Applied Category Theory at NIST (Part 2)

18 April, 2018

Here are links to the slides and videos for most of the talks from this workshop:

Applied Category Theory: Bridging Theory & Practice, March 15–16, 2018, NIST, Gaithersburg, Maryland, USA. Organized by Spencer Breiner and Eswaran Subrahmanian.

They give a pretty good picture of what went on. Spencer Breiner put them up here; what follows is just a copy of what’s on his site.

Unfortunately, the end of Dusko Pavlovic’s talk, as well as Ryan Wisnesky’s and Steve Huntsman’s were lost due to a technical error. You can also find a Youtube playlist with all of the videos here.

Introduction to NIST:

Ram Sriram – NIST and Category Theory

 

Spencer Breiner – Introduction

Invited talks:

Bob Coecke – From quantum foundations to cognition via pictures

 

Dusko Pavlovic – Security Science in string diagrams (partial video)

 

John Baez – Compositional design and tasking of networks (part 1)

 

John Foley – Compositional design and tasking of networks (part 2)

 

David Spivak – A higher-order temporal logic for dynamical systems

 

Lightning Round Talks:

Ryan Wisnesky – Categorical databases (no video)

Steve Huntsman – Towards an operad of goals (no video)

 

Bill Regli – Disrupting interoperability (no slides)

 

Evan Patterson – Applied category theory in data science

 

Brendan Fong – data structures for network languages

 

Stephane Dugowson – A short introduction to a general theory of interactivity

 

Michael Robinson – Sheaf methods for inference

 

Cliff Joslyn – Seeking a categorical systems theory via the category of hypergraphs

 

Emilie Purvine – A category-theoretical investigation of the type hierarchy for heterogeneous sensor integration

 

Helle Hvid Hansen – Long-term values in Markov decision processes, corecursively

 

Alberto Speranzon – Localization and planning for autonomous systems via (co)homology computation

 

Josh Tan – Indicator frameworks (no slides)

Breakout round report


Applied Category Theory 2018 Schedule

13 April, 2018

Here’s the schedule of the ACT2018 workshop:

Click to enlarge!

They put me on last, either because my talk will be so boring that it’s okay everyone will have left, or because my talk will be so exciting that nobody will want to leave. I haven’t dared ask the organizers which one.

On the other hand, they’ve put me on first for the “school” which occurs one week before the workshop. Here’s the schedule for the ACT 2018 Adjoint School:


Applied Category Theory Course

26 March, 2018

It just became a lot easier to learn about applied category theory, thanks to this free book:

• Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.

I’ve started an informal online course based on this book on the Azimuth Forum. I’m getting pretty sick of the superficial quality of my interactions on social media. This could be a way to do something more interesting.

The idea is that you can read chapters of this book, discuss them, try the exercises in the book, ask and answer questions, and maybe team up to create software that implements some of the ideas. I’ll try to keep things moving forward. For example, I’ll explain some stuff and try to help answer questions that people are stuck on. I may also give some talks or run discussions on Google Hangouts or similar software—but only when I have time: I’m more of a text-based guy. I may get really busy some times, and leave the rest of you alone for a while. But I like writing about math for at least 15 minutes a day, and more when I have time. Furthermore, I’m obsessed with applied category theory and plan to stay that way for at least a few more years.

If this sounds interesting, let me know here—and please visit the Azimuth Forum and register! Use your full real name as your username, with no spaces. I will add spaces and that will become your username. Use a real working email address. If you don’t, the registration process may not work.

Over 70 people have registered so far, so this process will take a while.

The main advantage of the Forum over this blog is that you can initiate new threads and edit your comments. Like here you can write equations in LaTeX. Like here, that ability is severely limited: for example you can’t define macros, and you can’t use TikZ. (Maybe someone could fix that.) But equations are better typeset over there—and more importantly, the ability to edit comments makes it a lot easier to correct errors in your LaTeX.

Please let me know what you think.

What follows is the preface to Fong and Spivak’s book, just so you can get an idea of what it’s like.

Preface

Category theory is becoming a central hub for all of pure mathematics. It is unmatched in its ability to organize and layer abstractions, to find commonalities between structures of all sorts, and to facilitate communication between different mathematical communities. But it has also been branching out into science, informatics, and industry. We believe that it has the potential to be a major cohesive force in the world, building rigorous bridges between disparate worlds, both theoretical and practical. The motto at MIT is mens et manus, Latin for mind and hand. We believe that category theory—and pure math in general—has stayed in the realm of mind for too long; it is ripe to be brought to hand.

Purpose and audience

The purpose of this book is to offer a self-contained tour of applied category theory. It is an invitation to discover advanced topics in category theory through concrete real-world examples. Rather than try to give a comprehensive treatment of these topics—which include adjoint functors, enriched categories, proarrow equipments, toposes, and much more–we merely provide a taste. We want to give readers some insight into how it feels to work with these structures as well as some ideas about how they might show up in practice.

The audience for this book is quite diverse: anyone who finds the above description intriguing. This could include a motivated high school student who hasn’t seen calculus yet but has loved reading a weird book on mathematical logic they found at the library. Or a machine learning researcher who wants to understand what vector spaces, design theory, and dynamical systems could possibly have in common. Or a pure mathematician who wants to imagine what sorts of applications their work might have. Or a recently-retired programmer who’s always had an eerie feeling that category theory is what they’ve been looking for to tie it all together, but who’s found the usual books on the subject impenetrable.

For example, we find it something of a travesty that in 2018 there seems to be no introductory material available on monoidal categories. Even beautiful modern introductions to category theory, e.g. by Riehl or Leinster, do not include anything on this rather central topic. The basic idea is certainly not too abstract; modern human intuition seems to include a pre-theoretical understanding of monoidal categories that is just waiting to be formalized. Is there anyone who wouldn’t correctly understand the basic idea being communicated in the following diagram?

Many applied category theory topics seem to take monoidal categories as their jumping off point. So one aim of this book is to provide a reference—even if unconventional—for this important topic.

We hope this book inspires both new visions and new questions. We intend it to be self-contained in the sense that it is approachable with minimal prerequisites, but not in the sense that the complete story is told here. On the contrary, we hope that readers use this as an invitation to further reading, to orient themselves in what is becoming a large literature, and to discover new applications for themselves.

This book is, unashamedly, our take on the subject. While the abstract structures we explore are important to any category theorist, the specific topics have simply been chosen to our personal taste. Our examples are ones that we find simple but powerful, concrete but representative, entertaining but in a way that feels important and expansive at the same time. We hope our readers will enjoy themselves and learn a lot in the process.

How to read this book

The basic idea of category theory—which threads through every chapter—is that if one pays careful attention to structures and coherence, the resulting systems will be extremely reliable and interoperable. For example, a category involves several structures: a collection of objects, a collection of morphisms relating objects, and a formula for combining any chain of morphisms into a morphism. But these structures need to cohere or work together in a simple commonsense way: a chain of chains is a chain, so combining a chain of chains should be the same as combining the chain. That’s it!

We will see structures and coherence come up in pretty much every definition we give: “here are some things and here are how they fit together.” We ask the reader to be on the lookout for structures and coherence as they read the book, and to realize that as we layer abstraction on abstraction, it is the coherence that makes everything function like a well-oiled machine.

Each chapter in this book is motivated by a real-world topic, such as electrical circuits, control theory, cascade failures, information integration, and hybrid systems. These motivations lead us into and through various sorts of category-theoretic concepts.

We generally have one motivating idea and one category-theoretic purpose per chapter, and this forms the title of the chapter, e.g. Chapter 4 is “Collaborative design: profunctors, categorification, and monoidal categories.” In many math books, the difficulty is roughly a monotonically-increasing function of the page number. In this book, this occurs in each chapter, but not so much in the book as a whole. The chapters start out fairly easy and progress in difficulty.

The upshot is that if you find the end of a chapter very difficult, hope is certainly not lost: you can start on the next one and make good progress. This format lends itself to giving you a first taste now, but also leaving open the opportunity for you to come back at a later date and get more deeply into it. But by all means, if you have the gumption to work through each chapter to its end, we very much encourage that!

We include many exercises throughout the text. Usually these exercises are fairly straightforward; the only thing they demand is that the reader’s mind changes state from passive to active, rereads the previous paragraphs with intent, and puts the pieces together. A reader becomes a student when they work the exercises; until then they are more of a tourist, riding on a bus and listening off and on to the tour guide. Hey, there’s nothing wrong with that, but we do encourage you to get off the bus and make contact with the natives as often as you can.


Hypergraph Categories of Cospans

11 March, 2018

 

Two students in the Applied Category Theory 2018 school wrote a blog article about Brendan Fong’s theory of decorated cospans:

• Jonathan Lorand and Fabrizio Genovese, Hypergraph categories of cospans, The n-Category Café, 28 February 2018.

Jonathan Lorand is a math grad student at the University of Zurich working on symplectic and Poisson geometry with Alberto Cattaneo. Fabrizio Genovese is a grad student in computer science at the University of Oxford, working with Bob Coecke and Dan Marsden on categorical quantum mechanics, quantum field theory and the like.

Brendan was my student, so it’s nice to see newer students writing a clear summary of some of his thesis work, namely this paper:

• Brendan Fong, Decorated cospans, Theory and Applications of Categories 30 (2015), 1096–1120.

I wrote a summary of it myself, so I won’t repeat it here:

• John Baez, Decorated cospans, Azimuth, 1 May 2015.

What’s especially interesting to me is that both Jonathan and Fabrizio know some mathematical physics, and they’re part of a group who will be working with me on some problems as part of the Applied Category Theory 2018 school! Brendan and Blake Pollard and I used symplectic geometry and decorated cospans to study the black-boxing of electrical circuits and Markov processes… maybe we should try to go further with that project!


Coarse-Graining Open Markov Processes

4 March, 2018

Kenny Courser and I have been working hard on this paper for months:

• John Baez and Kenny Courser, Coarse-graining open Markov processes.

It may be almost done. So, it would be great if people here could take a look and comment on it! It’s a cool mix of probability theory and double categories. I’ve posted a similar but non-isomorphic article on the n-Category Café, where people know a lot about double categories. But maybe some of you here know more about Markov processes!

‘Coarse-graining’ is a standard method of extracting a simple Markov process from a more complicated one by identifying states. We extend coarse-graining to open Markov processes. An ‘open’ Markov process is one where probability can flow in or out of certain states called ‘inputs’ and ‘outputs’. One can build up an ordinary Markov process from smaller open pieces in two basic ways:

• composition, where we identify the outputs of one open Markov process with the inputs of another,

and

• tensoring, where we set two open Markov processes side by side.

A while back, Brendan Fong, Blake Pollard and I showed that these constructions make open Markov processes into the morphisms of a symmetric monoidal category:

A compositional framework for Markov processes, Azimuth, January 12, 2016.

Here Kenny and I go further by constructing a symmetric monoidal double category where the 2-morphisms include ways of coarse-graining open Markov processes. We also extend the previously defined ‘black-boxing’ functor from the category of open Markov processes to this double category.

But before you dive into the paper, let me explain all this stuff a bit more….

Very roughly speaking, a ‘Markov process’ is a stochastic model describing a sequence of transitions between states in which the probability of a transition depends only on the current state. But the only Markov processes talk about are continuous-time Markov processes with a finite set of states. These can be drawn as labeled graphs:

where the number labeling each edge describes the probability per time of making a transition from one state to another.

An ‘open’ Markov process is a generalization in which probability can also flow in or out of certain states designated as ‘inputs’ and outputs’:

Open Markov processes can be seen as morphisms in a category, since we can compose two open Markov processes by identifying the outputs of the first with the inputs of the second. Composition lets us build a Markov process from smaller open parts—or conversely, analyze the behavior of a Markov process in terms of its parts.

In this paper, Kenny extend the study of open Markov processes to include coarse-graining. ‘Coarse-graining’ is a widely studied method of simplifying a Markov process by mapping its set of states X onto some smaller set X' in a manner that respects the dynamics. Here we introduce coarse-graining for open Markov processes. And we show how to extend this notion to the case of maps p: X \to X' that are not surjective, obtaining a general concept of morphism between open Markov processes.

Since open Markov processes are already morphisms in a category, it is natural to treat morphisms between them as morphisms between morphisms, or ‘2-morphisms’. We can do this using double categories!

Double categories were first introduced by Ehresmann around 1963. Since then, they’ve used in topology and other branches of pure math—but more recently they’ve been used to study open dynamical systems and open discrete-time Markov chains. So, it should not be surprising that they are also useful for open Markov processes..

A 2-morphism in a double category looks like this:

While a mere category has only objects and morphisms, here we have a few more types of things. We call A, B, C and D ‘objects’, f and g ‘vertical 1-morphisms’, M and N ‘horizontal 1-cells’, and \alpha a ‘2-morphism’. We can compose vertical 1-morphisms to get new vertical 1-morphisms and compose horizontal 1-cells to get new horizontal 1-cells. We can compose the 2-morphisms in two ways: horizontally by setting squares side by side, and vertically by setting one on top of the other. The ‘interchange law’ relates vertical and horizontal composition of 2-morphisms.

In a ‘strict’ double category all these forms of composition are associative. In a ‘pseudo’ double category, horizontal 1-cells compose in a weakly associative manner: that is, the associative law holds only up to an invertible 2-morphism, the ‘associator’, which obeys a coherence law. All this is just a sketch; for details on strict and pseudo double categories try the paper by Grandis and Paré.

Kenny and I construct a double category \mathbb{M}\mathbf{ark} with:

  1. finite sets as objects,
  2. maps between finite sets as vertical 1-morphisms,
  3. open Markov processes as horizontal 1-cells,
  4. morphisms between open Markov processes as 2-morphisms.

I won’t give the definition of item 4 here; you gotta read our paper for that! Composition of open Markov processes is only weakly associative, so \mathbb{M}\mathbf{ark} is a pseudo double category.

This is how our paper goes. In Section 2 we define open Markov processes and steady state solutions of the open master equation. In Section 3 we introduce coarse-graining first for Markov processes and then open Markov processes. In Section 4 we construct the double category \mathbb{M}\mathbf{ark} described above. We prove this is a symmetric monoidal double category in the sense defined by Mike Shulman. This captures the fact that we can not only compose open Markov processes but also ‘tensor’ them by setting them side by side.

For example, if we compose this open Markov process:

with the one I showed you before:

we get this open Markov process:

But if we tensor them, we get this:

As compared with an ordinary Markov process, the key new feature of an open Markov process is that probability can flow in or out. To describe this we need a generalization of the usual master equation for Markov processes, called the ‘open master equation’.

This is something that Brendan, Blake and I came up with earlier. In this equation, the probabilities at input and output states are arbitrary specified functions of time, while the probabilities at other states obey the usual master equation. As a result, the probabilities are not necessarily normalized. We interpret this by saying probability can flow either in or out at both the input and the output states.

If we fix constant probabilities at the inputs and outputs, there typically exist solutions of the open master equation with these boundary conditions that are constant as a function of time. These are called ‘steady states’. Often these are nonequilibrium steady states, meaning that there is a nonzero net flow of probabilities at the inputs and outputs. For example, probability can flow through an open Markov process at a constant rate in a nonequilibrium steady state. It’s like a bathtub where water is flowing in from the faucet, and flowing out of the drain, but the level of the water isn’t changing.

Brendan, Blake and I studied the relation between probabilities and flows at the inputs and outputs that holds in steady state. We called the process of extracting this relation from an open Markov process ‘black-boxing’, since it gives a way to forget the internal workings of an open system and remember only its externally observable behavior. We showed that black-boxing is compatible with composition and tensoring. In other words, we showed that black-boxing is a symmetric monoidal functor.

In Section 5 of our new paper, Kenny and I show that black-boxing is compatible with morphisms between open Markov processes. To make this idea precise, we prove that black-boxing gives a map from the double category \mathbb{M}\mathbf{ark} to another double category, called \mathbb{L}\mathbf{inRel}, which has:

  1. finite-dimensional real vector spaces U,V,W,\dots as objects,
  2. linear maps f : V \to W as vertical 1-morphisms from V to W,
  3. linear relations R \subseteq V \oplus W as horizontal 1-cells from V to W,
  4. squares

    obeying (f \oplus g)R \subseteq S as 2-morphisms.

Here a ‘linear relation’ from a vector space V to a vector space W is a linear subspace R \subseteq V \oplus W. Linear relations can be composed in the usual way we compose relations. The double category \mathbb{L}\mathbf{inRel} becomes symmetric monoidal using direct sum as the tensor product, but unlike \mathbb{M}\mathbf{ark} it is a strict double category: that is, composition of linear relations is associative.

Our main result, Theorem 5.5, says that black-boxing gives a symmetric monoidal double functor

\blacksquare : \mathbb{M}\mathbf{ark} \to \mathbb{L}\mathbf{inRel}

As you’ll see if you check out our paper, there’s a lot of nontrivial content hidden in this short statement! The proof requires a lot of linear algebra and also a reasonable amount of category theory. For example, we needed this fact: if you’ve got a commutative cube in the category of finite sets:

and the top and bottom faces are pushouts, and the two left-most faces are pullbacks, and the two left-most arrows on the bottom face are monic, then the two right-most faces are pullbacks. I think it’s cool that this is relevant to Markov processes!

Finally, in Section 6 we state a conjecture. First we use a technique invented by Mike Shulman to construct symmetric monoidal bicategories \mathbf{Mark} and \mathbf{LinRel} from the symmetric monoidal double categories \mathbb{M}\mathbf{ark} and \mathbb{L}\mathbf{inRel}. We conjecture that our black-boxing double functor determines a functor between these symmetric monoidal bicategories. This has got to be true. However, double categories seem to be a simpler framework for coarse-graining open Markov processes.

Finally, let me talk a bit about some related work. As I already mentioned, Brendan, Blake and I constructed a symmetric monoidal category where the morphisms are open Markov processes. However, we formalized such Markov processes in a slightly different way than Kenny and I do now. We defined a Markov process to be one of the pictures I’ve been showing you: a directed multigraph where each edge is assigned a positive number called its ‘rate constant’. In other words, we defined it to be a diagram

where X is a finite set of vertices or ‘states’, E is a finite set of edges or ‘transitions’ between states, the functions s,t : E \to X give the source and target of each edge, and r : E \to (0,\infty) gives the rate constant for each transition. We explained how from this data one can extract a matrix of real numbers (H_{i j})_{i,j \in X} called the ‘Hamiltonian’ of the Markov process, with two properties that are familiar in this game:

H_{i j} \geq 0 if i \neq j,

\sum_{i \in X} H_{i j} = 0 for all j \in X.

A matrix with these properties is called ‘infinitesimal stochastic’, since these conditions are equivalent to \exp(t H) being stochastic for all t \ge 0.

In our new paper, Kenny and I skip the directed multigraphs and work directly with the Hamiltonians! In other words, we define a Markov process to be a finite set X together with an infinitesimal stochastic matrix (H_{ij})_{i,j \in X}. This allows us to work more directly with the Hamiltonian and the all-important ‘master equation’

\displaystyle{    \frac{d p(t)}{d t} = H p(t)  }

which describes the evolution of a time-dependent probability distribution

p(t) : X \to \mathbb{R}

Clerc, Humphrey and Panangaden have constructed a bicategory with finite sets as objects, ‘open discrete labeled Markov processes’ as morphisms, and ‘simulations’ as 2-morphisms. The use the word ‘open’ in a pretty similar way to me. But their open discrete labeled Markov processes are also equipped with a set of ‘actions’ which represent interactions between the Markov process and the environment, such as an outside entity acting on a stochastic system. A ‘simulation’ is then a function between the state spaces that map the inputs, outputs and set of actions of one open discrete labeled Markov process to the inputs, outputs and set of actions of another.

Another compositional framework for Markov processes was discussed by de Francesco Albasini, Sabadini and Walters. They constructed an algebra of ‘Markov automata’. A Markov automaton is a family of matrices with non-negative real coefficients that is indexed by elements of a binary product of sets, where one set represents a set of ‘signals on the left interface’ of the Markov automata and the other set analogously for the right interface.

So, double categories are gradually invading the theory of Markov processes… as part of the bigger trend toward applied category theory. They’re natural things; scientists should use them.


Cartesian Bicategories

1 March, 2018

Two students in the Applied Category Theory 2018 school have blogged about a classic paper in category theory:

• Daniel Cicala and Jules Hedges, Cartesian bicategories, The n-Category Café, 19 February 2018.

Jules Hedges is a postdoc in the computer science department at Oxford who is applying category theory to game theory and economics. Daniel Cicala is a grad student working with me on a compositional approach to graph rewriting, which is about stuff like this:

This picture shows four ‘open graphs’: graphs with inputs and outputs. The vertices are labelled with operations. The top of the picture shows a ‘rewrite rule’ where one open graph is turned into another: the operation of multiplying by 2 is replaced by the operation of adding something to itself. The bottom of the picture shows one way we can ‘apply’ this rule: this takes us from open graph at bottom left to the open graph at bottom right.

So, we can use graph rewriting to think about ways to transform a computer program into another, perhaps simpler, computer program that does the same thing.

How do we formalize this?

A computer program wants to be a morphism, since it’s a process that turns some input into some output. Rewriting wants to be a 2-morphism, since it’s a ‘meta-process’ that turns some program into some other program. So, there should be some bicategory with computer programs (or labelled open graphs!) as morphisms and rewrites as 2-morphisms. In fact there should be a bunch of such bicategories, since there are a lot of details that one can tweak.

Together with my student Kenny Courser, Daniel has been investigating these bicategories:

• Daniel Cicala, Spans of cospans, Theory and Applications of Categories 33 (2018), 131–147.

Abstract. We discuss the notion of a span of cospans and define, for them, horizonal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting.

• Daniel Cicala, Spans of cospans in a topos, Theory and Applications of Categories 33 (2018), 1–22.

Abstract. For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that MonicSp(Csp(T)) is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as 2-morphisms inside a compact closed sub-bicategory of MonicSp(Csp(Graph)).

This stuff sounds abstract and esoteric when they talk about it, but it’s really all about things like the picture above—and it’s an important part of network theory!

Recently Daniel Cicala has noticed that some of the bicategories he’s getting are ‘cartesian bicategories’ in the sense of this paper:

• Aurelio Carboni and Robert F. C. Walters, Cartesian bicategories I, Journal of Pure and Applied Algebra 49 (1987), 11–32.

And that’s the paper he’s blogging about now with Jules Hedges!


Complex Adaptive System Design (Part 7)

19 February, 2018

In March, I’ll be talking at Spencer Breiner‘s workshop on Applied Category Theory at the National Institute of Standards and Technology. I’ll be giving a joint talk with John Foley about our work using operads to design networks. This work is part of the Complex Adaptive System Composition and Design Environment project being done by Metron Scientific Solutions and managed by John Paschkewitz at DARPA.

I’ve written about this work before:

• Complex adaptive systems design: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6.

But we’ve done a lot more, and my blog articles are having trouble keeping up! So I’d like to sketch out the big picture as it stands today.

If I had to summarize, I’d say we’ve developed a formalism for step-by-step compositional design and tasking, using commitment networks. But this takes a while to explain.

Here’s a very simple example of a commitment network:

It has four nodes, which represent agents: a port, a helicopter, a UAV (an unmanned aerial vehicle, or drone) and a target. The edges between these notes describe relationships between these agents. Some of these relationships are ‘commitments’. For example, the edges labelled ‘SIR’ say that one agent should ‘search, intervene and rescue’ the other.

Our framework for dealing with commitment networks has some special features. It uses operads, but this isn’t really saying much. An ‘operad’ is a bunch of ways of sticking things together. An ‘algebra’ of the operad gives a particular collection of these things, and says what we get when we stick them together. These concepts are extremely general, so there’s a huge diversity of operads, each with a huge diversity of algebras. To say one is using operads to solve a problem is a bit like saying one is using math. What matters more is the specific kind of operad one is using, and how one is using it.

For our work, we needed to develop a new class of operads called network operads, which are described here:

• John Baez, John Foley, Joseph Moeller and Blake Pollard, Network models.

In this paper we mainly discuss communication networks. Subsequently we’ve been working on a special class of network operads that describe how to build commitment networks.

Here are some of key ideas:

• Using network operads we can build bigger networks from smaller ones by overlaying them. David Spivak’s operad of wiring diagrams only let us ‘wire together’ smaller networks to form bigger ones:

Here networks X1, X2 and X3 are being wired together to form Y.

Network operads also let us wire together networks, but in addition they let us take one network:

and overlay another:

to create a larger network:

This is a new methodology for designing systems. We’re all used to building systems by wiring together subsystems: anyone who has a home stereo system has done this. But overlaying systems lets us do more. For example, we can take two plans of action involving the same collection of agents, and overlay them to get a new plan. We’ve all done this, too: you tell a bunch of people to do things… and then tell the same people, or an overlapping set of people, to do some other things. But lots of problems can arise if you aren’t careful. A mathematically principled approach can avoid some of these problems.

• The nodes of our networks represent agents of various types. The edges represent various relationships between agents. For example, they can represent communication channels. But more interestingly, they can represent commitments. For example, we can have an edge from A to B saying that agent A has to go rescue agent B. We call this kind of network a commitment network.

• By overlaying commitment networks, we can not only build systems out of smaller pieces but also build complicated plans by overlaying smaller pieces of plans. Since ‘tasking’ means telling a system what to do, we call this compositional tasking.

• If one isn’t careful, overlaying commitment networks can produce conflicts. Suppose we have a network with an edge saying that agent A has to rescue agent B. On top of this we overlay a network with an edge saying that A has to rescue agent C. If A can’t do both of these tasks at once, what should A do? There are various choices. We need to build a specific choice into the framework, so we can freely overlay commitment networks and get a well-defined result that doesn’t overburden the agents involved. We call this automatic deconflicting.

• Our approach to automatic deconflicting uses an idea developed by the famous category theorist Bill Lawvere: graphic monoids. I’ll explain these later, along with some of their side-benefits.

• Networks operads should let us do step-by-step compositional tasking. In other words, they should let us partially automate the process of tasking networks of agents, both

1) compositionally: tasking smaller networks and then sticking them together, e.g. by overlaying them, to get larger networks,

and

2) in a step-by-step way, starting at a low level of detail and then increasing the amount of detail.

To do this we need not just operads but their algebras.

• Remember, a network operad is a bunch of ways to stick together networks of some kind, e.g. by overlaying them. An algebra of this operad specifies a particular collection of networks of this kind, and says what we actually get when we stick them together.

So, a network operad can have one algebra in which things are described in a bare-bones, simplified way, and another algebra in which things are described in more detail. Indeed it will typically have many algebras, corresponding to many levels of detail, but for simplicity let’s just think about two.

When we have a ‘less detailed’ algebra A and a ‘more detailed’ algebra A', they will typically be related by a map

f \colon A' \to A

which ‘forgets the extra details’. This sort of map is called a homomorphism of algebras. We give examples in our paper Network models.

But what we usually want to do, when designing a system, is not forget extra detail, but rather add extra detail to a rough specification. There is not always a systematic way to do this. If there is, then we may have a homomorphism

g \colon A \to A'

going back the other way. This lets us automate the process of filling in the details. But we can’t usually count on being able to do this. So, often we may have to start with an element of A and search for an element of A' that is mapped to it by f : A' \to A. And typically we want this element to be optimal, or at least ‘good enough’, according to some chosen criteria. Expressing this idea formally helps us figure out how to automate the search. John Foley, in particular, has been working on this.

That’s an overview of our ideas.

Next, for the mathematically inclined, I want to give a few more details on one of the new elements not mentioned in our Network models paper: ‘graphic monoids’.

Graphic monoids

In our paper Network models we explain how the ‘overlay’ operation makes the collection of networks involving a given set of agents into a monoid. A monoid is a set M with a product that is associative and has an identity element 1:

(xy)z = x(yz)
1 x = x = x 1

In our application, this product is overlaying two networks.

A graphic monoid is one in which the graphic identity

x y x = x y

holds for all x,y.

To understand this identity, let us think of the elements of the monoid as “commitments”. The product x y means “first committing to do x, then committing to do y”. The graphic identity says that if we first commit to do x, then y, and then x again, it’s the same as first committing to do x and then y. Committing to do x again doesn’t change anything!

In particular, in any graphic monoid we have

xx = x 1 x = x 1 = x

so making the same commitment twice is the same as making it once. Mathematically we say every element x of a graphic monoid is idempotent:

x^2 = x

A commutative monoid obeying this law x^2 = x automatically obeys the graphic identity, since then

x y x = x^2 y = x y

But for a noncommutative monoid, the graphic identity is stronger than x^2 = x. It says that after committing to x, no matter what intervening commitments one might have made, committing to x again has no further effect. In other words: the intervening commitments did not undo the original commitment, so making the original commitment a second time has no effect! This captures the idea of how promises should behave.

As I said, for any network model, the set of all networks involving a fixed set of agents is a monoid. In a commitment network model, this monoid is required to be a graphic monoid. Joseph Moeller is writing a paper that shows how to construct a large class of commitment network models. We will follow this with a paper illustrating how to use these in compositional tasking.

For now, let me just mention a side-benefit. In any graphic monoid we can define a relation x \le y by

x \le y  \; \iff \; x a = y $ for some a

This makes the graphic monoid into a partially ordered set, meaning that these properties hold:

reflexivity: x \le x

transitivity: x \le y , y \le z \; \implies \; x \le z

antisymmetry: x \le y, y \le x \; \implies x = y

In the context of commitment networks, x \le y means that starting from x we can reach y by making some further commitment a: that is, x a = y for some a. So, as we ‘task’ a collection of agents by giving them more and more commitments, we move up in this partial order.