We discuss a novel extension of Dung's Argumentation Framework (AF) called Featured AF (FAF), where each argument has associated a set of features expressed by means of unary and binary facts [1]. Queries in FAF can be expressed by means of a conjunctive relational calculus formula, evaluated over the FAF extensions. We also discuss the Extended FAF (EFAF), a FAF extension where a first-order logic formula is used for reasoning over 'feasible' subframeworks that satisfy the formula and minimally difer from the original framework.
Various proposals have been made to extend the Dung’s framework with the aim of better modeling
the knowledge to be represented. The extensions include Bipolar AF [
Example 1. Consider a scenario in which three air pollution experts, Alice (), Bob (), and Carl (), are asked to propose short-, medium-, and long-term solutions to the problem. The experts have diferent general opinions: Alice supports industrial pollution regulation policies, Bob supports trafic limitation policies, and Carl promotes changes in human habits. Each expert expresses opinions for the short, medium, and long term, represented by the arguments 1, 2, 3, respectively, where denotes the ifrst letter of the experts’ name. These 9 arguments are summarized in Figure 1, while the following attacks can be derived from considerations made by the three experts.
Short-term: (1, 1), (1, 1), (1, 1), (1, 1); Medium-term: (2, 2), (2, 2), (2, 2), (2, 2); Long-term: (3, 3), (3, 3), (3, 3), (3, 3); Cross-timeframes: (1, 3), (3, 2), (2, 1), (3, 1). The derived AF Λ (Figure 1) has two stable/preferred extensions: 1={1, 2, 3} and 2 = {1, 2, 3}.
The previous example shows how possible air pollution policies can be defined using AF under stable (and preferred) semantics. However, by modeling the problem by means of AF, important information that can be very useful is neglected. In our example, useful information about the text associated with the argument, the author (and her sex and age), and the type of policy (short-, medium-, long-term) is disregarded. Generally, an argument is associated with a sentence with a proper meaning. This means that, in adopting the abstraction of AF, a lot of relevant information, such as the argument’s text, the author’s id and gender, the date it was introduced, the topic (e.g. what the argument is about), the polarity (e.g. whether the sentence conveys a positive, negative or neutral sentiment), the polarity rating, the sentence style (e.g. veracity, sarcasm, irony), and many others, are lost.
b1 1: Impose immediate fines and sanctions on industries exceeding pollution limits; c1 a1 32:: EMnafnordcaetesttrhiceteardionpdtuiosntroiaflgrreegeunlatteicohnnsowloitghiecsotmo palciahniecveemsuescthaainnaisbmles;industrial practices; b3 21:: IEmxppalenmdepnutbcliocntgreasntsiopnorcthatairognestoanreddruecsetrpicrtivaactceevseshfiocrlehuigshe-;polluting vehicles; a3 c3 13:: LRaeudnescihgnawciatrieesnetosspcraiomriptiaziegnpsedtoesetnricaonusr,acgyeclriesdtsu,caenddcealrecutsraicgeveahnidcleesn;ergy conservation; b2 c2 23:: IPnrtoemgroatteeseoncvieitraolnnmoernmtsalthedrouucgahtioinnceinnttoivsecshfooorlescaon-fdriwenodrklyplbaecheasv;ior and penalties for unsustaina2 able actions.
Figure 1: AF Λ of Example 1.
In [
Example 2 (cont’d). Each argument could have associated features such as author (e.g. author (1, Alice)), text (e.g. text (2, “integrate environmental education into schools and workplaces”)), supported policy, (e.g. policy (1, trafic_limitation) ), term time (e.g. term (3, long)). □
In such a context, users pose queries consisting of conjunctive relational calculus formulae such as checking whether there exists an extension containing only arguments concerning policies proposed by the same expert (2 is an extension of such kind in Example 1). Herein, a query is evaluated w.r.t. the extensions of the underlying AF, each consisting of a subset of arguments with the associated features.
We now discuss the Featured Argumentation Framework (FAF), an extension of Dung’s AF that consists
of adding features associated with the arguments [
Syntax. We assume to have distinct countable sets of arguments A, constants D, variables V, and (unary and binary) base predicates P. Constants can be either natural numbers or alphanumeric strings starting with a lower-case letter, whereas variables are denoted by alphanumeric strings starting with an upper-case letter. We also assume to have the predefined (or built-in) unary predicate . P denotes the set of all predicates (base and built-in). Arguments, constants and variables are terms. An atom is of the form (, ) (resp. ()), where is a binary (resp. unary) predicate symbol and , are terms. Unary atoms take values from A, (base) binary atoms take values from A × D. A variable-free base atom is called a fact. Given a set of arguments, is used to denote the set of built-in facts {() | ∈ }.
Definition 1 (FAF syntax [
For an FAF ⟨, , ⟩, the set of base and built-in facts will be denoted by = ∪ .
Semantics. Given a set of (base and built-in) facts ℱ and a set of arguments , we use ℱ↓ = {() |
() ∈ ℱ ∧ ∈ } ∪ {(, ) | (, ) ∈ ℱ ∧ ∈ } to denote the projection of ℱ over . That is,
ℱ↓ contains only the facts in ℱ referring to the arguments in . For any AF ⟨, ⟩ and AF semantics
(e.g. grounded, complete, stable, and preferred), (⟨, ⟩) denotes the set of -extensions for ⟨, ⟩.
Definition 2 (FAF Semantics [
The set of -extensions for Ω is denoted by (Ω). Thus, (⟨, , ⟩) = (⟨, ⟩).
A query is a conjunctive relational calculus query defined over the database , that is, it only uses
the predicate symbols contained in . A query is said to be boolean if all variables are (existentially)
quantified. For instance, the boolean query () checks the acceptance of argument .
Definition 3 (Query acceptance [
Consider the running example, and assume now the case where there are two politicians, the former is interested only in short-term policies, the latter is interested only in policies regarding pollution regulations and trafic limitations. For reasoning about the first context (concerning short-term policies), we should focus on the subframework containing only the arguments 1, 1 and 1, whereas for the second context we should consider the subframework containing only the arguments whose only policies are pollution_regulation and trafic_limitation .
To this end, an extension of FAF called Extended Featured Argumentation Framework (EFAF) has been
proposed [
Example 4 (cont’d). The subframework containing only short-term policies can be defined by means of the FOL formula ∀.(, short), whereas the subframework considering only arguments denoting policies regarding both pollution regulation and trafic limitation can be defined by means of the FOL formula ∀.(, pollution_regulation) ∨ (, trafic_limitation ). On the other side, the formula ∀.(, pollution_regulation) ∨ ∀.(, trafic_limitation ) allows determining two alternative subframeworks.
In a multi-agent context, the FAF encodes shared knowledge, while each constraint reflects an individual agent’s perspective. The FOL formula can be viewed as an integrity constraint that guarantees that available arguments and features correctly model the outside world.
Syntax. We first recall the concept of inclusion for FAF. Given two FAFs Ω and Ω′, we write Ω′ ⊑ Ω (resp. Ω′ ⊏ Ω) if Ω′ can be obtained from Ω through the deletion of a possibly empty (resp. non-empty) set of arguments (as well as the related base facts) and attacks. That is, Ω′ ⊑ Ω if ′ ⊆ , ′ ⊆ , and ′ = ↓′ ; moreover, Ω′ ⊏ Ω if Ω′ ⊑ Ω and Ω′ is not equal to Ω.
Given an FAF Ω = ⟨, , ⟩, we assume to also have the built-in binary predicate , denoting attacks among arguments, and the derived binary predicates ℎ, -ℎ and -ℎ denoting paths over the argumentation graph ⟨, ⟩; all these predicates take values from × . Hereinafter, we use to denote the set {(, ) | (, ) ∈ ⟩, and P to denote the set of all predicates (base, built-in, and derived ones). We use to denote the first-order language using base, built-in and derived predicates, and terms (i.e. arguments, constants and variables).
Definition 4 (EFAF Syntax [
= ∪ ∪ is used to denote the set containing base feature facts and built-in facts describing the topology of the underlying graph.
Semantics. We start by defining the satisfaction of the formula with respect to the underlying FAF ⟨, , ⟩. A base or built-in fact (, ) (resp. ()) is satisfied w.r.t. a EFAF Δ = ⟨, , , ⟩ if it occurs in . Moreover, a derived fact ℎ(, ) (resp. -ℎ(, ), -ℎ(, )) is satisfied w.r.t. Δ if there exists a path (resp. even simple path, odd simple path) from to in the underlying AF ⟨, ⟩. The role of the logical formula in Δ is to represent a constraint (or a set of constraints) to be satisfied by . Intuitively, if the logical formula is not satisfied, then the underlying FAF should be revised by computing subframeworks, that is, by (minimally) modifying the topology of the AF through the deletion of arguments and attacks, so that the formula is satisfied. Clearly, when deleting arguments, related features and attacks are deleted as well. In the following, we assume that the formula is satisfiable, that is that there exists an FAF ⟨, , ⟩ such that |= . The semantics of EFAF is based on the notion of subframework, which is formally recalled next.
Definition 5 (Subframework [
Thus, a set-subframework (s-subframework for short) is obtained by deleting a minimal set of
arguments and attacks, whereas a cardinality subframework (c-subframework) is obtained by deleting
a minimum number of arguments and attacks. Observe that, in computing subframeworks, feature
facts are implicitly deleted through the deletion of the related arguments. This is in line with repairing
strategies exploited in databases, where the inconsistency of an attribute leads to the deletion of an
entire tuple [
Example 5. Consider the EFAF Δ = ⟨, , , ⟩ obtained from the FAF ⟨, , ⟩ of Example 3, where the FOL formula = ∀.(, short), captures the subframeworks containing only short-term policies. The unique subframework (under set or cardinality semantics) is obtained by deleting every argument such that (, short) is false, and its unique stable extension is {1}.
Assume we want to fix exactly one term policy (either short, medium, or long), independently of the actual term’s value. The FOL formula = ∀, , , ′ (, ) ∧ (, ′) ⇒ = ′ allows to identify three s-subframeworks obtained by keeping only the arguments sharing the same term’s value.
Problems and Complexity. Observe that a given EFAF may admit zero, one, or multiple
subframeworks. Subframework existence, verification, and acceptance problems can be defined analogously to
those defined for iAF [
Relationships with incomplete AF. Notice that EFAF without base predicates is similar to
correlated/constrained incomplete AF [
Declaration on Generative AI. The author(s) have not employed any Generative AI tools. [24] J. Mailly, Possible controllability of control argumentation frameworks, in: Proc. of COMMA Conf., 2020, pp. 283–294.