A Logical Theory about Dynamics in Abstract
Argumentation
1
Richard Booth
aSouhila Kaci
bTjitze Rienstra
b,cLeon van der Torre
ca
SnT, University of Luxembourg
bLIRMM, University of Montpellier
c
CSC, University of Luxembourg
1
Introduction
In Dung-style argumentation [3] an argumentation framework (AF for short) is usually assumed to be static. There are, however, many scenarios where argumentation is a dynamic process: Agents may learn that an AF must have a certain outcome and may learn about new arguments/attacks. We address these issues by answering the following research questions: How can we model an agent’s belief about the outcome of an AF?and How can we characterize the effects of an agent learning that the AF should have a certain outcome, or learning about new arguments/attacks?
The basis of our approach is a logical labeling language, interpreted by labelings that assign to each argument a label indicating whether it is accepted, rejected or undecided [2]. Formulas in this language are statements about the acceptance of the arguments of an AF. This allows us to reason about the outcome of an AF in terms of beliefs, rather than extensions or labelings.
2
Argumentation and belief states
An argumentation framework [3] (AF for short) is a pair (A, R) where A is a finite set of arguments and R ⊆ A × A is an attack relation. An argument labeling for an AF is a function assigning to each argument in A a label I, O or U (i.e., in, out or undecided) indicating whether the argument is respectively accepted, rejected or neither [2]. Complete labelings represent fully rational points of view and set an argument I iff all its attackers are O, and O iff it has at least one attacker labeled I. Conflict-free labelings enforce the weaker requirement that every argument that either attacks or is attacked by an argument labeled I must be O.
We reason about the outcome of an AF by using a logical labeling language where formulas assign labels to arguments or are boolean combinations of such assignments. The language, given an AF F = (A, R), is denoted by LF and is generated by the BNF φ := inx | outx | ux | ¬φ | φ ∨ φ | > | ⊥ where x ranges
over A. The formulas of LF are evaluated true or false w.r.t. any labelling of F in the natural way, with
[φ] denoting the set of labelings that make φ true and |= denoting the corresponding model-based inference relation. An agent’s belief state is then made up of an AF plus a formula constraining the AF’s outcome. Definition 1. A belief state is a pair S = (F, K), where F is an AF and K ∈ LF the agent’sconstraint. We
defineBel(S) by [Bel(S)] = {L ∈ [K] | L is a complete labeling of F }. We say that the agent believes ψ iffBel(S) |= ψ and that S is coherent iff Bel(S) 6|= ⊥.
1This paper originally appears in the proceedings of the 7th International Conference on Scalable Uncertainty Management
We define two expansion operators for belief states: one that strengthens the agent’s constraint and one that expands the AF. The constraint expansion operator takes as input a belief state and a formula φ representing a constraint that is to be incorporated into the new belief state.
Definition 2. Let F be an AF, S = (F, K) a belief state and φ ∈ LF. Theconstraint expansion of S by φ,
denotedS ⊕ φ is defined by S ⊕ φ = (F, K ∧ φ).
As to expanding the AF, we make two assumptions (cf. normal expansions of [1]): (1) elements can be added to an AF but not removed, and (2) no new attacks can be added between arguments already present. Definition 3. Let F = (A, R) be an AF. An AF update for F is a pair F∗= (A∗, R∗) where A∗is a set of
added arguments, such that A ∩ A∗= ∅ and R∗⊆ ((A ∪ A∗) × (A ∪ A∗)) \ (A × A) a set of added attacks. LetS = (F, K) be a belief state and F∗ = (A∗, R∗) an AF update for F . The AF expansion of S by F∗, denoted byS ⊗ F∗is defined byS ⊗ F∗= ((A ∪ A∗, R ∪ R∗), K).
3
Two ways of handling incoherence
A belief state may be incoherent (Bel(S) |= ⊥). How can an agent reason in such a state? Our first main result says that coherence can be restored via AF expansion, provided the constraint of S is conflict-free. Theorem 1. Let (F, K) be an incoherent belief state where K is conflict-free. There exists a coherence-restoring AF update (i.e., an Af updateF∗forF such that (F, K) ⊗ F∗is coherent).
A second way to handle incoherence is to use fallback beliefs. Here we assign, to each AF F , a total pre-order Fover the conflict-free labelings of F with L F L0meaning that L is at least as rational as L0.
The minimal labelings in F are exactly the complete labelings of F . The fallback beliefs in state (F, K)
are then given by [Bel∗(F, K)] = minF{L ∈ [K] | L is a conflict-free labeling of F }.
How can we define F? We can do this by regarding conflict-free labelings that require less impact to be
turned into a complete labeling to be more rational. This impact can be equated with the sets of arguments illegally outor undecided [2]. Let us denote the set of arguments illegally out (resp. undecided) in a conflict-free labeling L, i.e. x ∈ A, L(x) = O (resp. L(x) = U ) without an y ∈ A s.t. (y, x) ∈ R and L(y) = I (resp. L(y) = U ), by ZO
F(L) (resp. ZFU(L)). We use the cardinality of ZFO(L) and ZFU(L) as the criterion to
define Fand define a faithful assignment as follows: L F L0iff |ZFO(L)∪ZFU(L)| ≤ |ZFO(L0)∪ZFU(L0)|.
The following theorem motivates fallback belief defined via the faithful assignment just defined. It states that fallback belief is the belief that would hold after a coherence restoring AF update that is minimal w.r.t. the number of existing arguments attacked by new arguments. For this we use the notion of the attack degree δ(A,R)((A∗, R∗)) of an AF update (A∗, R∗), defined by δ(A,R)((A∗, R∗)) =| {x ∈ A | ∃y ∈ A∗, (y, x) ∈
R∗} |.
Theorem 2. If S is an incoherent belief state and F1∗minimal coherence restoring(i.e., S ⊗ F1∗is coherent and there is noF2∗such thatS ⊗ F2∗is coherent andδF(F2∗) < δF(F1∗)) then Bel(S ⊗ F1∗) |= Bel∗(S).
As well as all this, in the full paper we provide an answer-set program for computing the fallback belief, i.e., for determining whether or not some formula is a fallback belief in a particular belief state. We also consider additional semantics.
References
[1] Ringo Baumann and Gerhard Brewka. Expanding argumentation frameworks: Enforcing and mono-tonicity results. In COMMA, pages 75–86, 2010.
[2] Martin Caminada. On the issue of reinstatement in argumentation. In JELIA, pages 111–123, 2006. [3] Phan Minh Dung. On the acceptability of arguments and its fundamental role in nonmonotonic
