yeah. The the whole process of structure mapping or of case based solution can actually be very rich. I can combine, for example, two big pieces, but like that's one example.

Yeah.

and and deduction or determine you know, determinism because in my mind, you know, I can set the temperature to GPT to zero and it still hallucinates. And I can I can have a I can have a poor deductive system that hallucinates all kinds of things? So to me, like, those are separate separate problems.

Like where what was I just kind of misunderstood. No. Very good.

So precisely the problem or one of the problems with GPT is that it it hallucinates even when you set the temperature to zero. What the hell? Right?

I want to have a mode. Yeah. Yeah.

Right? Not I, but like every Fortune 500 company, if it's going to use AI needs to have a mode where the logic of the business is just to bathe. The security isn't violated, the customer doesn't get light to it, We got to have that orient in that they will not take off, right?

And transformers can't do that. Tensor logic can do that precisely because in this reasoning innovating space mode that I just described, if you set the temperature to zero it does purely deductive reasoning. And by the way, the temperature can be different for each rule.

And I think this is what almost all applications are going to have is like there are some rules that are the mathematical truths or logic that you must guarantee will not be violated, they are the laws, right? And those have infinite temperature. And then there's all these others that are more qualitative reasoning and like more accumulating evidence, maybe stuff that you mined from the web, and those you know, those will have lower, those will have higher temperature.

And that temperature parameter you know can be learned in some rules and not others, right? So now you have this whole spectrum between the deductive and the more you know, or even fantasizing like truly hallucinating at the far end of the high temperature, right? But precisely the point that I'm making in the paper is that, you know, LLMs, the best thing you can get at zero temperature is still a lot of hallucinations, then there's things like rag, but all they do is retrieve and even then you still hallucinate, right?

Compare, you know, tensor logic in this mode with rag, right? And it doesn't just retrieve things, it computes the deductive closure of your knowledge which is an exponentially more powerful thing to have. Right?

And with zero hallucinations.

But let's let's make some distinctions here, right? The only claim I'm making here, because it's the only one you can make, is that sensor logic at zero temperature in this mode will give you the soundness properties that logic has, right? Soundness in the technical sense of soundness.

All that means is that you only reach conclusions that truly logically follow from the premises. You don't say anything about whether the premises are valid or not. If the premises were hallucinated, so will the conclusions be, right?

There's like, there's no magic there, right? But that is a very important property to have. Again, if I give to a transformer a bunch of, you know, true facts, it still hallucinates.

And that's what I can guarantee will not happen in tensor logic. Now coming up with the true facts, well that's a different part of the game. You can write them down, you can learn them, you can refine them, you never know for sure if you have the perfect model, and of course that's more the machine learning and knowledge acquisition part, right?

So I do I think have a very important guarantee here of non hallucination but it's not a guarantee that the, you know, that the model that you're working on came from the real world. That's a whole other neck of the woods. Who's going to adopt this first?

How are we going to bootstrap this as a community? How do you see this progressing? Very good.

So you know the last section in the paper is discussing adoption and what needs to happen and things like that. Let's suppose that everybody agrees that Tensor Logic is a beautiful perfect language and what we need for AI. Just for that reason, that would not be enough to make it take off, sadly, right?

Because you know there's a very, you know, people are still using COBOL these days, right? I risked my case, right? So legacy, there's this irony in computer science or in the information technology, it moves faster than anything else, but at the same time, you know, things never die, right?

You can't kill them, you can't kill COBOL, right? And I really do believe, I like Python, I program in Python, it's very nice in many ways, better than Fortran for some things, etc, right? Even though it was never or NumPy if you will, but you get the point, right?

It's like, for AI it's just a terrible thing. But like, I'm a Python programmer, like, I, you know, general is like, okay, your tensor logic is nice, I'm not going to rewrite all my code, Right? Forget that.

Right? So what is going to make it happen? Right?

But now, we can look at what has made this happen in the past. Right? And it's several things.

One is that, for example, look at how Java took off. Right? Java took off at the time of the Internet because it was the language of networking.

Allegedly, you could debate that, but like people wanted to do things that it was very hard to do with you know things like C and blah blah and C, right? And so Java took off. Right?

And we are in exactly It was the language of embedded programs and web That was the only option we had. Exactly, right? So, and you know, there's big arguments about this, but not relevant to us here.

The point I'm trying to make here is, we are precisely at a also very relevant. Why did languages like Lis and ProGod, you know, fall out, right? Because they were better for AI than, you know, Fortran or C or whatever, right?

Or Java, is that like, they were niche languages. And the network effects of the more widely used languages and all their aspects just completely overrode that, right? We understand that very well now, people did in the eighties.

But now, we're in a different ballgame now, now the big technology, the center of everything is AI, Right? If you have a better language for AI, that is the one that is gonna, you know, have the biggest users. And moreover, if you have a language that solves the big pains, right?

To adopt a new language or a new anything, right? You know, like a new app, right? It needs solve some big pain, right?

Is there a big pain that Tensor Logic solves? Well hell yeah! It solves hallucination, potentially, okay?

It solves all this is subject to empirical verification, but it potentially solves hallucination. It solves the opacity. Right?

Like, we're in this world right now where there's like multi billion corporations and systems that are like, they're driven by this black box. And nobody I've talked with CEOs of big tech companies that say like, you know, I can't sleep at night I don't know what this thing is going to do and the people who trained it have left the company and who knows it, right? So if we can make a dent in that, people will converge to it very very quickly.

Also I think when people have the experience of how easy it is to use Tensor Logic compared to the big pile of stuff that lies under PyTorch and whatnot, I think they will actually be very motivated to migrate very quickly. And then you know like, there are several things, there's like developing the open source community and vendor competition and whatnot. But, you know, like, there's a couple of other important things here, one of which is the following: Tensor logic is ideally suited for AI education.

It's one language in which you which has very little, you know, extraneous stuff and you can just and can teach the entire gamut of AI very well and do the exercise. It'll be a language that the professors, the TAs, and the students will like. Right?

And history shows, you know, going back to things like Unix that if you have something like that that ticks off in computer science education, then people go to industry and say like, I want to use this because it's what's good, it's what I like. And a generation later it's what everybody is using. And one more thing is the following.

The transition to tensor logic from Python doesn't have to happen all at once, right? You can have, for example, and I already have actually, another half, again because it's very easy to do, right? You read the paper and you do that in the next whatever thirty minutes.

You can write a preprocessor that just converts tensor equations into Python. And again, all it does is a one to one mapping between the syntax of tensor logic and EINSUM, right? Then making things efficiently as we discussed is another matter, but from this point of view of developer, you know, uptake, right, all it does, and there's a long history of people doing this with different languages, right, it's like you have a preprocessor that lets you write in some equations, but then it converts you, it converts those equations into into PyTorch or Python or just NumPy, let's say Python, right, and then you do everything else in Python that you did before.

Tensor logic like Prolog and Datalog has actually both declarative and procedural semantics. You can look at a tensor equation, that's actually the whole beauty of logic programming in some sense is that like, you can look at a tensor logic equation as like it's an equation. It's like Einstein's equation, it's the statement about the world.

But you can also look at it and treat it as a function call. The left hand side is the call and the right hand side is the body which is a bunch of other calls and the way to combine them. Right?

So so you can and in fact, you know, most of the time, I mean, my experience that I've used Tensor Logic so far, tend to use it in procedural mode, right? It's a set of equations, it's a bunch of statements, like you would have in any imperative language. Right?

So, you know, very important to have, to bear that in mind. Now, but to the heart of your question, which I think is very important, when you're teaching people something, I mean I would actually say this is the tragedy of computer science education. From high school to intro courses to the most advanced things, is that you want to teach them the beauty of what you can do and the essence of the algorithms and so on.

But then you and particularly they, the students, they spend all their time bogged down in all this crap. All these details where you get the semicolon wrong and the program doesn't work anymore, and they hate it. And they decide that computer science is not for them, or at best they waste 10 times more time than they should, right?

So precisely the whole point of having the right abstraction is to avoid that. Now, so I would say this is one of the best features of tensor logic is to do that for AI. Now, but you also say correctly that like, well a lot of the time you need to go beyond that level of abstraction and for example, from a point of view of inefficiency and so on, and a lot of things, right?

But I would say, and again, you know, we won't know until Tensor Logic is used widely and we'll see what happens, but tensor logic is a language that at some level, it's C, right? It's very low level. The beauty in my mind, again this gets back to multiple as a like, you can use it to say very high level things.

You can also to express the lowest level possible computations, right? Like a tensor equation is something that you can map onto a GPU with almost no change, right? And then optimize the heck out of, right?

Like in fact, you know, I've joked like with folks at NVIDIA that you know, CUDA is a nice mode, but Tensor Logic could be the end of that mode.

this is super exciting, right? This is what we've worked all our lives towards, right? It's like the dream is happening.

I used to tell people, you know, when I went into grad school, one day machine learning is going to take over the world and be like, what? And I'm like, see, it is taking over the world, there, take that. So, on a more serious note, like, transformers are a great leap forward and you know, anybody who's used a chatbot is like, wow, look at the things this can do, This is great.

But at the same time, the sheer amount of like wastefulness and stupidity and ignorance going on is just unbelievable. It's like, why are you reinventing Why? For example, I've talked with people at, for example, OpenAI that do the reasoning.

And many of them are very good people, so I'm not trying to, you know, pick on anybody. But it's like, oh, what is reasoning? We need to figure that out.

And like, and then they say a bunch of stuff that is completely wrong. And I'm thinking to myself, why don't you spend an afternoon reading a couple of chapters of Russell and Norvig and save a $100,000,000,000 in wasted compute? Please, just do that, right?

And in a way, you know, part of what I'm trying to do with Tensor is make things go direction because the current direction is just too damn painful. And it's not just that it's painful, this is going to end badly. Right?

People are spending all right? In Oh a way, like, you know, spending all this money on data centers is not wasted because it's not like the fiber that went dark. Right?

We in AI have an appetite for unlimited compute. Right? But they're spending all this money prematurely on stuff that isn't ready for that yet.

Right? The demand is probably not going to be there. And we're going to look back onto that and go like, wow, 99.

9% of that compute was completely wasted because of a lot of the reasons that we've been talking about including like, you didn't know how to do reasoning, so you brute forced it, etcetera etcetera. So like, know, we got to change the direction of this ship.

Exactly.