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# 1. Introduction

Any classical physical system (by which we simply mean any deterministic
function^{}) can be taken as a measuring apparatus or input/output device.
For example, a thermometer takes inputs from the atmosphere and outputs
numbers on a digital display. The thermometer categorizes inputs by
temperature 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^{} $X$, a
*measuring device* is a function $f:X\rightarrow{\mathbb{R}}$.
The output $r\in{\mathbb{R}}$ 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,
$T:X\rightarrow{\mathbb{R}}$ and $B:X\rightarrow{\mathbb{R}}$, mapping the state space
$X$ of configurations^{} (positions and momenta) of atmospheric particles
to real numbers. When the thermometer outputs $2^{\circ}$, it specifies that
the atmospheric configuration was in the pre-image $T^{{-1}}(2^{\circ})$ which,
assuming the thermometer perfectly measures temperature, is *exactly*
characterized as atmospheric configurations with temperature $2^{\circ}$.
Similarly, the pre-images generated by the barometer group atmospheric
configurations by pressure.

The classical definition of measurement takes a thermometer as a monolithic object described by a single function from atmospheric configurations to real numbers. The internal structure of the thermometer – that is composed of 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 by
*distributed* systems. We do so by adapting Definition 1
to a large class of systems that contains networks of Boolean functions
[10], Conway’s game of life [7]
and Hopfield networks [9, 2] as special cases.

Our motivation comes from prior work investigating information processing
in discrete neural networks [4, 5]. The brain $X$ can be
thought of as an enormously complicated measuring device $S\times X\xrightarrow{f}X$ mapping sensory states $s\in S$ and prior brain states
$x\in X$ to subsequent brain states. Analyzing the functional^{} dependencies
implicit in cortical computations reduces to analyzing how the measurements
performed by the brain are composed out of submeasurements by subdevices
such as individual neurons and neuronal assemblies. The cortex is of
particular interest since it seemingly effortlessly integrates diverse
contextual data into a unified gestalt that determines behavior. The
measurements performed by different neurons appear to interact in such
a way that they generate more information jointly than separately. To
improve our understanding of how the cortex integrates information we need
to a formal language for analyzing how context affects measurements in
distributed systems.

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

We sketch the contents of the paper. Section^{} §2
lays the groundwork by introducing the category of stochastic maps ${\mathtt{Stoch}}$.
Our goal is to study finite set valued functions and conditional probability
distributions^{} on finite sets. However, rather than work with sets, functions
and conditional^{} distributions, we prefer to study stochastic maps (Markov
matrices) between function spaces on sets. We therefore introduce the
faithful functor ${\mathcal{V}}$ taking functions on sets to Markov matrices:

$\Big[f:X\rightarrow Y\Big]\mapsto\Big[{\mathcal{V}}f:{\mathcal{V}}X\rightarrow% {\mathcal{V}}Y\Big],$ |

where ${\mathcal{V}}X$ is functions from $X$ to ${\mathbb{R}}$. Conditional probability distributions $p(y|x)$ can also be represented using stochastic maps.

Working with linear operators instead of set-valued functions is convenient
for two reasons. First, it unifies the deterministic and probabilistic cases
in a single language^{}. Second, the dual $T^{\natural}$ of a stochastic map $T$
provides a symmetric^{} treatment of functions and their corresponding inverse
image^{} functions. Recall the inverse^{} of function $f:X\rightarrow Y$ is
$f^{{-1}}:Y\rightarrow\underline{2}^{X}$, which takes values in the
*powerset* of $X$, rather than $X$ itself. Dualizing a stochastic map
flips the domain and range of the original map, without introducing any new
objects:

$\Big[f^{{-1}}:Y\rightarrow\underline{2}^{X}\Big]\,\mbox{ corresponds to }\,% \Big[({\mathcal{V}}f)^{\natural}:{\mathcal{V}}Y\rightarrow{\mathcal{V}}X\Big],$ | (1) |

see Corollary 2

Section §3
introduces distributed dynamical systems. These extend
probabilistic cellular automata by replacing cells (space
coordinates) with occasions (spacetime coordinates: cell $k$
at time $t$). Inspired by [8, 1], we
treat distributed systems as collections of stochastic maps
between function spaces so that processes (stochastic maps)
take center stage, rather than their outputs. framework bares a formal resemblance to the categorical
approach to quantum mechanics developed in [1].
Although the setting is abstract, it has the advantage that it
is *scalable*: using a coarse-graining procedure
introduced in [3] we can analyze distributed
systems at any spatiotemporal granularity.

Distributed dynamical systems provide a rich class of toy
universes^{}. However, since these toy universes do not contain
conscious observers we confront Bell’s problem [6]:
“What exactly qualifies some physical [system] to play the
role of ‘measurer’?” In our setting, where we do not have to
worry about collapsing wave-functions or the distinction
between macroscopic and microscopic processes, the solution is
simple: *every* physical system plays the role of
measurer. More precisely, we track measurers via the
category ${\mathtt{Sys}}_{{\mathbf{D}}}$ of subsystems of ${\mathbf{D}}$. Each
subsystem ${\mathbf{C}}$ is equipped with a mechanism ${\mathfrak{m}}_{{\mathbf{C}}}$
which is constructed by gluing together the mechanisms of the
occasions in ${\mathbf{C}}$ and averaging over extrinsic noise.

Measuring devices are typically analyzed by varying their
inputs and observing the effect on their outputs. By contrast
this paper fixes the output and *varies the device over
all its subdevices* to obtain a family of submeasurements
parametrized by all subsystems in ${\mathtt{Sys}}_{{\mathbf{D}}}$. The internal
structure of the measurement performed by ${\mathbf{D}}$ is then
studied by comparing submeasurements.

We keep track of submeasurements by observing that they are
sections of a suitably defined presheaf^{}. Sheaf theory provides
a powerful machinery for analyzing relationships between
objects and subobjects [11], which we adapt to
our setting by introducing the structure presheaf ${\mathcal{F}}$,
a contravariant functor from ${\mathtt{Sys}}_{{\mathbf{D}}}$ to the category
of measuring devices ${\mathtt{Meas}}_{{\mathbf{D}}}$ on ${\mathbf{D}}$. Importantly,
${\mathcal{F}}$ is *not* a sheaf: although the gluing axiom
holds, uniqueness fails, see Theorem 4.
This is because the restriction^{} operator in ${\mathtt{Meas}}$ is
(essentially) marginalization, and of course there are
infinitely many joint distributions^{} $p(x,y)$ that yield
marginals $p(x)$ and $p(y)$.

Section §4
adapts Definition 1 to distributed systems and
introduces the simplest quantity associated with a measurement:
effective information, which quantifies its precision, see
Proposition 5.
Crucially, effective information is *context-dependent* –
it is computed relative to a baseline which may be completely
uninformative (the so-called null system) or provided by a
subsystem.

Finally entanglement, introduced in §5,
quantifies the obstruction (in bits) to decomposing a
measurement into independent submeasurements. It turns out,
see discussion after Theorem 10,
that entanglement quantifies the extent to which a measurement
is context-dependent – the extent to which contextual
information provided by one submeasurement is useful in
understanding another. Theorem 9
shows that a measurement is more precise than the sum of its
submeasurements *only if* entanglement is non-zero.
Precision is thus inextricably bound to context-dependence
and indecomposability. The failure of unique descent is thus
a feature, not a bug, since it provides “elbow room” to
build measuring devices that are *not* products^{} of
subdevices.

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

$f:X\rightarrow Y\,\,\,\mbox{ and }\,\,\,g:X\times Y\rightarrow Z$ |

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

# References

- 1
Samson Abramsky & Bob Coecke
(2009):
*Categorical Quantum Mechanics*. In K Engesser, D M Gabbay & D Lehmann, editors: Handbook of Quantum Logic and Quantum Structures: Quantum Logic, Elsevier. - 2
DJ Amit (1989):
*Modelling brain function: the world of attractor neural networks*. Cambridge University Press. - 3
David Balduzzi (2011):
*Detecting emergent processes in cellular automata with excess information*. preprint . - 4
David Balduzzi & Giulio Tononi
(2008):
*Integrated Information in Discrete Dynamical Systems: Motivation and Theoretical Framework.*PLoS Comput Biol 4(6), p. e1000091, doi:10.1371/journal.pcbi.1000091. - 5
David Balduzzi & Giulio Tononi
(2009):
*Qualia: the geometry of integrated information*. PLoS Comput Biol 5(8), p. e1000462, doi:10.1371/journal.pcbi.1000462. - 6
J S Bell (1990):
*Against ‘Measurement’*. Physics World August, pp. 33–40. - 7
Martin Gardner (1970):
*Mathematical Games - The Fantastic Combinations of John Conway’s New Solitaire Game, Life*. Scientific American 223, pp. 120–123. - 8
G ’t Hooft (1999):
*Quantum gravity as a dissipative deterministic system*. Classical and Quantum Gravity 16(10). - 9
JJ Hopfield (1982):
*Neural networks and physical systems with emergent computational properties*. Proc. Nat. Acad. Sci. 79, pp. 2554–2558. - 10
Stuart Kauffman, Carsten Peterson,
Björn Samuelsson & Carl Troein
(2003):
*Random Boolean network models and the yeast transcriptional network*. Proc Natl Acad Sci U S A 100(25), pp. 14796–9, doi:10.1073/pnas.2036429100. - 11
S MacLane & Ieke Moerdijk
(1992):
*Sheaves in Geometry and Logic: A First Introduction to Topos Theory*. Springer.

## Mathematics Subject Classification

94A17*no label found*60J20

*no label found*81P15

*no label found*18F20

*no label found*

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