Dung's Argumentation Framework (AF) has been extended in several directions, including the possibility of representing uncertainty about the existence of arguments and attacks. In this regard, two main proposals have been introduced in the literature: Probabilistic Argumentation Framework (PrAF) and Incomplete Argumentation Framework (iAF). PrAF is an extension of AF with probability theory, thus representing quantified uncertainty. In contrast, iAF represents unquantified uncertainty, that is it can be seen as a special case where we only know that some elements (arguments or attacks) are uncertain. We discuss the problem of computing the probability that a given argument is accepted in PrAF, which is based on the concept of probabilistic explanation for any given (probabilistic) extension [1]. Our approach can be extended to iAF, as it can be viewed as a special case of PrAF where uncertain elements have associated a probability equal to 1/2.
The abstract Argumentation Framework (AF) is a simple, yet powerful formalism for modeling disputes
between two or more agents [
Recently, there has been an increasing interest in extending argumentation frameworks to manage
uncertain information. This has been carried out by either considering quantified uncertainty about
the existence of arguments and attacks, thus combining formal argumentation with probability theory,
or considering unquantified uncertainty by explicitly denoting the elements (arguments and attacks)
which are uncertain. In fact, Probabilistic Argumentation [
One of the most popular approaches based on probability theory for modeling the uncertainty is the so called constellations approach [8, 9, 10, 11, 12], where alternative scenarios, called possible meat white w1 w3 white worlds, are associated with probabilities. In particular, in a Probabilistic Argumentation Framework (PrAF) [12, 13, 14, 15, 16, 17, 18] a probability distribution function (PDF) on the set of possible worlds is entailed by the probabilities that are associated with arguments and attacks.
Example 1. Consider a PrAF Δ = ⟨{fish, meat, white, red}, {(fish, meat), (meat, fish), (meat, white), (white, red), (red, white)}, {fish/0.6, white/ 0.8}⟩, whose corresponding graph is shown in Figure 1, where nodes and edges represent arguments and attacks, respectively, and probabilities diferent from 1 are specified nearby them. For the sake of brevity, we do not specify the probabilities of certain elements in Δ (all the elements diferent from fish and white have probability 1). Intuitively, Δ describes what a person is going to have for lunch as follows. They will have either fish or meat, and will drink either white wine or red wine. However, if they will have meat, then they will not drink white wine. Furthermore, the probability that fish is available is 0.6, whereas the probability that white wine is available is 0.8. 2
Intuitively, PrAF is a combination of two powerful approaches to reasoning and decision making: probabilistic reasoning and abstract argumentation. Probabilities are assigned to arguments and attacks to indicate their degree of uncertainty. One of the benefits of probabilistic abstract argumentation is its ability to handle quantified uncertainty in the analysis. In fact, PrAF can help to model and analyze situations where there is uncertainty by capturing both the relationships between arguments and the uncertainty degrees of arguments and attacks.
Several argumentation semantics—e.g. grounded (gr), complete (co), preferred (pr), stable (st), and semi-stable (sst)—have been defined for AFs, leading to the characterization of σ -extensions, which intuitively consist of the sets of arguments that can be collectively accepted under semantics σ . Consider for instance the deterministic version of the PrAF in Example 1, obtained by assuming that all arguments are certain (i.e. they have probability 1). Considering the preferred semantics, the pr-extensions are E1 = {fish, white}, E2 = {fish, red}, and E3 = {meat, red}.
The semantics of a PrAF is given by considering all possible worlds (i.e. AFs) obtained by removing consistent subsets of the probabilistic elements. Here, for consistent subset we mean any subset of probabilistic elements (arguments and attacks) whose deletion from the initial framework results in an AF (for instance we cannot delete an argument without also deleting the attacks towards or from that argument). Every possible world has associated a probability value derived from the probabilities of the elements that have been kept or removed. Moreover, every possible world admits a set of σ -extensions. The probability of a possible world w is computed by multiplying the probabilities of the elements occurring in w and the complement to 1 of the probabilities of the elements not occurring in w. Example 2. Continuing with Example 1, the possible worlds of Δ are shown in Figure 2. The probability of a possible world wi is obtained by multiplying the probabilities P (a) of each argument a occurring in wi and the probabilities (1 − P (b)) of every argument b not occurring in wi. Since P (fish) = 0.6, P (white) = 0.8, and P (meet) = P (red) = 1, the probabilities of w1, w2, w3, and w4 are 0.6 · 1 · 0, 8 · 1 = 0.48, 0.6 · 1 · 0.2 · 1 = 0.12, 0.4 · 1 · 0.8 · 1 = 0.32, and 0.4 · 1 · 0.2 · 1 = 0.08. Since w1 coincides with the deterministic version of Δ, its pr-extensions are E1, E2, and E3 given earlier. The pr-extensions of w2 are E2 and E3, while w3 and w4 admit only E3 as their preferred extension. 2
Interesting problems recently investigated in the context of probabilistic argumentation are probabilistic credulous acceptance (PrCA) and probabilistic skeptical acceptance (PrSA) [19, 15]. In particular, given a PrAF Δ whose set of arguments is A, a goal argument g ∈ A and a semantics σ , PrCA is the problem of computing the probability P rCAσΔ(g) that the goal g is credulously accepted, that is, there is a possible world w of Δ such that g belongs to a σ -extension of w. Moreover, PrSA is the problem of computing the probability P rSAσΔ(g) that the goal g is skeptically accepted, that is, g is credulously accepted and belongs to all σ -extensions of w.
However, the answer to these problems does not reflect our intuition of probability that a goal argument is accepted under a given semantics. For instance, considering the PrAF Δ of Figure 1, the probability that meat is credulously accepted under preferred semantics is 1, whereas the probability that meat is skeptically accepted under preferred semantics is 0.4. However, the fact that P rCApΔr(meat) = 1 does not mean that the person in our example will surely have meat in any scenario (i.e. possible world). In fact, even if meat belongs to at least one preferred extension of every world of Δ, we expect that the probability of acceptance of meat should be lower than 1. Indeed, in any possible world, the presence of multiple extensions is an additional source of uncertainty that should be taken into account.
To better grasp the issue behind the probability of credulous acceptance, consider the following AF (where all elements are certain): Λ = ⟨{fish, meat}, {(fish, meat), (meat, fish)}⟩ saying that fish and meat are mutually exclusive. Again, the probability that a person will have meat is 1, under probabilistic credulous acceptance, when considering the preferred semantics, whereas we believe that the expected answer should be 0.5. Moreover, if we consider AF w1 of Example 2 (that can be obtained from Λ by adding arguments white and red and attacks (white, red), (red, white) and (meat, white)) we expect that the probability of having meat does not change.
With the aim of providing more intuitive answers for probabilistic acceptance, a new problem called
Probabilistic Acceptance (denoted as PrA, or PrA[σ ] when considering a given semantics σ ) has beed
investigated [
In general, in abstract argumentation an explanation for an extension E can be viewed as a (possibly minimal) subset S ⊆ E such that, by assuming that the elements in S are acceptable, it turns out that all elements in E \ S are “univocally” determined as acceptable (w.r.t. the underlying semantics). For instance, considering AF w1 of Example 2, for the preferred extension E = {meat, red}, the set S1 = {meat} is an explanation for E, whereas the set S2 = {red} is not. In our perspective, explanations are sequences of “choices” to be made to justify how an extension is obtained and they provide a tool to assign probabilities to extensions. Integrating explanations in argumentation systems is important for enhancing the argumentation and persuasion capabilities of software agents [21, 22, 23]. For this reasons, several researchers have explored how to deal with explanations in formal argumentation [24, 25, 26].
An instantiation of PrA[σ ] where the PDF over the set of σ -extensions of a world relies on the concept of explanation is called Explanation-based Probabilistic Acceptance problem, and denoted by PrEA (and PrEA[σ ] for a specific semantics σ ). Intuitively, an explanation for an σ -extension E is a sequence of arguments occurring in E that “justify” E. Every explanation is associated with a probability entailed by the possible choices that can be made when building it. These choices must be consistent with an ordering entailed by the strongly connected components of the given AF, and they are used to guide the construction of an extension. The sum of the probabilities of the explanations for an extension E gives the probability of E. Thus, we still assign to each possible world w of Δ a probability as in the standard way, but in addition propose to distinguish among extensions of a given world w by associating with them a probability based on explanations.
Example 3. Continuing with Example 1, take for instance the possible world w1 having probability
0.48. As shown in Example 2, w1 has three pr-extensions, namely E1, E2 and E3. As shown in [
The definition of Explanation-based Probabilistic Acceptance has been also carried out to another argumentation framework extending AF that has received an increasing attention in the last years and is tightly related to PrAF, that is, to incomplete AF (iAF) [27, 28]. This follows from the fact that iAF can be viewed as a special case of PrAF where uncertain elements have associated a probability equal to 1/2.
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