On

the

relevance

of

utterances

in

formal

inter-agent

dialogues

Simon

Peter

Department

of

Computer

Information

Science

Brooklyn

College,

City

University

of

New

York

Brooklyn

NY

USA

{parsons,sklar}@sci.brooklyn.cuny.edu

Elizabeth

Michael

Department

of

Computer

Science

University

of

Liverpool

Liverpool

UK

{p.j.mcburney,m.j.wooldridge}@csc.liv.ac.uk

ABSTRACT

Work

on

argumentation-based

dialogue

has

defined

frameworks

within

which

dialogues

can

be

carried

out,

established

protocols

that

govern

dialogues,

and

studied

different

properties

of

dialogues.

This

work

has

established

the

space

in

which

agents

are

permitted

to

interact

through

dialogues.

Recently,

there

has

been

increasing

interest

in

the

mechanisms

agents

might

use

to

choose

how

to

act

the

rhetorical

manoeuvring

that

they

use

to

navigate

through

the

space

defined

by

the

rules

of

the

dialogue.

Key

in

such

considerations

is

the

idea

of

relevance,

since

a

usual

requirement

is

that

agents

stay

focussed

on

the

subject

of

the

dialogue

and

only

make

relevant

remarks.

Here

we

study

several

notions

of

relevance,

showing

how

they

can

be

related

to

both

the

rules

for

carrying

out

dialogues

and

to

rhetorical

manoeuvring.

Categories

and

Subject

Descriptors

[Artificial

Intelligence]:

Distributed

Artificial

Intelligence:

Coherence

co-ordination;

languages

structures;

multiagent

systems.

General

Terms

Design,

languages,

theory.

INTRODUCTION

Finding

ways

for

agents

to

reach

agreements

in

multiagent

systems

is

an

area

of

active

research.

One

mechanism

for

achieving

agreement

is

through

the

use

of

argumentation

where

one

agent

tries

to

convince

another

agent

of

something

during

the

course

of

some

dialogue.

Early

examples

of

argumentation-based

approaches

to

multiagent

agreement

include

the

work

of

Dignum

et

al.

Kraus

Parsons

and

Jennings

Reed

Schroeder

et

al.

and

Sycara

The

work

of

Walton

and

Krabbe

popularised

in

the

multiagent

systems

community

by

Reed

has

been

particularly

influential

in

the

field

of

argumentation-based

dialogue.

This

work

influenced

the

field

in

a

number

of

ways,

perhaps

most

deeply

in

framing

multi-agent

interactions

as

dialogue

games

in

the

tradition

of

Hamblin

Viewing

dialogues

in

this

way,

as

in

provides

a

powerful

framework

for

analysing

the

formal

properties

of

dialogues,

and

for

identifying

suitable

protocols

under

which

dialogues

can

be

conducted

The

dialogue

game

view

overlaps

with

work

on

conversation

policies

(see,

for

example,

but

differs

in

considering

the

entire

dialogue

rather

than

dialogue

segments.

In

this

paper,

we

extend

the

work

of

by

considering

the

role

of

relevance

the

relationship

between

utterances

in

a

dialogue.

Relevance

is

a

topic

of

increasing

interest

in

argumentation-based

dialogue

because

it

relates

to

the

scope

that

an

agent

has

for

applying

strategic

manoeuvering

to

obtain

the

outcomes

that

it

requires

Our

work

identifes

the

limits

on

such

rhetorical

manoeuvering,

showing

when

it

can

and

cannot

have

an

effect.

BACKGROUND

We

begin

by

introducing

the

formal

system

of

argumentation

that

underpins

our

approach,

as

well

as

the

corresponding

terminology

and

notation,

all

taken

from

A

dialogue

is

a

sequence

of

messages

passed

between

two

or

more

members

of

a

set

of

agents

A.

An

agent

α

maintains

a

knowledge

base,

Σα,

containing

formulas

of

a

propositional

language

L

and

having

no

deductive

closure.

Agent

α

also

maintains

the

set

of

its

past

utterances,

called

the

commitment

store,

CSα.

We

refer

to

this

as

an

agent"s

public

knowledge,

since

it

contains

information

that

is

shared

with

other

agents.

In

contrast,

the

contents

of

Σα

are

private

to

α.

Note

that

in

the

description

that

follows,

we

assume

that

is

the

classical

inference

relation,

that

stands

for

logical

equivalence,

and

we

use

Δ

to

denote

all

the

information

available

to

an

agent.

Thus

in

a

dialogue

between

two

agents

α

and

β,

Δα

Σα

CSα

CSβ,

so

the

commitment

store

CSα

can

be

loosely

thought

of

as

a

subset

of

Δα

consisting

of

the

assertions

that

have

been

made

public.

In

some

dialogue

games,

such

as

those

in

anything

in

CSα

is

either

in

Σα

or

can

be

derived

from

it.

In

other

dialogue

games,

such

as

(RPS)

c

IFAAMAS

those

in

CSα

may

contain

things

that

cannot

be

derived

from

Σα.

Definition

An

argument

A

is

a

pair

(S,

p)

where

p

is

a

formula

of

L

and

S

a

subset

of

Δ

such

that

(i)

S

is

consistent;

(ii)

S

p;

and

(iii)

S

is

minimal,

so

no

proper

subset

of

S

satisfying

both

and

exists.

S

is

called

the

support

of

A,

written

S

Support(A)

and

p

is

the

conclusion

of

A,

written

p

Conclusion(A).

Thus

we

talk

of

p

being

supported

by

the

argument

(S,

p).

In

general,

since

Δ

may

be

inconsistent,

arguments

in

A(Δ),

the

set

of

all

arguments

which

can

be

made

from

Δ,

may

conflict,

and

we

make

this

idea

precise

with

the

notion

of

undercutting:

Definition

Let

and

be

arguments

in

A(Δ).

undercuts

iff

∃¬p

such

that

p

In

other

words,

an

argument

is

undercut

if

and

only

if

there

is

another

argument

which

has

as

its

conclusion

the

negation

of

an

element

of

the

support

for

the

first

argument.

To

capture

the

fact

that

some

beliefs

are

more

strongly

held

than

others,

we

assume

that

any

set

of

beliefs

has

a

preference

order

over

it.

We

consider

all

information

available

to

an

agent,

Δ,

to

be

stratified

into

non-overlapping

subsets

Δn

such

that

beliefs

in

Δi

are

all

equally

preferred

and

are

preferred

over

elements

in

Δj

where

i

j.

The

preference

level

of

a

nonempty

subset

S

Δ,

where

different

elements

s

S

may

belong

to

different

layers

Δi,

is

valued

at

the

highest

numbered

layer

which

has

a

member

in

S

and

is

referred

to

as

level(S).

In

other

words,

S

is

only

as

strong

as

its

weakest

member.

Note

that

the

strength

of

a

belief

as

used

in

this

context

is

a

separate

concept

from

the

notion

of

support

discussed

earlier.

Definition

Let

and

be

arguments

in

A(Δ).

is

preferred

to

according

to

Pref

Pref

iff

If

is

preferred

to

we

say

that

is

stronger

than

We

can

now

define

the

argumentation

system

we

will

use:

Definition

An

argumentation

system

is

a

triple:

A(Δ),

Undercut,

Pref

such

that:

A(Δ)

is

a

set

of

the

arguments

built

from

Δ,

Undercut

is

a

binary

relation

representing

the

defeat

relationship

between

arguments,

Undercut

A(Δ)

A(Δ),

and

Pref

is

a

pre-ordering

on

A(Δ)

A(Δ).

The

preference

order

makes

it

possible

to

distinguish

different

types

of

relations

between

arguments:

Definition

Let

be

two

arguments

of

A(Δ).

If

undercuts

then

defends

itself

against

iff

Pref

Otherwise,

does

not

defend

itself.

A

set

of

arguments

A

defends

iff

for

every

that

undercuts

where

does

not

defend

itself

against

then

there

is

some

A

such

that

undercuts

and

does

not

defend

itself

against

We

write

AUndercut,Pref

to

denote

the

set

of

all

non-undercut

arguments

and

arguments

defending

themselves

against

all

their

undercutting

arguments.

The

set

A(Δ)

of

acceptable

arguments

of

the

argumentation

system

A(Δ),

Undercut,

Pref

is

the

least

fixpoint

of

a

function

F:

A

A(Δ)

F(A)

{(S,

p)

A(Δ)

(S,

p)

is

defended

by

A}

Definition

The

set

of

acceptable

arguments

for

an

argumentation

system

A(Δ),

Undercut,

Pref

is

recursively

defined

as:

A(Δ)

AUndercut,Pref

h[

i

An

argument

is

acceptable

if

it

is

a

member

of

the

acceptable

set,

and

a

proposition

is

acceptable

if

it

is

the

conclusion

of

an

acceptable

argument.

An

acceptable

argument

is

one

which

is,

in

some

sense,

proven

since

all

the

arguments

which

might

undermine

it

are

themselves

undermined.

Definition

If

there

is

an

acceptable

argument

for

a

proposition

p,

then

the

status

of

p

is

accepted,

while

if

there

is

not

an

acceptable

argument

for

p,

the

status

of

p

is

not

accepted.

Argument

A

is

said

to

affect

the

status

of

another

argument

A

if

changing

the

status

of

A

will

change

the

status

of

A

DIALOGUES

Systems

like

those

described

in

lay

down

sets

of

locutions

that

agents

can

make

to

put

forward

propositions

and

the

arguments

that

support

them,

and

protocols

that

define

precisely

which

locutions

can

be

made

at

which

points

in

the

dialogue.

We

are

not

concerned

with

such

a

level

of

detail

here.

Instead

we

are

interested

in

the

interplay

between

arguments

that

agents

put

forth.

As

a

result,

we

will

consider

only

that

agents

are

allowed

to

put

forward

arguments.

We

do

not

discuss

the

detail

of

the

mechanism

that

is

used

to

put

these

arguments

forward

we

just

assume

that

arguments

of

the

form

(S,

p)

are

inserted

into

an

agent"s

commitment

store

where

they

are

then

visible

to

other

agents.

We

then

have

a

typical

definition

of

a

dialogue:

Definition

A

dialogue

D

is

a

sequence

of

moves:

mn.

A

given

move

mi

is

a

pair

α,

Ai

where

Ai

is

an

argument

that

α

places

into

its

commitment

store

CSα.

Moves

in

an

argumentation-based

dialogue

typically

attack

moves

that

have

been

made

previously.

While,

in

general,

a

dialogue

can

include

moves

that

undercut

several

arguments,

in

the

remainder

of

this

paper,

we

will

only

consider

dialogues

that

put

forward

moves

that

undercut

at

most

one

argument.

For

now

we

place

no

additional

constraints

on

the

moves

that

make

up

a

dialogue.

Later

we

will

see

how

different

restrictions

on

moves

lead

to

different

kinds

of

dialogue.

The

Sixth

Intl.

Joint

Conf.

on

Autonomous

Agents

and

Multi-Agent

Systems

(AAMAS

The

sequence

of

arguments

put

forward

in

the

dialogue

is

determined

by

the

agents

who

are

taking

part

in

the

dialogue,

but

they

are

usually

not

completely

free

to

choose

what

arguments

they

make.

As

indicated

earlier,

their

choice

is

typically

limited

by

a

protocol.

If

we

write

the

sequence

of

n

moves

mn

as

mn,

and

denote

the

empty

sequence

as

then

we

can

define

a

profocol

in

the

following

way:

Definition

A

protocol

P

is

a

function

on

a

sequence

of

moves

mi

in

a

dialogue

D

that,

for

all

i

identifies

a

set

of

possible

moves

from

which

the

move

may

be

drawn:

P

mi

In

other

words,

for

our

purposes

here,

at

every

point

in

a

dialogue,

a

protocol

determines

a

set

of

possible

moves

that

agents

may

make

as

part

of

the

dialogue.

If

a

dialogue

D

always

picks

its

moves

m

from

the

set

M

identified

by

protocol

P,

then

D

is

said

to

conform

to

P.

Even

if

a

dialogue

conforms

to

a

protocol,

it

is

typically

the

case

that

the

agent

engaging

in

the

dialogue

has

to

make

a

choice

of

move

it

has

to

choose

which

of

the

moves

in

M

to

make.

This

excercise

of

choice

is

what

we

refer

to

as

an

agent"s

use

of

rhetoric

(in

its

oratorical

sense

of

influencing

the

thought

and

conduct

of

an

audience).

Some

of

our

results

will

give

a

sense

of

how

much

scope

an

agent

has

to

exercise

rhetoric

under

different

protocols.

As

arguments

are

placed

into

commitment

stores,

and

hence

become

public,

agents

can

determine

the

relationships

between

them.

In

general,

after

several

moves

in

a

dialogue,

some

arguments

will

undercut

others.

We

will

denote

the

set

of

arguments

Aj}

asserted

after

moves

mj

of

a

dialogue

to

be

Aj

the

relationship

of

the

arguments

in

Aj

can

be

described

as

an

argumentation

graph,

similar

to

those

described

in,

for

example,

Definition

An

argumentation

graph

AG

over

a

set

of

arguments

A

is

a

directed

graph

(V,

E)

such

that

every

vertex

v,

v

V

denotes

one

argument

A

A,

every

argument

A

is

denoted

by

one

vertex

v,

and

every

directed

edge

e

E

from

v

to

v

denotes

that

v

undercuts

v

We

will

use

the

term

argument

graph

as

a

synonym

for

argumentation

graph.

Note

that

we

do

not

require

that

the

argumentation

graph

is

connected.

In

other

words

the

notion

of

an

argumentation

graph

allows

for

the

representation

of

arguments

that

do

not

relate,

by

undercutting

or

being

undercut,

to

any

other

arguments

(we

will

come

back

to

this

point

very

shortly).

We

adapt

some

standard

graph

theoretic

notions

in

order

to

describe

various

aspects

of

the

argumentation

graph.

If

there

is

an

edge

e

from

vertex

v

to

vertex

v

then

v

is

said

to

be

the

parent

of

v

and

v

is

said

to

be

the

child

of

v.

In

a

reversal

of

the

usual

notion,

we

define

a

root

of

an

argumentation

as

follows:

Definition

A

root

of

an

argumentation

graph

AG

(V,

E)

is

a

node

v

V

that

has

no

children.

Thus

a

root

of

a

graph

is

a

node

to

which

directed

edges

may

be

connected,

but

from

which

no

directed

edges

connect

to

other

nodes.

Thus

a

root

is

a

node

representing

an

Note

that

we

talk

of

a

root

rather

than

the

root

as

defined,

an

argumentation

graph

need

not

be

a

tree.

v

v"

Figure

An

example

argument

graph

argument

that

is

undercut,

but

which

itself

does

no

undercutting.

Similarly:

Definition

A

leaf

of

an

argumentation

graph

AG

(V,

E)

is

a

node

v

V

that

has

no

parents.

Thus

a

leaf

in

an

argumentation

graph

represents

an

argument

that

undercuts

another

argument,

but

does

no

undercutting.

Thus

in

Figure

v

is

a

root,

and

v

is

a

leaf.

The

reason

for

the

reversal

of

the

usual

notions

of

root

and

leaf

is

that,

as

we

shall

see,

we

will

consider

dialogues

to

construct

argumentation

graphs

from

the

roots

(in

our

sense)

to

the

leaves.

The

reversal

of

the

terminology

means

that

it

matches

the

natural

process

of

tree

construction.

Since,

as

described

above,

argumentation

graphs

are

allowed

to

be

not

connected

(in

the

usual

graph

theory

sense),

it

is

helpful

to

distinguish

nodes

that

are

connected

to

other

nodes,

in

particular

to

the

root

of

the

tree.

We

say

that

node

v

is

connected

to

node

v

if

and

only

if

there

is

a

path

from

v

to

v

Since

edges

represent

undercut

relations,

the

notion

of

connectedness

between

nodes

captures

the

influence

that

one

argument

may

have

on

another:

Proposition

Given

an

argumentation

graph

AG,

if

there

is

any

argument

A,

denoted

by

node

v

that

affects

the

status

of

another

argument

A

denoted

by

v

then

v

is

connected

to

v

The

converse

does

not

hold.

Proof.

Given

Definitions

and

the

only

ways

in

which

A

can

affect

the

status

of

A

is

if

A

either

undercuts

A

or

if

A

undercuts

some

argument

A

that

undercuts

A

or

if

A

undercuts

some

A

that

undercuts

some

A

that

undercuts

A

and

so

on.

In

all

such

cases,

a

sequence

of

undercut

relations

relates

the

two

arguments,

and

if

they

are

both

in

an

argumentation

graph,

this

means

that

they

are

connected.

Since

the

notion

of

path

ignores

the

direction

of

the

directed

arcs,

nodes

v

and

v

are

connected

whether

the

edge

between

them

runs

from

v

to

v

or

vice

versa.

Since

A

only

undercuts

A

if

the

edge

runs

from

v

to

v

we

cannot

infer

that

A

will

affect

the

status

of

A

from

information

about

whether

or

not

they

are

connected.

The

reason

that

we

need

the

concept

of

the

argumentation

graph

is

that

the

properties

of

the

argumentation

graph

tell

us

something

about

the

set

of

arguments

A

the

graph

represents.

When

that

set

of

arguments

is

constructed

through

a

dialogue,

there

is

a

relationship

between

the

structure

of

the

argumentation

graph

and

the

protocol

that

governs

the

dialogue.

It

is

the

extent

of

the

relationship

between

structure

and

protocol

that

is

the

main

subject

of

this

paper.

To

study

this

relationship,

we

need

to

establish

a

correspondence

between

a

dialogue

and

an

argumentation

graph.

Given

the

definitions

we

have

so

far,

this

is

simple:

Definition

A

dialogue

D,

consisting

of

a

sequence

of

moves

mn,

and

an

argument

graph

AG

(V,

E)

correspond

to

one

another

iff

∀m

mn,

the

argument

Ai

that

The

Sixth

Intl.

Joint

Conf.

on

Autonomous

Agents

and

Multi-Agent

Systems

(AAMAS

is

advanced

at

move

mi

is

represented

by

exactly

one

node

v

V

and

∀v

V

v

represents

exactly

one

argument

Ai

that

has

been

advanced

by

a

move

m

mn.

Thus

a

dialogue

corresponds

to

an

argumentation

graph

if

and

only

if

every

argument

made

in

the

dialogue

corresponds

to

a

node

in

the

graph,

and

every

node

in

the

graph

corresponds

to

an

argument

made

in

the

dialogue.

This

one-toone

correspondence

allows

us

to

consider

each

node

v

in

the

graph

to

have

an

index

i

which

is

the

index

of

the

move

in

the

dialogue

that

put

forward

the

argument

which

that

node

represents.

Thus

we

can,

for

example,

refer

to

the

third

node

in

the

argumentation

graph,

meaning

the

node

that

represents

the

argument

put

forward

in

the

third

move

of

the

dialogue.

RELEVANCE

Most

work

on

dialogues

is

concerned

with

what

we

might

call

coherent

dialogues,

that

is

dialogues

in

which

the

participants

are,

as

in

the

work

of

Walton

and

Krabbe

focused

on

resolving

some

question

through

the

To

capture

this

coherence,

it

seems

we

need

a

notion

of

relevance

to

constrain

the

statements

made

by

agents.

Here

we

study

three

notions

of

relevance:

Definition

Consider

a

dialogue

D,

consisting

of

a

sequence

of

moves

mi,

with

a

corresponding

argument

graph

AG.

The

move

i

is

said

to

be

relevant

if

one

or

more

of

the

following

hold:

Making

will

change

the

status

of

the

argument

denoted

by

the

first

node

of

AG.

Making

will

add

a

node

that

is

connected

to

the

first

node

of

AG.

Making

will

add

a

node

that

is

connected

to

the

last

node

to

be

added

to

AG.

is

the

form

of

relevance

defined

by

in

their

study

of

strategic

and

tactical

was

suggested

by

the

notion

used

in

and

though

it

differs

somewhat

from

that

suggested

there,

we

believe

it

captures

the

essence

of

its

predecessor.

Note

that

we

only

define

relevance

for

the

second

move

of

the

dialogue

onwards

because

the

first

move

is

taken

to

identify

the

subject

of

the

dialogue,

that

is,

the

central

question

that

the

dialogue

is

intended

to

answer,

and

hence

it

must

be

relevant

to

the

dialogue,

no

matter

what

it

is.

In

assuming

this,

we

focus

our

attention

on

the

same

kind

of

dialogues

as

We

can

think

of

relevance

as

enforcing

a

form

of

parsimony

on

a

dialogue

it

prevents

agents

from

making

statements

that

do

not

bear

on

the

current

state

of

the

dialogue.

This

promotes

efficiency,

in

the

sense

of

limiting

the

number

of

moves

in

the

dialogue,

and,

as

in

prevents

agents

revealing

information

that

they

might

better

keep

hidden.

Another

form

of

parsimony

is

to

insist

that

agents

are

not

allowed

to

put

forward

arguments

that

will

be

undercut

by

arguments

that

have

already

been

made

during

the

dialogue.

We

therefore

distinguish

such

arguments.

See

for

examples

of

dialogues

where

this

is

not

the

case.

We

consider

such

reasoning

sub-types

of

rhetoric.

Definition

Consider

a

dialogue

D,

consisting

of

a

sequence

of

moves

mi,

with

a

corresponding

argument

graph

AG.

The

move

and

the

argument

it

puts

forward,

are

both

said

to

be

pre-empted,

if

is

undercut

by

some

A

Ai.

We

use

the

term

pre-empted

because

if

such

an

argument

is

put

forward,

it

can

seem

as

though

another

agent

anticipated

the

argument

being

made,

and

already

made

an

argument

that

would

render

it

useless.

In

the

rest

of

this

paper,

we

will

only

deal

with

protocols

that

permit

moves

that

are

relevant,

in

any

of

the

senses

introduced

above,

and

are

not

allowed

to

be

pre-empted.

We

call

such

protocols

basic

protocols,

and

dialogues

carried

out

under

such

protocols

basic

dialogues.

The

argument

graph

of

a

basic

dialogue

is

somewhat

restricted.

Proposition

Consider

a

basic

dialogue

D.

The

argumentation

graph

AG

that

corresponds

to

D

is

a

tree

with

a

single

root.

Proof.

Recall

that

Definition

requires

only

that

AG

be

a

directed

graph.

To

show

that

it

is

a

tree,

we

have

to

show

that

it

is

acyclic

and

connected.

That

the

graph

is

connected

follows

from

the

construction

of

the

graph

under

a

protocol

that

enforces

relevance.

If

the

notion

of

relevance

is

each

move

adds

a

node

that

is

connected

to

the

previous

node.

If

the

notion

of

relevance

is

then

every

move

adds

a

node

that

is

connected

to

the

root,

and

thus

is

connected

to

some

node

in

the

graph.

If

the

notion

of

relevance

is

then

every

move

has

to

change

the

status

of

the

argument

denoted

by

the

root.

Proposition

tells

us

that

to

affect

the

status

of

an

argument

A

the

node

v

representing

the

argument

A

that

is

effecting

the

change

has

to

be

connected

to

v

the

node

representing

A

and

so

it

follows

that

every

new

node

added

as

a

result

of

an

move

will

be

connected

to

the

argumentation

graph.

Thus

AG

is

connected.

Since

a

basic

dialogue

does

not

allow

moves

that

are

preempted,

every

edge

that

is

added

during

construction

is

directed

from

the

node

that

is

added

to

one

already

in

the

graph

(thus

denoting

that

the

argument

A

denoted

by

the

added

node,

v,

undercuts

the

argument

A

denoted

by

the

node

to

which

the

connection

is

made,

v

rather

than

the

other

way

around).

Since

every

edge

that

is

added

is

directed

from

the

new

node

to

the

rest

of

the

graph,

there

can

be

no

cycles.

Thus

AG

is

a

tree.

To

show

that

AG

has

a

single

root,

consider

its

construction

from

the

initial

node.

After

the

graph

has

one

node,

that

is

both

a

root

and

a

leaf.

After

the

graph

is

two

nodes

connected

by

an

edge,

and

is

now

a

root

and

not

a

leaf.

is

a

leaf

and

not

a

root.

However

the

third

node

is

added,

the

argument

earlier

in

this

proof

demonstrates

that

there

will

be

a

directed

edge

from

it

to

some

other

node,

making

it

a

leaf.

Thus

will

always

be

the

only

root.

The

ruling

out

of

pre-empted

moves

means

that

will

never

cease

to

be

a

root,

and

so

the

argumentation

graph

will

always

have

one

root.

Since

every

argumentation

graph

constructed

by

a

basic

dialogue

is

a

tree

with

a

single

root,

this

means

that

the

first

node

of

every

argumentation

graph

is

the

root.

Although

these

results

are

straightforward

to

obtain,

they

allow

us

to

show

how

the

notions

of

relevance

are

related.

The

Sixth

Intl.

Joint

Conf.

on

Autonomous

Agents

and

Multi-Agent

Systems

(AAMAS

Proposition

Consider

a

basic

dialogue

D,

consisting

of

a

sequence

of

moves

mi,

with

a

corresponding

argument

graph

AG.

Every

move

that

is

is

The

converse

does

not

hold.

Every

move

that

is

is

The

converse

does

not

hold.

Not

every

move

that

is

is

and

not

every

move

that

is

is

Proof.

For

consider

how

move

can

satisfy

Proposition

tells

us

that

if

can

change

the

status

of

the

argument

denoted

by

the

root

(which,

as

observed

above,

is

the

first

node)

of

AG,

then

must

be

connected

to

the

root.

This

is

precisely

what

is

required

to

satisfy

and

the

relatiosnhip

is

proved

to

hold.

To

see

that

the

converse

does

not

hold,

we

have

to

consider

what

it

takes

to

change

the

status

of

r

(since

Proposition

tells

us

that

connectedness

is

not

enough

to

ensure

a

change

of

status

if

it

did,

and

relevance

would

coincide).

For

to

change

the

status

of

the

root,

it

will

have

to

make

the

argument

A

represented

by

r

either

unacceptable,

if

it

were

acceptable

before

the

move,

or

acceptable

if

it

were

unacceptable

before

the

move.

Given

the

definition

of

acceptability,

it

can

achieve

either

by

directly

undercutting

the

argument

represented

by

r,

in

which

case

will

be

directly

connected

to

r

by

some

edge,

or

by

undercutting

some

argument

A

that

is

part

of

the

set

of

non-undercut

arguments

defending

A.

In

the

latter

case,

will

be

directly

connected

to

the

node

representing

A

and

by

Proposition

to

r.

To

achieve

will

have

to

undercut

an

argument

A

that

is

either

currently

undercutting

A,

or

is

undercutting

an

argument

that

would

otherwise

defend

A.

Now,

further

consider

that

puts

forward

an

argument

that

undercuts

the

argument

denoted

by

some

node

v

but

this

latter

argument

defends

itself

against

In

such

a

case,

the

set

of

acceptable

arguments

will

not

change,

and

so

the

status

of

Ar

will

not

change.

Thus

a

move

that

is

need

not

be

For

consider

that

can

satisfy

simply

by

adding

a

node

that

is

connected

to

vi,

the

last

node

to

be

added

to

AG.

By

Proposition

it

is

connected

to

r

and

so

is

To

see

that

the

converse

does

not

hold,

consider

that

an

move

can

connect

to

any

node

in

AG.

The

first

part

of

follows

by

a

similar

argument

to

that

we

just

used

an

move

does

not

have

to

connect

to

vi,

just

to

some

v

that

is

part

of

the

graph

and

the

second

part

follows

since

a

move

that

is

may

introduce

an

argument

that

undercuts

the

argument

Ai

put

forward

by

the

previous

move

(and

so

is

connected

to

vi),

but

finds

that

Ai

defends

itself

against

preventing

a

change

of

status

at

the

root.

What

is

most

interesting

is

not

so

much

the

results

but

why

they

hold,

since

this

reveals

some

aspects

of

the

interplay

between

relevance

and

the

structure

of

argument

graphs.

For

example,

to

restate

a

case

from

the

proof

of

Proposition

a

move

that

is

by

definition

has

to

add

a

node

to

the

argument

graph

that

is

connected

to

the

last

node

that

was

added.

Since

a

move

that

is

can

add

a

node

that

connects

anywhere

on

an

argument

graph,

any

move

that

is

will

be

but

the

converse

does

not

hold.

It

turns

out

that

we

can

exploit

the

interplay

between

structure

and

relevance

that

Propositions

and

have

started

to

illuminate

to

establish

relationships

between

the

protocols

that

govern

dialogues

and

the

argument

graphs

constructed

during

such

dialogues.

To

do

this

we

need

to

define

protocols

in

such

a

way

that

they

refer

to

the

structure

of

the

graph.

We

have:

Definition

A

protocol

is

single-path

if

all

dialogues

that

conform

to

it

construct

argument

graphs

that

have

only

one

branch.

Proposition

A

basic

protocol

P

is

single-path

if,

for

all

i,

the

set

of

permitted

moves

Mi

at

move

i

are

all

The

converse

does

not

hold.

Proof.

requires

that

every

node

added

to

the

argument

graph

be

connected

to

the

previous

node.

Starting

from

the

first

node

this

recursively

constructs

a

tree

with

just

one

branch,

and

the

relationship

holds.

The

converse

does

not

hold

because

even

if

one

or

more

moves

in

the

protocol

are

or

it

may

be

the

case

that,

because

of

an

agent"s

rhetorical

choice

or

because

of

its

knowledge,

every

argument

that

is

chosen

to

be

put

forward

will

undercut

the

previous

argument

and

so

the

argument

graph

is

a

one-branch

tree.

Looking

for

more

complex

kinds

of

protocol

that

construct

more

complex

kinds

of

argument

graph,

it

is

an

obvious

move

to

turn

to:

Definition

A

basic

protocol

is

multi-path

if

all

dialogues

that

conform

to

it

can

construct

argument

graphs

that

are

trees.

But,

on

reflection,

since

any

graph

with

only

one

branch

is

also

a

tree:

Proposition

Any

single-path

protocol

is

an

instance

of

a

multi-path

protocol.

and,

furthermore:

Proposition

Any

basic

protocol

P

is

multi-path.

Proof.

Immediate

from

Proposition

So

the

notion

of

a

multi-path

protocol

does

not

have

much

traction.

As

a

result

we

distinguish

multi-path

protocols

that

permit

dialogues

that

can

construct

trees

that

have

more

than

one

branch

as

bushy

protocols.

We

then

have:

Proposition

A

basic

protocol

P

is

bushy

if,

for

some

i,

the

set

of

permitted

moves

Mi

at

move

i

are

all

or

Proof.

From

Proposition

we

know

that

if

all

moves

are

then

we"ll

get

a

tree

with

one

branch,

and

from

Proposition

we

know

that

all

basic

protocols

will

build

an

argument

graph

that

is

a

tree,

so

providing

we

exclude

moves,

we

will

get

protocols

that

can

build

multi-branch

trees.

The

Sixth

Intl.

Joint

Conf.

on

Autonomous

Agents

and

Multi-Agent

Systems

(AAMAS

Of

course,

since,

by

Proposition

any

move

that

is

is

and

can

quite

possibly

be

(all

that

Proposition

tells

us

is

that

there

is

no

guarantee

that

it

will

be),

all

that

Proposition

tells

us

is

that

dialogues

that

conform

to

bushy

protocols

may

have

more

than

one

branch.

All

we

can

do

is

to

identify

a

bound

on

the

number

of

branches:

Proposition

Consider

a

basic

dialogue

D

that

includes

m

moves

that

are

not

and

has

a

corresponding

argumentation

graph

AG.

The

number

of

branches

in

AG

is

less

than

or

equal

to

m

Proof.

Since

it

must

connect

a

node

to

the

last

node

added

to

AG,

an

move

can

only

extend

an

existing

branch.

Since

they

do

not

have

the

same

restriction,

and

moves

may

create

a

new

branch

by

connecting

to

a

node

that

is

not

the

last

node

added.

Every

such

move

could

create

a

new

branch,

and

if

they

do,

we

will

have

m

branches.

If

there

were

moves

before

any

of

these

new-branch-creating

moves,

then

these

m

branches

are

in

addition

to

the

initial

branch

created

by

the

moves,

and

we

have

a

maximum

of

m

possible

branches.

We

distinguish

bushy

protocols

from

multi-path

protocols,

and

hence

and

from

because

of

the

kinds

of

dialogue

that

enforces.

In

a

dialogue

in

which

all

moves

must

be

the

argumentation

graph

has

a

single

branch

the

dialogue

consists

of

a

sequence

of

arguments

each

of

which

undercuts

the

previous

one

and

the

last

move

to

be

made

is

the

one

that

settles

the

dialogue.

This,

as

we

will

see

next,

means

that

such

a

dialogue

only

allows

a

subset

of

all

the

moves

that

would

otherwise

be

possible.

COMPLETENESS

The

above

discussion

of

the

difference

between

dialogues

carried

out

under

single-path

and

bushy

protocols

brings

us

to

the

consideration

of

what

called

predeterminism,

but

we

now

prefer

to

describe

using

the

term

completeness.

The

idea

of

predeterminism,

as

described

in

captures

the

notion

that,

under

some

circumstances,

the

result

of

a

dialogue

can

be

established

without

actually

having

the

dialogue

the

agents

have

sufficiently

little

room

for

rhetorical

manoeuver

that

were

one

able

to

see

the

contents

of

all

the

Σi

of

all

the

αi

A,

one

would

be

able

to

identify

the

outcome

of

any

dialogue

on

a

given

We

develop

this

idea

by

considering

how

the

argument

graphs

constructed

by

dialogues

under

different

protocols

compare

to

benchmark

complete

dialogues.

We

start

by

developing

ideas

of

what

complete

might

mean.

One

reasonable

definition

is

that:

Definition

A

basic

dialogue

D

between

the

set

of

agents

A

with

a

corresponding

argumentation

graph

AG

is

topic-complete

if

no

agent

can

construct

an

argument

A

that

undercuts

any

argument

A

represented

by

a

node

in

AG.

The

argumentation

graph

constructed

by

a

topic-complete

dialogue

is

called

a

topic-complete

argumentation

graph

and

is

denoted

AG(D)T

Assuming

that

the

Σi

do

not

change

during

the

dialogue,

which

is

the

usual

assumption

in

this

kind

of

dialogue.

A

dialogue

is

topic-complete

when

no

agent

can

add

anything

that

is

directly

connected

to

the

subject

of

the

dialogue.

Some

protocols

will

prevent

agents

from

making

moves

even

though

the

dialogue

is

not

topic-complete.

To

distinguish

such

cases

we

have:

Definition

A

basic

dialogue

D

between

the

set

of

agents

A

with

a

corresponding

argumentation

graph

AG

is

protocol-complete

under

a

protocol

P

if

no

agent

can

make

a

move

that

adds

a

node

to

the

argumentation

graph

that

is

permitted

by

P.

The

argumentation

graph

constructed

by

a

protocol-complete

dialogue

is

called

a

protocol-complete

argumentation

graph

and

is

denoted

AG(D)P

Clearly:

Proposition

Any

dialogue

D

under

a

basic

protocol

P

is

protocol-complete

if

it

is

topic-complete.

The

converse

does

not

hold

in

general.

Proof.

If

D

is

topic-complete,

no

agent

can

make

a

move

that

will

extend

the

argumentation

graph.

This

means

that

no

agent

can

make

a

move

that

is

permitted

by

a

basic

protocol,

and

so

D

is

also

protocol

complete.

The

converse

does

not

hold

since

some

basic

dialogues

(under

a

protocol

that

only

permits

moves,

for

example)

will

not

permit

certain

moves

(like

the

addition

of

a

node

that

connects

to

the

root

of

the

argumentation

graph

after

more

than

two

moves)

that

would

be

allowed

in

a

topiccomplete

dialogue.

Corollary

For

a

basic

dialogue

D,

AG(D)P

is

a

sub-graph

of

AG(D)T

Obviously,

from

the

definition

of

a

sub-graph,

the

converse

of

Corollary

does

not

hold

in

general.

The

important

distinction

between

topic-

and

protocolcompleteness

is

that

the

former

is

determined

purely

by

the

state

of

the

dialogue

as

captured

by

the

argumentation

graph

and

is

thus

independent

of

the

protocol,

while

the

latter

is

determined

entirely

by

the

protocol.

Any

time

that

a

dialogue

ends

in

a

state

of

protocol-completeness

rather

than

topic

completeness,

it

is

ending

when

agents

still

have

things

to

say

but

can"t

because

the

protocol

won"t

allow

them

to.

With

these

definitions

of

completeness,

our

task

is

to

relate

topic-completeness

the

property

that

ensures

that

agents

can

say

everything

that

they

have

to

say

in

a

dialogue

that

is,

in

some

sense,

important

to

the

notions

of

relevance

we

have

developed

which

determine

what

agents

are

allowed

to

say.

When

we

need

very

specific

conditions

to

make

protocol-complete

dialogues

topic-complete,

it

means

that

agents

have

lots

of

room

for

rhetorical

maneouver

when

those

conditions

are

not

in

force.

That

is

there

are

many

ways

they

can

bring

dialogues

to

a

close

before

everything

that

can

be

said

has

been

said.

Where

few

conditions

are

required,

or

conditions

are

absent,

then

dialogues

between

agents

with

the

same

knowledge

will

always

play

out

the

same

way,

and

rhetoric

has

no

place.

We

have:

Proposition

A

protocol-complete

basic

dialogue

D

under

a

protocol

which

only

allows

moves

will

be

topic-complete

only

when

AG(D)T

has

a

single

branch

in

which

the

nodes

are

labelled

in

increasing

order

from

the

root.

The

Sixth

Intl.

Joint

Conf.

on

Autonomous

Agents

and

Multi-Agent

Systems

(AAMAS

Proof.

Given

what

we

know

about

the

condition

on

AG(D)P

having

a

single

branch

is

obvious.

This

is

not

a

sufficient

condition

on

its

own

because

certain

protocols

may

prevent

through

additional

restrictions,

like

strict

turn-taking

in

a

multi-party

dialogue

all

the

nodes

in

AG(D)T

which

is

not

subject

to

such

restrictions,

being

added

to

the

graph.

Only

when

AG(D)T

includes

the

nodes

in

the

exact

order

that

the

corresponding

arguments

are

put

forward

is

it

necessary

that

a

topic-complete

argumentation

graph

be

constructed.

Given

Proposition

these

are

the

conditions

under

which

dialogues

conducted

under

the

notion

of

will

always

be

predetermined,

and

given

how

restrictive

the

conditions

are,

such

dialogues

seem

to

have

plenty

of

room

for

rhetoric

to

play

a

part.

To

find

similar

conditions

for

dialogues

composed

of

moves,

we

first

need

to

distinguish

between

them.

We

can

do

this

in

terms

of

the

structure

of

the

argumentation

graph:

Proposition

Consider

a

basic

dialogue

D,

with

argumentation

graph

AG

which

has

root

r

denoting

an

argument

A.

If

argument

A

denoted

by

node

v

is

an

an

move

m,

m

is

not

if

and

only

if:

there

are

two

nodes

v

and

v

on

the

path

between

v

and

r,

and

the

argument

denoted

by

v

defends

itself

against

the

argument

denoted

by

v

or

there

is

an

argument

A

denoted

by

node

v

that

affects

the

status

of

A,

and

the

path

from

v

to

r

has

one

or

more

nodes

in

common

with

the

path

from

v

to

r.

Proof.

For

the

first

condition,

consider

that

since

AG

is

a

tree,

v

is

connected

to

r.

Thus

there

is

a

series

of

undercut

relations

between

A

and

A

and

this

corrresponds

to

a

path

through

AG.

If

this

path

is

the

only

branch

in

the

tree,

then

A

will

affect

the

status

of

A

unless

the

chain

of

affect

is

broken

by

an

undercut

that

can"t

change

the

status

of

the

undercut

argument

because

the

latter

defends

itself.

For

the

second

condition,

as

for

the

first,

the

only

way

that

A

cannot

affect

the

status

of

A

is

if

something

is

blocking

its

influence.

If

this

is

not

due

to

defending

against,

it

must

be

because

there

is

some

node

u

on

the

path

that

represents

an

argument

whose

status

is

fixed

somehow,

and

that

must

mean

that

there

is

another

chain

of

undercut

relations,

another

branch

of

the

tree,

that

is

incident

at

u.

Since

this

second

branch

denotes

another

chain

of

arguments,

and

these

affect

the

status

of

the

argument

denoted

by

u,

they

must

also

affect

the

status

of

A.

Any

of

these

are

the

A

in

the

condition.

So

an

move

m

is

not

if

either

its

effect

is

blocked

because

an

argument

upstream

is

not

strong

enough,

or

because

there

is

another

line

of

argument

that

is

currently

determining

the

status

of

the

argument

at

the

root.

This,

in

turn,

means

that

if

the

effect

is

not

due

to

defending

against,

then

there

is

an

alternative

move

that

is

a

move

that

undercuts

A

in

the

second

condition

We

can

now

show

Though

whether

the

agent

in

question

can

make

such

a

move

is

another

question.

Proposition

A

protocol-complete

basic

dialogue

D

will

always

be

topic-complete

under

a

protocol

which

only

includes

moves

and

allows

every

move

to

be

made.

The

restriction

on

rules

is

exactly

that

for

topiccompleteness,

so

a

dialogue

that

has

only

moves

will

continue

until

every

argument

that

any

agent

can

make

has

been

put

forward.

Given

this,

and

what

we

revealed

about

in

Proposition

we

can

see

that:

Proposition

A

protocol-complete

basic

dialogue

D

under

a

protocol

which

only

includes

moves

will

be

topic-complete

if

AG(D)T

includes

no

path

with

adjacent

nodes

v,

denoting

A,

and

v

denoting

A

such

that

A

undercuts

A

and

A

is

stronger

that

A;

and

is

such

that

the

nodes

in

every

branch

have

consecutive

indices

and

no

node

with

degree

greater

than

two

is

an

odd

number

of

arcs

from

a

leaf

node.

Proof.

The

first

condition

rules

out

the

first

condition

in

Proposition

and

the

second

deals

with

the

situation

that

leads

to

the

second

condition

in

Proposition

The

second

condition

ensures

that

each

branch

is

constructed

in

full

before

any

new

branch

is

added,

and

when

a

new

branch

is

added,

the

argument

that

is

undercut

as

part

of

the

addition

will

be

acceptable,

and

so

the

addition

will

change

the

status

of

the

argument

denoted

by

that

node,

and

hence

the

root.

With

these

conditions,

every

move

required

to

construct

AG(D)T

will

be

permitted

and

so

the

dialogue

will

be

topic-complete

when

every

move

has

been

completed.

The

second

part

of

this

result

only

identifies

one

possible

way

to

ensure

that

the

second

condition

in

Proposition

is

met,

so

the

converse

of

this

result

does

not

hold.

However,

what

we

have

is

sufficient

to

answer

the

question

about

predetermination

that

we

started

with.

For

dialogues

to

be

predetermined,

every

move

that

is

must

be

made.

In

such

cases

every

dialogue

is

topic

complete.

If

we

do

not

require

that

all

moves

are

made,

then

there

is

some

room

for

rhetoric

the

way

in

which

alternative

lines

of

argument

are

presented

becomes

an

issue.

If

moves

are

forced

to

be

then

there

is

considerable

room

for

rhetorical

play.

SUMMARY

This

paper

has

studied

the

different

ideas

of

relevance

in

argumentation-based

dialogue,

identifying

the

relationship

between

these

ideas,

and

showing

how

they

can

impact

the

extent

to

which

the

way

that

agents

choose

moves

in

a

dialogue

what

some

authors

have

called

the

strategy

and

tactics

of

a

dialogue.

This

extends

existing

work

on

relvance,

such

as

by

showing

how

different

notions

of

relevance

can

have

an

effect

on

the

outcome

of

a

dialogue,

in

particular

when

they

render

the

outcome

predetermined.

This

connection

extends

the

work

of

which

considered

dialogue

outcome,

but

stopped

short

of

identifying

the

conditions

under

which

it

is

predetermined.

There

are

two

ways

we

are

currently

trying

to

extend

this

work,

both

of

which

will

generalise

the

results

and

extend

its

applicability.

First,

we

want

to

relax

the

restrictions

that

The

Sixth

Intl.

Joint

Conf.

on

Autonomous

Agents

and

Multi-Agent

Systems

(AAMAS

we

have

imposed,

the

exclusion

of

moves

that

attack

several

arguments

(without

which

the

argument

graph

can

be

mulitply-connected)

and

the

exclusion

of

pre-empted

moves,

without

which

the

argument

graph

can

have

cycles.

Second,

we

want

to

extend

the

ideas

of

relevance

to

cope

with

moves

that

do

not

only

add

undercutting

arguments,

but

also

supporting

arguments,

thus

taking

account

of

bipolar

argumentation

frameworks

Acknowledgments

The

authors

are

grateful

for

financial

support

received

from

the

EC,

through

project

and

from

the

NSF

under

grants

and

NSF

They

are

also

grateful

to

Peter

Stone

for

a

question,

now

several

years

old,

which

this

paper

has

finally

answered.

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C.

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Dialogue

frames

in

agent

communications.

In

Y.

Demazeau,

editor,

Proceedings

of

the

Third

International

Conference

on

Multi-Agent

Systems,

pages

IEEE

Press,

M.

Rovatsos,

I.

Rahwan,

F.

Fisher,

and

G.

Weiss.

Adaptive

strategies

for

practical

argument-based

negotiation.

In

I.

Rahwan,

P.

Moraitis,

and

C.

Reed,

editors,

Proceedings

of

the

International

Workshop

on

Argumentation

in

Multiagent

Systems,

New

York,

NY,

M.

Schroeder,

D.

A.

Plewe,

and

A.

Raab.

Ultima

ratio:

should

Hamlet

kill

Claudius.

In

Proceedings

of

the

International

Conference

on

Autonomous

Agents,

pages

K.

Sycara.

Argumentation:

Planning

other

agents"

plans.

In

Proceedings

of

the

Eleventh

Joint

Conference

on

Artificial

Intelligence,

pages

D.

N.

Walton

and

E.

C.

W.

Krabbe.

Commitment

in

Dialogue:

Basic

Concepts

of

Interpersonal

Reasoning.

State

University

of

New

York

Press,

Albany,

NY,

USA,

The

Sixth

Intl.

Joint

Conf.

on

Autonomous

Agents

and

Multi-Agent

Systems

(AAMAS

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

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