Public Static: What is Abstraction?
post by johnswentworth · 20200609T18:36:49.838Z · LW · GW · 17 commentsContents
Formalization: Starting Point Information About Things “Far Away” Systems View Causality Exact Abstraction Summary Appendix: System Formulation Proof None 17 comments
Author’s Note: Most of the posts in this sequence [? · GW] are essentially a log of workinprogress. This post is intended as a more presentable (“public”) and higherconfidence (“static”) writeup of some formalizations of abstraction. Much of the material has appeared in other posts; the first two sections in particular are drawn almost verbatim from the opening “What is Abstraction?” post.
Let's start with a few examples (borrowed from here [LW · GW]) 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 groundlevel “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 worldmodels [? · GW], we're talking about multilevel 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 and/or macroscopic measurements, 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 lowlevel model and a set of queries . From these, we construct a minimal highlevel model by keeping exactly the information relevant to the queries, and throwing away all other information. By the minimal map theorems [LW · GW], we can represent directly by the full set of probabilities ; and contain exactly the same information. Of course, in practical examples, the probabilities will usually have some more compact representation, and will usually contain some extraneous information as well.
To illustrate a bit, let's identify the lowlevel model, class of queries, and highlevel model for a few of the examples from earlier.
 Ideal Gas:
 Lowlevel model is the full set of molecules, their interaction forces, and a distribution representing our knowledge about their initial configuration.
 Class of queries 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 highlevel model 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 nearequilibrium initial configuration distribution, we can predict the other macroscopic measurables in the queries (e.g. pressure).
 Tennis:
 Lowlevel model 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 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.
 Highlevel model is just the sentence "I am going to play tennis".
 Street Map:
 Lowlevel model is the physical city streets
 Class of queries 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.
 Highlevel model 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 a "highlevel map" of any "lowlevel territory", with the queries taken to be the events/statements about the territory which the map actually has some information about  not a very useful definition!
Information About Things “Far Away”
In order for abstraction to actually be useful, we need some efficient way to know which queries the abstract model can accurately answer, without having to directly evaluate each query within the lowlevel model.
In practice, we usually seem to have a notion of which variables are “far apart”, in the sense that any interactions between the two are mediated by many inbetween variables.
The mediating variables are noisy, so they wipe out most of the “finegrained” information present in the variables of interest. We can therefore ignore that finegrained information when making predictions about things far away. We just keep around whatever highlevel signal makes it past the noise of mediating variables, and throw out everything else, so long as we’re only asking questions about faraway variables.
An example: when I type “4+3” in a python shell, I think of that as adding two numbers, not as a bunch of continuous voltages driving electric fields and current flows in little patches of metal and doped silicon. Why? Because, if I’m thinking about what will show up on my monitor after I type “4+3” and hit enter, then the exact voltages and current flows on the CPU are not relevant. This remains true even if I’m thinking about the voltages driving individual pixels in my monitor  even at a fairly low level, the exact voltages in the arithmeticlogic unit on the CPU aren’t relevant to anything more than a few microns away  except for the highlevel information contained in the “numbers” passed in and out. Information about exact voltages in specific wires is quickly wiped out by noise within the chip.
Another example: if I’m an astronomer predicting the trajectory of the sun, then I’m presumably going to treat other stars as pointmasses. At such long distances, the exact mass distribution within the star doesn’t really matter  except for the highlevel information contained in the total mass, momentum and centerofmass location.
Formalizing this in the same language as the previous section:
 We have some variables and in the lowlevel model.
 Interactions between and are mediated by noisy variables .
 Noise in wipes out most finegrained information about , so only the highlevel summary is relevant to .
Mathematically: for any which is “not too close” to  i.e. any which do not overlap with (or with itself). Our highlevel model replaces with , and our set of valid queries is the whole joint distribution of given .
Now that we have two definitions, it’s time to start the Venn diagram of definitions of abstraction.
So far, we have:
 A highlevel model throws out information from a lowlevel model in such a way that some set of queries can still be answered correctly: .
 A highlevel model throws out information from some variable in such a way that all information about “far away” variables is kept: .
Systems View
The definition in the previous section just focuses on abstracting a single variable . In practice, we often want to take a systemlevel view, abstracting a whole bunch of lowlevel variables (or sets of lowlevel variables) all at once. This doesn’t involve changing the previous definition, just applying it to many variables in parallel.
Rather than just one variable of interest , we have many lowlevel variables (or nonoverlapping sets of variables) and their highlevel summaries . For each of the , we have some set of variables “nearby” , which mediate its interactions with everything else. Our “faraway” variables Y are now any faraway ’s, so we want
for any sets of indices and which are “far apart”  meaning that does not overlap any or .
(Notation: I will use lowercase indices like for individual variables, and uppercase indices like to represent sets of variables. I will also treat any single index interchangeably with the set containing just that index.)
For instance, if we’re thinking about wires and transistors on a CPU, we might look at separate chunks of circuitry. Voltages in each chunk of circuitry are , and summarizes the binary voltage values. are voltages in any components physically close to chunk on the chip. Anything physically far away on the chip will depend only on the binary voltage values in the components, not on the exact voltages.
The main upshot of all this is that we can rewrite the math in a cleaner way: as a (partial) factorization. Each of the lowlevel components are conditionally independent given the highlevel summaries, so:
This condition only needs to hold when picks out indices such that (i.e. we pick out a subset of the ’s such that no two are “close together”). Note that we can pick any set of indices which satisfies this condition  so we really have a whole family of factorizations of marginal distributions in which no two variables are “close together”. See the appendix to this post for a proof of the formula.
In English: any set of lowlevel variables which are all “far apart” are independent given their highlevel summaries . Intuitively, the picture looks like this:
We pick some set of lowlevel variables which are all far apart, and compute their summaries . By construction, we have a model in which each of the highlevel variables is a leaf in the graphical model, determined only by the corresponding lowlevel variables. But thanks to the abstraction condition, we can independently swap any subset of the summaries with their corresponding lowlevel variables  assuming that all of them are “far apart”.
Returning to the digital circuit example: if we pick any subset of the wires and transistors on a chip, such that no two are too physically close together, then we expect that their exact voltages are roughly independent given the highlevel summary of their digital values.
We’ll add this to our Venn diagram as an equivalent formulation of the previous definition.
I have found this formulation to be the most useful starting point in most of my own thinking, and it will be the jumpingoff point for our last two notions of abstraction in the next two sections.
Causality
So far we’ve only talked about “queries” on the joint distribution of variables. Another natural step is to introduce causal structure into the lowlevel model, and require interventional queries to hold on far apart variables.
There are some degrees of freedom in which interventional queries hold on far apart variables. One obvious answer is “all of them”:
… with the same conditions on as before, plus the added condition that the indices in and also be far apart. This is the usual requirement in math/programming abstraction, but it’s too strong for many realworld applications. For instance, when thinking about fluid dynamics, we don’t expect our abstractions to hold when all the molecules in a particular cell of space are pushed into the corner of that cell. Instead, we could weaken the lowlevel intervention to sample from lowlevel states compatible with the highlevel intervention:
We could even have lowlevel interventions sample from some entirely different distribution, to reflect e.g. a physical machine used to perform the interventions.
Another post will talk more about this, but it turns out that we can say quite a bit about causal abstraction while remaining agnostic to the details of the lowlevel interventions. Any of the above interventional query requirements have qualitativelysimilar implications, though obviously some are stronger than others.
In daytoday life, causal abstraction is arguably more common than noncausal. In fully deterministic problems, validity of interventional queries is essentially the only constraint (though often in guises which do not explicitly mention causality, e.g. functional behavior or logic). For instance, suppose I want to write a python function to sort a list. The only constraint is the abstract input/output behavior, i.e. the behavior of the designated “output” under interventions on the designated “inputs”. The lowlevel details  i.e. the actual steps performed by the algorithm  are free to vary, so long as those highlevel interventional constraints are satisfied.
This generalizes to other design/engineering problems: the desired behavior of a system is usually some abstract, highlevel behavior under interventions. Lowlevel details are free to vary so long as the highlevel constraints are satisfied.
Exact Abstraction
Finally, one important special case. In math and programming, we typically use abstractions with sharper boundaries than most of those discussed here so far. Prototypical examples:
 A function in programming: behavior of everything outside the function is independent of the function’s internal variables, given a highlevel summary containing only the function’s inputs and outputs. Same for private variables/methods of a class.
 Abstract algebra: many properties of mathematical objects hold independent of the internal details of the object, given certain highlevel summary properties  e.g. the group axioms, or the ring axioms, or …
 Interfaces for abstract data structures: the internal organization of the data structure is irrelevant to external users, given the abstract "interface"  a highlevel summary of the object's behavior under different inputs (a.k.a. different interventions).
In these cases, there’s no noisy intermediate variables, and no notion of “far away” variables. There’s just a hard boundary: the internal details of highlevel abstract objects do not interact with things of interest “outside” the object except via the highlevel summaries.
We can easily cast this as a special case of our earlier notion of abstraction: the set of noisy intermediate variables is empty. The “highlevel summary” of the lowlevel variables contains all information relevant to any variables outside of themselves.
Of course, exact abstraction overlaps quite a bit with causal abstraction. Exact abstractions in math/programming are typically deterministic, so they’re mainly constrained by interventional predictions rather than distributional predictions.
Summary
We started with a very general notion of abstraction: we take some lowlevel model and abstract it into a highlevel model by throwing away information in such a way that we can still accurately answer some queries. This is extremely general, but in order to actually be useful, we need some efficient way to know which queries are and are not supported by the abstraction.
That brought us to our next definition: abstraction keeps information relevant to “far away” variables. We imagine that interactions between the variabletobeabstracted and things far away are mediated by some noisy “nearby” variables , which wipe out most of the information in . So, we can support all queries on things far away by keeping only a relatively small summary .
Applying this definition to a whole system, rather than just one variable, we find a clean formulation: all sets of farapart lowlevel variables are independent given the corresponding highlevel summaries.
Next, we extended this to causal abstraction by requiring that interventional queries also be supported.
Finally, we briefly mentioned the special case in which there are no noisy intermediate variables, so the abstraction boundary is sharp: there’s just the variables to be abstracted, and everything outside of them. This is the usual notion of abstraction in math and programming.
Appendix: System Formulation Proof
We start with two pieces. By construction, is calculated entirely from , so
(construction)
… without any restriction on which subsets of the variables we look at. Then we also have the actual abstraction condition
(abstraction)
… as long as does not overlap or .
We want to show that
… for any set of nonnearby variables (i.e. ). In English: sets of farapart lowlevel variables are independent given their highlevel counterparts.
Let’s start with definitions of “farapart” and “nearby”, so we don’t have to write them out every time:
 Two sets of indices and are “far apart” if and do not overlap , and viceversa. Individual indices can be treated as sets containing one element for purposes of this definition  so e.g. two indices or an index and a set of indices could be “far apart”.
 Indices and/or sets of indices are “nearby” if they are not far apart.
As before, I will use capital letters for sets of indices and lowercase letters for individual indices, and I won’t distinguish between a single index and the set containing just that index.
With that out of the way, we’ll prove a lemma:
… for any far apart from , both far apart from and (though and need not be far apart from each other). This lets us swap highlevel with lowlevel given variables as we wish, so long as they’re all far apart from each other and from the query variables. Proof:
(by construction)
(by abstraction)
(by construction)
By taking and then marginalizing out unused variables, this becomes
That’s the first half of our lemma. Other half:
(by Bayes)
(by first half)
(by Bayes)
That takes care of the lemma.
Armed with the lemma, we can finish the main proof by iterating through the variables inductively:
(by Bayes)
(by construction)
(by lemma)
(by Bayes)
(by lemma & cancellation)
(by Bayes)
Here , , and are all far apart. Starting with empty and applying this formula to each variable , onebyone, completes the proof.
17 comments
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Planned summary for the Alignment Newsletter:
If we are to understand embedded agency, we will likely need to understand abstraction (see <@here@>(@Embedded Agency via Abstraction@)). This post presents a view of abstraction in which we abstract a lowlevel territory into a highlevel map that can still make reliable predictions about the territory, for some set of queries (whether probabilistic or causal).
For example, in an ideal gas, the lowlevel configuration would specify the position and velocity of _every single gas particle_. Nonetheless, we can create a highlevel model where we keep track of things like the number of molecules, average kinetic energy of the molecules, etc which can then be used to predict things like pressure exerted on a piston.
Given a lowlevel territory L and a set of queries Q that we’d like to be able to answer, the minimalinformation highlevel model stores P(Q  L) for every possible Q and L. However, in practice we don’t start with a set of queries and then come up with abstractions, we instead develop crisp, concise abstractions that can answer many queries. One way we could develop such abstractions is by only keeping information that is visible “far away”, and throwing away information that would be wiped out by noise. For example, when typing 3+4 into a calculator, the exact voltages in the circuit don’t affect anything more than a few microns away, except for the final result 7, which affects the broader world (e.g. via me seeing the answer).
If we instead take a systems view of this, where we want abstractions of multiple different lowlevel things, then we can equivalently say that two faraway lowlevel things should be independent of each other _when given their highlevel summaries_, which are supposed to be able to quantify all of their interactions.
Planned opinion:
I really like the concept of abstraction, and think it is an important part of intelligence, and so I’m glad to get better tools for understanding it. I especially like the formulation that lowlevel components should be independent given highlevel summaries  this corresponds neatly to the principle of encapsulation in software design, and does seem to be a fairly natural and elegant description, though of course abstractions in practice will only approximately satisfy this property.
I wonder whether it'd be useful to distinguish between the following things.
Consider the example of the street map. is the exact same thing as except that there is detail removed (in a loose sense at least; in practice the map will probably have small differences compared to the territory).
Now consider the example of an ideal gas. throws away all of the stuff in and replaces it with summary statistics.
Both fit the definition of abstraction because you're removing information but maintaining the ability to answer questions, but in the ideal gas example you're adding something new in. Namely the summary statistics.
(Well, maybe "new" isn't the right word. Maybe it'd be better to say "adding summary statistics". I guess the thing I'm really trying to point at is the fact that something is being added.)
For the street map example, is the physical city streets  which means it's the molecules/atoms/fields which comprise the streets. When we represent the streets as lines on paper, those lines are summary statistics of molecule positions, just like the ideal gas example. The only difference is that in the ideal gas example, it's a lot easier to express the relevant distribution which the statistics summarize.
That said, I do think you're pointing to something interesting. There is a sense in which a highlevel model adds something in.
Look at the factorizations in the "Systems View" section. They are factorizations of a joint distribution over both the highlevel and lowlevel variables. We have a single model which includes both sets of variables. The highlevel variables are quite literally added into the lowlevel model as new variables computed from the old. The highlevel model then keeps those new variables, and throws away all the original lowlevel variables.
Ah, I think I see what you mean. That makes sense that the high level model of the street map is also a summary statistic, not just the low level model with stuff thrown away. Let my try to refine my comment.
For the ideal gas example, I think of the low level model as looking something like this:
class LowLevelGas {
Particle[] particles;
}
class Particle {
String compound;
int speed;
int direction;
int mass;
// whatever else
}
And I think of the high level model as looking like this:
class HighLevelGas {
int pressure;
int volume;
int temperature;
}
LowLevelGas
and HighLevelGas
just look like there's a big difference between the two. On the other hand, LowLevelStreetMap
and HighLevelStreetMap
wouldn't look as different. It'd be analogous to a sketch vs a photograph, where the difference is sort of a matter of resolution. But with LowLevelGas
and HighLevelGas
, it seems like they are different in a more fundamental way. They have different properties, not the same properties at different resolutions.
I wonder if this "resolution" idea can be made more formal. Something along the lines of looking at the high level variables and low level variables and seeing how... similar?... they are.
Elizer's idea of Thingspace [LW · GW] comes to mind. In theory maybe you could look at how close they are in Thingspace, but in practice that seems really difficult.
I'll explain what I think you're pointing to here, then let me know if it sounds like what you're imagining.
In the street map example, everything is spatially delineated, at both the high and low level. The abstraction can be interpreted as blurring the spatial resolution. The spatial resolution is a conserved structure between the two. But in the gas example, everything is blurred "all the way", so there's just a single highlevel object which doesn't retain any structure which is obviouslysimilar to the lowlevel.
(An intermediate between these two would be Navier Stokes: it applies basically the same abstraction as the ideal gas, but only within small spatiallydelineated cells.)
So there's potentially a difference between abstractions which throw away basically all the structure vs abstractions which retain some.
This points toward a more general class of questions: when, and to what extent, does it all add up to normality [LW · GW]? We learned the highlevel ideal gas laws long before we learned the lowlevel molecular theory, but we knew the lowlevel had to at least be consistent with that highlevel structure. What lowlevel structures did that constraint exclude? More generally: to what extent does our knowledge of the highlevel model structure constrain the possible lowlevel structures?
One good class of structure for these sorts of questions is causal structure: to what extent does highlevel causal structure constrain the possible lowlevel causal structures? I'll probably have a post on that soonish.
So there's potentially a difference between abstractions which throw away basically all the structure vs abstractions which retain some.
Yeah, that's what I'm getting at.
This points toward a more general class of questions: when, and to what extent, does it all add up to normality [LW · GW]? We learned the highlevel ideal gas laws long before we learned the lowlevel molecular theory, but we knew the lowlevel had to at least be consistent with that highlevel structure. What lowlevel structures did that constraint exclude? More generally: to what extent does our knowledge of the highlevel model structure constrain the possible lowlevel structures?
One good class of structure for these sorts of questions is causal structure: to what extent does highlevel causal structure constrain the possible lowlevel causal structures? I'll probably have a post on that soonish.
Doesn't highlevel structure entail statistical averages and not necessarily Boltzmann brains in the lowlevel structure? Like  what of the nonequilibrium statistical mechanics?
Problem is, we didn't know beforehand (i.e. in 1800) that the highlevel things we saw (like temperatures, heat flow, etc) had anything to do with statistical averages. One could imagine an alternative universe running on different physics, where heat really is a fluid and yet macroscopically it behaves a lot like heat in our universe. If we imagine all the difference ways things could have turned out to work, given only what we knew in 1800, where does that leave us? What lowlevel structure is implied by the highlevel structure?
(I have a sense that the answer to this question is in the post but I'm having trouble extracting it out.)
There's something that I think of as composition, and I'm not sure if this fits the definition of abstraction. Consider in the context of programming a User
that has email
and password
properties. We think of User
as an abstraction. It's a thing that is composed of an email and password. I'm not seeing how this fits the definition of abstraction though. In particular, what information is being thrown away? What is the low level model, and what is the high level model?
The User
example demonstrates composition of properties, but you could also have composition of instructions. For example, setPassword
might consist of 1) saltPassword
, 2) hashPassword
and then 3) savePassword
. Here we'd say that setPassword
is an abstraction, but what is the information that is being thrown away?
This is a great question.
Months ago, I had multiple drafts of posts formulating abstraction in terms of transformations on causal models. The two main transformations were:
 Glom together some variables
 Throw out information from a variable
What you're calling composition is, I believe, the first one.
Eventually, I came to the view that it's information throwaway specifically which really characterizes abstraction  it's certainly the part where most of the interesting properties come from. But in the majority of usecases, at least some degree of composition takes place as a sort of preprocessing step.
Looking at your specific examples:
 User looks like it's just a composition, although we could talk about it as a degenerate case of abstraction where all of the information is potentially relevant to things outside the User object itself, so we throw away nothing. That said, a lot of what we do with a User object actually does involve ignoring the information in its fields  e.g. we don't think about emails and passwords when declaring a List<User> type or working with that list. So maybe there's a case to be made that it's an abstraction in the informationthrowaway sense.
 setPassword does throw away information: it throws away the individual steps. To an outside caller, it's just a blackbox function which sets the password; they don't know anything about the internal steps.
So to the extent that these are both "throwing away information", it's in the sense that large chunks of our code treat them as blackboxes and don't look at their internal details. When things do look at their internal details, those things are "close to" the objects, so the abstraction breaks down/leaks  e.g. if something tried to reconstruct the internal steps of setPassword, that would definitely be an example of leaky abstraction.
Hm, it seems to me that there is a distinction between 1) hiding information (or encapsulating it, or making it private), 2) ignoring it, and 3) getting rid of it all together.

For
setPassword
perhaps a programmer who uses this method can't see the internals of what is actually happening (the salting, hashing and storing). They just calluser.setPassword(form.password)
and it does what they need it to do. 
For
User
, in the example you give withList<User>
, maybe we want to count how many users there are, and in doing so we don't care about what properties users have. It could beemail
andpassword
, or it could beusername
anddob
, in the context of counting how many users there are you don't care. However, the inner details aren't actually hidden, you're just choosing to ignore it. 
For ideal gasses, we're getting rid of the information about particles. It's not that it's hidden/private/encapsulated, it's just not even there after we replace it with the summary statistics.
What do you think? Am I misunderstanding something?
And in the case that I am correct about the distinction, I wonder if it's something worth pointing out.
Sounds like the distinction is about where/how we're drawing the abstraction boundaries.
 "Hiding information" suggests that there's some object X with a boundary (i.e. a Markov blanket), and only the summary information is visible outside that boundary.
 "Ignoring information" suggests that there's some other object(s) Y with a boundary around them, and only the summary information about X is visible inside that boundary.
So basically we're defining which variables are "far away" by exclusion in one case (i.e. "everything except blah is far away") and inclusion in the other case (i.e. "only blah is far away"). I could definitely imagine the two having different algorithmic implications and different applications.
As for "getting rid of information", I think that's hiding information plus somehow eliminating our own ability to observe the hidden part. Again, I could definitely imagine that having additional algorithmic implications or applications. (Though this one feels weird for me to think about at all; I usually imagine everything from an external perspective where everything is always observable and immutable.)
Yeah I think your descriptions match what I was getting at.
(Typo: First bullet point under "Ideal Gas" should use instead of .)
Thanks a lot for this compressed summary! As someone who tries to understand your work, but is sometimes lost within your sequence, this helps a lot.
I cannot comment the maths in any interesting way, but I feel that your restricted notion of abstraction  where the highlevel summary capture what's relevant to "far way" variables"  works very well with my intuition. I like that it works with "far away" in time too, for example in abstracting current events as memories for future use.