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.
11/16/2018
ARGMAS 2008

3. Dung and AGM Formal foundations of both theories Argument revision
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 Pascal’s wager
Strategic argumentation
E.g., use conventional wisdom to persuade
Thus, a common framework is useful
11/16/2018
ARGMAS 2008

4. Dung – Non-Monotonic Logic - AGM
Explanatory non-monotonic logic
Non-monotonic logic – AGM
Shoham – KLM tradition
“Relating two kinds of NML is open problem”
11/16/2018
ARGMAS 2008

5. The Intuition: Dung and AGM are Related
Dung’s 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 (K*p)*q = K*q
DW2: If q ² : p, then (K*p)*q = K*q
DW3: If p 2 K*q, then p 2 (K*p)*q
DW4: If : p 2 K*q, then : p 2 (K*p)*q
In this presentation, we focus on DW2
11/16/2018
ARGMAS 2008

6. The Problem: How to Formalize Relation?
Use of arguments / propositions
Propositional argumentation
In Dung’s 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
11/16/2018
ARGMAS 2008

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
11/16/2018
ARGMAS 2008

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.
11/16/2018
ARGMAS 2008

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).''
11/16/2018
ARGMAS 2008

10. A Formal Definition Let U be the universe of arguments.
An acceptance function ES:U x 2UxU ->22U is
a partial function which is defined for each argumentation framework h A, ! i with finite A 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?)
11/16/2018
ARGMAS 2008

11. Do Baroni and Giacomin extend Dung’s?
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.
11/16/2018
ARGMAS 2008

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 .”
11/16/2018
ARGMAS 2008

13. Weak reinstatement, definition 13+16
Given an argumentation framework AF=h 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
11/16/2018
ARGMAS 2008

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
11/16/2018
ARGMAS 2008

15. Literal Argumentation
We associate with argument a set of literals
prop:U! 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
11/16/2018
ARGMAS 2008

16. Argument Runs Minimal attack
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
11/16/2018
ARGMAS 2008

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.
11/16/2018
ARGMAS 2008

18. Lemma 1: Reinstatement ! DW2If
reinstatement
expansion, persistence, new are better, minimality
constructible
Then
DW2: If q ² : p, then (K*p)*q = K*q
Proof sketch: extension is uniquely determined
11/16/2018
ARGMAS 2008

19. Lemma 2: DW2 ! ReinstatementIf
DW2: If q ² : p, then (K*p)*q = K*q
expansion, persistence, new are better, minimality
constructible
trivial reinstatement: if no attackers, then accepted
Then
reinstatement
11/16/2018
ARGMAS 2008

20. A Theorem and Our Research ProblemIf
expansion, persistence, new are better, minimality
constructible
trivial reinstatement: if no attackers, then accepted
Then
reinstatement iff DW2: If q ² : p, then (K*p)*q = K*q
Cycle-free frameworks are not very interesting
Our problem: how to generalize this result?
11/16/2018
ARGMAS 2008

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)
11/16/2018
ARGMAS 2008

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.)
11/16/2018
ARGMAS 2008

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)
11/16/2018
ARGMAS 2008

24. Other Formal Foundations?
Success postulate
11/16/2018
ARGMAS 2008

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)
11/16/2018
ARGMAS 2008

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
11/16/2018
ARGMAS 2008

27. Summary Dung reinstatement – AGM recovery
Intuition, example result for cycle free
Problem is how to generalize
Minimality, constructability: new principles needed
Other formal foundations of both theories?
Argument revision?
Arguing about revision, strategic arguing?
A common framework for Dung and AGM?
11/16/2018
ARGMAS 2008