## Higher-Dimensional Rewriting in Warsaw

18 February, 2015

This summer there will be a conference on higher-dimensional algebra and rewrite rules in Warsaw. They want people to submit papers! I’ll give a talk about presentations of symmetric monoidal categories that arise in electrical engineering and control theory. This is part of the network theory program, which we talk about so often here on Azimuth.

There should also be interesting talks about combinatorial algebra, homotopical aspects of rewriting theory, and more:

Higher-Dimensional Rewriting and Applications, 28-29 June 2015, Warsaw, Poland. Co-located with the RDP, RTA and TLCA conferences. Organized by Yves Guiraud, Philippe Malbos and Samuel Mimram.

### Description

Over recent years, rewriting methods have been generalized from strings and terms to richer algebraic structures such as operads, monoidal categories, and more generally higher dimensional categories. These extensions of rewriting fit in the general scope of higher-dimensional rewriting theory, which has emerged as a unifying algebraic framework. This approach allows one to perform homotopical and homological analysis of rewriting systems (Squier theory). It also provides new computational methods in combinatorial algebra (Artin-Tits monoids, Coxeter and Garside structures), in homotopical and homological algebra (construction of cofibrant replacements, Koszulness property). The workshop is open to all topics concerning higher-dimensional generalizations and applications of rewriting theory, including

• higher-dimensional rewriting: polygraphs / computads, higher-dimensional generalizations of string/term/graph rewriting systems, etc.

• homotopical invariants of rewriting systems: homotopical and homological finiteness properties, Squier theory, algebraic Morse theory, coherence results in algebra and higher-dimensional category theory, etc.

• linear rewriting: presentations and resolutions of algebras and operads, Gröbner bases and generalizations, homotopy and homology of algebras and operads, Koszul duality theory, etc.

• applications of higher-dimensional and linear rewriting and their interactions with other fields: calculi for quantum computations, algebraic lambda-calculi, proof nets, topological models for concurrency, homotopy type theory, combinatorial group theory, etc.

• implementations: the workshop will also be interested in implementation issues in higher-dimensional rewriting and will allow demonstrations of prototypes of existing and new tools in higher-dimensional rewriting.

### Submitting

Important dates:

• Submission: April 15, 2015

• Final version: May 20, 2015

• Conference: 28-29 June, 2015

Submissions should consist of an extended abstract, approximately 5 pages long, in standard article format, in PDF. The page for uploading those is here. The accepted extended abstracts will be made available electronically before the
workshop.

### Organizers

Program committee:

• Vladimir Dotsenko (Trinity College, Dublin)

• Yves Guiraud (INRIA / Université Paris 7)

• Jean-Pierre Jouannaud (École Polytechnique)

• Philippe Malbos (Université Claude Bernard Lyon 1)

• Paul-André Melliès (Université Paris 7)

• Samuel Mimram (École Polytechnique)

• Tim Porter (University of Wales, Bangor)

• Femke van Raamsdonk (VU University, Amsterdam)

## Sensing and Acting Under Information Constraints

30 October, 2014

I’m having a great time at a workshop on Biological and Bio-Inspired Information Theory in Banff, Canada. You can see videos of the talks online. There have been lots of good talks so far, but this one really blew my mind:

• Naftali Tishby, Sensing and acting under information constraints—a principled approach to biology and intelligence, 28 October 2014.

Tishby’s talk wasn’t easy for me to follow—he assumed you already knew rate-distortion theory and the Bellman equation, and I didn’t—but it was great!

It was about the ‘action-perception loop':

This is the feedback loop in which living organisms—like us—take actions depending on our goals and what we perceive, and perceive things depending on the actions we take and the state of the world.

How do we do this so well? Among other things, we need to balance the cost of storing information about the past against the payoff of achieving our desired goals in the future.

Tishby presented a detailed yet highly general mathematical model of this! And he ended by testing the model on experiments with cats listening to music and rats swimming to land.

It’s beautiful stuff. I want to learn it. I hope to blog about it as I understand more. But for now, let me just dive in and say some basic stuff. I’ll start with the two buzzwords I dropped on you. I hate it when people use terminology without ever explaining it.

### Rate-distortion theory

Rate-distortion theory is a branch of information theory which seeks to find the minimum rate at which bits must be communicated over a noisy channel so that the signal can be approximately reconstructed at the other end without exceeding a given distortion. Shannon’s first big result in this theory, the ‘rate-distortion theorem’, gives a formula for this minimum rate. Needless to say, it still requires a lot of extra work to determine and achieve this minimum rate in practice.

For the basic definitions and a statement of the theorem, try this:

• Natasha Devroye, Rate-distortion theory, course notes, University of Chicago, Illinois, Fall 2009.

One of the people organizing this conference is a big expert on rate-distortion theory, and he wrote a book about it.

• Toby Berger, Rate Distortion Theory: A Mathematical Basis for Data Compression, Prentice–Hall, 1971.

Unfortunately it’s out of print and selling for \$259 used on Amazon! An easier option might be this:

• Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, Chapter 10: Rate Distortion Theory, Wiley, New York, 2006.

### The Bellman equation

The Bellman equation reduces the task of finding an optimal course of action to choosing what to do at each step. So, you’re trying to maximize the ‘total reward’—the sum of rewards at each time step—and Bellman’s equation says what to do at each time step.

If you’ve studied physics, this should remind you of how starting from the principle of least action, we can get a differential equation describing the motion of a particle: the Euler–Lagrange equation.

And in fact they’re deeply related. The relation is obscured by two little things. First, Bellman’s equation is usually formulated in a context where time passes in discrete steps, while the Euler–Lagrange equation is usually formulated in continuous time. Second, Bellman’s equation is really the discrete-time version not of the Euler–Lagrange equation but a more or less equivalent thing: the ‘Hamilton–Jacobi equation’.

Ah, another buzzword to demystify! I was scared of the Hamilton–Jacobi equation for years, until I taught a course on classical mechanics that covered it. Now I think it’s the greatest thing in the world!

Briefly: the Hamilton–Jacobi equation concerns the least possible action to get from a fixed starting point to a point $q$ in space at time $t.$ If we call this least possible action $W(t,q),$ the Hamilton–Jacobi equation says

$\displaystyle{ \frac{\partial W(t,q)}{\partial q_i} = p_i }$

$\displaystyle{ \frac{\partial W(t,q)}{\partial t} = -E }$

where $p$ is the particle’s momentum at the endpoint of its path, and $E$ is its energy there.

If we replace derivatives by differences, and talk about maximizing total reward instead of minimizing action, we get Bellman’s equation:

Bellman equation, Wikipedia.

### Markov decision processes

Bellman’s equation can be useful whenever you’re trying to figure out an optimal course of action. An important example is a ‘Markov decision process’. To prepare you for Tishby’s talk, I should say what this is.

In a Markov process, something randomly hops around from state to state with fixed probabilities. In the simplest case there’s a finite set $S$ of states, and time proceeds in discrete steps. At each time step, the probability to hop from state $s$ to state $s'$ is some fixed number $P(s,s').$

This sort of thing is called a Markov chain, or if you feel the need to be more insistent, a discrete-time Markov chain.

A Markov decision process is a generalization where an outside agent gets to change these probabilities! The agent gets to choose actions from some set $A.$ If at a given time he chooses the action $\alpha \in A,$ the probability of the system hopping from state $s$ to state $s'$ is $P_\alpha(s,s').$ Needless to say, these probabilities have to sum to one for any fixed $s.$

That would already be interesting, but the real fun is that there’s also a reward $R_\alpha(s,s').$ This is a real number saying how much joy or misery the agent experiences if he does action $\alpha$ and the system hops from $s$ to $s'.$

The problem is to choose a policy—a function from states to actions—that maximizes the total expected reward over some period of time. This is precisely the kind of thing Bellman’s equation is good for!

If you’re an economist you might also want to ‘discount’ future rewards, saying that a reward $n$ time steps in the future gets multiplied by $\gamma^n,$ where $0 < \gamma \le 1$ is some discount factor. This extra tweak is easily handled, and you can see it all here:

Markov decision process, Wikipedia.

### Partially observable Markov decision processes

There’s a further generalization where the agent can’t see all the details of the system! Instead, when he takes an action $\alpha \in A$ and the system hops from state $s$ to state $s',$ he sees something: a point in some set $O$ of observations. He makes the observation $o \in O$ with probability $\Omega_\alpha(o,s').$

(I don’t know why this probability depends on $s'$ but not $s.$ Maybe it ultimately doesn’t matter much.)

Again, the goal is to choose a policy that maximizes the expected total reward. But a policy is a bit different now. The action at any time can only depend on all the observations made thus far.

Partially observable Markov decision processes are also called POMPDs. If you want to learn about them, try these:

Partially observable Markov decision process, Wikipedia.

• Tony Cassandra, Partially observable Markov decision processes.

The latter includes an introduction without any formulas to POMDPs and how to choose optimal policies. I’m not sure who would study this subject and not want to see formulas, but it’s certainly a good exercise to explain things using just words—and it makes certain things easier to understand (though not others, in a way that depends on who is trying to learn the stuff).

### The action-perception loop

I already explained the action-perception loop, with the help of this picture from the University of Bielefeld’s Department of Cognitive Neuroscience:

Nafthali Tishby has a nice picture of it that’s more abstract:

We’re assuming time comes in discrete steps, just to keep things simple.

At each time $t$

• the world has some state $W_t,$ and
• the agent has some state $M_t.$

Why the letter $M$? This stands for memory: it can be the state of the agent’s memory, but I prefer to think of it as the state of the agent.

At each time

• the agent takes an action $A_t,$ which affects the world’s next state, and

• the world provides a sensation $S_t$ to the agent, which affect’s the agent’s next state.

This is simplified but very nice. Note that there’s a symmetry interchanging the world and the agent!

We could make it fancier by having lots of agents who all interact, but there are a lot of questions already. The big question Tishby focuses on is optimizing how much the agent should remember about the past if they

• get a reward depending on the action taken and the resulting state of the world

but

• pay a price for the information stored from sensations.

Tishby formulates this optimization question as something like a partially observed Markov decision process, uses rate-distortion theory to analyze how much information needs to be stored to achieve a given reward, and uses Bellman’s equation to solve the optimization problem!

So, everything I sketched fits together somehow!

I hope what I’m saying now is roughly right: it will take me more time to get the details straight. If you’re having trouble absorbing all the information I just threw at you, don’t feel bad: so am I. But the math feels really natural and good to me. It involves a lot of my favorite ideas (like generalizations of the principle of least action, and relative entropy), and it seems ripe to be combined with network theory ideas.

For details, I highly recommend this paper:

• Naftali Tishby and Daniel Polani, Information theory of decisions and actions, in Perception-Reason-Action Cycle: Models, Algorithms and System. Vassilis, Hussain and Taylor, Springer, Berlin, 2010.

I’m going to print this out, put it by my bed, and read it every night until I’ve absorbed it.

## Entropy and Information in Biological Systems (Part 2)

4 July, 2014

John Harte, Marc Harper and I are running a workshop! Now you can apply here to attend:

Information and entropy in biological systems, National Institute for Mathematical and Biological Synthesis, Knoxville Tennesee, Wednesday-Friday, 8-10 April 2015.

Financial support for travel, meals, and lodging is available for workshop attendees who need it. We will choose among the applicants and invite 10-15 of them.

### The idea

Information theory and entropy methods are becoming powerful tools in biology, from the level of individual cells, to whole ecosystems, to experimental design, model-building, and the measurement of biodiversity. The aim of this investigative workshop is to synthesize different ways of applying these concepts to help systematize and unify work in biological systems. Early attempts at “grand syntheses” often misfired, but applications of information theory and entropy to specific highly focused topics in biology have been increasingly successful. In ecology, entropy maximization methods have proven successful in predicting the distribution and abundance of species. Entropy is also widely used as a measure of biodiversity. Work on the role of information in game theory has shed new light on evolution. As a population evolves, it can be seen as gaining information about its environment. The principle of maximum entropy production has emerged as a fascinating yet controversial approach to predicting the behavior of biological systems, from individual organisms to whole ecosystems. This investigative workshop will bring together top researchers from these diverse fields to share insights and methods and address some long-standing conceptual problems.

So, here are the goals of our workshop:

• To study the validity of the principle of Maximum Entropy Production (MEP), which states that biological systems – and indeed all open, non-equilibrium systems – act to produce entropy at the maximum rate.

• To familiarize all the participants with applications to ecology of the MaxEnt method: choosing the probabilistic hypothesis with the highest entropy subject to the constraints of our data. We will compare MaxEnt with competing approaches and examine whether MaxEnt provides a sufficient justification for the principle of MEP.

• To clarify relations between known characterizations of entropy, the use of entropy as a measure of biodiversity, and the use of MaxEnt methods in ecology.

• To develop the concept of evolutionary games as “learning” processes in which information is gained over time.

• To study the interplay between information theory and the thermodynamics of individual cells and organelles.

For more details, go here.

If you’ve got colleagues who might be interested in this, please let them know. You can download a PDF suitable for printing and putting on a bulletin board by clicking on this:

## Quantum Frontiers in Network Science

6 May, 2014

guest post by Jacob Biamonte

There’s going to be a workshop on quantum network theory in Berkeley this June. The event is being organized by some of my collaborators and will be a satellite of the biggest annual network science conference, NetSci.

A theme of the Network Theory series here on Azimuth has been to merge ideas appearing in quantum theory with other disciplines. Remember the first post by John which outlined the goal of a general theory of networks? Well, everyone’s been chipping away at this stuff for a few years now and I think you’ll agree that this workshop seems like an excellent way to push these topics even further, particularly as they apply to complex networks.

The event is being organized by Mauro Faccin, Filippo Radicchi and Zoltán Zimborás. You might recall when Tomi Johnson first explained to us some ideas connecting quantum physics with the concepts of complex networks (see Quantum Network Theory Part 1 and Part 2). Tomi’s going to be speaking at this event. I understand there is even still a little bit of space left to contribute talks and/or to attend. I suspect that those interested can sort this out by emailing the organizers or just follow the instructions to submit an abstract.

They have named their event Quantum Frontiers in Network Science or QNET for short. Here’s their call.

Quantum Frontiers in Network Science

This year the biggest annual network science conference, NetSci will take place in Berkeley California on 2-6 June. We are organizing a one-day Satellite Workshop on Quantum Frontiers in Network Science (QNET).

A grand challenge in contemporary complex network science is to reconcile the staple “statistical mechanics based approach” with a theory based on quantum physics. When considering networks where quantum coherence effects play a non-trivial role, the predictive power of complex network science has been shown to break down. A new theory is now being developed which is based on quantum theory, from first principles. Network theory is a diverse subject which developed independently in several disciplines to rely on graphs with additional structure to model complex systems. Network science has of course played a significant role in quantum theory, for example in topics such as tensor network states, chiral quantum walks on complex networks, categorical tensor networks, and categorical models of quantum circuits, to name only a few. However, the ideas of complex network science are only now starting to be united with modern quantum theory. From this respect, one aim of the workshop is to put in contact two big and generally not very well connected scientific communities: statistical and quantum physicists.

The topic of network science underwent a revolution when it was realized that systems such as social or transport networks could be interrelated through common network properties, but what are the relevant properties to consider when facing quantum systems? This question is particularly timely as there has been a recent push towards studying increasingly larger quantum mechanical systems, where the analysis is only beginning to undergo a shift towards embracing the concepts of complex networks.

For example, theoretical and experimental attention has turned to explaining transport in photosynthetic complexes comprising tens to hundreds of molecules and thousands of atoms using quantum mechanics. Likewise, in condensed matter physics using the language of “chiral quantum walks”, the topological structure of the interconnections comprising complex materials strongly affects their transport properties.

An ultimate goal is a mathematical theory and formal description which pinpoints the similarities and differences between the use of networks throughout the quantum sciences. This would give rise to a theory of networks augmenting the current statistical mechanics approach to complex network structure, evolution, and process with a new theory based on quantum mechanics.

Topics of special interest to the satellite include

• Quantum transport and chiral quantum walks on complex networks
• Detecting community structure in quantum systems
• Tensor algebra and multiplex networks
• Quantum information measures (such as entropy) applied to complex networks
• Quantum critical phenomena in complex networks
• Quantum models of network growth
• Quantum techniques for reaction networks
• Quantum algorithms for problems in complex network science
• Foundations of quantum theory in relation to complex networks and processes thereon
• Quantum inspired mathematics as a foundation for network science

Info

QNET will be held at the NetSci Conference venue at the Clark Kerr Campus of the University of California, on June 2nd in the morning (8am-1pm).

• Main conference page: NetSci2014
Call for abstracts and the program

It sounds interesting! You’ll notice that the list of topics seems reminiscent of some of the things we’ve been talking about right here on Azimuth! A general theme of the Network Theory Series has been geared towards developing frameworks to describe networked systems through a common language and then to map the use of tools and results across disciplines. It seems like a great place to talk about these ideas. Oh, and here’s a current list of the speakers:

Leonardo Banchi (UCL, London)
Ginestra Bianconi (London)
Silvano Garnerone (IQC, Waterloo)
Laetitia Gauvin (ISI Foundation)
Marco Javarone (Sassari)
Tomi Johnson (Oxford)

and again, the organizers are

Mauro Faccin (ISI Foundation)
Zoltán Zimborás (UCL)

From the call, we can notice that a central discussion topic at QNET will be about contrasting stochastic and quantum mechanics. Here on Azimuth we like this stuff. You might remember that stochastic mechanics was formulated in the network theory series to mathematically resemble quantum theory (see e.g. Part 12). This formalism was then employed to produce several results, including a stochastic version of Noether’s theorem by John and Brendan in Parts 11 and 13—recently Ville has also written Noether’s Theorem: Quantum vs Stochastic. Several other results were produced by relating quantum field theory to Petri nets from population biology and to chemical reaction networks in chemistry (see the Network Theory homepage). It seems to me that people attending QNET will be interested in these sorts of things, as well as other related topics.

One of the features of complex network science is that it is often numerically based and geared directly towards interesting real-world applications. I suspect some interesting results should stem from the discussions that will take place at this workshop.

By the way, here’s a view of downtown San Francisco at dusk from Berkeley Hills California from the NetSci homepage:

## Programming with Chemical Reaction Networks

23 March, 2014

There will be a 5-day workshop on Programming with Chemical Reaction Networks: Mathematical Foundation at BIRS from Sunday, June 8 to Friday June 13, 2014 It’s being organized by

Anne Condon (University of British Columbia)
David Doty (California Institute of Technology)
Chris Thachuk (University of Oxford).

BIRS is the Banff International Research Station, in the mountains west of Calgary, in Alberta, Canada.

### Description

Here’s the workshop proposal on the BIRS website. It’s a pretty interesting proposal, especially if you’ve already read Luca Cardelli’s description of computing with chemical reaction networks, at the end of our series of posts on chemical reaction networks. The references include a lot of cool papers, so I’ve created links to those to help you get ahold of them.

This workshop will explore three of the most important research themes concerning stochastic chemical reaction networks (CRNs). Below we motivate each theme and highlight key questions that the workshop will address. Our main objective is to bring together distinct research communities in order to consider new problems that could not be fully appreciated in isolation. It is also our aim to determine commonalities between different disciplines and bodies of research. For example, research into population protocols, vector addition systems, and Petri networks provide a rich body of theoretical results that may already address contemporary problems arising in the study of CRNs.

#### Computational power of CRNs

Before designing robust and practical systems, it is useful to know the limits to computing with a chemical soup. Some interesting theoretical results are already known for stochastic chemical reaction networks. The computational power of CRNs depend upon a number of factors, including: (i) is the computation deterministic, or probabilistic, and (ii) does the CRN have an initial context — certain species, independent of the input, that are initially present in some exact, constant count.

In general, CRNs with a constant number of species (independent of the input length) are capable of Turing universal computation [17], if the input is represented by the exact (unary) count of one molecular species, some small probability of error is permitted and an initial context in the form of a single-copy leader molecule is used.

Could the same result hold in the absence of an initial context? In a surprising result based on the distributed computing model of population protocols, it has been shown that if a computation must be error-free, then deterministic computation with CRNs having an initial context is limited to computing semilinear predicates [1], later extended to functions outputting natural numbers encoded by molecular counts [5].

Furthermore, any semilinear predicate or function can be computed by that class of CRNs in expected time polylogarithmic in the input length. Building on this result, it was recently shown that by incurring an expected time linear in the input length, the same result holds for “leaderless” CRNs [8] — CRNs with no initial context. Can this result be improved to sub-linear expected time? Which class of functions can be computed deterministically by a CRN without an initial context in expected time polylogarithmic in the input length?

While (restricted) CRNs are Turing-universal, current results use space proportional to the computation time. Using a non-uniform construction, where the number of species is proportional to the input length and each initial species is present in some constant count, it is known that any S(n) space-bounded computation can be computed by a logically-reversible tagged CRN, within a reaction volume of size poly(S(n)) [18]. Tagged CRNs were introduced to model explicitly the fuel molecules in physical realizations of CRNs such as DNA strand displacement systems [6] that are necessary to supply matter and energy for implementing reactions such as X → X + Y that violate conservation of mass and/or energy.

Thus, for space-bounded computation, there exist CRNs that are time-efficient or are space-efficient. Does there exist time- and space-efficient CRNs to compute any space-bounded function?

#### Designing and verifying robust CRNs

While CRNs provide a concise model of chemistry, their physical realizations are often more complicated and more granular. How can one be sure they accurately implement the intended network behaviour? Probabilistic model checking has already been employed to find and correct inconsistencies between CRNs and their DNA strand displacement system (DSD) implementations [9]. However, at present, model checking of arbitrary CRNs is only capable of verifying the correctness of very small systems. Indeed, verification of these types of systems is a difficult problem: probabilistic state reachability is undecidable [17, 20] and general state reachability is EXPSPACE-hard [4].

How can larger systems be verified? A deeper understanding of CRN behaviour may simplify the process of model checking. As a motivating example, there has been recent progress towards verifying that certain DSD implementations correctly simulate underlying CRNs [16, 7, 10]. This is an important step to ensuring correctness, prior to experiments. However, DSDs can also suffer from other errors when implementing CRNs, such as spurious hybridization or strand displacement. Can DSDs and more generally CRNs be designed to be robust to such predictable errors? Can error correcting codes and redundant circuit designs used in traditional computing be leveraged in these chemical computers? Many other problems arise when implementing CRNs. Currently, unique types of fuel molecules must be designed for every reaction type. This complicates the engineering process significantly. Can a universal type of fuel be designed to smartly implement any reaction?

#### Energy efficient computing with CRNs

Rolf Landauer showed that logically irreversible computation — computation as modeled by a standard Turing machine — dissipates an amount of energy proportional to the number of bits of information lost, such as previous state information, and therefore cannot be energy efficient [11]. However, Charles Bennett showed that, in principle, energy efficient computation is possible, by proposing a universal Turing machine to perform logically-reversible computation and identified nucleic acids (RNA/DNA) as a potential medium to realize logically-reversible computation in a physical system [2].

There have been examples of logically-reversible DNA strand displacement systems — a physical realization of CRNs — that are, in theory, capable of complex computation [12, 19]. Are these systems energy efficient in a physical sense? How can this argument be made formally to satisfy both the computer science and the physics communities? Is a physical experiment feasible, or are these results merely theoretical footnotes?

#### References

[1] D. Angluin, J. Aspnes, and D. Eisenstat. Stably computable predicates are semilinear. In PODC, pages 292–299, 2006.

[2] C. H. Bennett. Logical reversibility of computation. IBM Journal of Research and Development, 17 (6):525–532, 1973.

[3] L. Cardelli and A. Csikasz-Nagy. The cell cycle switch computes approximate majority. Scientific Reports, 2, 2012.

[4] E. Cardoza, R. Lipton, A. R. Meyer. Exponential space complete problems for Petri nets and commutative semigroups (Preliminary Report). Proceedings of the Eighth Annual ACM Symposium on Theory of Computing, pages 507–54, 1976.

[5] H. L. Chen, D. Doty, and D. Soloveichik. Deterministic function computation with chemical reaction networks. DNA Computing and Molecular Programming, pages 25–42, 2012.

[6] A. Condon, A. J. Hu, J. Manuch, and C. Thachuk. Less haste, less waste: on recycling and its limits in strand displacement systems. Journal of the Royal Society: Interface Focus, 2 (4):512–521, 2012.

[7] Q. Dong. A bisimulation approach to verification of molecular implementations of formal chemical reaction network. Master’s thesis. SUNY Stony Brook, 2012.

[8] D. Doty and M. Hajiaghayi. Leaderless deterministic chemical reaction networks. In Proceedings of the 19th International Meeting on DNA Computing and Molecular Programming, 2013.

[9] M. R. Lakin, D. Parker, L. Cardelli, M. Kwiatkowska, and A. Phillips. Design and analysis of DNA strand displacement devices using probabilistic model checking. Journal of The Royal Society Interface, 2012.

[10] M. R. Lakin, D. Stefanovic and A. Phillips. Modular Verification of Two-domain DNA Strand Displacement Networks via Serializability Analysis. In Proceedings of the 19th Annual conference on DNA computing, 2013.

[11] R. Landauer. Irreversibility and heat generation in the computing process. IBM Journal of research and development, 5 (3):183–191, 1961.

[12] L. Qian, D. Soloveichik, and E. Winfree. Efficient Turing-universal computation with DNA polymers (extended abstract) . In Proceedings of the 16th Annual conference on DNA computing, pages 123–140, 2010.

[13] L. Qian and E. Winfree. Scaling up digital circuit computation with DNA strand displacement cascades. Science, 332 (6034):1196–1201, 2011.

[14] L. Qian, E. Winfree, and J. Bruck. Neural network computation with DNA strand displacement cascades. Nature, 475 (7356):368–372, 2011.

[15] G. Seelig, D. Soloveichik, D.Y. Zhang, and E. Winfree. Enzyme-free nucleic acid logic circuits. Science, 314 (5805):1585–1588, 2006.

[16] S. W. Shin. Compiling and verifying DNA-based chemical reaction network implementations. Master’s thesis. California Insitute of Technology, 2011.

[17] D. Soloveichik, M. Cook, E. Winfree, and J. Bruck. Computation with finite stochastic chemical reaction networks. Natural Computing, 7 (4):615–633, 2008.

[18] C. Thachuk. Space and energy efficient molecular programming. PhD thesis, University of British Columbia, 2012.

[19] C. Thachuk and A. Condon. Space and energy efficient computation with DNA strand displacement systems. In Proceedings of the 18th Annual International Conference on DNA computing and Molecular Programming, 2012.

[20] G. Zavattaro and L. Cardelli. Termination Problems in Chemical Kinetics. In Proceedings of the 2008 Conference on Concurrency Theory, pages 477–491, 2008.

## Network Theory Overview

22 February, 2014

Here’s a video of a talk I gave yesterday, made by Brendan Fong. You can see the slides here—and then click the items in blue, and the pictures, for more information!

The idea: nature and the world of human technology are full of networks! People like to draw diagrams of networks. Mathematical physicists know that in principle these diagrams can be understood using category theory. But why should physicists have all the fun? This is the century of understanding living systems and adapting to life on a finite planet. Math isn’t the main thing we need for this, but it’s got to be part of the solution… so one thing we should do is develop a unified and powerful theory of networks.

We are still far from doing this. In this overview, I briefly described three parts of the jigsaw puzzle, and invited everyone to help fit them together:

• electrical circuits and signal-flow graphs.

• stochastic Petri nets, chemical reaction networks and Feynman diagrams.

• Bayesian networks, information and entropy.

In my talks coming up, I’ll go into more detail on each of these.﻿ With luck, you’ll be able to see videos here.

But if you’re near Oxford, you might as well actually attend! You can see dates, times, locations, my slides, and the talks themselves as they show up by going here.

## Network Theory Talks at Oxford

7 February, 2014

I’m giving some talks at Oxford:

### Network Theory

Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. After an overview, we will look at three portions of the jigsaw puzzle in three separate talks:

I. Electrical circuits and signal-flow graphs.

II. Stochastic Petri nets, chemical reaction networks and Feynman diagrams.

III. Bayesian networks, information and entropy.

If you’re nearby I hope you can come! All these talks will take place in Lecture Theatre B in the Computer Science Department—see the map below. Here are the times:

• Friday 21 February 2014, 2 pm: Network Theory: overview. See the slides or watch a video.

• Tuesday 25 February, 3:30 pm: Network Theory I: electrical circuits and signal-flow graphs. See the slides or watch a video.

• Tuesday 4 March, 3:30 pm: Network Theory II: stochastic Petri nets, chemical reaction networks and Feynman diagrams. See the slides or watch a video.

• Tuesday 11 March, 3:30 pm: Network Theory III: Bayesian networks, information and entropy. See the slides or watch a video

The first talk will be part of the OASIS series, meaning the “Oxford Advanced Seminar on Informatic Structures”.

I thank Samson Abramsky, Bob Coecke and Jamie Vicary of the Computer Science Department for inviting me, and Ulrike Tillmann and Minhyong Kim of the Mathematical Institute for helping me get set up. I also thank all the people who helped do the work I’ll be talking about, most notably Jacob Biamonte, Jason Erbele, Brendan Fong, Tobias Fritz, Tom Leinster, Tu Pham, and Franciscus Rebro.

Ulrike Tillmann has also kindly invited me to give a topology seminar:

### Operads and the Tree of Life

Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists use increasingly sophisticated mathematical techniques to reconstruct the most likely “phylogenetic tree” describing how these species evolved from earlier ones. In their work on this subject, they have encountered an interesting example of an operad, which is obtained by applying a variant of the Boardmann–Vogt “W construction” to the operad for commutative monoids. The operations in this operad are labelled trees of a certain sort, and it plays a universal role in the study of stochastic processes that involve branching. It also shows up in tropical algebra. This talk is based on work in progress with Nina Otter.

I’m not sure exactly where this will take place, but surely somewhere in the Mathematical Institute building:

• Monday 24 February, 3:30 pm, Operads and the Tree of Life. See the slides.

The Computer Science Department is shown in the map here:

The Mathematical Institute is a bit to the west: