Dispute Trees as Explanations in Quantitative (Bipolar) Argumentation
Kristijonas Čyras1, Timotheus Kampik2, Qingtao Weng1,
1: Ericsson; 2: Umeå University
#### Idea
* Dispute trees: given a target argument, determine tree of *proponents* and *opponents*
* We introduce:
* Dispute trees for quantitative (bipolar) argumentation
* Functions for determining an argument's *contribution* to the final strength of another argument
* Desirable properties of contribution functions and argumentation semantics that guarantee the existence of (sufficient) DTs for argument strength explainability
#### How to Construct Dispute Trees
* According to the topological order of the graph, we start with the argument we are interested in
* Potentially, *contribution* functions can be used to check whether an argument is a proponent or opponent of another argument
Dung, Kowalski, Toni. *Dialectic Proof Procedures for Assumption-Based, Admissible Argumentation*. 2006.
Delobelle & Villata. *Interpretability of Gradual Semantics in Abstract Argumentation*. 2019.
#### Quantitative Bipolar Argumentation I
* Interval $\mathbb{I}$, typically real interval, e.g. $\mathbb{I} \in [0, 1]$
* A Quantitative Bipolar Argumentation Graph (QBAG) is a quadruple
$(Args, \tau, Att, Supp)$ consisting of:
* a set of arguments
$Args$,
* an attack relation
$Att \subseteq Args \times Args$, a support relation
$Supp \subseteq Args \times Args$
* a total function
$\tau : Args \to \mathbb{I}$
that assigns the initial strength $\tau(x)$ to every $x \in Args$.
Potyka. *Extending Modular Semantics for Bipolar Weighted Argumentation*. 2019.
Baroni, Rago, Toni. *From Fine-Grained Properties to Broad Principles for Gradual Argumentation: A Principled Spectrum*. 2019.
#### Quantitative Bipolar Argumentation II
* A **gradual semantics** $\sigma$ defines for $G = (Args, \tau, Att, Supp)$ a (possibly partial) **strength function** $\sigma_{G} : Args \to \mathbb{I} \cup \{ \perp \}$ that assigns the **final strength** $\sigma_{G}(x)$ to each $x \in Args$,
where $\perp$ is a reserved symbol meaning `undefined`.
* Given a QBAG $G = (Args, \tau, Att, Supp)$ and a set of arguments $A \subseteq Args$, we define
the **restriction of $G$ to $A$** as the QBAG $G\downarrow_{A} := \left(A, \tau \cap (A \times \mathbb{I}), Att \cap (A \times A), Supp \cap (A \times A) \right)$.
Potyka. *Extending Modular Semantics for Bipolar Weighted Argumentation*. 2019.
Baroni, Rago, Toni. *From Fine-Grained Properties to Broad Principles for Gradual Argumentation: A Principled Spectrum*. 2019.
#### Contribution Functions I
Given $G = (Args, \tau, Att, Supp)$, $a, x \in Args$, $\sigma$
a contribution function $ctrb_G: Args \times Args \rightarrow \mathbb{I}$
quantifies the contribution of an argument $x$ to the final strength of $a$.
#### Contribution Functions II
Counterfactual contribution -- effect of absence:
$ctrb_G(a,x) = \sigma_G(a) - \sigma_{G\downarrow_{Args \setminus \{x\}}}(a)$
#### Contribution Functions III
Intrinsic counterfactual contribution:
$ctrb_G(a, x) =$
$\sigma_{(Args, \tau, Att \setminus Args \times \\{x\\}), Supp \setminus Args \times \\{x\\})}(a)$
$ - $
$\sigma_{G\downarrow_{Args \setminus \{x\}}}(a)$
#### Contribution Functions IV
##### Shapley value:
$ctrb_G(a, x)=$
$\sum_{X \subseteq Args \setminus \\{ x, a \\}} \frac{|X|! \cdot (|Args \setminus \\{a\\}|-|X|-1)!}{|Args \setminus \\{a\\}|!}\left(\sigma_{G\downarrow_{Args \setminus X}}(a) - \sigma_{G\downarrow_{Args \setminus (X \cup \\{ x \\})}}(a)\right)$
#### Contribution Functions V
Derivative -- effect of initial strength change:
$ctrb_G(a, x)=$
$\left.\frac{\partial f}{\partial \tau(x)} \right|_{(\tau(a), \tau(b_1), \ldots, \tau(x), \ldots, \tau(b_K))}$,
where we traverse $G$ in its topological order and write $\sigma(a) = f(\tau(a), \tau(b_1), \ldots, \tau(b_K))$, where $\{ b_1, \ldots, b_K \}$ are all the arguments from which there is a path to $a$ in $G$.
#### Contribution Functions - Principles wrt. Semantics $\sigma$
* **Contribution Existence:** for every QBAG $G = (Args, \tau, Att, Supp)$, for every $a \in Args$ it holds that whenever $\sigma_G(a) \neq \tau(a)$, then $\exists x \in Args$ s.t. $ctrb(a, x) \neq 0$.
* **Contribution Directionality:** for every QBAG $G = (Args, \tau, Att, Supp)$, for each $a, x \in Args$ it holds that whenever $a$ is not reachable from $x$, then $ctrb(a, x) = 0$.
#### Quantiative (Biploar) Dispute Trees (QDTs)
A **quantitative dispute tree** (QDT) for the topic argument $a \in Args$ in a QBAG $G = (Args, \tau, Att, Supp)$ is an in-tree $T$ s.t.:
1. Every node of $T$ is of the form $L: x$, with $L \in \{P, O\}$ and $x \in Args$: the node holds argument named $x$ and is:
* either a proponent ($P$) node, denoted $P: x$, if $ctrb(x) \geqslant 0$,
* or an opponent ($O$) node, denoted $O: x$, if $ctrb(x) \< 0$;
2. The root of $T$ is $P: a$ -- a $P$ node holding $a$.
#### QDT Sufficiency
A QDT $T$ is **sufficient** for $a$ iff **for any** super-tree $T'$ of $T$ satisfying the QDT conditions it holds that:
* $\sigma_{T'}(a) \geqslant \tau(a)$, in case $\sigma_{G}(a) \geqslant \tau(a)$;
* $\sigma_{T'}(a) \< \tau(a)$, in case $\sigma_{G}(a) < \tau(a)$.
QDT Sufficiency - Example
#### Guarantees I - Directional Connectedness Principle
A gradual semantics $\sigma$ satisfies the **directional connectedness principle** iff for every QBAG $G = (Args, \tau, Att, Supp)$, for any $x \in Args$ it holds that $\sigma_G(x) = \sigma_{G \downarrow_{Args'}}(x)$, where $Args'$ entails all arguments in $Args$ that can reach $x$ in $G$.
#### Guarantees II
* For every acyclic QBAG, there is a QDT.
* Given the three principles (contribution existence, contribution directionality, directional connectedness principle), for every acyclic QBAG, for every argument, there is a sufficient QDT.
#### Future Work
* Cover cyclic QBAGs
* What are the principles that distinguish between our different contribution functions?
* Use dispute trees to explain why an argument falls below a certain threshold
* Use dispute trees to compare the final strengths of several arguments
* QBAG dispute trees as generalizations of 'abstract' dispute trees
#### Thank you! Questions?
