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: 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 X_{1}, X_{2} and X_{3} 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 and a ‘more detailed’ algebra they will typically be related by a map

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

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 and search for an element of that is mapped to it by 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

A **monoid** is a set M with an binary product that is associative and has an identity element 1:

Thanks to the associative law we don’t need to write parentheses when we multiply a string of elements in a monoid.

A **graphic monoid** is a monoid in which the **graphic identity**

holds for all

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

In particular, in any graphic monoid we have

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

A commutative monoid obeying this law automatically obeys the graphic identity, since then

But for a noncommutative monoid, the graphic identity is stronger than . It says that *after committing to no matter what intervening commitments one might have made, committing to 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.

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. In a **commitment network** 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 networks. 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 by

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

transitivity:

antisymmetry:

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