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# Lecture 18 - Chapter 2: Resource Theories

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51.

Concerning $$x\leq x$$, what is the canonical way in order to get $$x$$ from $$x$$? You take $$x$$ and do nothing to it! So the simplest resource flow diagram that turns $$x$$ into itself is the one that only consists of a wire labelled $$x$$ and no boxes.

Of course, there may also be other ways to get $$x$$ from itself, which may be described by some nontrivial flow diagrams. For example, for many kinds of resources $$x$$, you can turn $$x$$ into itself by using the process "wait for 1 min", and you will get $$x$$ out again.

It may seem like monoidal posets are really an impoverished version of working with flow diagrams. That's true, and it's exactly the point! If we describe the resource theory as a monoidal poset, then we can analyze what we aim for: is it possible to get the thing that we want from the things that we have? If so, at what yield? These are the kinds of questions that can be addressed by working with monoidal posets. But if we also want to know how to achieve these things once we know that they are possible, then we need to go to the level of flow diagrams (or equivalently monoidal categories). In many cases, I think that it's helpful to first figure out what resource target or yield we can hope to get, and we do this by analyzing the monoidal poset. And then we try and figure out how to find a process that achieves it. This is historically what happened in information theory, where Shannon's theorem tells us the maximal yield for communicating across a noisy channel. Figuring out how to concretely achieve this yield took several decades longer!

Comment Source:Right, I've fixed the link. Concerning \$$x\leq x\$$, what is the canonical way in order to get \$$x\$$ from \$$x\$$? You take \$$x\$$ and do nothing to it! So the simplest resource flow diagram that turns \$$x\$$ into itself is the one that only consists of a wire labelled \$$x\$$ and no boxes. Of course, there may also be other ways to get \$$x\$$ from itself, which may be described by some nontrivial flow diagrams. For example, for many kinds of resources \$$x\$$, you can turn \$$x\$$ into itself by using the process "wait for 1 min", and you will get \$$x\$$ out again. It may seem like monoidal posets are really an impoverished version of working with flow diagrams. That's true, and it's exactly the point! If we describe the resource theory as a monoidal poset, then we can analyze what we aim for: is it *possible* to get the thing that we want from the things that we have? If so, at what yield? These are the kinds of questions that can be addressed by working with monoidal posets. But if we also want to know *how* to achieve these things once we know that they are possible, then we need to go to the level of flow diagrams (or equivalently monoidal categories). In many cases, I think that it's helpful to first figure out what resource target or yield we can hope to get, and we do this by analyzing the monoidal poset. And then we try and figure out how to find a process that achieves it. This is historically what happened in information theory, where [Shannon's theorem](https://en.wikipedia.org/wiki/Noisy-channel_coding_theorem) tells us the maximal yield for communicating across a noisy channel. Figuring out how to concretely achieve this yield [took several decades longer](https://pdfs.semanticscholar.org/5624/35787bcbdbcfe750a61a90a0df60a0932eec.pdf)!
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52.
edited May 2018

@Tobias Fritz thanks for the interesting words.. @Keith E. Peterson yes economics deserves a topic..good idea

@John Baez..yes Prof Baez I believe that in future humanity will laugh about our actual 1D money...in close analogy to flat the earth ...for some years i was on the "front" of implementation of environmental policies..mine ultimate conclusion is that indeed the "divide to conquer" effect intrinsically induced by the math model of the 1D money is the the stuff to be handled...in some sense it seems to guarantee it owns resilience and sucess ... perhaps some analogy to DNA self repair holds ?Low energy requirements? Indeed a puzzle for me ...

In other hand, possible the with out borders air polution /climate change is a clear nature's flag/warning pointing out how naive the 1d money is on the big picture.. Perhaps some multiplicative adimensional factors could improve it a little bit...this factors could bring some kind of cloud recomendation database to the table correcting an aparent currency value owned by each person to a realistic one concerning sustainability.... then transfering the task of capture the complexity of the "earth system" to this multiplicative adimensionals...it would relies at least on a vast and well developed/intelinked personal id system..but i also feel that the best solution would be to use higher dimensions... For me this is simply THE MOST important problem to be solved nowadays ..some ideas and intuitions for discussion..

It would be great if you could bring this theme again, for sure .Best regards

Comment Source:@Tobias Fritz thanks for the interesting words.. @Keith E. Peterson yes economics deserves a topic..good idea @John Baez..yes Prof Baez I believe that in future humanity will laugh about our actual 1D money...in close analogy to flat the earth ...for some years i was on the "front" of implementation of environmental policies..mine ultimate conclusion is that indeed the "divide to conquer" effect intrinsically induced by the math model of the 1D money is the the stuff to be handled...in some sense it seems to guarantee it owns resilience and sucess ... perhaps some analogy to DNA self repair holds ?Low energy requirements? Indeed a puzzle for me ... In other hand, possible the with out borders air polution /climate change is a clear nature's flag/warning pointing out how naive the 1d money is on the big picture.. Perhaps some multiplicative adimensional factors could improve it a little bit...this factors could bring some kind of cloud recomendation database to the table correcting an aparent currency value owned by each person to a realistic one concerning sustainability.... then transfering the task of capture the complexity of the "earth system" to this multiplicative adimensionals...it would relies at least on a vast and well developed/intelinked personal id system..but i also feel that the best solution would be to use higher dimensions... For me this is simply THE MOST important problem to be solved nowadays ..some ideas and intuitions for discussion.. It would be great if you could bring this theme again, for sure .Best regards 
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53.

Bartosz - thanks for catching that typo. Fixed!

Comment Source:Bartosz - thanks for catching that typo. Fixed!
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54.
edited May 2018

I copied and pasted $$x \leq x$$ and somehow did not notice that it did not say what I meant. I plead temporary blindness or something...

What I meant to say was, in $$x \leq y$$ - like, how (by what process, method, or function) do you get y from x?

Is that what you were referring to when you wrote

It may seem like monoidal posets are really an impoverished version of working with flow diagrams.

?

Comment Source:[Tobias](https://forum.azimuthproject.org/discussion/comment/18033/#Comment_18033) I had a typo in my comment, now fixed, and repeated here (and I also removed the bit about your link). I copied and pasted \$$x \leq x\$$ and somehow did not notice that it did not say what I meant. I plead temporary blindness or something... What I meant to say was, in \$$x \leq y\$$ - like, how (by what process, method, or function) do you get y from x? Is that what you were referring to when you wrote > It may seem like monoidal posets are really an impoverished version of working with flow diagrams. ?
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55.

@Bob: Oh, I see. Yes, that's exactly what I meant! If you prove that $$x \leq y$$ for your favourite $$x$$ and $$y$$ in your favourite resource theory, then this only tells us that you can get $$y$$ from $$x$$, but not by which process. [Or, now that we've fixed the opposite convention for which direction we write the order, $$x\leq y$$ does not tell you how you can get $$x$$ from $$y$$.] This may sound like it makes the whole formalism useless in actual applications, since the mere knowledge of being able to do it does not enable us to do it. This is correct, but if very often helps us to figure out how to do it.

So what I was trying to explain above is that it's very often useful to separate the resource analysis into two steps: first figure out what is possible and what you can hope to achieve, and then think about how to achieve that. The first thing is useless without the second, but the second is often much easier once you've done the first. What we've been discussing here so far is the first step, and the formalism of resource theories provides some general tools and methods for how to approach it.

Comment Source:@Bob: Oh, I see. Yes, that's exactly what I meant! If you prove that \$$x \leq y\$$ for your favourite \$$x\$$ and \$$y\$$ in your favourite resource theory, then this only tells us *that* you can get \$$y\$$ from \$$x\$$, but not *by which process*. [Or, now that we've [fixed the opposite convention](https://forum.azimuthproject.org/discussion/comment/18042/#Comment_18042) for which direction we write the order, \$$x\leq y\$$ does not tell you how you can get \$$x\$$ from \$$y\$$.] This may sound like it makes the whole formalism useless in actual applications, since the mere knowledge of being able to do it does not enable us to do it. This is correct, but if very often helps us *to figure out* how to do it. So what I was trying to explain [above](https://forum.azimuthproject.org/discussion/comment/18033/#Comment_18033) is that it's very often useful to separate the resource analysis into two steps: *first* figure out what is possible and what you can hope to achieve, and *then* think about how to achieve that. The first thing is useless without the second, but the second is often much easier once you've done the first. What we've been discussing here so far is the first step, and the formalism of resource theories provides some general tools and methods for how to approach it.
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56.
edited May 2018

I took a couple passes at chapter 2 during the time off, and tried my hand at making a $$\mathscr{V}$$-category. In making the preorder, I took the book's wiring diagrams to mean that, in a diagram of the pre-order, the 'earliest prerequisites' would be lower in the diagram. When it came time to identify hom-objects for the objects in the category (which, in this case, I was thinking of states of a particular game as the objects) it seemed like everything was somehow backwards, like I was supposed to identify an element of the preorder that would get me from a 'later state' to an 'earlier state,' which didn't seem right at all.

My point is, I'm delighted that this lecture just pointed out that the preorder should have actually been such that (outcome) $$\leq$$ (prerequisites), which means I drew my preorder upside-down. (...which, I suppose, means I may have drawn the preorder's opposite.)

So, rather than the 'upward flow' of the preorder's diagram reflecting some kind of flow of prerequisites-to-final-outcome, eg. [INCORRECT] $$x \leq y \rightarrow$$ x is prerequisite to y, it seems I want it to reflect outcome-to-its-various-prerequisites: $$x \leq y \rightarrow$$ x has y as one of its prerequisites.

That seems more in line with what I interpreted the hom-objects needing to do. So that's a relief; I wasn't sure what I'd misunderstood when I first got tripped up by this, but I'm glad to hear that it was something easy to identify and remedy.

Comment Source:I took a couple passes at chapter 2 during the time off, and tried my hand at making a \$$\mathscr{V}\$$-category. In making the preorder, I took the book's wiring diagrams to mean that, in a diagram of the pre-order, the 'earliest prerequisites' would be lower in the diagram. When it came time to identify hom-objects for the objects in the category (which, in this case, I was thinking of states of a particular game as the objects) it seemed like everything was somehow backwards, like I was supposed to identify an element of the preorder that would get me from a 'later state' to an 'earlier state,' which didn't seem right at all. My point is, I'm delighted that this lecture just pointed out that the preorder should have actually been such that (outcome) \$$\leq \$$ (prerequisites), which means I drew my preorder upside-down. (...which, I suppose, means I may have drawn the preorder's opposite.) So, rather than the 'upward flow' of the preorder's diagram reflecting some kind of flow of prerequisites-to-final-outcome, eg. [INCORRECT] \$$x \leq y \rightarrow \$$ x is prerequisite to y, it seems I want it to reflect outcome-to-its-various-prerequisites: \$$x \leq y \rightarrow \$$ x has y as one of its prerequisites. That seems more in line with what I interpreted the hom-objects needing to do. So that's a relief; I wasn't sure what I'd misunderstood when I first got tripped up by this, but I'm glad to hear that it was something easy to identify and remedy. 
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57.

Bob wrote:

What I meant to say was, in x≤y - like, how (by what process, method, or function) do you get y from x?

If you care about how you get from one thing to another, you need categories. If you only care whether you can get from one thing to another, you need preorders (or posets, a special case). Right now we are doing preorders, but this will ultimately be a course on category theory.

Comment Source:Bob wrote: > What I meant to say was, in x≤y - like, how (by what process, method, or function) do you get y from x? If you care about _how_ you get from one thing to another, you need _categories_. If you only care _whether_ you can get from one thing to another, you need preorders (or posets, a special case). Right now we are doing preorders, but this will ultimately be a course on category theory. 
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58.

Thanks a lot. Got it. (I think.) And I will patiently await those categories.

Comment Source:Re [preorders vs categories](https://forum.azimuthproject.org/discussion/comment/18154/#Comment_18154): Thanks a lot. Got it. (I think.) And I will patiently await those categories.
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59.

Comment Source:Hi. How can I learn more about the connection between lenses and supervised learning? Also, how does etiquette work on these forums. Going back a couple of comments, I see links like @Bob, which feel like they should point to profiles *and* ideally alert folks that someone responded to them, but the links are broken.