Dung-style Argumentation and AGM-style Belief Revision Guido - - PowerPoint PPT Presentation

Dung-style Argumentation and AGM-style Belief Revision

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

5/24/2008 ARGMAS 2008 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.

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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

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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”

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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

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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

• ptimality is no longer tenable with respect to the dominance relation and

new concepts have to be developed to take over its function of singling

• ut 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

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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

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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.

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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).''

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A Formal Definition

Let U be the universe of arguments. An acceptance function ES:U x 2UxU ->22U is

• 1. a partial function which is defined for each

argumentation framework h A, → i with finite A ⊆ U and → ⊆ AxA, and

• 2. which maps an argumentation framework

hA,→i to sets of subsets of A: ES (hA,→i)⊆ 2A (Do we need A in argumentation framework?)

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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.

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Reinstatement, [BG07, definition 15]

A semantics S satisfies the reinstatement criterion if ∀ AF ∈ DS, ∀ E ∈ ES(AF) it holds that (∀ β ∈ parAF(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

• f α.”

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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 ∈ parAF(α) ∃γ ∈ S \ {α}: γ → β & sd(γ,S \ {α}) A semantics S satisfies the weak reinstatement criterion if ∀ AF ∈ DS, ∀ E ∈ ES(AF) it holds that sd(α,E) ⇒ α ∈ E

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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

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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

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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

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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.

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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

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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

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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?

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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)

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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.)

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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)

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Other Formal Foundations?

Success postulate

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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

• f 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?):

• ne can attack the fact that the father is more reliable than the brother

(maybe the kid heard mom said so while quarreling with father)

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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

• f 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

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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?