Dung-style Argumentation and AGM-style Belief Revision Guido Boella, Celia da Costa Pereira, Andrea Tettamanzi and Leon van der Torre.
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. 5/24/2008 ARGMAS 2008 2
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 Pascal’s wager � Strategic argumentation � E.g., use conventional wisdom to persuade � Thus, a common framework is useful 5/24/2008 ARGMAS 2008 3
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” 5/24/2008 ARGMAS 2008 4
The Intuition: Dung and AGM are Related � Dung’s reinstatement � If α attacked by β & β attacked by γ , then α reinstated � AGM recovery, Darwiche and Pearl, etc � If p ∈ 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 ∈ K*q, then p ∈ (K * p )* q � DW4: If ¬ p ∈ K*q, then ¬ p ∈ (K * p )* q � In this presentation, we focus on DW2 5/24/2008 ARGMAS 2008 5
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 5/24/2008 ARGMAS 2008 6
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 5/24/2008 ARGMAS 2008 7
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. 5/24/2008 ARGMAS 2008 8
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 E S ( AF ).'' 5/24/2008 ARGMAS 2008 9
A Formal Definition � Let U be the universe of arguments. � An acceptance function E S : U x 2 U x U ->2 2 U is 1. a partial function which is defined for each argumentation framework h A , → i with finite A ⊆ U and → ⊆ A x A , and 2. which maps an argumentation framework h A , → i to sets of subsets of A : E S ( h A , → i ) ⊆ 2 A � (Do we need A in argumentation framework?) 5/24/2008 ARGMAS 2008 10
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. 5/24/2008 ARGMAS 2008 11
Reinstatement, [BG07, definition 15] � A semantics S satisfies the reinstatement criterion if ∀ AF ∈ D S , ∀ E ∈ E S ( AF ) it holds that ( ∀ β ∈ par AF ( α ) E → β ) ⇒ α ∈ E � “Intuitively, an argument α 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 α .” 5/24/2008 ARGMAS 2008 12
Weak reinstatement, definition 13+16 � Given an argumentation framework AF = h A , → i , α ∈ A and S ⊆ A , we say that α is strongly defended by S , denoted as sd( α , S ), iff ∀ β ∈ par AF ( α ) ∃ γ ∈ S \ { α }: γ → β & sd( γ , S \ { α }) � A semantics S satisfies the weak reinstatement criterion if ∀ AF ∈ D S , ∀ E ∈ E S ( AF ) it holds that sd( α , E ) ⇒ α ∈ E 5/24/2008 ARGMAS 2008 13
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( α ) | α ∈ S } � Problems: 1. Argument extensions, unique belief set � Solutions for non-deterministic belief revision 2. Consistency of belief set difficult to ensure 5/24/2008 ARGMAS 2008 14
Literal Argumentation � We associate with argument a set of literals � prop: U → Lit, where Lit set of literals built from atoms � ∀ α , β ∈ U , if prop( α ) ∧ prop( β ) inconsistent, � (i.e., α and β contain a complementary literal), � then either α attacks β or β attacks α (or both) � K ( S ) = { prop( α ) | α ∈ S } � Property: for a set S , if each pair of S is consistent, then S is consistent 5/24/2008 ARGMAS 2008 15
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 5/24/2008 ARGMAS 2008 16
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. 5/24/2008 ARGMAS 2008 17
Lemma 1: Reinstatement → DW2 � If � 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 5/24/2008 ARGMAS 2008 18
Lemma 2: DW2 → Reinstatement � If � 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 5/24/2008 ARGMAS 2008 19
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 (K * p )* q = K * q � Cycle-free frameworks are not very interesting � Our problem: how to generalize this result? 5/24/2008 ARGMAS 2008 20
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: � ∀ α , β ∈ A , whether α attacks β depends only on α and β , not on the other arguments � (Compare, e.g., the language independence principle of Baroni and Giacomin) 5/24/2008 ARGMAS 2008 21
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.) 5/24/2008 ARGMAS 2008 22
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) 5/24/2008 ARGMAS 2008 23
Other Formal Foundations? � Success postulate 5/24/2008 ARGMAS 2008 24
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