A collective reply to comments so far.

All the posted answers to the exercises so far are correct.

1. Warming the thermostat with a candle will depress the room temperature while leaving the thermostat temperature constant.

2. Pressing the brake when the cruise control does not disengage will leave the car speed constant while the accelerator pedal goes down -- until something breaks.

3. The effect of raising a piece-rate worker's hourly rate will depend on what the worker wants (and not on what the employer intended to happen).

4. The doctor's target will be met while patients will still have to wait just as long, they just won't be able to book more than four weeks ahead. (This is an actual example from the British National Health Service.)

Does no-one want to tackle 5 or 6? Anyone who knows the derivative of exp(a t) knows enough to do number 6.

Thank you, kpreid, for linking to the very article that I knew, even while writing the original post, I would be invoking in response to the comments. Anyone who has not come across it before, please read it, and then I will talk about the concept that (it turns out) we are all talking about, when we talk about models, except for the curious fit that comes over some of us when contemplating the simple thermostat.

i77: As you say, the Smith predictor contains a model, and the subsystem C does not. Likewise the MRAC. In the PID case, the engineer has a model. But don't slide from that to attributing a model to the PID system. There isn't one there.

Vladimir_Nesov, pretty much all the concepts listed in the first three sections of that article are special cases of what is here meant by the word. As for the rest, I think we can all agree that we are not talking about a professional clothes horse or a village in Gmina Pacyna. I don't believe I have committed any of these offences (another article I'd recommend to anyone who has only just now had the good fortune to encounter it), but let those call foul who see any.

So, what are we talking about, when we talk about models? What I am talking about -- I'll come to the "we" part -- I said in a comment of mine on my first post:

What is a model? A model is a piece of mathematics in which certain quantities correspond to certain properties of the thing modelled, and certain mathematical relationships between these correspond to certain physical relationships.

and more briefly in the current post:

signals ... that are designed to relate to each other in the same way as do corresponding properties of the world outside

This is exactly what is meant by the word in model-based control theory. I linked to one paper where models in precisely this sense appear, and I am sure Google Books or Amazon will show the first chapters of any number of books on the subject, all using the word in exactly the same way. There is a definite thing here, and that is the thing I am talking of when I talk of a model.

This is not merely a term of art from within some branch of engineering, in which no-one outside it need be interested. Overcoming Bias has an excellent feature, a Google search box specialised to OB. When I search for "model", I get 523 hits. The first five (as I write -- I daresay the ranking may change from time to time) all use it in the above sense, some with less mathematical content but still with the essential feature of one thing being similar in structure to another, especially for the purpose of predicting how that other thing will behave. Here they are:

"So rather than your model for cognitive bias being an alternative model to self-deception..." (The model here is an extended analogy of the brain to a political bureaucracy.)

"Data-based model checking is a powerful tool for overcoming bias" (The writer is talking about statistical models, i.e. "a set of mathematical equations which describe the behavior of an object of study in terms of random variables and their associated probability distributions.")

"the model predicts much lower turnout than actually occurs" (The model is "the Rational Choice Model of Voting Participation, which is that people will vote if p times B > C".)

"I don't think student reports are a very good model for this kind of cognitive bias." (I.e. a system that behaves enough like another system to provide insight about that other.)

The 5th is a duplicate of the 2nd.

Those are enough examples to quote, but I inspected the rest of the first ten and sampled a few other hits at random (nos. 314, 159, 265, and 358, in fact), and except for a mention of a "role model", which could be arguable but not in any useful way, found no other senses in use.

When I googlesearch LW, excluding my own articles and the comments on them, the first two hits are to this, and this. These are also using the word in the same sense. The models are not as mathematical as they would have to be for engineering use, but they are otherwise of the same form: stuff here (the model) which is similar in structure to stuff there (the thing modelled), such that the model can be used to predict properties of the modelled.

In other words, what I am talking about, when I talk about models, is exactly what we on OB and LW are all talking about, when we talk about models, every time we talk about models. There is a definite thing here that has an easily understood shape in thingspace, we all call it a model, and to a sufficiently good approximation we call nothing else a model.

Until, strangely, we contemplate some very simple devices that reliably produce certain results, yet contain nothing drawn from that region of thingspace. Suddenly, instead of saying, "well well, no models here, fancy that", the definition of "model" is immediately changed to mean nothing more than mere entanglement, most explicitly by SilasBarta:

"A controller has a model (explicit or implicit) of it's environment iff there is mutual information between the controller and the environment."

Or the model in the designer's head is pointed to, and some sort of contagion invoked to attribute it to the thing he designed. No, this is butter spread over too much bread. That is not what is called a model anywhere on OB or LW except in these comment threads; it is not what is called a model, period.

You can consider the curvature of a bimetallic strip a model of the temperature if you like. It's a trivial model with one variable and no other structure, but there it is. However, a thermometer and a thermostat both have that model of the temperature, but only the thermostat controls it. You can also consider the thermostat's reference input to be a model of the position of the control dial, and the signal to the relay a model of the state of the relay, and the relay state a model of the heater state, but none of these trivial models explain the thermostat's functioning. What does explain the thermostat's functioning is the relation "turn on if below T1, turn off if above T2". That relation is not a model of anything. It is what the thermostat does; it does not map to anything else.

Exercise 7. How can you discover someone's goals? Assume you either cannot ask them, or would not trust their answers.