1. Introduction

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

→

ℝ

.
The output

r

∈

ℝ

is the

reading

and the pre-image

f

-

1

⁢

(

r

)

⊂

X

is the

measurement

.

From this point of view a thermometer and a barometer are two functions,

T

:

X

→

ℝ

and

B

:

X

→

ℝ

,

mapping

the state space

X

of

configurations

(positions and momenta) of atmospheric particles
to real numbers. When the thermometer outputs

2

∘

, it specifies that
the atmospheric configuration was in the pre-image

T

-

1

⁢

(

2

∘

)

which,
assuming the thermometer perfectly

measures

temperature, is

exactly

characterized as atmospheric configurations with temperature

2

∘

.
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

,

berlekamp:82

]

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

×

X

→

𝑓

X

mapping sensory states

s

∈

S

and prior brain states

x

∈

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 (

http://planetmath.org/2stochasticmaps

)
lays the groundwork by introducing the

category

of

stochastic

maps

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

𝒱

taking functions on sets to Markov matrices:

[

f

:

X

→

Y

]

↦

[

𝒱

f

:

𝒱

X

→

𝒱

Y

]

,

where

𝒱

⁢

X

is functions from

X

to

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

♮

of a stochastic map

T

provides a

symmetric

treatment of functions and their corresponding

inverse
image

functions. Recall the

inverse

of function

f

:

X

→

Y

is

f

-

1

:

Y

→

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:

[

f

-

1

:

Y

→

2

¯

X

]

corresponds to

[

(

𝒱

f

)

♮

:

𝒱

Y

→

𝒱

X

]

,

(1)

see Corollary 2 (

http://planetmath.org/2stochasticmaps#Thmthm2

)

Section §3 (

http://planetmath.org/3distributeddynamicalsystems

)
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

𝚂𝚢𝚜

𝐃

of subsystems of

𝐃

. Each
subsystem

𝐂

is equipped with a mechanism

𝔪

𝐂

which is constructed by gluing together the mechanisms of the
occasions in

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

𝚂𝚢𝚜

𝐃

. The internal
structure of the measurement performed by

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

ℱ

,
a

contravariant functor

from

𝚂𝚢𝚜

𝐃

to the category
of measuring devices

𝙼𝚎𝚊𝚜

𝐃

on

𝐃

. Importantly,

ℱ

is

not

a sheaf: although the gluing axiom
holds, uniqueness fails, see

Theorem

4 (

http://planetmath.org/3distributeddynamicalsystems#Thmthm4

).
This is because the

restriction

operator

in

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 (

http://planetmath.org/4measurement

)
adapts Definition

1

to distributed systems and
introduces the simplest quantity associated with a measurement:

effective

information, which quantifies its precision, see

Proposition

5 (

http://planetmath.org/4measurement#Thmthm5

).
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 (

http://planetmath.org/5entanglement

),
quantifies the obstruction (in bits) to decomposing a
measurement into

independent

submeasurements. It turns out,
see discussion after Theorem 10 (

http://planetmath.org/5entanglement#Thmthm10

),
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 (

http://planetmath.org/5entanglement#Thmthm9

)
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

→

Y

⁢

and

⁢

g

:

X

×

Y

→

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.

Title

1. Introduction

Canonical name

1Introduction

Date of creation

2014-04-22 16:27:47

Last modified on

2014-04-22 16:27:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

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Author

rspuzio (6075)

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