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单词 3DistributedDynamicalSystems
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3. Distributed dynamical systems


Probabilistic cellular automata provide useful toy models of awide range of physical and biological systems. A cellularautomatonMathworldPlanetmath consists of a collectionMathworldPlanetmath of cells, each equipped withneighbors. Two important important examples are

Example 2 (Conway’s game of life).

The cellular automaton is a grid of deterministicMathworldPlanetmath cellswith outputs {0,1}. A cell outputs 1 at time t iff:(i) three of its neighbors outputted 1s at time t-1 or(ii) it and two neighbors outputted 1s at t-1. 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 {0,1}. Cell nk fires with probability proportional to

p(nk,t=1|n,t-1)exp[1Tjkαjknj,t-1].

Temperature T controls network stochasticity. Attractors{ξ1,,ξN} are embedded into a network by settingthe connectivity matrix as αjk=μ=1N(2ξjμ-1)(2ξkμ-1).

It is useful to take a finer perspective on cellular automataby decomposing them into spacetime coordinates or occasions[2]. An occasion vl=ni,t is a cellni at a time point t. Two occasions are linked vkvl if there is a connection from vk’scell to vl’s (because they are neighbors or the same cell)and their time coordinates are t-1 and t respectively forsome t, so occasions form a directed graph. More generally:

Definition 4.

A distributed dynamical systemMathworldPlanetmathPlanetmath 𝐃 consists ofthe following data:

  1. 𝐃1.

    Directed graph.A graph G𝐃=(V𝐃,E𝐃) with a finite setMathworldPlanetmath ofvertices or occasions V𝐃={v1vn} andedges E𝐃V𝐃×V𝐃.

  2. 𝐃2.

    Alphabets.Each vertex vlV𝐃 has finite alphabetAl of outputs and finite alphabet Sl:=ksrc(l)Ak of inputs, where src(l)={vk|vkvl}.

  3. 𝐃3.

    Mechanisms.Each vertex vl is equipped with stochastic map𝔪l:𝒱Sl𝒱Al.

Figure 1: Mapping a cellular automaton to a distributeddynamical system.

Taking any cellular automaton over a finite time interval[tα,tω] initializing the mechanisms at timetα with fixed values (initial conditions) orprobability distributions (noise sources) yields a distributeddynamical system, see Fig. 1. Each cell of theoriginal automatonMathworldPlanetmath 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 sectionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath below investigate the compositional structureMathworldPlanetmath ofmeasurements: how they are built out of submeasurements.Technology for tracking subsystems and submeasurements istherefore necessary. We introduce two closely relatedcategoriesMathworldPlanetmath:

Definition 5.

The category of subsystems 𝚂𝚢𝚜𝐃 of 𝐃is a Boolean lattice with objects given by sets of orderedpairsMathworldPlanetmath of vertices 𝐂2¯V𝐃×V𝐃 and arrows given by inclusions i12:𝐂1𝐂2. The initial and terminalobjects are 𝐃= and 𝐃=V𝐃×V𝐃.

Remark 2.

Subsystems are defined as ordered pairs of vertices,rather than subgraphs of the directed graph of 𝐃.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 src(𝐂)={vk|(vk,vl)𝐂} and similarly for trg(𝐂). Set the input alphabet of 𝐂 as the productPlanetmathPlanetmathPlanetmathPlanetmath of the output alphabets of its source occasions S𝐂=src(𝐂)Ak and similarly the output alphabet of 𝐂 as the product of the output alphabets of its target occasions A𝐂=trg(𝐂)Al.

Definition 6.

The category of measuring devices 𝙼𝚎𝚊𝚜𝐃 on 𝐃 has objects Hom𝚂𝚝𝚘𝚌𝚑(𝒱A𝐂,𝒱S𝐂) for 𝐂2¯V𝐃×V𝐃. For 𝐂1𝐂2 define arrow

r21:Hom(𝒱A𝐂2,𝒱S𝐂2)Hom(𝒱A𝐂1,𝒱S𝐂1)
[𝒱A𝐂2𝑇𝒱S𝐂2][𝒱A𝐂1πA𝒱A𝐂2𝑇𝒱S𝐂2πS𝒱S𝐂1],

where πA and πS are shorthands for projectionsas in Example 1 (http://planetmath.org/2stochasticmaps#Thmeg1)

The reason for naming 𝙼𝚎𝚊𝚜𝐃 the category of measuringdevices will become clear in §4 (http://planetmath.org/4measurement)below. The two categories are connected by contravariantfunctorMathworldPlanetmath :

Theorem 4 (structure presheaf).

The structure presheafPlanetmathPlanetmathPlanetmath F taking

𝐃:𝚂𝚢𝚜𝐃𝚘𝚙𝙼𝚎𝚊𝚜𝐃:𝐂Hom(𝒱A𝐂,𝒱S𝐂) and i12r21

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{𝐂j}j=1l of subsystems and sections𝔪j𝐃(𝐂j) such thatrj,ji(𝔪j)=ri,ji(𝔪i)for all i, j there exists section 𝔪𝐃(j=1l𝐂j) such thatri(𝔪)=𝔪i for all i. This reducesto finding a conditional distribution that causes diagram

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

where p(x|w)=v,zp(x,z|v,w)pmaxH(v) and similarlyfor the vertical arrow. It is easy to see that

p(x,y,z|u,v,w):=p(x,y|u,w)p(x,z|v,w)p(x|w)

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{𝐂j}j=1l with compatible restrictionsPlanetmathPlanetmath but mustalso be unique. Descent in is not uniquebecause there are many distributionsPlanetmathPlanetmath satisfying the requirementabove: strictly speaking r is a marginalization operatorrather than restriction. For example, there are manydistributions p(y,z) that marginalize to give p(y) andp(z) besides the product distribution p(y)p(z).

The structure presheaf depends on the graphstructure and alphabets; mechanisms play no role. We nowconstruct a family of sections of using themechanisms of 𝐃’s occasions. Specifically, given asubsystem 𝐂𝚂𝚢𝚜𝐃, we show how to glue itsoccasions’ mechanisms together to form joint mechanism𝔪𝐂. The mechanism 𝔪𝐃=𝔪 of theentire system 𝐃 is recovered as a special case.

In general, subsystem 𝐂 is not isolated: it receivesinputs along edges contained in 𝐃 but not in𝐂. Inputs along these edges cannot be assigned a fixedvalue since in general there is no preferred element ofAl. They also cannot be ignored since 𝔪l is definedas receiving inputs from all its sources. Nevertheless, themechanism of 𝐂 should depend on 𝐂 alone. Wetherefore treat edges not in 𝐂 as sources of extrinsicnoise by marginalizing with respect to the uniform distributionMathworldPlanetmathas in Corollary 3 (http://planetmath.org/2stochasticmaps#Thmthm3).

For each vertex vltrg(𝐂) let Sl𝐂=ksrc(l)src(C)Ak. We then have projectionπl:𝒱Sl𝒱Sl𝐂.Define

𝔪l𝐂:=[𝒱Sl𝐂πl𝒱Sl𝔪l𝒱Al].(1)

It follows immediately that 𝐂 is itself a distributeddynamical system defined by its graph, whose alphabets areinherited from 𝐃 and whose mechanisms are constructedby marginalizing.

Next, we tensor the mechanisms of individual occasions and gluethem together using the diagonal map Δ:S𝐂vltrg(𝐂)Sl𝐂. Thediagonal map used here11which is surjectivePlanetmathPlanetmath in thesense that all rows contain non-zero entries generalizesXΔX×X and removes redundancies inlSl𝐂, which may, for example, include thesame source alphabets many times in different factors.

Let mechanism 𝔪𝐂 be

𝔪𝐂:=[𝒱S𝐂ιΔvltrg(𝐂)𝒱Sl𝐂vltrg(𝐂)𝔪l𝐂𝒱A𝐂].(2)

The dual of 𝔪𝐂 is

𝔪𝐂:=[𝒱A𝐂𝒱S𝐂].(3)

Finally, we find that we have constructed a family of sectionsof :

Definition 7.

The quale 𝐪𝐃 is the family ofsections of constructed inEqs. (1), (2) and(3)

𝐪𝐃:={𝔪𝐂(𝐂)=Hom(𝒱A𝐂,𝒱S𝐂)|𝐂𝚂𝚢𝚜𝐃}.

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.
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