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Every good regulator of a system must be a model of that system (1970) [pdf] (vub.ac.be)
92 points by ismail on Aug 22, 2019 | hide | past | favorite | 31 comments

This paper is so popular all over the web but nobody has ever managed to follow the proof of its 'theorem'. This guy is a math professor and quantum physicist and he was unable to extract any sense from it: https://johncarlosbaez.wordpress.com/2016/01/27/the-good-reg... He points out that the proof uses terms that aren't even defined. The fact that the authors of the paper are a psychiatrist and, I think a humanities academic, doesn't fill me with confidence that the 'theorem' and its 'proof' are actually those things. It's disquieting to see so many people taking the paper's claim and running with it because it 'resonates'. I know there is also an 'internal model principle' paper which is technically correct but that applies to a specific set of linear systems under specific conditions.

I felt like I followed the proof. The claim is actually very narrow: that to be a perfect regulator of a system you need an exact model. Obviously, lots of devices in our world can regulate things reasonably well with simple models.

For instance, a standard thermostat keeps your house at 20 +/- 2 degrees C. In order to keep it at 20.00000000 exactly, it might need to model every air molecule bouncing around.

The issue I'm having with the claims is that their use of the word "model" seems to connote exactly of the opposite of what someone reading the paper without seeing their definition would suggest, and indeed what other commenters here are taking it to mean.

The colloquial restatement of the theorem suggests a regulator "is a model of [a] system" if the regulator's action is a function of the system, and nothing else. As in, a constant regulator that has only one state still meets the criteria for being a "model". The possibility that is excluded is that the regulator has additional details that don't correspond to states of the system.

In your thermostat example, the theorem doesn't even begin to say that, even though that's what the title suggests, and what everyone wants to believe. In fact, the theorem merely says a thermostat that only depends on the temperature of the room is simpler than one that also cares if the walls are blue.

That's a problem, because we don't actually use "perfect" regulators for any system. I'm very willing to believe those things don't even exist.

So, it may be a correct proof, but it's a useless one.

The implication being what, exactly?

Well, except for the fact that the article is trivially wrong, and that there is no reason to trust its conclusions, I don't think there are any.

But you didn't need my comment for that, the fact that there is an entire engineering discipline thriving on doing exactly what the article claims to be impossible should suffice.

I kind of get the sense Baez is being polite and humble there, not wanting to say something is nonsense if others seem to find value in it.

That said, I think I agree with Baez about the actual mathematical content of the paper. That is, the proof is fine, but the way they try to formalize "model" seems so weak as to be useless. The definition used in the proof is that R is a model of S if the state of S determines the state of R. There's nothing about preserving the fidelity of those states.

edit: Maybe the clearest way I can put this confusion is that a regulator based on measurement of the system it's regulating is a "model" per the paper. The authors choose a formalism which sounds like it was more common at the time that describes the regulation as a function of "disturbances"/inputs only. Through this lens their definition of model makes more sense to me, but only in those cases where there is no measurement or feedback.

This is very interesting in more modern fields of reinforcement learning where debates rages between "model-free" and "model-based" approaches. I'd no idea that there was actual any proofs around this. However, I have to say the theorem goes against my intuition. If you want to control a heating system, you don't really need to build the model of heater which could be very complex requiring the knowledge of how it works internally. Instead, you observe the temperature for given action and successfully control it without ever knowing how your heater actually works. You do build a model but its model of observations and actions, not the heater itself. In fact, for vast majority of things we control such as bicycle or fan or oven, people rarely knows the model of these very complex systems (in terms physical principals and internals).

Well, the paper doesn't say that that doesn't work, just that it's not optimal. If you set the aircon to a given temperature it will cycle on when the temperature dips below the desired value and cycle off when it exceeds it. You're essentially regulating it by the error-states which the paper says doesn't minimize H(Z), so while it works it's only 'partial successfuly'. I guess that means the aircon doesn't keep the temperature rock-solid, but rather it fluctuates it within an acceptable range. I mean, it's a fine result for every-day life, but I guess theoretically according to this paper a better outcome is actually achievable??

> You do build a model but its model of observations and actions, not the heater itself. In fact, for vast majority of things we control such as bicycle or fan or oven, people rarely knows the model of these very complex systems (in terms physical principals and internals).

I think you're mistaking the term model for a replica. You don't need a replica, but you do need a model for your relevant tasks.

If the heater needs 30 mins to warm up before it starts pumping heat, you'll need to model that. If it overheats after running for 3hrs straight you'll need to model that too. You don't need every molecule of the physical item replicated but you do need to model all of the relevant behaviors.

This sounds a little circular. If a "model" is defined as a representation of a system of sufficient fidelity to control it, then of course all controllers contain models. Deciding which behaviors are "relevant" would appear to be the key issue here.

You have a point, but I think it is useful to have a unifying general principle to use in analyzing the design of regulators, and having a theoretical basis makes it harder to dismiss.

I can think of a couple of places where it could have avoided accidents (one tragic and one expensive.) The 1974 DC10 crash in Paris was supposed to be impossible because the airplane could not be pressurized unless the cargo door was properly latched, but the mechanism depended on the position of the handle rather than the latching pins. At Three Mile Island, the operators were trained to use the pressurizer water level as a primary indicator of the state of the system, and turned off the emergency cooling feed as a consequence.

A model is in practice mostly an approximation rather than a replica. Think of a water bottle that is about to fall off a table. You brain does not model the trajectory of 10^26 atoms it consists of, but it has some abstract representations of bottles and the way bottles propagate through space, approximately by Newton's laws. Not only does this suffice for catching the bottle before it falls, but the computational capacity of the brain and the resolution of our sensory organs and of light rays fwiw. are also way too limited for us to model the bottle entirely.

> The peculiarities of cybernetics. Many a book has borne the title “Theory of Machines”, but it usually contains information about mechanical things, about levers and cogs. Cybernetics, too, is a “theory of machines”, but it treats, not things but ways of behaving. It does not ask “what is this thing?” but “what does it do?” Thus it is very interested in such a statement as “this variable is undergoing a simple harmonic oscillation”, and is much less concerned with whether the variable is the position of a point on a wheel, or a potential in an electric circuit. It is thus essentially functional and behaviouristic.

~W. Ross Ashby "Introduction to Cybernetics", http://pespmc1.vub.ac.be/ASHBBOOK.html

I worked at a place that had some sort of Just-in-Time heater for the sinks in the bathroom. One of them oscillated too hot too cold too hot too cold, etc. Eventually someone adjusted some setting and it worked properly. That setting was (part of) the "model" of the heater+sink+bathroom system.

For many simple systems a PID-style controller does well enough, but for more complex systems some kind of state-based model tends to work out better, whether the model is built on knowledge of the system or by observing its behavior.

For the latter, there's an entire field devoted to this problem called "system identification", one popular textbook seems to be online nowadays at http://user.it.uu.se/~ts/sysidbook.pdf

Just looked up model-based RL to understand the difference and found this article to be surprisingly elucidating:


I guess you could make it more intuitive by thinking about the system in terms of the units of your feedback (e.g. force on an actuator like a bicycle handle bar in your example) rather than the deeper, underlying physics. In the end it doesn't matter which you use as long as it works.

It's funny, because I initially thought the title was about regulators as in "government regulators", not about control theory.

Even more interesting is the fact that this idea from control theory in fact completely transposes to government regulation, as in: it'd be really good if regulators had the faintest idea what they're doing (aka in a perfect world, they would have a somewhat decent model of the system they're trying to regulate).

Except ... that's very probably not the case ... (I'm looking at you economists).

> Except ... that's very probably not the case ... (I'm looking at you economists).

I'm not sure I agree. From what I've seen, it's usually politicians that ignore unintended consequences and incentives that lead to poor outcomes. Not sure what that has to do with economists ... they don't run the country.

One of the main results of Cybernetics. You can get a PDF of Ashby's book "Introduction to Cybernetics" from here, as well as a lot of other information:


(edit: Ha! I just realized it's the same Principia Cybernetica Web!)

In the management sciences this is one argument for diversity. If you have a complex environment, then your team has to mirror that complexity. But the whole thing based on system theory, so the basic assumption is that there is no direct access to the internal parts of the system, that is difficult if one assume that they are human actors.

Why would "diversity" help modelling the complexity of the environment? Would a chess-playing program get better if it also contained a go-engine, checker-engine, scissor-paper-rocks rules and tiddly-winks?

I think what they mean is that your market is the system, and your team is the controller, so if the market is diverse, your team should be diverse too.

I think it's still cargo-cult control theory. You can't just make your team as diverse as your market and expect results to magically improve.

I agree that that's what they mean.

What I'm not seeing is how the "requisite variety" that control theory talks about, relates to "diversity" in modern usage (which for practical purposes is coeval with the political demand of hiring fewer members of certain demographics).

Let's use a less loaded analogy, intellectual diversity: the Manhattan Project, which needed to adapt to a most complex and adversarial environment (world war), hired a lot of STEM scientists, in particular mathematicians, physicists and chemists, and mostly from a few top universities, in other words, a highly homogeneous group that were educated on the same few core subjects (linear algebra, real and complex anaysis, Newtonian and quantum physics, special and general relativity). I do not believe that the Manhattan Project would have been better by hiring fewer STEM graduates, and more psychologists, historians, yoga teachers and so forth. How would modern management science convince me that I'm wrong?

The heart of the issue is the conflation of "diversity" (a political concept) with control theory's "requisite variety". The two are related, but not the same.

I partly agree that it's cargo-culting, but that's not the whole story: the Melanesians after whom Cargo Cult is modelled, would probably not have tried to fire / doxx / cancel anyone who wasn't on board with the belief that building of an airplane runway will by itself bring desirable Western goods.

You’re missing the part where they say that diversity is needed because the target audience is diverse. The goal of the Manhattan project was to deliver a working nuclear weapon. Consequently its hiring practices were in line with this goal. Consequently your example of yoga instructors and psychologists (I’m sure there were psychologists consulting for the Manhattan project, however), is something of a strawman.

What was being discussed was the idea that a company or organization that wants to appeal to the populace at large should have a workforce representative of that population at large. I think that’s debatable, but you can imagine that having someone on staff who can say “this is offensive (to my group of people)” can be useful to an organization. In this case, “I find this offensive” would not be a political or cultural message, but an actionable piece of data that says “For our goal of appealing to the populace at large, this statement/product may have lower, or negative, utility in appealing to this part of the population we’d like to like us/like to sell widgets to.”

correct. the Manhattan project's complexity is generated by a natural, to be distinct from a social problem. and it is not just about offence, but also the inherent complexity of subsidiaries, that can be controlled by having a diverse workforce in the headquarters, that has cultural or organizational knowledge.

The Manhattan Project was headed by a Jewish polymath intellectual political radical with very strong Communist ties at a time when all of those factors were serious impediments to mainstream acceptance. Nevertheless, Oppenheimer was given the job, and delivered.

If we wish to be good regulators of the environment, we must be a model of the environment.

What does this mean?

You must figure out the causal structure of the system you are trying to regulate and incorporate it into the regulator. Using some generic and simple model to observe variables will not do.

When the system becomes very complex this involves simulation. Think the game of chess as a system you try to regulate towards win. The only working solution is to simulate gameplay, dong sequences of moves and evaluating their outcomes (search). Chess computer replicates the system it regulates.

That if we wish to control some figure of merit, such as the average global temperature or sea level, then we must understand which variables influence them.

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