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Title: Dungstyle Argumentation and AGMstyle Belief Revision

1
Dung-style Argumentation and AGM-style Belief
Revision

Guido Boella, Celia da Costa Pereira, Andrea
Tettamanzi and Leon van der Torre.

2
Position Statement

Formal study of Dung-style argumentation and
AGM-style belief revision is useful
Reinstatement in argumentation can formally be
related to recovery-related principles in
revision
This has been suggested also by Guillermo Simari,
Tony Hunter, Fabio Paglieri, and others
This presentation explains the problem, all
comments or references are highly appreciated.

3
Dung and AGM

Formal foundations of both theories
E.g., reinstatement and recovery
Argument revision
E.g., politics we should increase taxes (for the
rich)
Arguing about revision
E.g., you should believe in God, given Pascals
wager
Strategic argumentation
E.g., use conventional wisdom to persuade
Thus, a common framework is useful

4
Dung Non-Monotonic Logic - AGM

Dung Non-monotonic logic
Explanatory non-monotonic logic
Non-monotonic logic AGM
Shoham KLM tradition
Relating two kinds of
NML is open problem

5
The Intuition Dung and AGM are Related

Dungs reinstatement
If ? attacked by ? ? attacked by ?, then ?
reinstated
AGM recovery, Darwiche and Pearl, etc
If p 2 K , then (K-p)p K
DW1 If q ² p, then (Kp)q Kq
DW2 If q ² p, then (Kp)q Kq
DW3 If p 2 Kq, then p 2 (Kp)q
DW4 If p 2 Kq, then p 2 (Kp)q
In this presentation, we focus on DW2

6
The Problem How to Formalize Relation?

Use of arguments / propositions
Propositional argumentation
In Dungs approach, reinstatement is built in
Take a more general theory, like dominance theory
The dominance relation need not generally be
transitive and may even contain cycles. This
makes that the common concept of maximality or
optimality is no longer tenable with respect to
the dominance relation and new concepts have to
be developed to take over its function of
singling out elements that are in some sense
primary. Von Neumann and Morgenstern considered
this phenomenon as one of the most fundamental
problems the mathematical social sciences have to
cope with (see von Neumann and Morgenstern, 1947,
Ch. 1). BH08
No dynamics in argumentation / dominance
Dynamics in dialogue proof theories

7
Baroni and Giacomin, AIJ 2007

Framework for the evaluation of extension-based
argumentation semantics.
Solves the latter two problems
Definitions of reinstatement in this framework
Dynamics, because A arguments produced by a
reasoner at a given instant of time

8
Baroni and Giacomin, AIJ 2007

h A,! i is Dung argumentation framework
A is finite,
independently of the fact that the underlying
mechanism of argument generation admits the
existence of infinite sets of arguments.
We make the set of all arguments explicit
U is set of arguments which can be generated,
U for the universe of arguments.

9
Baroni and Giacomin, AIJ 2007

An extension-based argumentation semantics is
defined by specifying the criteria for deriving,
for a generic argumentation framework, a set of
extensions, where each extension represents a set
of arguments considered to be acceptable
together. Given a generic argumentation semantics
S, the set of extensions prescribed by S for a
given argumentation framework AF is denoted as
ES(AF).''

10
A Formal Definition

Let U be the universe of arguments.
An acceptance function ESU x 2UxU -gt22U is
a partial function which is defined for each
argumentation framework h A, ! i with finiteA µ
U and ! µ AxA, and
which maps an argumentation framework hA,!i to
sets of subsets of A ES (hA,!i)µ 2A
(Do we need A in argumentation framework?)

11
Do Baroni and Giacomin extend Dungs?

Baroni and Giacomin do not present their
framework as a generalization of Dung's,
Many papers claim to generalize Dung's,
for example with support relations, preferences,
values, nested attack relations, etc.
Implicitly, Baroni and Giacomin define
argumentation at another abstraction level.

12
Reinstatement, BG07, definition 15

A semantics S satisfies the reinstatement
criterion if 8 AF 2 DS, 8 E 2 ES(AF) it holds
that (8 ? 2 parAF(?) E! ?) ) ? 2 E
Intuitively, an argument ? is reinstated if its
defeaters are in turn defeated and, as a
consequence, one may assume that they should have
no effect on the justification state of ?.

13
Weak reinstatement, definition 1316

Given an argumentation framework AFh A,!i, ? 2 A
and S µ A, we say that ? is strongly defended by
S, denoted as sd(?,S), iff 8? 2 parAF(?) 9? 2 S
\ ? ? ! ? sd(?,S \ ?)
A semantics S satisfies the weak reinstatement
criterion if 8 AF 2 DS, 8 E 2 ES(AF) it holds
that sd(?,E) ) ? 2 E

14
Propositional argumentation

We associate proposition with each argument
prop A ! L, where L is propositional language
Belief set propositions of justified arguments
K(S) prop(?) j ? 2 S
Problems
Argument extensions, unique belief set
Solutions for non-deterministic belief revision
Consistency of belief set difficult to ensure

15
Literal Argumentation

We associate with argument a set of literals
propU! Lit, where Lit set of literals built from
atoms
8 ?, ? 2 U, if prop(?) Æ prop(?) inconsistent,
(i.e., ? and ? contain a complementary literal),
then either ? attacks ? or ? attacks ? (or both)
K(S) prop(?) j ? 2 S
Property for a set S, if each pair of S is
consistent, then S is consistent

16
Argument Runs

Run Sequence of argumentation frameworks
Abstraction of dialogue among players
Expansion based argumentation run
Only add arguments and attack relations
Persistence of relation among arguments
Only add attack relations involving newly added
argument
New is better
Only add attacks from new arguments to older ones
Minimal attack
New attack old argument if and only if conflicting

17
Constructability

Constructible argumentation framework
framework which can be reached from empty
framework in a finite number of steps
New is better leads to cycle free frameworks
See S. Kaci, L. van der Torre and E. Weydert, On
the acceptability fof conflicting arguments.
Proceedings of ECSQARU07, Springer, 2007.

18
Lemma 1 Reinstatement ! DW2

If
reinstatement
expansion, persistence, new are better,
minimality
constructible
Then
DW2 If q ² p, then (Kp)q Kq
Proof sketch extension is uniquely determined

19
Lemma 2 DW2 ! Reinstatement

If
DW2 If q ² p, then (Kp)q Kq
expansion, persistence, new are better,
minimality
constructible
trivial reinstatement if no attackers, then
accepted
Then
reinstatement

20
A Theorem and Our Research Problem

If
expansion, persistence, new are better,
minimality
constructible
trivial reinstatement if no attackers, then
accepted
Then
reinstatement iff DW2 If q ² p, then (Kp)q
Kq
Cycle-free frameworks are not very interesting
Our problem how to generalize this result?

21
Generalization 1 Minimality in Attack

Suppose a new argument can attack arguments which
are not conflicting
E.g., in assumption based reasoning
Additional independence assumption
8 ?,? 2 A, whether ? attacks ? depends only on ?
and ?, not on the other arguments
(Compare, e.g., the language independence
principle of Baroni and Giacomin)

22
Generalization 2 Constructability

Suppose an argumentation framework does not have
to be constructible
E.g., for general argumentation frameworks
Additional (strong) abstraction assumption
If an argument is not in any extension, then if
we abstract from it, then the extensions remain
the same
(Compare, e.g., the directionality criterion of
Baroni and Giacomin.)

23
Generalization 3 Constructability

Suppose a framework can contain cycles
Revise the constructability assumption
An argumentation framework is constructed in a
proponent opponent game (TPI)
(compare, e.g., the dialogue games of Prakken and
Vreeswijk)

24
Other Formal Foundations?

Success postulate

25
Argument Revision

For example, a kid does not want to go upstairs
since he is afraid of a monster - clearly you -
the father - do not believe this. you can say to
him that there is daylight (which is true), since
the kid believes monsters do not like daylight.
Alternatively you can say that upstairs is safe,
and the child has to give up the argument that
there are monsters (ie remove the argument).
If his brother said there are monsters and dad
says otherwise, the argument of the father is a
motivation for canceling the first argument,
since dad is more reliable (until I discover how
much he cheated to me).
Maybe if, instead, mom said to him that there are
monsters - rather than his brother - he just
overshadows (it is defeated but not cancelled)
the argument pro monsters, till she adds more
information.
However the reliability issue of brother vs
mother is relative and it could become subject to
another level of argumentation (like Sanjay
proposes?) one can attack the fact that the
father is more reliable than the brother (maybe
the kid heard mom said so while quarreling with
father)

26
Common Framework

Arguing about revision, strategic argumentation
When an agent uses an argument to persuade
another one, he must consider not only the
proposition supported by the argument, but also
the overall impact of the argument on the beliefs
of the addressee. Different arguments lead to
different belief revisions by the addressee. We
propose an approach whereby the best argument is
defined as the one which is both rational and the
most appealing to the addressee.
G. Boella, C. da Costa Pereira, A. Tettamanzi and
L/ van der Torre. Making Others Believe What They
Want. Proceedings of IFIP-AI 2008

27
Summary