Let's start with a few examples (borrowed from here) to illustrate what we're talking about:

• We have a gas consisting of some huge number of particles. We throw away information about the particles themselves, instead keeping just a few summary statistics: average energy, number of particles, etc. We can then make highly precise predictions about things like e.g. pressure just based on the reduced information we've kept, without having to think about each individual particle. That reduced information is the "abstract layer" - the gas and its properties.
• We have a bunch of transistors and wires on a chip. We arrange them to perform some logical operation, like maybe a NAND gate. Then, we throw away information about the underlying details, and just treat it as an abstract logical NAND gate. Using just the abstract layer, we can make predictions about what outputs will result from what inputs. Note that there’s some fuzziness - 0.01 V and 0.02 V are both treated as logical zero, and in rare cases there will be enough noise in the wires to get an incorrect output.
• I tell my friend that I'm going to play tennis. I have ignored a huge amount of information about the details of the activity - where, when, what racket, what ball, with whom, all the distributions of every microscopic particle involved - yet my friend can still make some reliable predictions based on the abstract information I've provided.
• When we abstract formulas like "1+1=2*1" and "2+2=2*2" into "n+n=2*n", we're obviously throwing out information about the value of n, while still making whatever predictions we can given the information we kept. This is what abstraction is all about in math and programming: throw out as much information as you can, while still maintaining the core "prediction" - i.e. the theorem or algorithm.
• I have a street map of New York City. The map throws out lots of info about the physical streets: street width, potholes, power lines and water mains, building facades, signs and stoplights, etc. But for many questions about distance or reachability on the physical city streets, I can translate the question into a query on the map. My query on the map will return reliable predictions about the physical streets, even though the map has thrown out lots of info.

The general pattern: there’s some ground-level “concrete” model (or territory), and an abstract model (or map). The abstract model throws away or ignores information from the concrete model, but in such a way that we can still make reliable predictions about some aspects of the underlying system.

Notice that the predictions of the abstract models, in most of these examples, are not perfectly accurate. We're not dealing with the sort of "abstraction" we see in e.g. programming or algebra, where everything is exact. There are going to be probabilities involved.

In the language of embedded world-models, we're talking about multi-level models: models which contain both a notion of "table", and of all the pieces from which the table is built, and of all the atoms from which the pieces are built. We want to be able to use predictions from one level at other levels (e.g. predict bulk material properties from microscopic structure, or predict from material properties whether it's safe to sit on the table), and we want to move between levels consistently.

Formalization: Starting Point

To repeat the intuitive idea: an abstract model throws away or ignores information from the concrete model, but in such a way that we can still make reliable predictions about some aspects of the underlying system.

So to formalize abstraction, we first need some way to specify which "aspects of the underlying system" we wish to predict, and what form the predictions take. The obvious starting point for predictions is probability distributions. Given that our predictions are probability distributions, the natural way to specify which aspects of the system we care about is via a set of events or logic statements for which we calculate probabilities. We'll be agnostic about the exact types for now, and just call these "queries".

That leads to a rough construction. We start with some concrete model $M_C$ and a set of queries $Q$. From these, we construct a minimal abstract model $M_A$ by keeping exactly the information relevant to the queries, and throwing away all other information. By the minimal map theorems, we can represent $M_A$ directly by the full set of probabilities $P\left[Q|M_C\right]$; $M_A$ and $P\left[Q|M_C\right]$ contain exactly the same information. Of course, in practical examples, the probabilities $P\left[Q|M_C\right]$ will usually have some more compact representation, and $M_A$ will usually contain some extraneous information as well.

To illustrate a bit, let's identify the concrete model, class of queries, and abstract model for a few of the examples from earlier.

• Ideal Gas:
• Concrete model $M_C$ is the full set of molecules, their interaction forces, and a distribution representing our knowledge about their initial configuration.
• Class of queries $Q$ consists of combinations of macroscopic measurements, e.g. one query might be "pressure = 12 torr & volume = 1 m^3 & temperature = 110 K".
• For an ideal gas, the abstract model $M_A$ can be represented by e.g. temperature, number of particles (of each type if the gas is mixed), and container volume. Given these values and assuming a near-equilibrium initial configuration distribution, we can predict the other macroscopic measurables in the queries (e.g. pressure).
• Tennis:
• Concrete model $M_C$ is the full microscopic configuration of me and the physical world around me as I play tennis (or whatever else I do).
• Class of queries $Q$ is hard to sharply define at this point, but includes things like "John will answer his cell phone in the next hour", "John will hold a racket and hit a fuzzy ball in the next hour", "John will play Civ for the next hour", etc - all the things whose probabilities change on hearing that I'm going to play tennis.
• Abstract model $M_A$ is just the sentence "I am going to play tennis".
• Street Map:
• Concrete model $M_C$ is the physical city streets
• Class of queries $Q$ includes things like "shortest path from Times Square to Central Park starts by following Broadway", "distance between the Met and the Hudson is less than 1 mile", etc - all the things we can deduce from a street map.
• Abstract model $M_A$ is the map. Note that the physical map also includes some extraneous information, e.g. the positions of all the individual atoms in the piece of paper/smartphone.

Already with the second two examples there seems to be some "cheating" going on in the model definition: we just define the query class as all the events/logic statements whose probabilities change based on the information in the map. But if we can do that, then anything can be an "abstract map" of any "concrete territory", with the queries $Q$ taken to be the events/statements about the territory which the map actually has some information about - not a very useful definition!

Natural Abstractions

Intuitively, It seems like there exist "natural abstractions" - large sets of queries on a given territory which all require roughly the same information. Statistical mechanics is a good source of examples - from some macroscopic initial conditions, we can compute whatever queries we want about any macroscopic measurements later on. Note that such natural abstractions are a property of the territory - it's the concrete-level model which determines what large classes of queries can be answered with relatively little information.

For now, I'm interested primarily in abstraction of causal dags - i.e. cases in which both the concrete and abstract models are causal dags, and there is some reasonable correspondence between counterfactuals in the two. In this case, the set of queries should include counterfactuals, i.e. do() operations in Pearl's language. (This does require updating definitions/notation a bit, since our queries are no longer purely events, but it's a straightforward-if-tedious patch.) That's the main subject I'm researching in the short term: what are the abstractions which support large classes of causal counterfactuals? Expect more posts on the topic soon.