Reasoning in Abstract Argumentation is the activity of understanding subsets of arguments that resolve conflicts or can be used as a justification for a thesis. Several semantics have been proposed so far to identify subsets of arguments showing properties based on self-consistency or defence against attackers. Real-world scenarios and peculiarities of many problems gave rise to the introduction of more complex frameworks that also specify degrees of certainty of arguments and attacks. However, semantics barely take into account the fuzziness of these “beliefs” properly. In this work, we propose a strategy to compute arguments' beliefs based on the initial degrees of certainty of arguments and propose a new semantics, most-probable, that, starting from a target argument we aim to justify, captures the set of admissible arguments containing the target argument which maximizes the minimum belief of the whole set. The introduction of this new semantics can facilitate the understanding of the consequences of supporting a claim, neglecting those that may result in low belief.
The field of argumentation can be traced back to what the philosopher Pollock in [
Despite the simplicity of the framework, there is great potential and much demand from the world,
especially in the field of law where it goes without saying that for each case there are conflicting
facts that need to be sorted out in order to find the most plausible set of claims. An overview of the
interconnection between the two fields is provided by Prakken et al. [
The representation of conflicting arguments is not an end in itself. Mining arguments represents the possibility of studying whether claims are somehow supported. To this extent, claims are proofs themselves, used as justification for other claims. In principle, there cannot be a general approach to satisfy the needs of one to shed light on the facts, but it is strictly dependent on how we consider a claim to be supported. For this reason, diferent semantics are available, some more specific than others, that can capture arguments with more specific properties in the abstract framework. 8th Workshop on Advances in Argumentation in Artificial Intelligence, November 27, 2024, Bolzano, Italy ∗ Corresponding author. †These authors contributed equally. $ davide.dipierro@uniba.it (D. Di Pierro); stefano.ferilli@uniba.it (S. Ferilli) 0000-0002-8081-3292 (D. Di Pierro); 0000-0003-1118-0601 (S. Ferilli)
© 2024 Copyright © 2024 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
In order to better describe real-world scenarios, a certain degree of uncertainty should be introduced
in the matter of discourse. Even though Dung’s formalization is still the most used, extensions including
probabilities, beliefs or fuzziness are available. In the realm of model representation, many techniques are
borrowed from the Knowledge Representation and Reasoning field. To this extent, logic programming
frameworks provide comprehensive representation tools and strategies to store, manage and compute
argumentation semantics. They are particularly suited for this scope given their interpretable nature,
symbolic representation and textual (rather than numeric) information. These frameworks have already
been studied to support uncertainty with plenty of solutions available for dealing with probabilities,
fuzziness or degree of beliefs. In some languages, nonmonotonic reasoning is not inherently available
but techniques are possible to add or retract information. Prolog [
In this work, we propose a strategy to deal with a degree of beliefs and a new semantics focused on a target argument. Specifically, we suppose a given knowledge of arguments is available with previous beliefs (regardless of the others). After arguments are put into correlation with the attacks, the likelihood of arguments should vary, in correlation with the beliefs of the attackers. After updating beliefs, the most-probable extension will be computed. It can be considered as a variant of a very well-known one (shortly to be described) with the introduction of an optimisation filter to rule out arguments that reduce the overall likelihood of the set of accepted facts. The overall solution is provided with full ifrst-order logic frameworks with (mainly) Prolog for the representation of arguments and attacks, and ASP for the most-probable extension given its optimization functions. We see a natural use in the field of law when trying to identify which are all the facts consistent with a claim (e.g. “A is guilty”) with the highest possible belief.
This manuscript is structured as follows: Section 2 introduces theory on argumentation and semantics, Section 3 specifies the proposed strategy and semantics, Section 4 provides technical details about implementation of the proposed setting, Section 5 lists related works in the field, and finally Section 6 makes evidence of consequences and concludes the manuscript.
Although this is not intended to be a comprehensive review of the literature in the argumentation field, we provide basic definitions and concepts for this work. Definitions mainly have roots from the first Dung’s formalization and the proposed basic semantics.
Definition 1 An Argumentation Framework (AF) is a pair ⟨Args, Att⟩ where Args is a (finite) set of arguments and Att ⊆ Args × Args the attack relationship. If ⟨A, B⟩ ∈ Att we say that argument A attacks argument B (or that A is an attacker of B). For each A ∈ Args, we denote as A− = { B ∈ Args : ⟨B, A⟩ ∈ Att } the set of all attackers of A, and as A+ = { B ∈ Args : ⟨A, B⟩ ∈ Att } the set of all attacked by A. In this scope, we neglect self-attacking arguments, so A ∈ Args =⇒ A ̸∈ A− (A+). The concept of attack can also be extended to a set of arguments. Given S ⊆ Args, A ∈ Args attacks S if exists B ∈ S such that A attacks B. Conversely, a set S ⊆ Args attacks A ∈ Args if ∃B ∈ S such that B attacks A. Figure 1 shows an example of an argumentation graph.
A
B
C
D
Definition 2 Let an AF G = ⟨Args, Att⟩, a Probabilistic Argumentation Framework (PAF) ⟨Args, Att, P Args, P Att⟩ is a 4-tuple in which Args and Att are defined as above, P Args : Args → ]0, 1] and P Att : Att →]0, 1] functions respectively indicating the likelihood of arguments and attacks.
It is often the case that the terms “probability” and “belief” are confused. They both represent a
degree of certainty but have diferent meanings and implications, as indicated by Fagin et al. [
0.7
A B C D P (A) = 0.8 P (B) = 0.6 P (C) = 0.70.4P (D) = 0.5 0.6
Slightly modifying Definition 1, we can generalise the probabilistic framework with a generic weighted
framework. From Bistarelli et al. [
Definition 3 Let an AF G⟨Args, Att⟩, a Weighted Argumentation Framework ⟨Args, Att, W Att⟩ is a triple in which Args and Att are defined as above, and W Att : Att → R> a function indicating the weight of attacks.
Diferently from probabilities, the meaning of weights does not reflect how probable is the attack, but how much the arguments attack. Figure 3 shows an argumentation graph with weights attached to attacks.
A 7
B 4
C 2 2
D
The Argumentation Framework and its variants can be enriched with the support relationship, the opposite of the attack. In some contexts, it is advantageous to add support relationships, but these considerations are beyond the scope of this work.
Given an AF, we are interested in collecting sets of arguments that can be generally considered acceptable. The condition of being acceptable depends on the semantics.
Definition 4 Let an AF G = ⟨Args, Att⟩ be an AF , an extension-based semantics S associates AF with a subset of 2Args, denoted by ES(AF ). Every subset of 2Args can be considered a distinct semantics (even ∅), but only some of them are convenient for acceptability. The basic condition for a set of arguments to be generally acceptable is to show the conflict-free condition; Definition 6 Let an AF G = ⟨Args, Att⟩ be an AF , a set of arguments S ⊆ Args is conflict-free if ̸ ∃ ⟨A, B⟩ ∈ S such that ⟨A, B⟩ ∈ Att. The set of all conflict-free extensions is indicated as cf(G).
Referring to the example in Figure 1, in Figure 4 two ways of grouping arguments into conflict-free sets are shown.
This semantics guarantees to obtain groups of arguments at least consistent with each other. Many semantics have a hierarchical structure, from the least restrictive to the most. The conflict-free is the minimal result for acceptability. More interesting semantics start from the conflict-free one. Note that ∅ is always conflict-free, and, with the assumption of non-self-attacking arguments, singleton sets are conflict-free as well, but this result is not practically interesting.
For distinguishing some semantics it is necessary to define the concept of defense. Definition 5 Let an AF G = ⟨Args, Att⟩, S ⊆ Args , and α Args ⟨γ, α ⟩ ∈ Att =⇒ ∃B ∈ S such that ⟨B, γ ⟩ ∈ Att. ∈ Args, S defends α if ∀γ ∈ Definition 7 Let an AF G = ⟨Args, Att⟩ be an AF , a set of arguments S ⊆ Args is admissible if S is conflict-free and ∀γ ∈ Args γ attacks S =⇒ S attacks γ . In other words, S defends every A ∈ S. When this happens, we say that S is able to defend itself. The set of all admissible extensions is indicated as adm(G). In Figure 4 three pairs of arguments are conflict-free: {A, C}, {B, D}, {A, D}, but {B, D} is not admissible since A attacks B but neither B nor D attacks A; hence, it cannot defend itself. Figure 5 shows the two possible pairs of arguments, white nodes cannot be part of any admissible set.
In general, we are not only interested in admissibility but also in including as many arguments as possible in order to make the discovery process more interesting.
Definition 8 Let an AF G = ⟨Args, Att⟩ be an AF , a set of arguments S ⊆ Args is a complete extension if S is admissible and ∀α ∈ Args S defends α =⇒ α ∈ S. The set of all complete extensions is indicated as comp(G).
The completeness of semantics acts as a means of spreading the defence relationship. Starting from a singleton, in order for the set to be complete, it must include all the arguments it defends, and then progressively add all the others defended by the ones already included in the set.
Having collected all complete extensions, we can collect all the arguments available in every complete extension by intersection. Here, we define grounded semantics.
Definition 9 Let an AF G = ⟨Args, Att⟩ be an AF , a set of arguments S ⊆ Args is a grounded extension if S is complete and ̸ ∃T ∈ comp(G) such that t ⊂ S. The grounded extension is the minimal set of arguments belonging to the complete extensions. It is unique and always exists. In general scenarios, it may be preferable to select extensions that guarantee as many arguments as possible. To this extent, we can define preferred extensions.
Definition 10 Let an AF G = ⟨Args, Att⟩ be an AF , a set of arguments S ⊆ Args is a preferred extension if S is admissible and ̸ ∃T ∈ adm(G) such that S ⊂ T . The set of all preferred extensions is indicated as pref(G).
Finally, when computing extensions, it happens that not all the arguments outside preferred extensions are attacked. These are the so-called “undecided” arguments. Stable extensions split the set of arguments between arguments in the extension and attacked arguments.
Definition 11 Let an AF G = ⟨Args, Att⟩ be an AF , a set of arguments S ⊆ Args is a stable extension if S is conflict-free and ∀α ∈ Args α ̸∈ S =⇒ S attacks α . The set of all stable extensions is indicated as stb(G).
Given the variety of argumentation models and semantics, we define here our settings for the work. As previously said, research on argumentation semantics under uncertainty is still an underrepresented line of research. We start from a small variety of the PAF defined above. Specifically, we suppose probabilities only on arguments. Probabilities come under the assumption of non-additivity, meaning that it is not impossible that Δ ̸|= α and Δ ̸|= ¬α . Attacks always trigger with the same “intensity”. Probabilities of arguments may come from several external sources. In general, statistical approaches come into play when providing an initial estimation of a phenomenon. In this setting, probabilities of arguments do not take into account correlation with other arguments. Albeit this assumption may not hold in real scenarios, it facilitates an initial approximation of probabilities, which would be extremely demanding otherwise since statistical information of events barely can take into account all surrounding conditions.
Formally, we can define our version of PAF, namely Simplified PAF (SPAF) .
Definition 12 Let an AF G = ⟨Args, Att⟩, a Simplified Probabilistic Argumentation Framework (SPAF) ⟨Args, Att, P Args⟩ is a triple in which Args and Att are defined as above, P Args : Args → ]0, 1], a function indicating the likelihood of arguments.
Figure 6 shows an example of SPAF.
A B C D
P (A) = 0.8 P (B) = 0.4 P (C) = 0.7 P (D) = 0.3
Note that probabilities of arguments cannot be compared with Kolmogorov Axioms [
Probabilities of arguments are “blind” with respect to other arguments and attacks. For this reason, we need to update the probabilities of arguments, based on attacks and the initial probability of arguments. As a general rule, we expect the updated probability inversely proportional to the probability of the attacker and the number of attackers.
The update should consider using a hyperparameter α ∈]0, 1] to establish the pace at which values change. The hyperparameter is an indicator of how much attacks should impact the acceptability of attacked arguments.
The proposed formula is
P ′(A) = P (A) ·
Y where P ′(A) represents the probability of A after being attacked by all its attackers. The formula presents nice properties but has also some limitations or some assumptions. Nice properties are: • multiple attacks influence exponentially the updated belief, for this reason, a low α may bring less severe changes. • the product 1 − α · P (γ ) is always between 0 and 1; hence, the updated value keeps its original interpretation. • the initial belief of an argument is an upper bound for the update since multiplications of factors between 0 and 1 only reduce the number.
• many weak (low P(γ )) attackers are less influential than few strong (high P( γ )) attackers. Some limitations are: • if an attacker has probability 1, only with α = 1 the attack has the maximum efect (the updated probability of the attacked is 0). • the function is nonlinear, meaning that small changes have large efects. This is mitigated by the fact that we are more interested in relative probabilities among arguments rather than absolute numbers. • it is assumed attacks are independent, which is in general false but neglected in many contexts given the complexity of the dependencies.
We report in Figure 7 two argumentation graphs, the first one after updating probabilities with α = 0.5 and the second with α = 0.9.
Note that the whole discussion with probabilities does not change if we deal with generic “degrees of belief”. We chose probability values given their statistical interpretation and the availability of statistical data about phenomena.
With this setting, the possibility of performing an iteration process over probabilities can be considered. In this case, the iterative process of probability update will be the following: where
P n(A) = (P (A)
if n = 0,
P n− 1(A) · U n(A) otherwise U i(A) = P i− 1(A) ·
1 − α · P i− 1(γ ).
Y
However, we are currently interested in relative proportions among probabilities. The iterative
process keeps proportions without adding further information, and it is not trivial to design a stopping
criterion either. For these reasons, we only make use of a single update step in this work. The proposed
formula resembles one of those proposed by Gabbay et al. [
Given this argumentation framework, we are ready to define the most-probable extension. Diferently from other semantics, we define this semantics starting from an argument we call “target”. We need first to specify how we consider the probability of a set. Given {a1, a2, ..., an} ⊆ Args, P ({a1, a2, ..., an}) = min{P (a1), P (a2), ..., P (an)}.
Definition 13 Let ⟨Args, Att, P Args⟩ G a Simplified Probabilistic Argumentation Framework and t ∈ Args, a set of arguments S ⊆ Args is a most-probable extension for t, indicated as S ∈ most-probt(G) if t ∈ S, S is admissible and ̸ ∃Y ⊂ S such that: • t ∈ Y • Y is admissible • P (Y ∖ {t}) > P (S ∖ {t}). and ̸ ∃Z ⊆
• S ⊂ Z • Z is admissible • P (Z ∖ {t}) = P (S ∖ {t}).
In other words, S is most-probable with respect to t if it is admissible, includes t, including any other argument will make the set non-admissible or reduce the minimum probability of the set, and removing any argument will not increase the minimum probability while keeping admissibility. By construction, the target argument t always belongs to most-probable sets, and its probability is not evaluated when computing the minimum, saving us from a paradox of certainty. Suppose P (t) ≃ 1, then it is likely that the singleton {t} is the only most-probable set since adding any other arguments would probably reduce the minimum probability.
Lemma 1
Proof by construction, suppose G a SPAF with Args = {a1, a2, ..., an, s0, s1, t} where {a1, a2, ..., an, t} are not attacked by anyone, and s0, s1 attack each other. Then, both {a1, a2, ..., an, s0, t} and {a1, a2, ..., an, s1, t} are most-probt(G).
A variant of this semantics can be easily considered if, instead of getting the most probable admissible set, we are interested only in admissible sets showing at least a probability of δ .
Definition 14 Let ⟨Args, Att, P Args⟩ G a Simplified Probabilistic Argumentation Framework , t ∈ Args and δ ∈ R>, a set of arguments S ⊆ Args is a δ -most-probable extension with respect to t if S is most-probable with respect to t and P (S ∖ {t}) >= δ Note that δ = 0 would reduce threshold most-probable to most-probable.
In most cases, the variant version may be more advantageous in order to not only optimise as much as possible the likelihood of extracted arguments but also to accept (resp. restrict) the result set. This extension should be used as a further step after most-probable. Given the probability of one of the most-probable, any threshold above would result in having the set, while a lower threshold provides additional information on other sets.
We report in Figure 8 a more complex example of SPAF (after probability updates) in which green nodes represent one of its most-probable extensions with respect to D.
P (A) = 0.8
A
P (F ) = 0.4
F B
Note that F belongs to an admissible set in which there is also D but it reduced minimum probability. {C, D, E} is (with respect to D) most-probable and 0.6-most-probable, but not 0.8-most-probable.
The most-probable extension is integrated in a more general framework of argumentation proposed
in [
For implementation, we developed this probabilistic argumentation framework in a platform for argument reasoning called ARGuing Using Enhanced Reasoning (Arguer). It is an interactive Prolog system for argument reasoning in which the most common frameworks and semantics have been developed. The system makes use of the YAP1 compiler. Experiments of the proposed work have been performed on an Intel(R) Core(TM) i7-1065G7 CPU @ 1.30GHz-1.50 GHz processor with 16GB of RAM.
Arguer is a generic argumentation reasoning platform tailored for domain experts who can provide arguments in the form of propositional logic. The formalism to express arguments, attacks, or other components (e.g. weights) is predetermined. The knowledge about arguments and attacks can be fetched from a Prolog-like text file or by constructing facts directly from the interface. Based on the selected argumentation framework, several extensions can be computed.
Arguer provides the following argumentation frameworks: • Abstract Argumentation Framework (AF) • Value-Based AF • Bipolar AF • Weighted AF • Bipolar Weighted AF • Simplified Probabilistic AF
Abstract Argumentation Framework (AF). To express arguments and attacks, the predicates argument/1 and attack/2 are used. The first one specifies the name of the argument, and the second the relationship between the attacker (the first parameter) and the attacked (the second parameter). After loading a Prolog source made up of these facts, the following semantics can be computed: • admissible • complete • grounded • preferred • stable • semi-stable • eager • stage • naive.
Figure 10 shows the menu after loading an abstract AF knowledge source.
Figure 11 shows the admissible sets for the previously-loaded AF.
We report in Listing 1 the Prolog code for extracting admissible (and conflict-free) sets from arguments. Value-Based AF. It is an extension of an abstract AF in which arguments are associated with values, and values play a crucial role, as well as attacks, in choosing acceptable sets of arguments. Apart from the predicates above, we can find value/1 to express possible values and the val_mapping/2 term to express that the argument in the first position is associated with the value in the second. For this framework, the admissible, preferred and stable semantics can be computed. Bipolar AF. This extension provides the support/2 term to express the opposite of the attack relationship.
For this framework, the admissible, preferred and stable semantics can be computed. Weighted AF. In this extension, for each argument, we can specify its authority in a domain. This is supported by the terms domain/1 to specify domains and authority/3 in which the first item is the argument, the second a value of authority, and the third the domain.
Bipolar Weighted AF. In this extension, we have the same terms as the ones available in Bipolar AF, with the introduction of rel_weight/3 expressing attacks or support based on the positivity of the weight on relationships. The range is [− 1, 0[∪]0, 1].
For this framework, the strength-propagation ranking semantics can be computed. Simplified Probabilistic AF. In the proposed extension, the only added term is likelihood/2 expressing the probability of each argument, and having range ]0, 1].
For this framework, the proposed max-probable semantics can be computed.
Listing 1: Prolog Code for Conflict-Free Sets
Most-probable extension for a target argument is computed starting from the admissible sets, filtering those containing the target argument, and then an optimization function selects the arguments to be kept. Listing 2 shows the solution for probability updating after attacks, and how to normalize results for Answer Set Programming. ASP cannot manage floating numbers. For this reason, updated probabilities are normalized in the range ]0 − 100].
Listing 2: Prolog Code for Updating Likelihoods
Finally, Listing 3 shows the ASP code for computing most-prob extension. The listing shows the (normalized) example reported in Figure 8, and Listing 4 the result provided by ASP after optimization.
Listing 3: ASP Code for Most-probable Extension 1 argument(a;b;c;d;e;f). 2 target(d).
One of the first and most successful ways of formalizing probability dependencies is with the help
of Bayesian nets [
In this work, we proposed a new representation for probabilistic argumentation frameworks, introducing a new semantics, and providing management tools and interpretation of it. Much research can be pursued in the direction of new probability updates, statistical interpretation of arguments, and applications, but also in the direction of fully exploiting the more general framework and proposing a more comprehensive semantics for the other extensions.
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