A labelling framework for probabilistic argumentation

Abstract

The combination of argumentation and probability paves the way to new accounts of qualitative and quantitative uncertainty, thereby offering new theoretical and applicative opportunities. Due to a variety of interests, probabilistic argumentation is approached in the literature with different frameworks, pertaining to structured and abstract argumentation, and with respect to diverse types of uncertainty, in particular the uncertainty on the credibility of the premises, the uncertainty about which arguments to consider, and the uncertainty on the acceptance status of arguments or statements. Towards a general framework for probabilistic argumentation, we investigate a labelling-oriented framework encompassing a basic setting for rule-based argumentation and its (semi-) abstract account, along with diverse types of uncertainty. Our framework provides a systematic treatment of various kinds of uncertainty and of their relationships and allows us to back or question assertions from the literature.Definition 2.4 (Conflict relation) Given a set of literals Φ, a conflict relation 'conf lict' is a binary relation over Φ, i.e. conf lict ⊆ Φ × Φ, such that for any ϕ 1 , ϕ 2 ∈ Φ, if ϕ 1 and ϕ 2 are complementary, i.e. ϕ 1 = −ϕ 2 , then ϕ 1 and ϕ 2 are in conflict, i.e. (ϕ 1 , ϕ 2 ) ∈ conf lict.The conflict relation may be further specified. For example, the relation may be refined with asymmetric and symmetric conflicts to deal with contrary or contradictory literals as in [40]. However, treating in detail such sophistications is not necessary for our purposes.When two rules have heads in conflict, one rule may prevail over another one. Informally, a rule superiority relation r 1 ≻ r 2 states that the rule r 1 prevails over the rule r 2 .Definition 2.5 (Superiority relation) Let Rules be a set of rules. A superiority relation ≻ is a binary relation over Rules, i.e. ≻⊆ Rules × Rules, with r ≻ r ′ denoting that r is superior to r ′ .The superiority relation may enjoy some particular properties. For example, it may be antireflexive and antisymmetric, so that for a rule r it does not hold that r ≻ r and for two distinct rules r and r ′ , we cannot have both r ≻ r ′ and r ′ ≻ r. However, as for the conflict relation, treating in detail such sophistications is not necessary for our purposes.From a set of rules, a conflict relation and a superiority relation, we can define defeasible theories, cf. [22,34].Definition 2.6 (Defeasible theory) A defeasible theory is a tuple Rules, conf lict, ≻ where Rules is a set of rules, conf lict is a conflict relation over literals, and ≻ is a superiority relation over rules.

Neuro-Symbolic Probabilistic Argumentation Machines

“…We adopt the probabilistic semi-abstract argumentation framework taken from (Riveret et al 2018). First, a semiabstract argumentation framework is put forward, then the use of 'maxconsistent' labellings is proposed for our purposes, and the probabilistic development is exposed.…”

“…And indeed, the combination of formal argumentation and probability theory has received increasing attention in recent years, see e.g. (Verheij et al 2015;Hunter and Thimm 2017;Riveret et al 2018). Challenges in probabilistic argumentation include (i) reasoning upon the probability of argument and statement statuses (and with no particular assumptions on probabilistic dependencies), and (ii) learning the probability distribution of statuses from examples of argument or statement statuses.…”

“…To address learning and reasoning challenges in probabilistic argumentation, as well as explainability issues in Boltzmann machines (RBMs), we propose a novel probabilistic argumentation system called neurosymbolic argumentation machine (NSAM). We adopt probabilistic labellings in argumentation (Riveret et al 2018), and integrate the approach with the probabilistic neural network model of RBMs to learn and reason upon distributions of argument statuses. To do so, we develop the idea of confidence rules (Tran and d'Avila Garcez 2018): prior argumentation knowledge is captured as an argumentation graph, and constraints on labellings of the graph are expressed into weighted strict disjunctive normal forms (the confidence rules) which are then incorporated into the structure of an RBM to constitute a NSAM.…”

Polynomial-Time Updates of Epistemic States in a Fragment of Probabilistic Epistemic Argumentation

“…There is a large variety of other probabilistic argumentation approaches [9,26,37,14,7,28,41,24,38,42,39], which basically differ in the level of detail (e.g., structured or abstract argumentation), in the way how uncertainty is introduced (e.g. possible worlds correspond to argument interpretations or the graph structure) and in the nature of uncertainty (e.g., uncertainty about the acceptance state or uncertainty about the nature of a relation between arguments).…”

A Bayesian Argumentation Framework for Distributed Fault Diagnosis in Telecommunication Networks

“…In contrast, other recent works explore the application of probabilistic argumentation frameworks [44,45,46] for modelling uncertain logical arguments [47] extracting them from a Bayesian networks [48,49]. With respect to the application of probabilistic reasoning in argumentation processes, we can find in the literature many works from a mathematical point of view, such as the application of a probabilistic approach to modelling uncertain logical arguments [47,50], direct translations from Bayesian approach to argumentation [51], to a philosophical point of view, such as a Bayesian perspective of testimonies and arguments [52,53], or to a legal point of view, such as an argumentation supporting tool [54]. Following these approaches, we propose to extend the previous diagnosis system [25], which applied Bayesian reasoning to infer fault root causes with an argumentative capability.…”