## Corelations in Network Theory

2 February, 2016

Category theory reduces a large chunk of math to the clever manipulation of arrows. One of the fun things about this is that you can often take a familiar mathematical construction, think of it category-theoretically, and just turn around all the arrows to get something new and interesting!

In math we love functions. If we have a function

$f: X \to Y$

we can formally turn around the arrow to think of $f$ as something going back from $Y$ back to $X$. But this something is usually not a function: it’s called a ‘cofunction’. A cofunction from $Y$ to $X$ is simply a function from $X$ to $Y.$

Cofunctions are somewhat interesting, but they’re really just functions viewed through a looking glass, so they don’t give much new—at least, not by themselves.

The game gets more interesting if we think of functions and cofunctions as special sorts of relations. A relation from $X$ to $Y$ is a subset

$R \subseteq X \times Y$

It’s a function when for each $x \in X$ there’s a unique $y \in Y$ with $(x,y) \in R.$ It’s a cofunction when for each $y \in Y$ there’s a unique $x \in x$ with $(x,y) \in R.$

Just as we can compose functions, we can compose relations. Relations have certain advantages over functions: for example, we can ‘turn around’ any relation $R$ from $X$ to $Y$ and get a relation $R^\dagger$ from $Y$ to $X:$

$R^\dagger = \{(y,x) : \; (x,y) \in R \}$

If we turn around a function we get a cofunction, and vice versa. But we can also do other fun things: for example, since both functions and cofunctions are relations, we can compose a function and a cofunction and get a relation.

Of course, relations also have certain disadvantages compared to functions. But it’s utterly clear by now that the category $\mathrm{FinRel},$ where the objects are finite sets and the morphisms are relations, is very important.

So far, so good. But what happens if we take the definition of ‘relation’ and turn all the arrows around?

There are actually several things I could mean by this question, some more interesting than others. But one of them gives a very interesting new concept: the concept of ‘corelation’. And two of my students have just written a very nice paper on corelations:

• Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids.

Here’s why this paper is important for network theory: corelations between finite sets are exactly what we need to describe electrical circuits made of ideal conductive wires! A corelation from a finite set $X$ to a finite set $Y$ can be drawn this way:

I have drawn more wires than strictly necessary: I’ve drawn a wire between two points whenever I want current to be able to flow between them. But there’s a reason I did this: a corelation from $X$ to $Y$ simply tells us when current can flow from one point in either of these sets to any other point in these sets.

Of course circuits made solely of conductive wires are not very exciting for electrical engineers. But in an earlier paper, Brendan introduced corelations as an important stepping-stone toward more general circuits:

• John Baez and Brendan Fong, A compositional framework for passive linear circuits. (Blog article here.)

The key point is simply that you use conductive wires to connect resistors, inductors, capacitors, batteries and the like and build interesting circuits—so if you don’t fully understand the math of conductive wires, you’re limited in your ability to understand circuits in general!

In their new paper, Brendan teamed up with Brandon Coya, and they figured out all the rules obeyed by the category $\mathrm{FinCorel},$ where the objects are finite sets and the morphisms are corelations. I’ll explain these rules later.

This sort of analysis had previously been done for $\mathrm{FinRel},$ and it turns out there’s a beautiful analogy between the two cases! Here is a chart displaying the analogy:

 Spans Cospans extra bicommutative bimonoids special commutative Frobenius monoids Relations Corelations extraspecial bicommutative bimonoids extraspecial commutative Frobenius monoids

I’m sure this will be cryptic to the nonmathematicians reading this, and even many mathematicians—but the paper explains what’s going on here.

I’ll actually say what an ‘extraspecial commutative Frobenius monoid’ is later in this post. This is a terse way of listing all the rules obeyed by corelations between finite sets—and thus, all the rules obeyed by conductive wires.

But first, let’s talk about something simpler.

### What is a corelation?

Just as we can define functions as relations of a special sort, we can also define relations in terms of functions. A relation from $X$ to $Y$ is a subset

$R \subseteq X \times Y$

but we can think of this as an equivalence class of one-to-one functions

$i: R \to X \times Y$

Why an equivalence class? The image of $i$ is our desired subset of $X \times Y.$ The set $R$ here could be replaced by any isomorphic set; its only role is to provide ‘names’ for the elements of $X \times Y$ that are in the image of $i.$

Now we have a relation described as an arrow, or really an equivalence class of arrows. Next, let’s turn the arrow around!

There are different things I might mean by that, but we want to do it cleverly. When we turn arrows around, the concept of product (for example, cartesian product $X \times Y$ of sets) turns into the concept of sum (for example, disjoint union $X + Y$ of sets). Similarly, the concept of monomorphism (such as a one-to-one function) turns into the concept of epimorphism (such as an onto function). If you don’t believe me, click on the links!

So, we should define a corelation from a set $X$ to a set $Y$ to be an equivalence class of onto functions

$p: X + Y \to C$

Why an equivalence class? The set $C$ here could be replaced by any isomorphic set; its only role is to provide ‘names’ for the sets of elements of $X + Y$ that get mapped to the same thing via $p.$

In simpler terms, a corelation from $X$ to a set $Y$ is just a partition of the disjoint union $X + Y.$ So, it looks like this:

If we like, we can then draw a line connecting any two points that lie in the same part of the partition:

These lines determine the corelation, so we can also draw a corelation this way:

This is why corelations describe circuits made solely of wires!

### The rules governing corelations

The main result in Brandon and Brendan’s paper is that $\mathrm{FinCorel}$ is equivalent to the PROP for extraspecial commutative Frobenius monoids. That’s a terse way of the laws governing $\mathrm{FinCorel}.$

Let me just show you the most important laws. In each of these law I’ll draw two circuits made of wires, and write an equals sign asserting that they give the same corelation from a set $X$ to a set $Y.$ The inputs $X$ of each circuit are on top, and the outputs $Y$ are at the bottom. I’ll draw 3-way junctions as little triangles, but don’t worry about that. When we compose two corelations we may get a wire left in mid-air, not connected to the inputs or outputs. We draw the end of the wire as a little circle.

There are some laws called the ‘commutative monoid’ laws:

and an upside-down version called the ‘cocommutative comonoid’ laws:

Then we have ‘Frobenius laws’:

and finally we have the ‘special’ and ‘extra’ laws:

All other laws can be derived from these in some systematic ways.

Commutative Frobenius monoids obey the commutative monoid laws, the cocommutative comonoid laws and the Frobenius laws. They play a fundamental role in 2d topological quantum field theory. Special Frobenius monoids are also well-known. But the ‘extra’ law, which says that a little piece of wire not connected to anything can be thrown away with no effect, is less well studied. Jason Erbele and I gave it this name in our work on control theory:

• John Baez and Jason Erbele, Categories in control. (Blog article here.)

### For more

David Ellerman has spent a lot of time studying what would happen to mathematics if we turned around a lot of arrows in a certain systematic way. In particular, just as the concept of relation would be replaced by the concept of corelation, the concept of subset would be replaced by the concept of partition. You can see how it fits together: just as a relation from $X$ to $Y$ is a subset of $X \times Y,$ a corelation from $X$ to $Y$ is a partition of $X + Y.$

There’s a lattice of subsets of a set:

In logic these subsets correspond to propositions, and the lattice operations are the logical operations ‘and’ and ‘or’. But there’s also a lattice of partitions of a set:

In Ellerman’s vision, this lattice of partitions gives a new kind of logic. You can read about it here:

• David Ellerman, Introduction to partition logic, Logic Journal of the Interest Group in Pure and Applied Logic 22 (2014), 94–125.

As mentioned, the main result in Brandon and Brendan’s paper is that $\mathrm{FinCorel}$ is equivalent to the PROP for extraspecial commutative Frobenius monoids. After they proved this, they noticed that the result has also been stated in other language and proved in other ways by two other authors:

• Fabio Zanasi, Interacting Hopf Algebras—the Theory of Linear Systems, PhD thesis, École Normale Supériere de Lyon, 2015.

• K. Dosen and Z. Petrić, Syntax for split preorders, Annals of Pure and Applied Logic 164 (2013), 443–481.

Unsurprisingly, I prefer Brendan and Brandon’s approach to deriving the result. But it’s nice to see different perspectives!

## The Internal Model Principle

27 January, 2016

“Every good key must be a model of the lock it opens.”

That sentence states an obvious fact, but perhaps also a profound insight if we interpret it generally enough.

That sentence is also the title of a paper:

• Daniel L. Scholten, Every good key must be a model of the lock it opens (the Conant & Ashby Theorem revisited), 2010.

Scholten gives a lot of examples, including these:

• A key is a model of a lock’s keyhole.

• A city street map is a model of the actual city streets

• A restaurant menu is a model of the food the restaurant prepares and sells.

• Honey bees use a kind of dance to model the location of a source of nectar.

• An understanding of some phenomenon (for example a physicist’s understanding of lightning) is a mental model of the actual phenomenon.

This line of thought has an interesting application to control theory. It suggests that to do the best job of regulating some system, a control apparatus should include a model of that system.

Indeed, much earlier, Conant and Ashby tried to turn this idea into a theorem, the ‘good regulator theorem’:

• Roger C. Conant and W. Ross Ashby, Every good regulator of a system must be a model of that system), International Journal of Systems Science 1 (1970), 89–97.

Scholten’s paper is heavily based on this earlier paper. He summarizes it as follows:

What all of this means, more or less, is that the pursuit of a goal by some dynamic agent (Regulator) in the face of a source of obstacles (System) places at least one particular and unavoidable demand on that agent, which is that the agent’s behaviors must be executed in such a reliable and predictable way that they can serve as a representation (Model) of that source of obstacles.

It’s not clear that this is true, but it’s an appealing thought.

A particularly self-referential example arises when the regulator is some organism and the System is the world it lives in, including itself. In this case, it seems the regulator should include a model of itself! This would lead, ultimately, to self-awareness.

It all sounds great. But Scholten raises an obvious question: if Conant and Ashby’s theorem is so great, why isn’t more well-known? Scholten puts it quite vividly:

Given the preponderance of control-models that are used by humans (the evidence for this preponderance will be surveyed in the latter part of the paper), and especially given the obvious need to regulate that system, one might guess that the C&A theorem would be at least as famous as, say, the Pythagorean Theorem ($a^2 + b^2 = c^2$), the Einstein mass-energy equivalence ($E = mc^2,$ which can be seen on T-shirts and bumper stickers), or the DNA double helix (which actually shows up in TV crime dramas and movies about super heroes). And yet, it would appear that relatively few lay-persons have ever even heard of C&A’s important prerequisite to successful regulation.

There could be various explanations. But here’s mine: when I tried to read Conant and Ashby’s paper, I got stuck. They use some very basic mathematical notation in nonstandard ways, and they don’t clearly state the hypotheses and conclusion of their theorem.

Luckily, the paper is short, and the argument, while mysterious, seems simple. So, I immediately felt I should be able to dream up the hypotheses, conclusion, and argument based on the hints given.

Scholten’s paper didn’t help much, since he says:

Throughout the following discussion I will assume that the reader has studied Conant & Ashby’s original paper, possesses the level of technical competence required to understand their proof, and is familiar with the components of the basic model that they used to prove their theorem [….]

However, I have a guess about the essential core of Conant and Ashby’s theorem. So, I’ll state that, and then say more about their setup.

Needless to say, I looked around to see if someone else had already done the work of figuring out what Conant and Ashby were saying. The best thing I found was this:

• B. A. Francis and W. M. Wonham, The internal model principle of control theory, Automatica 12 (1976) 457–465.

This paper works in a more specialized context: linear control theory. They’ve got a linear system or ‘plant’ responding to some input, a regulator or ‘compensator’ that is trying to make the plant behave in a desired way, and a ‘disturbance’ that affects the plant in some unwanted way. They prove that to perfectly correct for the disturbance, the compensator must contain an ‘internal model’ of the disturbance.

I’m probably stating this a bit incorrectly. This paper is much more technical, but it seems to be more careful in stating assumptions and conclusions. In particular, they seem to give a precise definition of an ‘internal model’. And I read elsewhere that the ‘internal model principle’ proved here has become a classic result in control theory!

This paper says that Conant and Ashby’s paper provided “plausibility arguments in favor of the internal model idea”. So, perhaps Conant and Ashby inspired Francis and Wonham, and were then largely forgotten.

### My guess

My guess is that Conant and Ashby’s theorem boils down to this:

Theorem. Let $R$ and $S$ be finite sets, and fix a probability distribution $p$ on $S$. Suppose $q$ is any probability distribution on $R \times S$ such that

$\displaystyle{ p(s) = \sum_{r \in R} q(r,s) \; \textrm{for all} \; s \in S}$

Let $H(p)$ be the Shannon entropy of $p$ and let $H(q)$ be the Shannon entropy of $q.$ Then

$H(q) \ge H(p)$

and equality is achieved if there is a function

$h: S \to R$

such that

$q(r,s) = \left\{\begin{array}{cc} p(s) & \textrm{if} \; r = h(s) \\ 0 & \textrm{otherwise} \end{array} \right.$       █

Note that this is not an ‘if and only if’.

The proof of this is pretty easy to anyone who knows a bit about probability theory and entropy. I can restate it using a bit of standard jargon, which may make it more obvious to experts. We’ve got an $S$-valued random variable, say $\textbf{s}.$ We want to extend it to an $R \times S$-valued random variable $(\textbf{r}, \textbf{s})$ whose entropy is small as possible. Then we can achieve this by choosing a function $h: S \to R,$ and letting $\textbf{s} = h(\textbf{r}).$

Here’s the point: if we make $\textbf{s}$ be a function of $\textbf{r},$ we aren’t adding any extra randomness, so the entropy doesn’t go up.

What in the world does this have to do with a good regulator containing a model of the system it’s regulating?

Well, I can’t explain that as well as I’d like—sorry. But the rough idea seems to be this. Suppose that $S$ is a system with a given random behavior, and $R$ is another system, the regulator. If we want the combination of the system and regulator to behave as ‘nonrandomly’ as possible, we can let the state of the regulator be a function of the state of the system.

This theorem is actually a ‘lemma’ in Conant and Ashby’s paper. Let’s look at their setup, and the ‘good regulator theorem’ as they actually state it.

### Their setup

Conant and Ashby consider five sets and three functions. In a picture:

The sets are these:

• A set $Z$ of possible outcomes.

• A goal: some subset $G \subseteq Z$ of good outcomes

• A set $D$ of disturbances, which I might prefer to call ‘inputs’.

• A set $S$ of states of some system that is affected by the disturbances.

• A set $R$ of states of some regulator that is also affected by the disturbances.

The functions are these:

• A function $\phi : D \to S$ saying how a disturbance determines a state of the system.

• A function $\rho: D \to R$ saying how a disturbance determines a state of the regulator.

• A function $\psi: S \times R \to Z$ saying how a state of the system and a state of the regulator determines an outcome.

Of course we want some conditions on these maps. What we want, I guess, is for the outcome to be good regardless of the disturbance. I might say that as follows: for every $d \in D$ we have

$\psi(\phi(d), \rho(d)) \in G$

Unfortunately Conant and Ashby say they want this:

$\rho \subset [\psi^{-1}(G)]\phi$

I can’t parse this: they’re using math notation in ways I don’t recognize. Can you figure out what they mean, and whether it matches my guess above?

Then, after a lot of examples and stuff, they state their theorem:

Theorem. The simplest optimal regulator $R$ of a reguland $S$ produces events $R$ which are related to events $S$ by a mapping $h: S \to R.$

Clearly I’ve skipped over too much! This barely makes any sense at all.

Unfortunately, looking at the text before the theorem, I don’t see these terms being explained. Furthermore, their ‘proof’ introduces extra assumptions that were not mentioned in the statement of the theorem. It begins:

The sets $R, S,$ and $Z$ and the mapping $\psi: R \times S \to Z$ are presumed given. We will assume that over the set $S$ there exists a probability distribution $p(S)$ which gives the relative frequencies of the events in $S.$ We will further assume that the behaviour of any particular regulator $R$ is specified by a conditional distribution $p(R|S)$ giving, for each event in $S,$ a distribution on the regulatory events in $R.$

Get it? Now they’re saying the state of the regulator $R$ depends on the state of the system $S$ via a conditional probability distribution $p(r|s)$ where $r \in R$ and $s \in S.$ It’s odd that they didn’t mention this earlier! Their picture made it look like the state of the regulator is determined by the ‘disturbance’ via the function $\rho: D \to R.$ But okay.

They’re also assuming there’s a probability distribution on $S.$ They use this and the above conditional probability distribution to get a probability distribution on $R.$

In fact, the set $D$ and the functions out of this set seem to play no role in their proof!

It’s unclear to me exactly what we’re given, what we get to choose, and what we’re trying to optimize. They do try to explain this. Here’s what they say:

Now $p(S)$ and $p(R|S)$ jointly determine $p(R,S)$ and hence $p(Z)$ and $H(Z),$ the entropy in the set of outcomes:

$\displaystyle{ H(Z) = - \sum_{z \in Z} p(z) \log (p(z)) }$

With $p(S)$ fixed, the class of optimal regulators therefore corresponds to the class of optimal distributions $p(R|S)$ for which $H(Z)$ is minimal. We will call this class of optimal distributions $\pi.$

I could write a little essay on why this makes me unhappy, but never mind. I’m used to the habit of using the same letter $p$ to stand for probability distributions on lots of different sets: folks let the argument of $p$ say which set they have in mind at any moment. So, they’re starting with a probability distribution on $S$ and a conditional probability distribution on $r \in R$ given $s \in S.$ They’re using these to determine probability distribution on $R \times S.$ Then, presumably using the map $\psi: S \times R \to Z,$ they get a probability distribution on $Z.$ $H(Z)$ is the entropy of the probability distribution on $Z,$ and for some reason they are trying to minimize this.

(Where did the subset $G \subseteq Z$ of ‘good’ outcomes go? Shouldn’t that play a role? Oh well.)

I believe the claim is that when this entropy is minimized, there’s a function $h : S \to R$ such that

$p(r|s) = 1 \; \textrm{if} \; r = h(s)$

This says that the state of the regulator should be completely determined by the the state of the system. And this, I believe, is what they mean by

Every good regulator of a system must be a model of that system.

I hope you understand: I’m not worrying about whether the setup is a good one, e.g. sufficiently general for real-world applications. I’m just trying to figure out what the setup actually is, what Conant and Ashby’s theorem actually says, and whether it’s true.

I think I’ve just made a lot of progress. Surely this was no fun to read. But it I found it useful to write it.

## Biology, Networks and Control Theory

13 September, 2015

The Institute for Mathematics and its Applications (or IMA, in Minneapolis, Minnesota), is teaming up with the Mathematical Biosciences Institute (or MBI, in Columbus, Ohio). They’re having a big program on control theory and networks:

### At the IMA

Here’s what’s happening at the Institute for Mathematics and its Applications:

Concepts and techniques from control theory are becoming increasingly interdisciplinary. At the same time, trends in modern control theory are influenced and inspired by other disciplines. As a result, the systems and control community is rapidly broadening its scope in a variety of directions. The IMA program is designed to encourage true interdisciplinary research and the cross fertilization of ideas. An important element for success is that ideas flow across disciplines in a timely manner and that the cross-fertilization takes place in unison.

Due to the usefulness of control, talent from control theory is drawn and often migrates to other important areas, such as biology, computer science, and biomedical research, to apply its mathematical tools and expertise. It is vital that while the links are strong, we bring together researchers who have successfully bridged into other disciplines to promote the role of control theory and to focus on the efforts of the controls community. An IMA investment in this area will be a catalyst for many advances and will provide the controls community with a cohesive research agenda.

In all topics of the program the need for research is pressing. For instance, viable implementations of control algorithms for smart grids are an urgent and clearly recognized need with considerable implications for the environment and quality of life. The mathematics of control will undoubtedly influence technology and vice-versa. The urgency for these new technologies suggests that the greatest impact of the program is to have it sooner rather than later.

First trimester (Fall 2015): Networks, whether social, biological, swarms of animals or vehicles, the Internet, etc., constitute an increasingly important subject in science and engineering. Their connectivity and feedback pathways affect robustness and functionality. Such concepts are at the core of a new and rapidly evolving frontier in the theory of dynamical systems and control. Embedded systems and networks are already pervasive in automotive, biological, aerospace, and telecommunications technologies and soon are expected to impact the power infrastructure (smart grids). In this new technological and scientific realm, the modeling and representation of systems, the role of feedback, and the value and cost of information need to be re-evaluated and understood. Traditional thinking that is relevant to a limited number of feedback loops with practically unlimited bandwidth is no longer applicable. Feedback control and stability of network dynamics is a relatively new endeavor. Analysis and control of network dynamics will occupy mostly the first trimester while applications to power networks will be a separate theme during the third trimester. The first trimester will be divided into three workshops on the topics of analysis of network dynamics and regulation, communication and cooperative control over networks, and a separate one on biological systems and networks.

The second trimester (Winter 2016) will have two workshops. The first will be on modeling and estimation (Workshop 4) and the second one on distributed parameter systems and partial differential equations (Workshop 5). The theme of Workshop 4 will be on structure and parsimony in models. The goal is to explore recent relevant theories and techniques that allow sparse representations, rank constrained optimization, and structural constraints in models and control designs. Our intent is to blend a group of researchers in the aforementioned topics with a select group of researchers with interests in a statistical viewpoint. Workshop 5 will focus on distributed systems and related computational issues. One of our aims is to bring control theorists with an interest in optimal control of distributed parameter systems together with mathematicians working on optimal transport theory (in essence an optimal control problem). The subject of optimal transport is rapidly developing with ramifications in probability and statistics (of essence in system modeling and hence of interest to participants in Workshop 4 as well) and in distributed control of PDE’s. Emphasis will also be placed on new tools and new mathematical developments (in PDE’s, computational methods, optimization). The workshops will be closely spaced to facilitate participation in more than one.

The third trimester (Spring 2016) will target applications where the mathematics of systems and control may soon prove to have a timely impact. From the invention of atomic force microscopy at the nanoscale to micro-mirror arrays for a next generation of telescopes, control has played a critical role in sensing and imaging of challenging new realms. At present, thanks to recent technological advances of AFM and optical tweezers, great strides are taking place making it possible to manipulate the biological transport of protein molecules as well as the control of individual atoms. Two intertwined subject areas, quantum and nano control and scientific instrumentation, are seen to blend together (Workshop 6) with partial focus on the role of feedback control and optimal filtering in achieving resolution and performance at such scales. A second theme (Workshop 7) will aim at control issues in distributed hybrid systems, at a macro scale, with a specific focus the “smart grid” and energy applications.

For more information on individual workshops, go here:

• Workshop 1, Distributed Control and Decision Making Over Networks, 28 September – 2 October 2015.

• Workshop 2, Analysis and Control of Network Dynamics, 19-23 October 2015.

• Workshop 3, Biological Systems and Networks, 11-16 November 2015.

• Workshop 4, Optimization and Parsimonious Modeling, 25-29 January 2016.

• Workshop 5, Computational Methods for Control of Infinite-dimensional Systems, 14-18 March 2016.

• Workshop 6, Quantum and Nano Control, 11-15 April 2016.

• Workshop 7, Control at Large Scales: Energy Markets and Responsive Grids, 9-13 March 2016.

### At the MBI

Here’s what’s going on at the Mathematical Biology Institute:

The MBI network program is part of a yearlong cooperative program with IMA.

Networks and deterministic and stochastic dynamical systems on networks are used as models in many areas of biology. This underscores the importance of developing tools to understand the interplay between network structures and dynamical processes, as well as how network dynamics can be controlled. The dynamics associated with such models are often different from what one might traditionally expect from a large system of equations, and these differences present the opportunity to develop exciting new theories and methods that should facilitate the analysis of specific models. Moreover, a nascent area of research is the dynamics of networks in which the networks themselves change in time, which occurs, for example, in plasticity in neuroscience and in up regulation and down regulation of enzymes in biochemical systems.

There are many areas in biology (including neuroscience, gene networks, and epidemiology) in which network analysis is now standard. Techniques from network science have yielded many biological insights in these fields and their study has yielded many theorems. Moreover, these areas continue to be exciting areas that contain both concrete and general mathematical problems. Workshop 1 explores the mathematics behind the applications in which restrictions on general coupled systems are important. Examples of such restrictions include symmetry, Boolean dynamics, and mass-action kinetics; and each of these special properties permits the proof of theorems about dynamics on these special networks.

Workshop 2 focuses on the interplay between stochastic and deterministic behavior in biological networks. An important related problem is to understand how stochasticity affects parameter estimation. Analyzing the relationship between stochastic changes, network structure, and network dynamics poses mathematical questions that are new, difficult, and fascinating.

In recent years, an increasing number of biological systems have been modeled using networks whose structure changes in time or which use multiple kinds of couplings between the same nodes or couplings that are not just pairwise. General theories such as groupoids and hypergraphs have been developed to handle the structure in some of these more general coupled systems, and specific application models have been studied by simulation. Workshop 3 will bring together theorists, modelers, and experimentalists to address the modeling of biological systems using new network structures and the analysis of such structures.

Biological systems use control to achieve desired dynamics and prevent undesirable behaviors. Consequently, the study of network control is important both to reveal naturally evolved control mechanisms that underlie the functioning of biological systems and to develop human-designed control interventions to recover lost function, mitigate failures, or repurpose biological networks. Workshop 4 will address the challenging subjects of control and observability of network dynamics.

#### Events

Workshop 1: Dynamics in Networks with Special Properties, 25-29 January, 2016.

Workshop 2: The Interplay of Stochastic and Deterministic Dynamics in Networks, 22-26 February, 2016.

Workshop 3: Generalized Network Structures and Dynamics, 21-15 March, 2016.

Workshop 4: Control and Observability of Network Dynamics, 11-15 April, 2016.

You can get more schedule information on these posters:

## A Compositional Framework for Markov Processes

4 September, 2015

This summer my students Brendan Fong and Blake Pollard visited me at the Centre for Quantum Technologies, and we figured out how to understand open continuous-time Markov chains! I think this is a nice step towards understanding the math of living systems.

Admittedly, it’s just a small first step. But I’m excited by this step, since Blake and I have been trying to get this stuff to work for a couple years, and it finally fell into place. And we think we know what to do next.

Here’s our paper:

• John C. Baez, Brendan Fong and Blake S. Pollard, A compositional framework for open Markov processes.

And here’s the basic idea…

### Open detailed balanced Markov processes

A continuous-time Markov chain is a way to specify the dynamics of a population which is spread across some finite set of states. Population can flow between the states. The larger the population of a state, the more rapidly population flows out of the state. Because of this property, under certain conditions the populations of the states tend toward an equilibrium where at any state the inflow of population is balanced by its outflow.

In applications to statistical mechanics, we are often interested in equilibria such that for any two states connected by an edge, say $i$ and $j,$ the flow from $i$ to $j$ equals the flow from $j$ to $i.$ A continuous-time Markov chain with a chosen equilibrium having this property is called ‘detailed balanced‘.

I’m getting tired of saying ‘continuous-time Markov chain’, so from now on I’ll just say ‘Markov process’, just because it’s shorter. Okay? That will let me say the next sentence without running out of breath:

Our paper is about open detailed balanced Markov processes.

Here’s an example:

The detailed balanced Markov process itself consists of a finite set of states together with a finite set of edges between them, with each state $i$ labelled by an equilibrium population $q_i >0,$ and each edge $e$ labelled by a rate constant $r_e > 0.$

These populations and rate constants are required to obey an equation called the ‘detailed balance condition’. This equation means that in equilibrium, the flow from $i$ to $j$ equal the flow from $j$ to $i.$ Do you see how it works in this example?

To get an ‘open’ detailed balanced Markov process, some states are designated as inputs or outputs. In general each state may be specified as both an input and an output, or as inputs and outputs multiple times. See how that’s happening in this example? It may seem weird, but it makes things work better.

People usually say Markov processes are all about how probabilities flow from one state to another. But we work with un-normalized probabilities, which we call ‘populations’, rather than probabilities that must sum to 1. The reason is that in an open Markov process, probability is not conserved: it can flow in or out at the inputs and outputs. We allow it to flow both in and out at both the input states and the output states.

Our most fundamental result is that there’s a category $\mathrm{DetBalMark}$ where a morphism is an open detailed balanced Markov process. We think of it as a morphism from its inputs to its outputs.

We compose morphisms in $\mathrm{DetBalMark}$ by identifying the output states of one open detailed balanced Markov process with the input states of another. The populations of identified states must match. For example, we may compose this morphism $N$:

with the previously shown morphism $M$ to get this morphism $M \circ N$:

And here’s our second most fundamental result: the category $\mathrm{DetBalMark}$ is actually a dagger compact category. This lets us do other stuff with open Markov processes. An important one is ‘tensoring’, which lets us take two open Markov processes like $M$ and $N$ above and set them side by side, giving $M \otimes N$:

The so-called compactness is also important. This means we can take some inputs of an open Markov process and turn them into outputs, or vice versa. For example, using the compactness of $\mathrm{DetBalMark}$ we can get this open Markov process from $M$:

In fact all the categories in our paper are dagger compact categories, and all our functors preserve this structure. Dagger compact categories are a well-known framework for describing systems with inputs and outputs, so this is good.

### The analogy to electrical circuits

In a detailed balanced Markov process, population can flow along edges. In the detailed balanced equilibrium, without any flow of population from outside, the flow along from state $i$ to state $j$ will be matched by the flow back from $j$ to $i.$ The populations need to take specific values for this to occur.

In an electrical circuit made of linear resistors, charge can flow along wires. In equilibrium, without any driving voltage from outside, the current along each wire will be zero. The potentials will be equal at every node.

This sets up an analogy between detailed balanced continuous-time Markov chains and electrical circuits made of linear resistors! I love analogy charts, so this makes me very happy:

 Circuits Detailed balanced Markov processes potential population current flow conductance rate constant power dissipation

This analogy is already well known. Schnakenberg used it in his book Thermodynamic Network Analysis of Biological Systems. So, our main goal is to formalize and exploit it. This analogy extends from systems in equilibrium to the more interesting case of nonequilibrium steady states, which are the main topic of our paper.

Earlier, Brendan and I introduced a way to ‘black box’ a circuit and define the relation it determines between potential-current pairs at the input and output terminals. This relation describes the circuit’s external behavior as seen by an observer who can only perform measurements at the terminals.

An important fact is that black boxing is ‘compositional’: if one builds a circuit from smaller pieces, the external behavior of the whole circuit can be determined from the external behaviors of the pieces. For category theorists, this means that black boxing is a functor!

Our new paper with Blake develops a similar ‘black box functor’ for detailed balanced Markov processes, and relates it to the earlier one for circuits.

When you black box a detailed balanced Markov process, you get the relation between population–flow pairs at the terminals. (By the ‘flow at a terminal’, we more precisely mean the net population outflow.) This relation holds not only in equilibrium, but also in any nonequilibrium steady state. Thus, black boxing an open detailed balanced Markov process gives its steady state dynamics as seen by an observer who can only measure populations and flows at the terminals.

### The principle of minimum dissipation

At least since the work of Prigogine, it’s been widely accepted that a large class of systems minimize entropy production in a nonequilibrium steady state. But people still fight about the the precise boundary of this class of systems, and even the meaning of this ‘principle of minimum entropy production’.

For detailed balanced open Markov processes, we show that a quantity we call the ‘dissipation’ is minimized in any steady state. This is a quadratic function of the populations and flows, analogous to the power dissipation of a circuit made of resistors. We make no claim that this quadratic function actually deserves to be called ‘entropy production’. Indeed, Schnakenberg has convincingly argued that they are only approximately equal.

But still, the ‘dissipation’ function is very natural and useful—and Prigogine’s so-called ‘entropy production’ is also a quadratic function.

### Black boxing

I’ve already mentioned the category $\mathrm{DetBalMark},$ where a morphism is an open detailed balanced Markov process. But our paper needs two more categories to tell its story! There’s the category of circuits, and the category of linear relations.

A morphism in the category $\mathrm{Circ}$ is an open electrical circuit made of resistors: that is, a graph with each edge labelled by a ‘conductance’ $c_e > 0,$ together with specified input and output nodes:

A morphism in the category $\mathrm{LinRel}$ is a linear relation $L : U \leadsto V$ between finite-dimensional real vector spaces $U$ and $V.$ This is nothing but a linear subspace $L \subseteq U \oplus V.$ Just as relations generalize functions, linear relations generalize linear functions!

In our previous paper, Brendan and I introduced these two categories and a functor between them, the ‘black box functor’:

$\blacksquare : \mathrm{Circ} \to \mathrm{LinRel}$

The idea is that any circuit determines a linear relation between the potentials and net current flows at the inputs and outputs. This relation describes the behavior of a circuit of resistors as seen from outside.

Our new paper introduces a black box functor for detailed balanced Markov processes:

$\square : \mathrm{DetBalMark} \to \mathrm{LinRel}$

We draw this functor as a white box merely to distinguish it from the other black box functor. The functor $\square$ maps any detailed balanced Markov process to the linear relation obeyed by populations and flows at the inputs and outputs in a steady state. In short, it describes the steady state behavior of the Markov process ‘as seen from outside’.

How do we manage to black box detailed balanced Markov processes? We do it using the analogy with circuits!

### The analogy becomes a functor

Every analogy wants to be a functor. So, we make the analogy between detailed balanced Markov processes and circuits precise by turning it into a functor:

$K : \mathrm{DetBalMark} \to \mathrm{Circ}$

This functor converts any open detailed balanced Markov process into an open electrical circuit made of resistors. This circuit is carefully chosen to reflect the steady-state behavior of the Markov process. Its underlying graph is the same as that of the Markov process. So, the ‘states’ of the Markov process are the same as the ‘nodes’ of the circuit.

Both the equilibrium populations at states of the Markov process and the rate constants labelling edges of the Markov process are used to compute the conductances of edges of this circuit. In the simple case where the Markov process has exactly one edge from any state $i$ to any state $j,$ the rule is this:

$C_{i j} = H_{i j} q_j$

where:

$q_j$ is the equilibrium population of the $j$th state of the Markov process,

$H_{i j}$ is the rate constant for the edge from the $j$th state to the $i$th state of the Markov process, and

$C_{i j}$ is the conductance (that is, the reciprocal of the resistance) of the wire from the $j$th node to the $i$th node of the resulting circuit.

The detailed balance condition for Markov processes says precisely that the matrix $C_{i j}$ is symmetric! This is just right for an electrical circuit made of resistors, since it means that the resistance of the wire from node $i$ to node $j$ equals the resistance of the same wire in the reverse direction, from node $j$ to node $i.$

### A triangle of functors

If you paid careful attention, you’ll have noticed that I’ve described a triangle of functors:

And if you know anything about how category theorists think, you’ll be wondering if this diagram commutes.

In fact, this triangle of functors does not commute! However, a general lesson of category theory is that we should only expect diagrams of functors to commute up to natural isomorphism, and this is what happens here:

The natural transformation $\alpha$ ‘corrects’ the black box functor for resistors to give the one for detailed balanced Markov processes.

The functors $\square$ and $\blacksquare \circ K$ are actually equal on objects. An object in $\mathrm{DetBalMark}$ is a finite set $X$ with each element $i \in X$ labelled a positive populations $q_i.$ Both functors map this object to the vector space $\mathbb{R}^X \oplus \mathbb{R}^X.$ For the functor $\square,$ we think of this as a space of population-flow pairs. For the functor $\blacksquare \circ K,$ we think of it as a space of potential-current pairs. The natural transformation $\alpha$ then gives a linear relation

$\alpha_{X,q} : \mathbb{R}^X \oplus \mathbb{R}^X \leadsto \mathbb{R}^X \oplus \mathbb{R}^X$

in fact an isomorphism of vector spaces, which converts potential-current pairs into population-flow pairs in a manner that depends on the $q_i.$ I’ll skip the formula; it’s in the paper.

But here’s the key point. The naturality of $\alpha$ actually allows us to reduce the problem of computing the functor $\square$ to the problem of computing $\blacksquare.$ Suppose

$M: (X,q) \to (Y,r)$

is any morphism in $\mathrm{DetBalMark}.$ The object $(X,q)$ is some finite set $X$ labelled by populations $q,$ and $(Y,r)$ is some finite set $Y$ labelled by populations $r.$ Then the naturality of $\alpha$ means that this square commutes:

Since $\alpha_{X,q}$ and $\alpha_{Y,r}$ are isomorphisms, we can solve for the functor $\square$ as follows:

$\square(M) = \alpha_Y \circ \blacksquare K(M) \circ \alpha_X^{-1}$

This equation has a clear intuitive meaning! It says that to compute the behavior of a detailed balanced Markov process, namely $\square(f),$ we convert it into a circuit made of resistors and compute the behavior of that, namely $\blacksquare K(f).$ This is not equal to the behavior of the Markov process, but we can compute that behavior by converting the input populations and flows into potentials and currents, feeding them into our circuit, and then converting the outputs back into populations and flows.

### What we really do

So that’s a sketch of what we do, and I hope you ask questions if it’s not clear. But I also hope you read our paper! Here’s what we actually do in there. After an introduction and summary of results:

• Section 3 defines open Markov processes and the open master equation.

• Section 4 introduces detailed balance for open Markov
processes.

• Section 5 recalls the principle of minimum power
for open circuits made of linear resistors, and explains how to black box them.

• Section 6 introduces the principle of minimum dissipation for open detailed balanced Markov processes, and describes how to black box these.

• Section 7 states the analogy between circuits and detailed balanced Markov processes in a formal way.

• Section 8 describes how to compose open Markov processes, making them into the morphisms of a category.

• Section 9 does the same for detailed balanced Markov processes.

• Section 10 describes the ‘black box functor’ that sends any open detailed balanced Markov process to the linear relation describing its external behavior, and recalls the similar functor for circuits.

• Section 11 makes the analogy between between open detailed balanced Markov processes and open circuits even more formal, by making it into a functor. We prove that together with the two black box functors, this forms a triangle that commutes up to natural isomorphism.

• Section 12 is about geometric aspects of this theory. We show that the linear relations in the image of these black box functors are Lagrangian relations between symplectic vector spaces. We also show that the master equation can be seen as a gradient flow equation.

• Section 13 is a summary of what we have learned.

Finally, Appendix A is a quick tutorial on decorated cospans. This is a key mathematical tool in our work, developed by Brendan in an earlier paper.

## Trends in Reaction Network Theory (Part 2)

1 July, 2015

Here in Copenhagen we’ll soon be having a bunch of interesting talks on chemical reaction networks:

Workshop on Mathematical Trends in Reaction Network Theory, 1-3 July 2015, Department of Mathematical Sciences, University of Copenhagen. Organized by Elisenda Feliu and Carsten Wiuf.

Looking through the abstracts, here are a couple that strike me.

First of all, Gheorghe Craciun claims to have proved the biggest open conjecture in this field: the Global Attractor Conjecture!

• Gheorge Craciun, Toric differential inclusions and a proof of the global attractor conjecture.

This famous old conjecture says that for a certain class of chemical reactions, the ones coming from ‘complex balanced reaction networks’, the chemicals will approach equilibrium no matter what their initial concentrations are. Here’s what Craciun says:

Abstract. In a groundbreaking 1972 paper Fritz Horn and Roy Jackson showed that a complex balanced mass-action system must have a unique locally stable equilibrium within any compatibility class. In 1974 Horn conjectured that this equilibrium is a global attractor, i.e., all solutions in the same compatibility class must converge to this equilibrium. Later, this claim was called the Global Attractor Conjecture, and it was shown that it has remarkable implications for the dynamics of large classes of polynomial and power-law dynamical systems, even if they are not derived from mass-action kinetics. Several special cases of this conjecture have been proved during the last decade. We describe a proof of the conjecture in full generality. In particular, it will follow that all detailed balanced mass action systems and all deficiency zero mass-action systems have the global attractor property. We will also discuss some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.

Manoj Gopalkrishnan wrote a great post explaining the concept of complex balanced reaction network here on Azimuth, so if you want to understand the conjecture you could start there.

Even better, Manoj is talking here about a way to do statistical inference with chemistry! His talk is called ‘Statistical inference with a chemical soup’:

Abstract. The goal is to design an “intelligent chemical soup” that can do statistical inference. This may have niche technological applications in medicine and biological research, as well as provide fundamental insight into the workings of biochemical reaction pathways. As a first step towards our goal, we describe a scheme that exploits the remarkable mathematical similarity between log-linear models in statistics and chemical reaction networks. We present a simple scheme that encodes the information in a log-linear model as a chemical reaction network. Observed data is encoded as initial concentrations, and the equilibria of the corresponding mass-action system yield the maximum likelihood estimators. The simplicity of our scheme suggests that molecular environments, especially within cells, may be particularly well suited to performing statistical computations.

It’s based on this paper:

• Manoj Gopalkrishnan, A scheme for molecular computation of maximum likelihood estimators for log-linear models.

I’m not sure, but this idea may exploit existing analogies between the approach to equilibrium in chemistry, the approach to equilibrium in evolutionary game theory, and statistical inference. You may have read Marc Harper’s post about that stuff!

David Doty is giving a broader review of ‘Computation by (not about) chemistry’:

Abstract. The model of chemical reaction networks (CRNs) is extensively used throughout the natural sciences as a descriptive language for existing chemicals. If we instead think of CRNs as a programming language for describing artificially engineered chemicals, what sorts of computations are possible for these chemicals to achieve? The answer depends crucially on several formal choices:

1) Do we treat matter as infinitely divisible (real-valued concentrations) or atomic (integer-valued counts)?

2) How do we represent the input and output of the computation (e.g., Boolean presence or absence of species, positive numbers directly represented by counts/concentrations, positive and negative numbers represented indirectly by the difference between counts/concentrations of a pair of species)?

3) Do we assume mass-action rate laws (reaction rates proportional to reactant counts/concentrations) or do we insist the system works correctly under a broader class of rate laws?

The talk will survey several recent results and techniques. A primary goal of the talk is to convey the “programming perspective”: rather than asking “What does chemistry do?”, we want to understand “What could chemistry do?” as well as “What can chemistry provably not do?”

I’m really interested in chemical reaction networks that appear in biological systems, and there will be lots of talks about that. For example, Ovidiu Radulescu will talk about ‘Taming the complexity of biochemical networks through model reduction and tropical geometry’. Model reduction is the process of simplifying complicated models while preserving at least some of their good features. Tropical geometry is a cool version of algebraic geometry that uses the real numbers with minimization as addition and addition as multiplication. This number system underlies the principle of least action, or the principle of maximum energy. Here is Radulescu’s abstract:

Abstract. Biochemical networks are used as models of cellular physiology with diverse applications in biology and medicine. In the absence of objective criteria to detect essential features and prune secondary details, networks generated from data are too big and therefore out of the applicability of many mathematical tools for studying their dynamics and behavior under perturbations. However, under circumstances that we can generically denote by multi-scaleness, large biochemical networks can be approximated by smaller and simpler networks. Model reduction is a way to find these simpler models that can be more easily analyzed. We discuss several model reduction methods for biochemical networks with polynomial or rational rate functions and propose as their common denominator the notion of tropical equilibration, meaning finite intersection of tropical varieties in algebraic geometry. Using tropical methods, one can strongly reduce the number of variables and parameters of biochemical network. For multi-scale networks, these reductions are computed symbolically on orders of magnitude of parameters and variables, and are valid in wide domains of parameter and phase spaces.

I’m talking about the analogy between probabilities and quantum amplitudes, and how this makes chemistry analogous to particle physics. You can see two versions of my talk here, but I’ll be giving the ‘more advanced’ version, which is new:

Abstract. Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, the master equation for a chemical reaction network describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this formalism turns out to involve creation and annihilation operators, coherent states and other well-known ideas—but with a few big differences.

Anyway, there are a lot more talks, but if I don’t have breakfast and walk over to the math department, I’ll miss those talks!

You can learn more about individual talks in the comments here (see below) and also in Matteo Polettini’s blog:

• Matteo Polettini, Mathematical trends in reaction network theory: part 1 and part 2, Out of Equilibrium, 1 July 2015.

## PROPs for Linear Systems

18 May, 2015

Eric Drexler likes to say: engineering is dual to science, because science tries to understand what the world does, while engineering is about getting the world to do what you want. I think we need a slightly less ‘coercive’, more ‘cooperative’ approach to the world in order to develop ‘ecotechnology’, but it’s still a useful distinction.

For example, classical mechanics is the study of what things do when they follow Newton’s laws. Control theory is the study of what you can get them to do.

Say you have an upside-down pendulum on a cart. Classical mechanics says what it will do. But control theory says: if you watch the pendulum and use what you see to move the cart back and forth correctly, you can make sure the pendulum doesn’t fall over!

Control theorists do their work with the help of ‘signal-flow diagrams’. For example, here is the signal-flow diagram for an inverted pendulum on a cart:

When I take a look at a diagram like this, I say to myself: that’s a string diagram for a morphism in a monoidal category! And it’s true. Jason Erbele wrote a paper explaining this. Independently, Bonchi, Sobociński and Zanasi did some closely related work:

• John Baez and Jason Erbele, Categories in control.

• Filippo Bonchi, Paweł Sobociński and Fabio Zanasi, Interacting Hopf algebras.

• Filippo Bonchi, Paweł Sobociński and Fabio Zanasi, A categorical semantics of signal flow graphs.

I’ll explain some of the ideas at the Turin meeting on the categorical foundations of network theory. But I also want to talk about this new paper that Simon Wadsley of Cambridge University wrote with my student Nick Woods:

• Simon Wadsley and Nick Woods, PROPs for linear systems.

This makes the picture neater and more general!

You see, Jason and I used signal flow diagrams to give a new description of the category of finite-dimensional vector spaces and linear maps. This category plays a big role in the control theory of linear systems. Bonchi, Sobociński and Zanasi gave a closely related description of an equivalent category, $\mathrm{Mat}(k),$ where:

• objects are natural numbers, and

• a morphism $f : m \to n$ is an $n \times m$ matrix with entries in the field $k,$

and composition is given by matrix multiplication.

But Wadsley and Woods generalized all this work to cover $\mathrm{Mat}(R)$ whenever $R$ is a commutative rig. A rig is a ‘ring without negatives’—like the natural numbers. We can multiply matrices valued in any rig, and this includes some very useful examples… as I’ll explain later.

Wadsley and Woods proved:

Theorem. Whenever $R$ is a commutative rig, $\mathrm{Mat}(R)$ is the PROP for bicommutative bimonoids over $R.$

This result is quick to state, but it takes a bit of explaining! So, let me start by bringing in some definitions.

### Bicommutative bimonoids

We will work in any symmetric monoidal category, and draw morphisms as string diagrams.

A commutative monoid is an object equipped with a multiplication:

and a unit:

obeying these laws:

For example, suppose $\mathrm{FinVect}_k$ is the symmetric monoidal category of finite-dimensional vector spaces over a field $k$, with direct sum as its tensor product. Then any object $V \in \mathrm{FinVect}_k$ is a commutative monoid where the multiplication is addition:

$(x,y) \mapsto x + y$

and the unit is zero: that is, the unique map from the zero-dimensional vector space to $V.$

Turning all this upside down, cocommutative comonoid has a comultiplication:

and a counit:

obeying these laws:

For example, consider our vector space $V \in \mathrm{FinVect}_k$ again. It’s a commutative comonoid where the comultiplication is duplication:

$x \mapsto (x,x)$

and the counit is deletion: that is, the unique map from $V$ to the zero-dimensional vector space.

Given an object that’s both a commutative monoid and a cocommutative comonoid, we say it’s a bicommutative bimonoid if these extra axioms hold:

You can check that these are true for our running example of a finite-dimensional vector space $V.$ The most exciting one is the top one, which says that adding two vectors and then duplicating the result is the same as duplicating each one, then adding them appropriately.

Our example has some other properties, too! Each element $c \in k$ defines a morphism from $V$ to itself, namely scalar multiplication by $c:$

$x \mapsto c x$

We draw this as follows:

These morphisms are compatible with the ones so far:

Moreover, all the ‘rig operations’ in $k$—that is, addition, multiplication, 0 and 1, but not subtraction or division—can be recovered from what we have so far:

We summarize this by saying our vector space $V$ is a bicommutative bimonoid ‘over $k$‘.

More generally, suppose we have a bicommutative bimonoid $A$ in a symmetric monoidal category. Let $\mathrm{End}(A)$ be the set of bicommutative bimonoid homomorphisms from $A$ to itself. This is actually a rig: there’s a way to add these homomorphisms, and also a way to ‘multiply’ them (namely, compose them).

Suppose $R$ is any commutative rig. Then we say $A$ is a bicommutative bimonoid over $R$ if it’s equipped with a rig homomorphism

$\Phi : R \to \mathrm{End}(A)$

This is a way of summarizing the diagrams I just showed you! You see, each $c \in R$ gives a morphism from $A$ to itself, which we write as

The fact that this is a bicommutative bimonoid endomorphism says precisely this:

And the fact that $\Phi$ is a rig homomorphism says precisely this:

So sometimes the right word is worth a dozen pictures!

What Jason and I showed is that for any field $k,$ the $\mathrm{FinVect}_k$ is the free symmetric monoidal category on a bicommutative bimonoid over $k.$ This means that the above rules, which are rules for manipulating signal flow diagrams, completely characterize the world of linear algebra!

Bonchi, Sobociński and Zanasi used ‘PROPs’ to prove a similar result where the field is replaced by a sufficiently nice commutative ring. And Wadlsey and Woods used PROPS to generalize even further to the case of an arbitrary commutative rig!

But what are PROPs?

### PROPs

A PROP is a particularly tractable sort of symmetric monoidal category: a strict symmetric monoidal category where the objects are natural numbers and the tensor product of objects is given by ordinary addition. The symmetric monoidal category $\mathrm{FinVect}_k$ is equivalent to the PROP $\mathrm{Mat}(k),$ where a morphism $f : m \to n$ is an $n \times m$ matrix with entries in $k,$ composition of morphisms is given by matrix multiplication, and the tensor product of morphisms is the direct sum of matrices.

We can define a similar PROP $\mathrm{Mat}(R)$ whenever $R$ is a commutative rig, and Wadsley and Woods gave an elegant description of the ‘algebras’ of $\mathrm{Mat}(R)$. Suppose $C$ is a PROP and $D$ is a strict symmetric monoidal category. Then the category of algebras of $C$ in $D$ is the category of strict symmetric monoidal functors $F : C \to D$ and natural transformations between these.

If for every choice of $D$ the category of algebras of $C$ in $D$ is equivalent to the category of algebraic structures of some kind in $D,$ we say $C$ is the PROP for structures of that kind. This explains the theorem Wadsley and Woods proved:

Theorem. Whenever $R$ is a commutative rig, $\mathrm{Mat}(R)$ is the PROP for bicommutative bimonoids over $R.$

The fact that an algebra of $\mathrm{Mat}(R)$ is a bicommutative bimonoid is equivalent to all this stuff:

The fact that $\Phi(c)$ is a bimonoid homomorphism for all $c \in R$ is equivalent to this stuff:

And the fact that $\Phi$ is a rig homomorphism is equivalent to this stuff:

This is a great result because it includes some nice new examples.

First, the commutative rig of natural numbers gives a PROP $\mathrm{Mat}.$ This is equivalent to the symmetric monoidal category $\mathrm{FinSpan},$ where morphisms are isomorphism classes of spans of finite sets, with disjoint union as the tensor product. Steve Lack had already shown that $\mathrm{FinSpan}$ is the PROP for bicommutative bimonoids. But this also follows from the result of Wadsley and Woods, since every bicommutative bimonoid $V$ is automatically equipped with a unique rig homomorphism

$\Phi : \mathbb{N} \to \mathrm{End}(V)$

Second, the commutative rig of booleans

$\mathbb{B} = \{F,T\}$

with ‘or’ as addition and ‘and’ as multiplication gives a PROP $\mathrm{Mat}(\mathbb{B}).$ This is equivalent to the symmetric monoidal category $\mathrm{FinRel}$ where morphisms are relations between finite sets, with disjoint union as the tensor product. Samuel Mimram had already shown that this is the PROP for special bicommutative bimonoids, meaning those where comultiplication followed by multiplication is the identity:

But again, this follows from the general result of Wadsley and Woods!

Finally, taking the commutative ring of integers $\mathbb{Z},$ Wadsley and Woods showed that $\mathrm{Mat}(\mathbb{Z})$ is the PROP for bicommutative Hopf monoids. The key here is that scalar multiplication by $-1$ obeys the axioms for an antipode—the extra morphism that makes a bimonoid into a Hopf monoid. Here are those axioms:

More generally, whenever $R$ is a commutative ring, the presence of $-1 \in R$ guarantees that a bimonoid over $R$ is automatically a Hopf monoid over $R.$ So, when $R$ is a commutative ring, Wadsley and Woods’ result implies that $\mathrm{Mat}(R)$ is the PROP for Hopf monoids over $R.$

Earlier, in their paper on ‘interacting Hopf algebras’, Bonchi, Sobociński and Zanasi had given an elegant and very different proof that $\mathrm{Mat}(R)$ is the PROP for Hopf monoids over $R$ whenever $R$ is a principal ideal domain. The advantage of their argument is that they build up the PROP for Hopf monoids over $R$ from smaller pieces, using some ideas developed by Steve Lack. But the new argument by Wadsley and Woods has its own charm.

In short, we’re getting the diagrammatics of linear algebra worked out very nicely, providing a solid mathematical foundation for signal flow diagrams in control theory!

## Resource Theories

12 May, 2015

by Brendan Fong

Hugo Nava-Kopp and I have a new paper on resource theories:

• Brendan Fong and Hugo Nava-Kopp, Additive monotones for resource theories of parallel-combinable processes with discarding.

A mathematical theory of resources is Tobias Fritz’s current big project. He’s already explained how ordered monoids can be viewed as theories of resource convertibility in a three part series on this blog.

Ordered monoids are great, and quite familiar in network theory: for example, a Petri net can be viewed as a presentation for an ordered commutative monoid. But this work started in symmetric monoidal categories, together with my (Oxford) supervisor Bob Coecke and Rob Spekkens.

The main idea is this: think of the objects of your symmetric monoidal category as resources, and your morphisms as ways to convert one resource into another. The monoidal product or ‘tensor product’ in your category allows you to talk about collections of your resources. So, for example, in the resource theory of chemical reactions, our objects are molecules like oxygen O2, hydrogen H2, and water H2O, and morphisms things like the electrolysis of water:

2H2O → O2 + 2H2

This is a categorification of the ordered commutative monoid of resource convertibility: we now keep track of how we convert resources into one another, instead of just whether we can convert them.

Categorically, I find the other direction easier to state: being a category, the resource theory is enriched over $\mathrm{Set}$, while a poset is enriched over the poset of truth values or ‘booleans’ $\mathbb{B} = \{0,1\}.$ If we ‘partially decategorify’ by changing the base of enrichment along the functor $\mathrm{Set} \to \mathbb{B}$ that maps the empty set to 0 and any nonempty set to 1, we obtain the ordered monoid corresponding to the resource theory.

But the research programme didn’t start at resource theories either. The starting point was ‘partitioned process theories’.

Here’s an example that guided the definitions. Suppose we have a bunch of labs with interacting quantum systems, separated in space. With enough cooperation and funding, they can do big joint operations on their systems, like create entangled pairs between two locations. For ‘free’, however, they’re limited to classical communication between the locations, although they can do the full range of quantum operations on their local system. So you’ve got a symmetric monoidal category with objects quantum systems and morphisms quantum operations, together with a wide (all-object-including) symmetric monoidal subcategory that contains the morphisms you can do with local quantum operations and classical communication (known as LOCC operations).

This general structure: a symmetric monoidal category (or SMC for short) with a wide symmetric monoidal subcategory, is called a partitioned process theory. We call the morphisms in the SMC processes, and those in the subSMC free processes.

There are a number of methods for building a resource theory (i.e. an SMC) from a partitioned process theory. The unifying idea though, is that your new SMC has the processes $f,g$ as objects, and morphisms $f \to g$ ways of using the free processes to build $g$ from $f.$

But we don’t have to go to fancy sounding quantum situations to find examples of partitioned process theories. Instead, just look at any SMC in which each object is equipped with an algebraic structure. Then the morphisms defining this structure can be taken as our ‘free’ processes.

For example, in a multigraph category every object has the structure of a ‘special commutative Frobenius algebra’. That’s a bit of a mouthful, but John defined it a while back, and examples include categories where morphisms are electrical circuits, and categories where morphisms are signal flow diagrams.

So these categories give partitioned process theories! This idea of partitioning the morphisms into ‘free’ ones and ‘costly’ ones is reminiscent of what I was saying earlier about the operad of wiring diagrams about it being useful to separate behavioural structure from interconnection structure.

This suggests that we can also view the free processes as generating some sort of operad, that describes the ways we allow ourselves to use free processes to turn processes into other processes. If we really want to roll a big machine out to play with this stuff, framed bicategories may also be interesting; Spivak is already using them to get at questions about operads. But that’s all conjecture, and a bit of a digression.

To get back to the point, this was all just to say that if you find yourself with a bunch of resistors, and you ask ‘what can I build?’, then you’re after the resource theory apparatus.