3. Distributed dynamical systems
Probabilistic cellular automata provide useful toy models of awide range of physical and biological systems. A cellularautomaton consists of a collection
of cells, each equipped withneighbors. Two important important examples are
Example 2 (Conway’s game of life).
The cellular automaton is a grid of deterministic cellswith outputs . A cell outputs 1 at time iff:(i) three of its neighbors outputted 1s at time or(ii) it and two neighbors outputted 1s at . Remarkably,a sufficiently large game of life grid can implement anydeterministic computation [3].
Example 3 (Hopfield networks).
These are probabilistic cellular automata [4, 1], again with outputs . Cell fires with probability proportional to
Temperature controls network stochasticity. Attractors are embedded into a network by settingthe connectivity matrix as .
It is useful to take a finer perspective on cellular automataby decomposing them into spacetime coordinates or occasions[2]. An occasion is a cell at a time point . Two occasions are linked if there is a connection from ’scell to ’s (because they are neighbors or the same cell)and their time coordinates are and respectively forsome , so occasions form a directed graph. More generally:
Definition 4.
A distributed dynamical system consists ofthe following data:
- 1.
Directed graph.A graph with a finite set
ofvertices or occasions andedges .
- 2.
Alphabets.Each vertex has finite alphabet of outputs and finite alphabet of inputs, where .
- 3.
Mechanisms.Each vertex is equipped with stochastic map.

Taking any cellular automaton over a finite time interval initializing the mechanisms at time with fixed values (initial conditions) orprobability distributions (noise sources) yields a distributeddynamical system, see Fig. 1. Each cell of theoriginal automaton corresponds to a series of occasions inthe distributed dynamical system, one per time step.
Cells with memory – i.e. whose outputs depend on theirneighbors outputs over multiple time steps – receive inputsfrom occasions more than one time step in the past. If a cell’smechanism changes (learns) over time then different mechanismsare assigned to the cell’s occasions at different time points.
The sections below investigate the compositional structure
ofmeasurements: how they are built out of submeasurements.Technology for tracking subsystems and submeasurements istherefore necessary. We introduce two closely relatedcategories
:
Definition 5.
The category of subsystems of is a Boolean lattice with objects given by sets of orderedpairs of vertices and arrows given by inclusions . The initial and terminalobjects are and .
Remark 2.
Subsystems are defined as ordered pairs of vertices,rather than subgraphs of the directed graph of .Pairs of occasions that are not connected by edges areineffective; they do not contribute to theinformation-processing performed by the system. We includethem in the formalism precisely to make their lack ofcontribution explicit, see Remark 3 (http://planetmath.org/4measurement#Thmrem3).
Let and similarly for . Set the input alphabet of as the product of the output alphabets of its source occasions and similarly the output alphabet of as the product of the output alphabets of its target occasions .
Definition 6.
The category of measuring devices on has objects for . For define arrow
where and are shorthands for projectionsas in Example 1 (http://planetmath.org/2stochasticmaps#Thmeg1)
The reason for naming the category of measuringdevices will become clear in §4 (http://planetmath.org/4measurement)below. The two categories are connected by contravariantfunctor :
Theorem 4 (structure presheaf).
The structure presheaf taking
satisfies the gluing axiom but has non-unique descent.
Proof: Functor is trivially a presheaf since it iscontravariant. It is an interesting presheaf becausethe gluing axiom holds.
For gluing we need to show that for any collection of subsystems and sections such thatfor all , there exists section such that for all . This reducesto finding a conditional distribution that causes diagram

in to commute. The vertices are conditional distributions and the arrows are marginalizations, so rewrite as

where and similarlyfor the vertical arrow. It is easy to see that
satisfies the requirement.
For to be a sheaf it would also have to satisfyunique descent: the section satisfying the gluing axiom mustnot only exist for any collection with compatible restrictions but mustalso be unique. Descent in is not uniquebecause there are many distributions
satisfying the requirementabove: strictly speaking is a marginalization operatorrather than restriction. For example, there are manydistributions that marginalize to give and besides the product distribution .
The structure presheaf depends on the graphstructure and alphabets; mechanisms play no role. We nowconstruct a family of sections of using themechanisms of ’s occasions. Specifically, given asubsystem , we show how to glue itsoccasions’ mechanisms together to form joint mechanism. The mechanism of theentire system is recovered as a special case.
In general, subsystem is not isolated: it receivesinputs along edges contained in but not in. Inputs along these edges cannot be assigned a fixedvalue since in general there is no preferred element of. They also cannot be ignored since is definedas receiving inputs from all its sources. Nevertheless, themechanism of should depend on alone. Wetherefore treat edges not in as sources of extrinsicnoise by marginalizing with respect to the uniform distributionas in Corollary 3 (http://planetmath.org/2stochasticmaps#Thmthm3).
For each vertex let . We then have projection.Define
(1) |
It follows immediately that is itself a distributeddynamical system defined by its graph, whose alphabets areinherited from and whose mechanisms are constructedby marginalizing.
Next, we tensor the mechanisms of individual occasions and gluethem together using the diagonal map . Thediagonal map used here11which is surjective in thesense that all rows contain non-zero entries generalizes and removes redundancies in, which may, for example, include thesame source alphabets many times in different factors.
Let mechanism be
(2) |
The dual of is
(3) |
Finally, we find that we have constructed a family of sectionsof :
Definition 7.
The quale is the family ofsections of constructed inEqs. (1), (2) and(3)
The construction used to glue together the mechanism of theentire system can also be used to construct the mechanism ofany subsystem, which provides a window – the quale – intothe compositional structure of distributed processes.
References
- 1 DJ Amit (1989):Modelling brain function: the world of attractor neuralnetworks. Cambridge University Press.
- 2 David Balduzzi (2011):Detecting emergent processes in cellular automata withexcess information. preprint .
- 3 ER Berlekamp, JC Conway &RK Guy (1982):Winning Ways for your Mathematical Plays. 2, Academic Press.
- 4 JJ Hopfield (1982):Neural networks and physical systems with emergentcomputational properties. Proc. Nat. Acad. Sci. 79,pp. 2554–2558.