Entropy

and

Self-Organization

in

Multi-Agent

Systems

Emergent

self-organization

in

multi-agent

systems

appears

to

contradict

the

second

law

of

thermodynamics

This

paradox

has

been

explained

in

terms

of

a

coupling

between

the

macro

level

that

hosts

self-organization

-LRB-

and

an

apparent

reduction

in

entropy

-RRB-

and

the

micro

level

-LRB-

where

random

processes

greatly

increase

entropy

-RRB-

Metaphorically

the

micro

level

serves

as

an

entropy

sink

permitting

overall

system

entropy

to

increase

while

sequestering

this

increase

from

the

interactions

where

self-organization

is

desired

We

make

this

metaphor

precise

by

constructing

a

simple

example

of

pheromone-based

coordination

defining

a

way

to

measure

the

Shannon

entropy

at

the

macro

-LRB-

agent

-RRB-

and

micro

-LRB-

pheromone

-RRB-

levels

and

exhibiting

an

entropy-based

view

of

the

coordination

Researchers

who

construct

multi-agent

systems

must

cope

with

the

world

's

natural

tendency

to

disorder

Many

applications

require

a

set

of

agents

that

are

individually

autonomous

-LRB-

in

the

sense

that

each

agent

determines

its

actions

based

on

its

own

state

and

the

state

of

the

environment

without

explicit

external

command

-RRB-

but

corporately

structured

We

want

individual

local

decisions

to

yield

coherent

global

behavior

Self-organization

in

natural

systems

-LRB-

e.g.

human

culture

insect

colonies

-RRB-

is

an

existence

proof

that

individual

autonomy

is

not

incompatible

with

global

order

However

widespread

human

experience

warns

us

that

building

systems

that

exhibit

both

individual

autonomy

and

global

order

is

not

trivial

Not

only

agent

researchers

but

humans

in

general

seek

to

impose

structure

and

organization

on

the

world

around

us

It

is

a

universal

experience

that

the

structure

we

desire

can

be

achieved

only

through

hard

work

and

that

it

tends

to

fall

apart

if

not

tended

This

experience

is

sometimes

summarized

informally

as

Murphy

's

Law

the

observation

that

anything

that

can

go

wrong

will

go

wrong

and

at

the

worst

possible

moment

At

the

root

of

the

ubiquity

of

disorganizing

tendencies

is

the

Second

Law

of

Thermodynamics

that

energy

spontaneously

tends

to

flow

only

from

being

concentrated

in

one

place

to

becoming

diffused

and

spread

out

-LSB-

-RSB-

Adding

energy

to

a

system

can

overcome

the

Second

Law

's

spontaneous

tendency

and

lead

to

increasing

structure

However

the

way

in

which

energy

is

added

is

critical

Gasoline

in

the

engines

of

construction

equipment

can

construct

a

building

out

of

raw

steel

and

concrete

while

the

same

gasoline

in

a

bomb

can

reduce

a

building

to

a

mass

of

raw

steel

and

concrete

Agents

are

not

immune

to

Murphy

The

natural

tendency

of

a

group

of

autonomous

processes

is

to

disorder

not

to

organization

Adding

information

to

a

collection

of

agents

can

lead

to

increased

organization

but

only

if

it

is

added

in

the

right

way

We

will

be

successful

in

engineering

agent-based

systems

just

to

the

degree

that

we

understand

the

interplay

between

disorder

and

order

The

fundamental

claim

of

this

paper

is

that

the

relation

between

self-organization

in

multi-agent

systems

and

thermodynamic

concepts

such

as

the

second

law

is

not

just

a

loose

metaphor

but

can

provide

quantitative

analytical

guidelines

for

designing

and

operating

agent

systems

We

explain

the

link

between

these

concepts

and

demonstrate

by

way

of

a

simple

example

how

they

can

be

applied

in

measuring

the

behavior

of

multi-agent

systems

Our

inspiration

is

a

model

for

self-organization

proposed

by

Kugler

and

Turvey

-LSB-

-RSB-

which

suggests

that

the

key

to

reducing

disorder

in

a

multi-agent

system

is

coupling

that

system

to

another

in

which

disorder

increases

Section

reviews

this

model

and

relates

it

to

the

problem

of

agent

coordination

Section

describes

a

test

scenario

that

we

have

devised

inspired

by

self-organization

in

pheromone

systems

and

outlines

a

method

for

measuring

entropy

in

this

scenario

Section

reports

our

experimental

results

Section

summarizes

our

conclusions

In

the

context

of

biomechanical

systems

Kugler

and

Turvey

-LSB-

-RSB-

suggest

that

self-organization

can

be

reconciled

with

second-law

tendencies

if

a

system

includes

multiple

coupled

levels

of

dynamic

activity

Purposeful

self-organizing

behavior

occurs

at

the

macro

level

By

itself

such

behavior

would

be

contrary

to

the

second

law

However

the

system

includes

a

micro

level

whose

dynamics

generate

increasing

disorder

Thus

the

system

as

a

whole

is

increasingly

disordered

over

time

Crucially

the

behavior

of

elements

at

the

macro

level

is

coupled

to

the

micro

level

dynamics

To

understand

this

model

we

begin

with

an

example

then

abstract

out

the

underlying

dynamics

and

finally

comment

on

the

legitimacy

of

identifying

processes

at

this

level

with

principles

from

thermodynamics

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AGENTS

May

Montr

al

Quebec

Canada

Copyright

ACM

An

Example

Pheromones

The

parade

example

of

such

a

system

is

the

self-organization

of

an

insect

colony

-LRB-

such

as

the

construction

of

minimal

spanning

tree

networks

among

nests

and

food

sources

by

ants

or

the

erection

of

multi-storied

structures

with

regularly

spaced

pillars

and

floors

by

tropical

termites

-RRB-

through

pheromone-based

coordination

-LSB-

-RSB-

Pheromones

are

scent

markers

that

insects

use

in

two

ways

First

they

deposit

pheromones

in

the

environment

to

record

their

state

For

example

a

foraging

ant

just

emerging

from

the

nest

in

search

of

food

might

deposit

nest

pheromone

while

an

ant

that

has

found

food

and

is

carrying

it

will

deposit

food

pheromone

-LRB-

-LSB-

-RSB-

documents

use

of

multiple

pheromones

by

insects

-RRB-

Second

they

orient

their

movements

to

the

gradient

of

the

pheromone

field

In

the

example

of

foraging

ants

those

seeking

food

climb

the

gradient

of

the

food

pheromone

while

those

carrying

food

climb

the

gradient

of

the

nest

pheromone

The

most

realistic

models

of

the

ants

pheromone-climbing

behavior

incorporates

a

stochastic

element

in

their

movement

That

is

they

do

not

follow

the

gradient

deterministically

but

use

its

strength

to

weight

a

roulette

wheel

from

which

they

determine

their

movement

The

environment

in

which

pheromones

are

deposited

plays

a

critical

role

in

such

a

system

It

is

not

passive

but

active

and

performs

three

information-processing

functions

with

the

pheromones

It

aggregates

deposits

of

the

same

flavor

of

pheromone

from

different

ants

thus

providing

a

form

of

data

fusion

across

multiple

agents

at

different

times

based

on

their

traversal

of

a

common

location

It

evaporates

pheromones

over

time

thus

forgetting

obsolete

information

This

dynamic

is

usefully

viewed

as

a

novel

approach

to

truth

maintenance

Conventional

knowledge

bases

remember

every

assertion

unless

there

is

cause

to

retract

it

and

execute

truth

maintenance

processes

to

detect

and

resolve

the

conflicts

that

result

when

inconsistent

assertions

coexist

Insect

systems

forget

every

assertion

unless

it

is

regularly

reinforced

Evaporation

provides

a

third

function

that

of

disseminating

information

from

the

location

at

which

it

was

deposited

to

nearby

locations

An

ant

does

not

have

to

stumble

across

the

exact

location

at

which

pheromone

was

deposited

in

order

to

access

the

information

it

represents

but

can

sense

the

direction

from

its

current

location

to

the

pheromone

deposit

in

the

form

of

the

gradient

of

evaporated

pheromone

molecules

The

Model

in

Detail

In

the

Kugler-Turvey

model

ants

and

their

movements

constitute

the

macro

level

of

the

system

while

pheromone

molecules

constitute

the

micro

level

The

purposeful

movement

of

ants

constructing

minimal

paths

among

their

nests

and

food

sources

achieve

a

reduction

in

disorder

at

the

macro

level

made

possible

because

the

agents

at

this

level

are

coupled

to

the

micro

level

where

the

evaporation

of

pheromone

molecules

under

Brownian

motion

results

in

an

overwhelming

growth

in

disorder

As

a

result

the

disorder

of

the

overall

system

increases

in

keeping

with

the

Second

Law

in

spite

of

the

emergence

of

useful

order

at

the

macro

level

Figure

illustrates

the

interplay

among

these

processes

and

how

this

model

of

agent

coordination

differs

from

more

classical

views

Classically

agents

are

considered

to

perceive

one

another

directly

reason

about

this

perception

and

then

take

rational

action

The

Kugler-Turvey

model

views

coordination

as

being

mediated

by

an

environment

that

agents

change

by

their

actions

-LRB-

e.g.

depositing

pheromones

-RRB-

a

process

known

as

stigmergy

-LSB-

-RSB-

Processes

in

the

environment

generate

structures

that

the

agents

perceive

thus

permitting

ordered

behavior

at

the

agent

level

At

the

same

time

these

processes

increase

disorder

at

the

micro

level

so

that

the

system

as

a

whole

becomes

less

ordered

over

time

Research

in

synthetic

pheromones

-LSB-

-RSB-

draws

directly

on

this

model

of

coordination

but

the

model

is

of

far

broader

applicability

In

a

multi-commodity

market

individual

agents

follow

economic

fields

generated

by

myriad

individual

transactions

and

self-organization

in

the

demand

and

supply

of

a

particular

commodity

is

supported

by

an

environment

that

distributes

resources

based

on

the

other

transactions

in

the

system

The

movement

of

currency

in

such

a

system

provides

similar

functions

to

those

of

pheromones

in

insect

systems

More

broadly

we

hypothesize

that

a

coupling

of

ordered

and

disordered

systems

is

ubiquitous

in

robust

self-organizing

systems

and

that

the

lack

of

such

a

coupling

correlates

with

architectures

that

do

not

meet

their

designers

expectations

for

emergent

cohesiveness

A

Caveat

At

this

point

readers

with

a

background

in

physics

and

chemistry

may

be

uneasy

These

disciplines

formulated

the

Second

Law

within

a

strict

context

of

processes

that

result

in

energy

changes

The

fundamental

physical

measures

associated

with

the

second

law

are

temperature

T

heat

Q

and

-LRB-

thermodynamic

-RRB-

entropy

S

related

by

the

definition

Equation

T

dQ

dS

Statistical

mechanics

identifies

this

macroscopic

measure

with

the

number

of

microscopically

defined

states

accessible

to

the

system

by

the

relation

Equation

S

k

ln

where

k

is

Boltzmann

's

constant

erg/deg

M

icro

New

to

n

ian

F

o

rce

F

ield

E

n

tro

p

y

F

lo

w

-LRB-

E

n

tro

p

y

-RRB-

F

lo

w

-LRB-

E

n

tro

p

y

-RRB-

M

acro

No

n

New

to

n

ian

F

lo

w

F

ield

Neg

en

tro

p

y

Pe

rcep

tio

n

Percep

tion

P

h

ero

m

o

n

e

Rational

Action

-LRB-

Entropy

-RRB-

Ra

tio

l

A

cti

on

-LRB-

Ent

ropy

-RRB-

P

h

ero

m

o

n

e

Dynam

ics

P

ercep

tio

n

Ratio

n

al

A

ctio

n

Ag

en

t

Ag

en

t

T

rad

itio

n

al

A

g

en

t

Dynam

ics

Key

M

icro

New

to

n

ian

F

o

rce

F

ield

E

n

tro

p

y

F

lo

w

-LRB-

E

n

tro

p

y

-RRB-

F

lo

w

-LRB-

E

n

tro

p

y

-RRB-

M

acro

No

n

New

to

n

ian

F

lo

w

F

ield

Neg

en

tro

p

y

Pe

rcep

tio

n

Percep

tion

P

h

ero

m

o

n

e

Rational

Action

-LRB-

Entropy

-RRB-

Ra

tio

l

A

cti

on

-LRB-

Ent

ropy

-RRB-

P

h

ero

m

o

n

e

Dynam

ics

P

ercep

tio

n

Ratio

n

al

A

ctio

n

Ag

en

t

Ag

en

t

T

rad

itio

n

al

A

g

en

t

Dynam

ics

Key

Figure

Comparison

of

Conventional

and

Pheromone-Based

Models

of

Coordination

Thus

defined

thermodynamic

entropy

has

strong

formal

similarities

-LSB-

-RSB-

to

information

entropy

-LSB-

-RSB-

Equation

i

i

i

p

p

S

log

where

i

ranges

over

the

possible

states

of

the

system

and

p

i

is

the

probability

of

finding

the

system

in

state

i

These

formal

similarities

have

led

to

a

widespread

appropriation

of

the

notion

of

entropy

as

a

measure

of

macro-level

disorder

and

of

the

Second

Law

as

describing

a

tendency

of

systems

to

become

more

chaotic

Our

approach

participates

to

this

appropriation

It

has

been

objected

-LSB-

-RSB-

that

such

appropriation

completely

ignores

the

role

of

energy

intrinsic

to

both

thermodynamic

definitions

-LRB-

via

T

and

dQ

in

the

macro

definition

and

k

in

the

micro

definition

-RRB-

Such

an

objection

implicitly

assumes

that

energy

is

logically

prior

to

the

definition

and

that

unless

information

processes

are

defined

in

terms

of

energy

changes

it

is

illegitimate

to

identify

their

changes

in

entropy

with

those

of

thermodynamics

An

alternative

approach

to

the

question

would

argue

that

in

fact

the

prior

concept

is

not

ergs

but

bits

the

universe

is

nothing

but

a

very

large

cellular

automaton

with

very

small

cells

-LSB-

-RSB-

and

physics

and

chemistry

can

in

principle

be

redefined

in

terms

of

information-theoretic

concepts

Our

approach

is

sympathetic

with

this

view

While

we

are

not

prepared

at

this

point

to

define

the

precise

correspondence

between

ergs

and

bits

we

believe

that

physical

models

are

an

under-exploited

resource

for

understanding

computational

systems

in

general

and

multi-agent

systems

in

particular

The

fact

that

the

thermodynamic

and

information

approaches

work

in

different

fundamental

units

-LRB-

ergs

vs.

bits

-RRB-

is

not

a

reason

to

separate

them

but

a

pole

star

to

guide

research

that

may

ultimately

bring

them

together

We

experiment

with

these

concepts

using

a

simple

model

of

pheromone-based

behavior

In

this

section

we

describe

the

experiment

and

how

one

measures

entropy

over

it

The

Coordination

Problem

Consider

two

agents

one

fixed

and

one

mobile

who

desire

to

be

together

Neither

knows

the

location

of

the

other

The

mobile

agent

or

walker

could

travel

to

the

destination

of

the

stationary

one

if

it

only

knew

where

to

go

The

stationary

agent

or

target

deposits

pheromone

molecules

at

its

location

As

the

pheromone

molecules

diffuse

through

the

environment

they

create

a

gradient

that

the

walker

can

follow

Initially

the

walker

is

at

-LRB-

-RRB-

and

the

target

is

at

-LRB-

-RRB-

in

a

field

Every

time

step

the

target

deposits

one

molecule

at

-LRB-

-RRB-

Both

the

walker

and

the

pheromone

molecules

move

by

computing

an

angle

-LSB-

-RSB-

relative

to

their

current

heading

and

taking

a

step

of

constant

length

-LRB-

for

the

walker

for

the

pheromone

molecule

-RRB-

in

the

resulting

direction

Thus

both

molecules

and

walkers

can

be

located

at

any

real-valued

coordinates

in

the

field

Molecules

move

every

cycle

of

the

simulation

and

the

walker

every

five

cycles

so

altogether

the

molecules

move

ten

times

as

fast

as

the

walker

Molecules

fall

off

of

the

field

when

they

reach

the

edge

while

the

walker

bounces

off

the

edges

Molecules

choose

the

heading

for

their

next

step

from

a

uniform

random

distribution

and

so

execute

an

unbiased

random

walk

The

walker

computes

its

heading

from

two

inputs

It

generates

a

gradient

vector

Gr

from

its

current

location

to

each

molecule

within

a

specified

radius

with

magnitude

Equation

i

r

i

r

g

G

r

where

r

i

is

the

distance

between

the

walker

and

the

ith

molecule

and

g

is

a

gravitational

constant

-LRB-

currently

-RRB-

It

generates

a

random

vector

Rr

with

random

heading

and

length

equal

to

a

temperature

parameter

T

The

vector

sum

R

G

r

r

normalized

to

the

walker

's

step

length

-LRB-

in

these

experiments

-RRB-

defines

the

walker

's

next

step

Including

Rr

in

the

computation

permits

us

to

explore

the

effectiveness

of

different

degrees

of

stochasticity

in

the

walker

's

movement

following

the

example

of

natural

pheromone

systems

The

state

of

the

walker

defines

the

macro

state

of

the

system

while

the

states

of

the

molecules

define

the

micro

state

This

model

can

easily

be

enhanced

in

a

number

of

directions

including

adding

multiple

walkers

and

multiple

targets

and

permitting

walkers

and

targets

to

deposit

pheromone

molecules

of

various

flavors

The

simple

configuration

is

sufficient

to

demonstrate

our

techniques

and

their

potential

for

understanding

how

the

walker

finds

the

target

Measuring

Entropy

Computing

the

Shannon

or

Information

Entropy

defined

in

Equation

requires

that

we

measure

the

set

of

states

accessible

to

the

system

and

the

probability

of

finding

the

system

in

each

of

those

states

Measuring

the

Number

of

System

States

In

most

computational

systems

the

discreteness

of

digital

computation

makes

counting

system

states

straightforward

-LRB-

though

the

number

of

possible

states

is

extremely

high

-RRB-

We

have

purposely

defined

the

movement

of

our

walker

and

molecules

in

continuous

space

to

highlight

the

challenge

of

counting

discrete

system

states

in

an

application

embedded

in

the

physical

world

-LRB-

such

as

a

robotic

application

-RRB-

Our

approach

is

to

superimpose

a

grid

on

the

field

and

define

a

state

on

the

basis

of

the

populations

of

the

cells

of

the

grid

We

can

define

state

and

thus

entropy

in

terms

either

of

location

or

direction

Location-based

state

is

based

on

a

single

snapshot

of

the

system

while

direction-based

state

is

based

on

how

the

system

has

changed

between

successive

snapshots

Each

approach

has

an

associated

gridding

technique

For

location-based

entropy

we

divide

the

field

with

a

grid

Figure

shows

a

grid

with

four

cells

one

spanning

each

quarter

of

the

field

The

state

of

this

system

is

a

four-element

vector

reporting

the

number

of

molecules

in

each

cell

-LRB-

in

the

example

reading

row-wise

from

upper

left

The

number

of

possible

states

in

an

nxn

grid

with

m

particles

is

n

The

parameters

in

location-based

gridding

are

the

number

of

divisions

in

each

direction

their

orientation

and

the

origin

of

the

grid

Rectangular

grids

are

easiest

to

manage

computationally

but

one

could

also

tile

the

plane

with

hexagons

For

direction-based

entropy

we

center

a

star

on

the

previous

location

of

each

particle

and

record

the

sector

of

the

star

into

which

the

particle

is

found

at

the

current

step

Figure

shows

a

four-rayed

star

with

a

two

particles

The

state

of

the

system

is

a

vector

with

one

element

for

each

particle

in

some

canonical

order

Counting

sectors

clockwise

from

the

upper

left

the

state

of

this

example

is

The

number

of

possible

states

with

an

n-pointed

star

and

m

particles

is

mn

The

parameters

in

direction-based

gridding

are

the

number

of

rays

in

the

star

and

the

rotation

of

the

star

about

its

center

In

both

techniques

the

analysis

depends

critically

on

the

resolution

of

the

grid

-LRB-

the

parameter

n

-RRB-

and

its

origin

and

orientation

-LRB-

for

location

-RRB-

or

rotation

-LRB-

for

direction

-RRB-

To

understand

the

dependency

on

n

consider

two

extremes

If

n

is

very

large

the

chance

of

two

distributions

of

particles

on

the

field

having

the

same

state

is

vanishingly

small

For

N

distributions

each

will

be

a

distinct

state

each

state

will

have

equal

probability

and

the

entropy

will

be

log

-LRB-

N

-RRB-

This

state

of

affairs

is

clearly

not

informative

At

the

other

extreme

n

all

distributions

represent

the

same

state

which

therefore

occurs

with

probability

yielding

entropy

again

not

informative

We

choose

the

gridding

resolution

empirically

by

observing

the

length

scales

active

in

the

system

as

it

operates

To

understand

the

dependency

on

origin/orientation

or

rotation

consider

two

particles

in

the

same

cell

After

they

move

will

they

still

be

in

the

same

cell

-LRB-

keeping

entropy

the

same

-RRB-

or

in

different

cells

-LRB-

increasing

entropy

-RRB-

Exactly

the

same

movements

of

the

two

particles

could

yield

either

result

depending

on

how

the

grid

is

registered

with

the

field

We

follow

Gutowitz

's

technique

-LSB-

-RSB-

of

measuring

the

entropy

with

several

different

origins

and

taking

the

minimum

thus

minimizing

entropy

contributions

resulting

from

the

discrete

nature

of

the

grid

Measuring

the

Probabilities

In

principle

one

could

compute

the

probability

of

different

system

states

analytically

This

approach

would

be

arduous

even

for

our

simple

system

and

completely

impractical

for

a

more

complex

system

We

take

a

Monte

Carlo

approach

instead

We

run

the

system

repeatedly

At

each

step

in

time

we

estimate

the

probability

of

each

observed

state

by

counting

the

number

of

replications

in

which

that

state

was

observed

The

results

reported

here

are

based

on

replications

Shannon

entropy

has

a

maximum

value

of

log

-LRB-

N

-RRB-

for

N

different

states

achieved

when

each

state

is

equally

probable

To

eliminate

this

dependence

on

N

we

normalize

the

entropies

we

report

by

dividing

by

log

-LRB-

N

-RRB-

-LRB-

in

our

case

log

-LRB-

-RRB-

-RRB-

incidentally

making

the

choice

of

base

of

logarithms

irrelevant

We

report

the

behavior

of

entropy

first

in

the

micro

system

then

in

the

unguided

and

guided

macro

system

and

finally

in

the

complete

system

Entropy

in

the

Micro

System

Figure

shows

locational

entropy

in

the

micro

system

-LRB-

the

pheromone

molecules

-RRB-

computed

from

a

grid

Entropy

increases

with

time

until

it

saturates

at

The

more

molecules

enter

the

system

and

the

more

they

disperse

throughout

the

field

the

higher

the

entropy

grows

Increasing

the

grid

resolution

has

no

effect

on

the

shape

of

this

increase

but

reduces

the

time

to

saturation

because

the

molecules

must

spread

out

from

a

single

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Figure

Location-based

gridding

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Figure

Direction-based

gridding

Time

o

r

c

i

My

p

o

r

t

n

E

Figure

Micro

Entropy

x

Time

-LRB-

Grid

-RRB-

location

and

the

finer

the

grid

the

sooner

they

can

generate

a

large

number

of

different

states

Directional

entropy

also

increases

with

time

to

saturation

This

result

-LRB-

not

plotted

-RRB-

can

be

derived

analytically

The

molecule

population

increases

linearly

with

time

until

molecules

start

reaching

the

edge

Then

the

growth

slows

and

eventually

reaches

Let

M

be

the

population

of

the

field

at

equilibrium

and

consider

all

M

molecules

being

located

at

-LRB-

-RRB-

through

the

entire

run

Initially

all

are

stationary

and

each

time

step

one

additional

molecule

is

activated

Then

the

total

number

of

possible

system

states

for

a

is

but

the

number

actually

sampled

during

the

period

of

linear

population

growth

is

since

the

stationary

molecules

do

not

generate

any

additional

states

Thus

the

entropy

during

the

linear

phase

is

log

-LRB-

-RRB-

log

-LRB-

-RRB-

As

the

growth

becomes

sublinear

the

entropy

asymptotically

approaches

as

with

locational

entropy

Entropy

in

the

Unguided

Macro

System

Figure

shows

the

path

of

a

walker

uncoupled

to

the

micro

system

-LRB-

when

the

target

is

emitting

no

pheromone

molecules

-RRB-

With

no

coupling

to

the

micro

field

the

walker

is

just

a

single

molecule

executing

a

random

walk

Figure

shows

that

locational

entropy

for

this

walker

increases

over

time

reflecting

the

increased

number

of

cells

accessible

to

the

walker

as

its

random

walk

takes

it

farther

from

its

base

The

grid

size

-LRB-

divisions

in

each

direction

-RRB-

is

chosen

on

the

basis

of

observations

of

the

guided

walker

discussed

in

the

next

section

The

directional

entropy

-LRB-

not

plotted

-RRB-

is

constant

at

since

the

walker

chooses

randomly

at

each

step

from

all

available

directions

Entropy

in

Guided

Macro

System

Now

we

provide

the

walker

with

a

micro

field

by

emitting

pheromone

molecules

from

the

target

Figure

shows

the

path

followed

by

a

typical

walker

with

radius

and

T

This

path

has

three

distinct

parts

Initially

the

walker

wanders

randomly

around

its

origin

at

-LRB-

-RRB-

until

the

wavefront

of

molecules

diffusing

from

-LRB-

-RRB-

encounters

its

radius

In

this

region

the

walker

has

no

guidance

because

no

molecules

are

visible

Once

the

walker

begins

to

sense

molecules

it

moves

rather

directly

from

the

vicinity

of

-LRB-

-RRB-

to

-LRB-

-RRB-

following

the

pheromone

gradient

When

it

arrives

at

-LRB-

-RRB-

it

again

receives

no

guidance

from

the

molecules

because

they

are

distributed

equally

in

all

directions

So

it

again

meanders

The

clouds

of

wandering

near

the

start

and

finish

have

diameters

in

the

range

of

to

suggesting

a

natural

grid

between

and

We

report

here

experiments

with

a

grid

Because

of

their

initial

random

walk

around

their

origin

walkers

in

different

runs

will

be

at

different

locations

when

they

start

to

move

and

will

follow

slightly

different

paths

to

the

target

-LRB-

Figure

-RRB-

x

y

Figure

Unguided

Walker

Path

Axes

are

location

in

the

-LRB-

-RRB-

field

Time

o

r

c

a

My

p

o

r

t

n

E

Figure

Unguided

Walker

Locational

Entropy

-LRB-

Grid

-RRB-

x

y

Figure

Guided

Walker

Path

-LRB-

T

-RRB-

x

y

Figure

Ensemble

of

Guided

Walkers

-LRB-

T

-RRB-

The

dots

in

Figure

and

Figure

show

the

directional

and

locational

entropies

across

this

ensemble

of

guided

walkers

as

a

function

of

time

The

solid

line

in

each

case

plots

the

normalized

median

distance

from

the

walkers

to

the

target

-LRB-

actual

maximum

-RRB-

while

the

dashed

line

plots

the

normalized

median

number

of

molecules

visible

to

the

walkers

-LRB-

actual

maximum

-RRB-

The

lines

show

how

changes

in

entropy

and

reduction

in

distance

to

the

target

are

correlated

with

the

number

of

molecules

that

the

walker

senses

at

any

given

moment

At

the

beginning

and

end

of

the

run

when

the

walkers

are

wandering

without

guidance

directional

entropy

is

corresponding

to

a

random

walk

During

the

middle

portion

of

the

run

when

the

walker

is

receiving

useful

guidance

from

the

micro

level

the

entropy

drops

dramatically

As

the

temperature

parameter

T

is

increased

in

the

range

to

the

bottom

of

the

entropy

well

rises

but

the

overall

shape

remains

the

same

-LRB-

plot

not

shown

-RRB-

The

locational

entropy

presents

a

different

story

The

minimization

method

for

avoiding

discreteness

artifacts

has

the

effect

of

selecting

at

each

time

step

the

offset

that

best

centers

the

cells

on

the

walkers

At

the

beginning

of

the

run

and

again

at

the

end

most

walkers

are

close

together

and

fall

within

the

same

cell

-LRB-

because

we

chose

a

cell

size

comparable

to

these

clouds

-RRB-

Walkers

leave

the

starting

cloud

at

different

times

since

those

closer

to

the

target

sense

the

pheromones

sooner

and

follow

different

paths

depending

on

where

they

were

when

the

pheromone

reached

them

Thus

they

spread

out

during

this

movement

phase

and

cluster

together

again

once

they

reach

the

target

The

effect

of

raising

T

to

on

locational

entropy

is

that

the

right

end

of

the

curve

rises

until

the

curve

assumes

a

similar

shape

-LRB-

plot

not

shown

-RRB-

to

Figure

Comparison

of

Figure

and

Figure

shows

that

though

the

directed

portion

of

the

walker

's

movement

has

higher

entropy

than

the

undirected

portions

coupling

the

walker

to

the

micro

level

does

reduce

the

walker

's

overall

entropy

Even

at

its

maximum

the

entropy

of

the

guided

walker

is

much

lower

than

that

of

the

random

one

demonstrating

the

basic

dynamics

of

the

Kugler-Turvey

model

The

different

behavior

of

locational

and

directional

entropy

is

instructive

Which

is

more

orderly

a

randomly

moving

walker

or

one

guided

by

pheromones

The

expected

location

of

a

random

walker

is

stationary

-LRB-

though

with

a

non-zero

variance

-RRB-

while

that

of

a

guided

walker

is

non-stationary

In

terms

of

location

the

random

walker

is

thus

more

regular

and

the

location

entropy

reflects

this

However

the

movement

of

the

guided

walker

is

more

orderly

than

that

of

the

random

walker

and

this

difference

is

reflected

in

the

directional

entropy

This

difference

highlights

the

importance

of

paying

attention

to

dynamical

aspects

of

agent

behavior

Our

intuition

that

the

guided

walker

is

more

orderly

than

the

random

one

is

really

an

intuition

about

the

movement

of

this

walker

not

its

location

Entropy

in

the

Overall

System

Central

to

the

Kugler-Turvey

model

is

the

assertion

that

entropy

increase

at

the

micro

level

is

sufficient

to

ensure

entropy

increase

in

the

overall

system

even

in

the

presence

of

self-organization

and

concomitant

entropy

reduction

at

the

micro

level

Our

experiment

illustrates

this

dynamic

As

illustrated

in

Figure

by

time

normalized

entropy

in

the

micro

system

has

reached

the

maximum

level

of

indicating

that

each

of

the

replications

of

the

experiment

results

in

a

distinct

state

If

each

replication

is

already

distinct

on

the

basis

of

the

locations

of

the

pheromone

molecules

alone

adding

additional

state

elements

-LRB-

such

as

the

location

of

the

walker

-RRB-

can

not

cause

two

replications

to

become

the

same

Thus

by

time

the

normalized

entropy

of

the

entire

system

must

also

be

at

a

maximum

In

particular

decreases

in

macro

entropy

such

as

the

decrease

in

locational

entropy

from

time

on

seen

in

Figure

do

not

reduce

the

entropy

of

the

overall

system

One

may

ask

whether

the

reduction

in

macro

-LRB-

walker

-RRB-

entropy

is

causally

related

to

the

increase

in

micro

entropy

or

just

coincidental

After

all

a

static

gradient

of

pheromone

molecules

would

guide

the

walker

to

the

target

just

as

effectively

but

would

be

identical

in

every

run

and

so

exhibit

zero

entropy

This

argument

neglects

whatever

process

generates

the

static

gradient

in

the

first

place

An

intelligent

observer

could

produce

the

gradient

but

then

the

behavior

of

the

system

would

hardly

be

self-organizing

In

our

scenario

the

gradient

emerges

as

a

natural

consequence

of

a

completely

random

process

the

random

walk

of

the

pheromone

molecules

emerging

from

the

target

The

gradient

can

then

reduce

the

entropy

of

a

walker

at

the

macro

level

but

the

price

paid

for

this

entropy

reduction

is

the

increase

in

entropy

generated

by

the

random

process

that

produces

and

maintains

the

gradient

Time

y

p

o

r

t

n

E

e

c

n

a

t

s

i

Dd

n

as

e

l

u

c

e

l

o

M

Figure

Guided

walker

dots

directional

entropy

-LRB-

star

-RRB-

solid

line

median

distance

to

target

-LRB-

max

-RRB-

dashed

line

median

visible

molecules

-LRB-

max

-RRB-

Time

o

r

c

a

My

p

o

r

t

n

E

e

c

n

a

t

s

i

Dd

n

as

e

l

u

c

e

l

o

M

Figure

Guided

walker

dots

locational

entropy

-LRB-

grid

-RRB-

solid

line

median

distance

to

target

-LRB-

max

-RRB-

dashed

line

median

visible

molecules

-LRB-

max

-RRB-

One

may

also

ask

whether

our

hypothesis

requires

a

quantitative

relation

between

entropy

loss

at

the

macro

level

and

entropy

gain

at

the

micro

level

A

strict

entropy

balance

is

not

required

the

micro

level

might

generate

more

entropy

than

the

macro

level

loses

In

operational

terms

the

system

may

have

a

greater

capacity

for

coordination

than

a

particular

instantiation

exploits

What

is

required

is

that

the

entropy

increase

at

the

micro

level

be

sufficient

to

cover

the

decrease

at

the

macro

level

and

this

we

have

shown

To

be

effective

multi-agent

systems

must

yield

coordinated

behavior

from

individually

autonomous

actions

Concepts

from

thermodynamics

-LRB-

in

particular

the

Second

Law

and

entropy

-RRB-

have

been

invoked

metaphorically

to

explain

the

conditions

under

which

coordination

can

emerge

Our

work

makes

this

metaphor

more

concrete

and

derives

several

important

insights

from

it

This

metaphor

can

be

made

quantitative

through

simple

state

partitioning

methods

and

Monte

Carlo

simulation

These

methods

show

how

coordination

can

arise

through

coupling

the

macro

level

-LRB-

in

which

we

desire

agent

self-organization

with

a

concomitant

decrease

in

entropy

-RRB-

to

an

entropy-increasing

process

at

a

micro

level

-LRB-

e.g.

pheromone

evaporation

-RRB-

Our

demonstration

focuses

on

synthetic

pheromones

for

the

sake

of

expositional

simplicity

but

we

believe

that

the

same

approach

would

be

fruitful

for

understanding

self-organization

with

other

mechanisms

of

agent

coordination

such

as

market

systems

This

confirmation

of

the

Kugler-Turvey

model

encourages

us

as

agent

designers

to

think

explicitly

in

terms

of

macro

and

micro

levels

with

agent

behaviors

at

the

macro

level

coupled

in

both

directions

-LRB-

causally

and

perceptually

-RRB-

to

entropy-increasing

processes

at

the

micro

level

Some

form

of

pheromone

or

currency

is

a

convenient

mechanism

for

creating

such

an

entropy-increasing

process

Researchers

must

distinguish

between

static

and

dynamic

order

in

a

multi-agent

system

We

have

exhibited

a

system

that

appears

intuitively

to

be

self-organizing

and

shown

that

the

measure

of

order

underlying

this

intuition

is

dynamic

rather

than

static

This

work

is

supported

in

part

by

the

DARPA

JFACC

program

under

contract

to

ERIM

CEC

The

views

and

conclusions

in

this

document

are

those

of

the

authors

and

should

not

be

interpreted

as

representing

the

official

policies

either

expressed

or

implied

of

the

Defense

Advanced

Research

Projects

Agency

or

the

US

Government

-LSB-

-RSB-

E.

Bonabeau

M.

Dorigo

and

G.

Theraulaz

Swarm

Intelligence

From

Natural

to

Artificial

Systems

New

York

Oxford

University

Press

-LSB-

-RSB-

S.

Brueckner

Return

from

the

Ant

Synthetic

Ecosystems

for

Manufacturing

Control

Thesis

at

Humboldt

University

Berlin

Department

of

Computer

Science

-LSB-

-RSB-

E.

Fredkin

Finite

Nature

In

Proceedings

of

The

XXVIIth

Recontre

de

Moriond

-LSB-

-RSB-

P.-P

Grass

La

Reconstruction

du

nid

et

les

Coordinations

Inter-Individuelles

chez

Bellicositermes

Natalensis

et

Cubitermes

sp.

La

th

orie

de

la

Stigmergie

Essai

d'interpr

tation

du

Comportement

des

Termites

Constructeurs

Insectes

Sociaux

-LSB-

-RSB-

H.

A.

Gutowitz

Complexity-Seeking

Ants

In

Proceedings

of

Third

European

Conference

on

Artifical

Life

-LSB-

-RSB-

B.

Hayes

Computational

Creationism

American

Scientist

-LRB-

-RRB-

-LSB-

-RSB-

P.

N.

Kugler

and

M.

T.

Turvey

Information

Natural

Law

and

the

Self-Assembly

of

Rhythmic

Movement

Lawrence

Erlbaum

-LSB-

-RSB-

F.

L.

Lambert

Shuffled

Cards

Messy

Desks

and

Disorderly

Dorm

Rooms

Examples

of

Entropy

Increase

Nonsense

Journal

of

Chemical

Education

-LSB-

-RSB-

F.

L.

Lambert

The

Second

Law

of

Thermodynamics

Web

Page

-LSB-

-RSB-

J.

Lukkarinen

Re

continuing

on

Entropy

Email

Archive

html

-LSB-

-RSB-

V.

D.

Parunak

Go

to

the

Ant

Engineering

Principles

from

Natural

Agent

Systems

Annals

of

Operations

Research

-LSB-

-RSB-

V.

D.

Parunak

and

S.

Brueckner

Ant-Like

Missionaries

and

Cannibals

Synthetic

Pheromones

for

Distributed

Motion

Control

In

Proceedings

of

Fourth

International

Conference

on

Autonomous

Agents

-LRB-

Agents

-RRB-

pages

-LSB-

-RSB-

Peeters

P.

Valckenaers

J.

Wyns

and

S.

Brueckner

Manufacturing

Control

Algorithm

and

Architecture

In

Proceedings

of

Second

International

Workshop

on

Intelligent

Manufacturing

Systems

pages

K.U.

Leuven

-LSB-

-RSB-

E.

Shannon

and

W.

Weaver

The

Mathematical

Theory

of

Communication

Urbana

IL

University

of

Illinois

-LSB-

-RSB-

ithsonian

Institution

Encyclopedia

Smithsonian

Pheromones

in

Insects

Web

Page

m.

B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B 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