A computational perspective on observation and cognition

Hello and welcome everyone. This is Active Inference Math Stream 9.1 on March 5th, 2024. We're here with Jonathan Gorard and we'll be discussing a variety of topics yet to be determined, or are they? So thank you for joining and to you for any introduction and we'll really look forward to everyone's comments and questions. So thanks again for joining, to you. Okay, well yeah, thanks very much, Daniel, for the introduction and for inviting me to be here on Active Inference. I'm looking forward to a very, very fun discussion. So I don't have anything especially prepared to talk about, which is probably a good thing because it means we'll be able to extend the unstructured part of this for as long as possible. But I think just to give a little bit of context, I want to talk about an area where I think some things that I've been working on, some collaborators that might have been working on, that might have some kind of intersection of interest with things that, you know, active inference type people might care about, right? So, and in particular, that concerns the relationship between kind of computation, observation and cognition, and specifically using methods that come from category theory and topos theory and some other kind of branches of mathematics and theoretical computer science, to understand the relationship between system, specifically the computational and algorithmic complexity of systems, versus the computational algorithmic complexity of observers of those systems, and how those things trade off between each other. So, so just to give a little bit of context to that, I want to show, these are just some visuals from a paper that I put out about a year ago now, and that this kind of really defines this research program that I've been working on for the last year and a half in some form or another, which is looking at exactly this trade off using category theory theory. So, here's a specification of a Turing machine. This is just a simple deterministic computation. It's saying, you know, you have a Turing machine that has this head state and this tape state, and on the next step, you're going to replace the tape state with something that looks like this, the head state with something that looks like that, and you're going to scroll the Turing machine head left, or in this case, scroll it right, etc. So, this is just a, you know, specification of a very simple computation. I think this is a two-state, two-color Turing machine on a simple, you know, one-dimensional tape. It's about as simple a computation as you could define. So, if you run that thing for some initial condition, you'll get an evolution that looks like this. And so, right now, this is just a purely deterministic, you know, single-path evolution. But from this, we can construct, we can build a mathematical structure, namely, we can build a category. So, and the rules for how we build that category are very simple. So, you know, each arrow here is some simple computation, some application of the Turing machine transition function. And then what we can do is we can say, well, anytime we have two arrows that are laid end-to-end like this, we can compose them together to create a third arrow that goes like that. I may even have a picture. Yes, like this. So, you know, we have a computation F that takes us from X to Y, a computation G that takes us from Y to Z, and then we obtain a composite computation G compose F that takes us directly from X to Z. And we also add some additional edges, some additional arrows on each state itself, a sort of identity, an identity operation that maps a computational state directly to itself. And so this, combined with some axioms of associativity and identity, forms a category of elementary computations. So this is a very, very simple example. But what I want to try and build up towards and kind of pump your intuition for is a category which I call comp, which is a category whose objects are essentially the class of all data…