1. Introduction

Any classical physical system (by which we simply mean any deterministicfunction) can be taken as a measuring apparatus or input/output device.For example, a thermometer takes inputs from the atmosphere and outputsnumbers on a digital display. The thermometer categorizes inputs bytemperature and is blind to, say, differences in air pressure.

Classical measurements are formalized as follows:

Definition 1.

Given a classical physical system with state space $X$, ameasuring device is a function $f:X\to ℝ$.The output $r\in ℝ$ is the reading and the pre-image$f^{-1}(r)\subset X$ is the measurement.

From this point of view a thermometer and a barometer are two functions,$T:X\to ℝ$ and $B:X\to ℝ$, mapping the state space$X$ of configurations (positions and momenta) of atmospheric particlesto real numbers. When the thermometer outputs $2^{\circ }$, it specifies thatthe atmospheric configuration was in the pre-image $T^{-1}(2^{\circ })$ which,assuming the thermometer perfectly measures temperature, is exactlycharacterized as atmospheric configurations with temperature $2^{\circ }$.Similarly, the pre-images generated by the barometer group atmosphericconfigurations by pressure.

The classical definition of measurement takes a thermometer as a monolithicobject described by a single function from atmospheric configurations toreal numbers. The internal structure of the thermometer – that is composedof countless atoms and molecules arranged in an extremely specific manner– is swept under the carpet (or, rather, into the function).

This paper investigates the structure of measurements performed bydistributed systems. We do so by adapting Definition 1to a large class of systems that contains networks of Boolean functions[10], Conway’s game of life [7, berlekamp:82]and Hopfield networks [9, 2] as special cases.

Our motivation comes from prior work investigating information processingin discrete neural networks [4, 5]. The brain $X$ can bethought of as an enormously complicated measuring device $S\times X\stackrel{𝑓}{\to }X$ mapping sensory states $s\in S$ and prior brain states$x\in X$ to subsequent brain states. Analyzing the functional dependenciesimplicit in cortical computations reduces to analyzing how the measurementsperformed by the brain are composed out of submeasurements by subdevicessuch as individual neurons and neuronal assemblies. The cortex is ofparticular interest since it seemingly effortlessly integrates diversecontextual data into a unified gestalt that determines behavior. Themeasurements performed by different neurons appear to interact in sucha way that they generate more information jointly than separately. Toimprove our understanding of how the cortex integrates information we needto a formal language for analyzing how context affects measurements indistributed systems.

As a first step in this direction, we develop methods for analyzing thegeometry of measurements performed by functions with overlapping domains.We propose, roughly speaking, to study context-dependence in terms of thegeometry of intersecting pre-images. However, since we wish to work withboth probabilistic and deterministic systems, things are a bit morecomplicated.

We sketch the contents of the paper. Section §2 (http://planetmath.org/2stochasticmaps)lays the groundwork by introducing the category of stochastic maps $\mathrm{𝚂𝚝𝚘𝚌𝚑}$.Our goal is to study finite set valued functions and conditional probabilitydistributions on finite sets. However, rather than work with sets, functionsand conditional distributions, we prefer to study stochastic maps (Markovmatrices) between function spaces on sets. We therefore introduce thefaithful functor $𝒱$ taking functions on sets to Markov matrices:

$$[f:X\to Y]\mapsto [𝒱f:𝒱X\to 𝒱Y],$$

where $𝒱X$ is functions from $X$ to $ℝ$. Conditional probabilitydistributions $p(y|x)$ can also be represented using stochastic maps.

Working with linear operators instead of set-valued functions is convenientfor two reasons. First, it unifies the deterministic and probabilistic casesin a single language. Second, the dual $T^{\mathrm{♮}}$ of a stochastic map $T$provides a symmetric treatment of functions and their corresponding inverseimage functions. Recall the inverse of function $f:X\to Y$ is$f^{-1}:Y\to {\underset{¯}{2}}^X$, which takes values in thepowerset of $X$, rather than $X$ itself. Dualizing a stochastic mapflips the domain and range of the original map, without introducing any newobjects:

$$[f^{-1}:Y\to {\underset{¯}{2}}^X]\text{ corresponds to }[{(𝒱f)}^{\mathrm{♮}}:𝒱Y\to 𝒱X],$$(1)

see Corollary 2 (http://planetmath.org/2stochasticmaps#Thmthm2)

Section §3 (http://planetmath.org/3distributeddynamicalsystems)introduces distributed dynamical systems. These extendprobabilistic cellular automata by replacing cells (spacecoordinates) with occasions (spacetime coordinates: cell $k$at time $t$). Inspired by [8, 1], wetreat distributed systems as collections of stochastic mapsbetween function spaces so that processes (stochastic maps)take center stage, rather than their outputs. framework bares a formal resemblance to the categoricalapproach to quantum mechanics developed in [1].Although the setting is abstract, it has the advantage that itis scalable: using a coarse-graining procedureintroduced in [3] we can analyze distributedsystems at any spatiotemporal granularity.

Distributed dynamical systems provide a rich class of toyuniverses. However, since these toy universes do not containconscious observers we confront Bell’s problem [6]:“What exactly qualifies some physical [system] to play therole of ‘measurer’?” In our setting, where we do not have toworry about collapsing wave-functions or the distinctionbetween macroscopic and microscopic processes, the solution issimple: every physical system plays the role ofmeasurer. More precisely, we track measurers via thecategory ${\mathrm{𝚂𝚢𝚜}}_{𝐃}$ of subsystems of $𝐃$. Eachsubsystem $𝐂$ is equipped with a mechanism ${𝔪}_{𝐂}$which is constructed by gluing together the mechanisms of theoccasions in $𝐂$ and averaging over extrinsic noise.

Measuring devices are typically analyzed by varying theirinputs and observing the effect on their outputs. By contrastthis paper fixes the output and varies the device overall its subdevices to obtain a family of submeasurementsparametrized by all subsystems in ${\mathrm{𝚂𝚢𝚜}}_{𝐃}$. The internalstructure of the measurement performed by $𝐃$ is thenstudied by comparing submeasurements.

We keep track of submeasurements by observing that they aresections of a suitably defined presheaf. Sheaf theory providesa powerful machinery for analyzing relationships betweenobjects and subobjects [11], which we adapt toour setting by introducing the structure presheaf $ℱ$,a contravariant functor from ${\mathrm{𝚂𝚢𝚜}}_{𝐃}$ to the categoryof measuring devices ${\mathrm{𝙼𝚎𝚊𝚜}}_{𝐃}$ on $𝐃$. Importantly,$ℱ$ is not a sheaf: although the gluing axiomholds, uniqueness fails, see Theorem 4 (http://planetmath.org/3distributeddynamicalsystems#Thmthm4).This is because the restriction operator in $\mathrm{𝙼𝚎𝚊𝚜}$ is(essentially) marginalization, and of course there areinfinitely many joint distributions $p(x,y)$ that yieldmarginals $p(x)$ and $p(y)$.

Section §4 (http://planetmath.org/4measurement)adapts Definition 1 to distributed systems andintroduces the simplest quantity associated with a measurement:effective information, which quantifies its precision, seeProposition 5 (http://planetmath.org/4measurement#Thmthm5).Crucially, effective information is context-dependent –it is computed relative to a baseline which may be completelyuninformative (the so-called null system) or provided by asubsystem.

Finally entanglement, introduced in §5 (http://planetmath.org/5entanglement),quantifies the obstruction (in bits) to decomposing ameasurement into independent submeasurements. It turns out,see discussion after Theorem 10 (http://planetmath.org/5entanglement#Thmthm10),that entanglement quantifies the extent to which a measurementis context-dependent – the extent to which contextualinformation provided by one submeasurement is useful inunderstanding another. Theorem 9 (http://planetmath.org/5entanglement#Thmthm9)shows that a measurement is more precise than the sum of itssubmeasurements only if entanglement is non-zero.Precision is thus inextricably bound to context-dependenceand indecomposability. The failure of unique descent is thusa feature, not a bug, since it provides “elbow room” tobuild measuring devices that are not products ofsubdevices.

Space constraints prevent us from providing concrete examples;the interested reader can find these in [4, 5, 3]. Our running examples are the deterministicset-valued functions

$$f:X\to Y\text{ and }g:X\times Y\to Z$$

which we use to illustrate the concepts as they are developed.