Strong Admissibility for Abstract Dialectical Frameworks

Abstract dialectical frameworks (ADFs) have been introduced as a formalism for modeling and evaluating argumentation allowing general logical satisfaction conditions. Different criteria used to settle the acceptance of arguments are called semantics. Semantics of ADFs have so far mainly been defined based on the concept of admissibility. However, the notion of strongly admissible semantics studied for abstract argumentation frameworks has not yet been introduced for ADFs. In the current work we present the concept of strong admissibility of interpretations for ADFs. Further, we show that strongly admissible interpretations of ADFs form a lattice with the grounded interpretation as top element.

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1. Introduction

Interest and attention in artificial intelligence-related areas in argumentation theory has been increasing, by the wide variety of formalisms of argumentation to model argumentation and by the variety of semantics that clarify the acceptance of arguments (Baroni et al., 2018; van Eemeren et al., 2014). Abstract argumentation frameworks (AFs) as introduced by Dung (Dung, 1995) are a core formalism in formal argumentation, (have proven successful in many applications related to multi-agent systems (McBurney et al., 2012)). Abstract dialectical frameworks (ADFs) were first introduced in (Brewka and Woltran, 2010), further refined in (Brewka et al., 2013, 2017). They are expressive generalizations of AFs in which the logical relations among arguments can be represented.

A key question in formal argumentation is ‘How is it possible to evaluate arguments in a given formalism’ Answering this question leads to the introduction of several types of semantics. Different semantics reflect different types of point of view about the acceptance or denial of arguments. Most of the semantics of AFs/ADFs are based on the concept of admissibility, in (Caminada and Amgoud, 2007) it is shown that admissibility plays an important role w.r.t. rationality postulates.

It is shown in (Brewka et al., 2017), that each AF can be represented as an ADF, further, it is shown that semantics defined for ADFs are proper generalizations of the semantics of AFs. However, some of the semantics of AFs have not yet been introduced for ADFs, namely strongly admissible semantics. In the current work we introduce strongly admissible semantics of ADFs.

In ADFs an interpretation is called admissible if it does not contain any unjustifiable information. An interpretation is called preferred if it is a maximal admissible interpretation. Thus, each admissible interpretation is contained in a preferred interpretation. That is, to answer the credulous decision problem under preferred semantics it is enough to answer the problem under admissible semantics. In addition, an interpretation is grounded if it collects all the information that is beyond any doubt.

In AFs the concept of strongly admissible semantics has first been defined in the work of Baroni and Giacomin (Baroni and Giacomin, 2007), based on the notion of strong defence. Later in (Caminada, 2014) this concept was introduced without referring to strong defence. Further, in (Caminada and Dunne, 2019) Caminada and Dunne presented a labelling account of strong admissibility to answer the credulous decision problem of AFs under grounded semantics. In (Caminada, 2018, 2014; Caminada and Dunne, 2019) it was shown that strong admissibility plays a critical role in discussion games for AFs under grounded semantics. That is, it has been shown that strongly admissible extensions/labellings make a lattice with the maximum element of the grounded extension of a given AF. Therefore, the concept of strong admissibility semantics of AFs relates to grounded semantics of AFs in a similar way as the relation between admissible semantics of AFs and preferred semantics of AFs. That is, to answer the credulous decision problem of AFs under grounded semantics it is enough to solve the decision problem for AFs under strongly admissible semantics.

In (Keshavarzi Zafarghandi et al., 2020), a discussion game was introduced to answer the credulous decision problem of ADFs under grounded semantics without constructing the full grounded interpretation of the given ADF. However, the concept of strongly admissible semantics of ADFs has not been introduced.

This was a motivation for us to present the notion of strongly admissible semantics for ADFs in this work. However, studying whether the game that is presented in (Keshavarzi Zafarghandi et al., 2020) is equivalent to constructing a strongly admissible interpretation that satisfies the claim, in the given ADF, is beyond the topic of this work and is left for future research.

Semantics of AFs are usually defined based on extensions using the notion of argument acceptability. In contrast, semantics of ADFs are defined in terms of three-valued interpretations using both argument acceptability and deniability. In this sense, there is a connection with the use of labelings for AFs using argument acceptability/deniability (e.g., [16]). However, by the use of general propositional formulas as argument acceptance conditions, ADFs allow for richer relations between arguments than AFs, which only allow attack.

As a result, because of the special structure of ADFs, the definition of strong admissibility semantics of AFs cannot be directly reused in ADFs. Thus, we first present the notion strong acceptability/deniability of arguments in an interpretation. Then, we present the concept of strong admissibility to characterise the properties of the grounded interpretation of ADFs.

The presented notion of strong admissibility for ADFs is closely related to strong admissibility for AFs in three ways. First strong admissibility is defined in terms of strongly acceptable/deniable arguments the truth value of which presented in a given interpretation. Second such strongly acceptable/deniable arguments are recursively reconstructed from their strongly acceptable/deniable parents. Third there is a close relation to the grounded semantics, in the formally precise sense that the maximal element of the lattice of strongly admissible sets is the grounded interpretation.

This paper is structured as follows. In Section 2, we present the relevant background. Then, in Section 3, the main contribution of our work is to introduce the concept of strongly admissible semantics for ADFs. Then we show that in each ADF, the set of strongly admissible interpretations form a lattice with the trivial interpretation as the unique minimal element and the grounded interpretation as the unique maximal element In Section 4, we present a conclusion of our work and we present some future research questions arising from this work.

2. Formal Preliminaries

In this section, we only briefly present the syntax of AFs (Dung, 1995). We present the concept of strongly admissible semantics of AFs due to (Baroni and Giacomin, 2007). Then, we present ADFs due to (Brewka and Woltran, 2010; Brewka et al., 2013, 2017).

2.1. Abstract Argumentation Frameworks

We start the preliminaries to our work by recalling the basic notion of Dung’s abstract argumentation frameworks (AFs) (Dung, 1995) and the concept of strong admissibility semantics of AFs due to Baroni and Giacomin (Baroni and Giacomin, 2007).

Definition 2.1 ().

(Dung, 1995) An abstract argumentation framework (AF) is a pair in which is a set of arguments and is a binary relation representing attacks among arguments.

Let be a given AF. For each , the relation is used to represent that a is an argument attacking the argument b. An argument is, on the other hand, defended by a set of arguments (alternatively, the argument is acceptable w.r.t. ) (in ) if for each argument , it holds that if then there is a such that ( is called a defender of ).

Example 2.2 ().

Let be an AF. In , means that argument attacks , and means that attacks . Here, argument is defended by set (alternatively, is acceptable with respect to ), since attacks the attacker of , namely .

Different semantics of AFs present which sets of arguments in a given AF can be accepted jointly. 111The interested reader in semantics of AFs can see (Dung, 1995). Let be an AF, then is a conflict-free set (extension), if there exists no such that . For instance, in Example 2.2, the set is a conflict-free set of . Further, a set of arguments is a grounded extension of an AF if (intuitively) there is no doubt on the acceptance of the arguments in the set. Every AF has a unique grounded extension. In Example 2.2, a unique grounded extension of is . We avoid here to present the formal definition of the grounded extension However, in Example 2.2, the intuition is that is not attacked by any argument, thus no one has any doubt to accept argument . Argument is attacked by , however, it is defended by which was accepted by everyone. Thus, is a unique grounded extension of . In Definition 2.4 we represent the notion of strongly admissible semantics of AFs.

Definition 2.3 ().

(Baroni and Giacomin, 2007) Given an argumentation framework, , and , it is said that is strongly defended by if and only if each attacker of is attacked by some such that is strongly defended by .

In other words, is strongly defended by if for any attacker of there exists a defender for in that is not equal to , i.e. , such that is strongly defended by . In Example 2.2, argument is strongly defended by set , since the attacker of , namely is attacked by and is strongly defended by . Actually, is strongly defended by , since is not attacked by any argument.

Definition 2.4 ().

Given an AF and set . It is said that is a strongly admissible extension of if every is strongly defended by .

In Example 2.2, sets , , and are strongly admissible extensions of ; all of them are subsets of the grounded extension of . However, set is not a strongly admissible extension of , since is not strongly defended by . Because argument is attacked by , however, no argument in attacks .

2.2. Abstract Dialectical Frameworks

We briefly restate some of the key concepts of abstract dialectical frameworks that are derived from those given in (Brewka and Woltran, 2010; Brewka et al., 2013, 2017).

Definition 2.5 ().

An abstract dialectical framework (ADF) is a tuple where:

  • is a finite set of arguments (statements, positions);

  • is a set of links among arguments;

  • is a collection of propositional formulas over arguments, called acceptance conditions.

An ADF can be represented by a graph in which nodes indicate arguments and links show the relation among arguments. Each argument in an ADF is labelled by a propositional formula, called acceptance condition, over such that, . The acceptance condition of each argument clarifies under which condition the argument can be accepted (Brewka and Woltran, 2010; Brewka et al., 2013, 2017). Further, acceptance conditions indicate the set of links implicitly, thus, there is no need of presenting in ADFs explicitly.

An argument is called an initial argument if . An interpretation (for ) is a function , that maps arguments to one of the three truth values true (), false (), or undecided (). Truth values can be ordered via the information ordering relation given by and and no other pair of truth values are related by . Relation is the reflexive and transitive closure of . The pair is a complete meet-semilattice with the meet operator , such that, , , and returns otherwise. The meet of two interpretations and is then defined as for all .

Further, is called trivial, and is denoted by , if for each . Further, is called a two-valued interpretation if for each either or . Interpretations can be ordered via with respect to their information content. Let be the set of all interpretations for an ADF . It is said that an interpretation is an extension of another interpretation , if for each , denoted by . Further, if and , then and are equivalent, denoted by .

For reasons of brevity, we will sometimes shorten the notion of three-valued interpretation with arguments and truth values as follows: . For instance, . We use this notation in Figure 3.

Semantics for ADFs can be defined via the characteristic operator which maps interpretations to interpretations. Given an interpretation (for ), the partial valuation of by , is , for .

Definition 2.6 ().

Let be an ADF and let be an interpretation of . Applying on leads to s.t. for each , is as follows:

Note that the operator is monotonic, that is, when for interpretations and , then . The semantics of ADFs are defined via the characteristic operator as follows.

Definition 2.7 ().

Given an ADF , an interpretation is:

  • conflict-free iff implies is satisfiable and implies is unsatisfiable;

  • admissible in iff ;

  • preferred in iff is -maximal admissible;

  • the grounded interpretation of iff is the least fixed point of .

The set of all interpretations for an ADF is denoted by , where abbreviates the different semantics in the obvious manner. The notion of an argument being accepted and the symmetric notion of an argument being denied in an interpretation are as follows.

Definition 2.8 ().

Let be an ADF and let be an interpretation of .

  • An argument is called acceptable with respect to if is irrefutable.

  • An argument is called deniable with respect to if is unsatisfiable.

Example 2.9 ().

An example of an ADF is shown in Figure 1. To each argument a propositional formula is associated, the acceptance condition of the argument. For instance, the acceptance condition of , namely , states that can be accepted in an interpretation where is denied and is accepted. In the interpretation is conflict-free. However, is not an admissible interpretation, because , that is, .

The interpretation on the other hand is an admissible interpretation. Since and . Further, in a unique grounded interpretation is a preferred interpretation of .

Figure 1. ADF of Examples 2.9 and 3.3

Given an ADF , an argument and a semantics , argument is credulously acceptable (deniable) under if there exists a interpretation of in which is acceptable ( is deniable, respectively).

In ADFs, relations between arguments can be classified into four types, reflecting the relationship of attack and/or support that exists between the arguments. These are listed in Definition

2.10. Further, we denote the update of an interpretation with a truth value for an argument by , i.e. and for .

Definition 2.10 ().

Let be an ADF. A relation is called

  • supporting (in ) if for every two-valued interpretation , implies ;

  • attacking (in ) if for every two-valued interpretation , implies ;

  • redundant (in ) if it is both attacking and supporting;

  • dependent (in ) if it is neither attacking nor supporting.

In the current work we say that the truth value of is presented in , if .

3. The Strongly Admissible semantics for ADFs

In the following, we first present the concept of strongly admissible semantics for ADFs. In ADFs, beside an argument being acceptable in an interpretation, there is a symmetric notion of an argument being deniable. Thus, in Definition 3.1 we introduce the notion of strong acceptability/deniability of an argument in an ADF with respect to a given interpretation. In Theorem 3.21, we show that in a given ADF, the set of strongly admissible interpretations of make a lattice, with the unique minimal element and the unique maximal element .

Note that in the following, is equal to for any , however, it assigns all other arguments that do not belong to to . Further, in Definition 3.1 set contains the ancestors of the truth value of which are presented in , that have an effect on the truth value of in . This is similar to Definition 2.3, in which set contains the defenders of . In the first item of Definition 3.1, set contains exactly those parents of , excluding , that satisfy and of which the truth value is presented in .

Definition 3.1 ().

Let be an ADF and let be an interpretation of . Argument is a strongly acceptable/deniable argument with respect to interpretation and set if the following conditions hold.

  • Let . There exists a subset of parents of excluding , namely such that if and if .

  • Each , with that satisfies the first item, is strongly acceptable/deniable with respect to interpretation and set such that .

Note that in Definition 3.1 to indicate whether an argument is strongly acceptable/deniable, we collect the set of ancestors of that affect the truth value of in set . If the set of parents of an argument, namely , is an empty set, then . In Definition 3.2 the concept of strong admissibility of an interpretation of a given ADF is introduced.

Definition 3.2 ().

Let be an ADF and let be an interpretation of . An interpretation is a strongly admissible interpretation if for each such that , then is a strongly acceptable/deniable argument with respect to and set .

These notions are clarified in Example 3.3. Note that set in Definitions 3.1 and 3.2 can be the empty set. Example 3.4 is an instance of strong acceptability of an argument with .

Example 3.3 ().

Let , depicted in Figure 1. Let . We show that is strongly deniable with respect to and set . To satisfy the first condition of Definition 3.1, we choose the subset of parents of excluding equal to . It is easy to check that . In this step . To check the second condition of Definition 3.1, we have to show that is also a strongly deniable argument. To this end, by the definition extends to . Further, clearly . Thus, is strongly deniable with respect to and set . In other words, set indicates a parent of , namely that has affect on the truth value of in .

On the other hand, is not strongly deniable with respect to and set . The reason is as follows. Although the first condition of Definition 3.1 is satisfiable, that is, , the second condition is not satisfiable, i.e.  is not strongly acceptable with respect to . Toward a contradiction, assume that is strongly acceptable w.r.t. . Thus, we have to choose a parent of that does not belong to , namely and we have to show that . However, . Therefore, is not strongly acceptable with respect to .

Note that is also strongly acceptable with respect to and . In other words, is the least subset of that satisfies the conditions of Definition 3.1 for .

Example 3.4 is an instance of ADFs with a redundant link.

Example 3.4 ().

Let be an ADF, depicted in Figure 2. We show that is a strongly admissible interpretation of . To this end, we show that is strongly acceptable with respect to and . It is clear that is the empty set and is irrefutable. Thus, satisfies the conditions of Definition 3.1 for . That is, is strongly acceptable with respect to and .

Figure 2. ADF of Examples 3.4

As we presented earlier, for instance, in Example 3.3, we are interested in finding a least set of ancestors of an argument in question that satisfies the conditions of Definition 3.1, presented in Definition 3.5.

Definition 3.5 ().

Let be an argument that is strongly acceptable/
deniable with respect to and . We say that is a least set that satisfies the conditions of Definition 3.1 for if there is no with such that is strongly acceptable/deniable with respect to and .

For instance, in Example 3.3, is the least set that satisfies the conditions of Definition 3.1 for . We define the maximum level of in a least set recursively, as follows.

Definition 3.6 ().

Let be an ADF and let be strongly acceptable/deniable with respect to and a least set , and let such that . The maximum level of with respect to a least set is:

  • If , then the maximum level of in is .

  • If and the maximum of the maximum level of an argument of in is , then the level of with respect to is .

For instance, in Example 3.3, the maximum level of with respect to is . This is because the maximum level of with respect to is .

Considering ADF of Example 3.4, by Definition 3.6 the maximum level of with respect to the least set is one. Lemma 3.7 shows that if is strongly acceptable/deniable with respect to and , then the maximum level of is finite in any given ADF.

Lemma 3.7 ().

Let be an ADF, let be an interpretation of and let be an argument that is strongly acceptable/deniable with respect to and a least set . Then has a finite maximum level in in .

Proof.

Toward a contradiction assume that is an argument with infinite maximum level in . Therefore, by Definition 3.6, the set of parents of , namely with is a non-empty set. Further, there exists an argument in with infinite maximum level. By the same reason has a parent with infinite maximum level that is neither equal to nor . Thus, has an infinite number of ancestors. This is a contradiction by the assumption that the is a finite ADF. Thus, the assumption that has an infinite maximum level is wrong. ∎

Lemma 3.8 ().

Let be an ADF. If is strongly acceptable/ deniable with respect to interpretation of and a least set and , then is also strongly acceptable/deniable with respect to and a least set .

Proof.

Since is strongly acceptable/deniable with respect to and , there exists that satisfies the first condition of Definition 3.1. Since the same set of parents of , namely guarantees that the first condition of Definition 3.1 holds for with respect to and .

Assume that is also a least set that satisfies the conditions of the current lemma. We show that the second condition of Definition 3.1 works by induction on the maximum level of argument in .

Base case: let be an argument of the maximum level one that is strongly acceptable/deniable with respect to and . Therefore, . Thus, is clearly strongly acceptable/deniable with respect to and .

Inductive step: Assuming that this property holds for each argument of the maximum level with in , i.e., if is an argument with the maximum level in that is strongly acceptable/deniable with respect to   and , then is strongly acceptable/deniable with respect to and . We show that this property also holds for arguments of level . Let be an argument of the maximum level . By Definition 3.1, there exists the set of parents of , namely , that satisfies the conditions of the definition with respect to and set . We claim that this also satisfies the conditions of the definition for w.r.t. and . By Definition 3.6, the maximum level of each is at most . Thus, by induction hypothesis is strongly acceptable/deniable with respect to and set . Therefore, the second condition of Definition 3.1 also holds. Thus, is strongly acceptable/deniable with respect to and . ∎

A sequence of interpretations, for a given ADF , is presented in Lemma 3.9, each member of which is strongly admissible. In Lemma 3.10 it is shown that the maximum element of this sequence is the grounded interpretation of .

Lemma 3.9 ().

Let be an ADF, let and let for . For each it holds that

  • ,

  • is a strongly admissible interpretation of .

Proof.
  • The first item holds because the characteristic operator is a monotonic function.

  • We show that each is a strongly admissible interpretation by induction on .

    Base case: For , it is clear that is a strongly admissible interpretation.

    Inductive step: Assume that for with is a strongly admissible interpretation. We show that is a strongly admissible interpretation. Let be an argument that is assigned to either or in . If , there is nothing to prove, since by the induction assumption is a strongly admissible interpretation. Assume that and . We show that is strongly acceptable with respect to and set . For the case that , the proof follows a similar method. Since , we can conclude that is irrefutable. Let be a subset of parents of the truth value of which appears in and . Otherwise, cannot be irrefutable. Thus, the first condition of Definition 3.1 holds.

    To show the second condition of Definition 3.1, assume that . Otherwise, there is nothing to prove. Let . By the induction assumption, is a strongly admissible interpretation. Since for each , is strongly acceptable/deniable with respect to and set . Thus, by the monotonicity of the characteristic operator, is strongly acceptable/deniable with respect to and . Thus, the second condition of Definition 3.1 holds, as well. That is, arbitrary argument is strongly acceptable with respect to and . Thus, is a strongly admissible interpretation. Hence, every interpretation in the sequence is a strongly admissible interpretation.

Lemma 3.10 ().

Let be an ADF.

  • has at least one strongly admissible interpretation.

  • The least strong admissible interpretation of , with respect to the ordering, is the trivial interpretation.

  • The biggest strongly admissible interpretation, with respect to the ordering, is the unique grounded interpretation of .

Proof.
  • The first and the second items of the lemma are clear by Lemma 3.9, which says that is a strongly admissible interpretation.

  • By Definition, the grounded interpretation of is the least fixed-point of the characteristic operator over with respect to the -ordering. By Lemma 3.9, each is a strongly admissible interpretation. Thus, the least fixed-point of is also a strongly admissible interpretation. Note that, the th power off is defined inductively, that is, .

In Theorem 3.11 we show that each strongly admissible interpretation is an admissible interpretation as well as conflict-free. However, the other direction of the following theorem does not work. For instance, let be a given ADF. The interpretation is an admissible interpretation of , however, neither nor is strongly admissible with respect to . Thus, is not a strongly admissible interpretation of . Further, is a conflict-free interpretation of that is neither an admissible nor a strongly admissible interpretation. The only strongly admissible interpretation of , which is also the grounded interpretation of , is the trivial interpretation.

Theorem 3.11 ().

Let be an ADF and let be a strongly admissible interpretation of . Then the following hold:

  • is an admissible interpretation of .

  • is a conflict-free interpretation of .

Proof.
  • Let be a strongly admissible interpretation of . We show that is an admissible interpretation. Toward a contradiction assume that is not an admissible interpretation, that is, . That is, there exists such that , but . By the assumption is a strongly admissible interpretation. That is, if , then is strongly acceptable/deniable with respect to and set . Thus, by the first item of Definition 3.1, there exists a subset of parents of , namely such that if , and if . However, implies that is irrefutable and implies that is unsatisfiable. The former implies if , than and the latter one implies that if , then . This is a contradiction by the assumption that there exists such that , and . Thus, the assumption that is not an admissible interpretation is wrong. Hence, if is a strongly admissible interpretation, then it is also an admissible interpretation.

  • If is a strongly admissible interpretation, then by the first item of this theorem it is an admissible interpretation. By the fact that in ADFs every admissible interpretation is a conflict-free interpretation, we conclude that is a conflict-free interpretation, as well.

3.1. The Strongly Admissible Interpretations of an ADF form a lattice

Although the sequence of interpretations presented in Lemma 3.9 produces a sequence of strongly admissible interpretations of a given ADF , this sequence does not contain all of the strongly admissible interpretations of . For instance, in Example 3.3, is a strongly admissible interpretation of . However, is not equal to any of the elements of the sequence for given in Example 3.3. However, Theorem 3.12, indicates that any strongly admissible interpretation of ADF is bounded by an element of the sequence of strongly admissible interpretations presented in Lemma 3.9.

Theorem 3.12 ().

Let be an ADF, let be an interpretation of , and let for be the sequence of interpretations presented in Lemma 3.9. If is a strongly admissible interpretation of , then there exists the least such that .

Proof.

Let be the set of arguments the truth values of which appear in . Further, assume that each is strongly acceptable/deniable with respect to and a least set . Let . Let be an argument with the greatest maximum level in . We claim that . We have to show that if , then . We show our claim by induction on the maximum level of arguments in .

Base case: If and the maximum level of in is , then it is clear that . Therefore, .

As induction hypothesis, assume that if and the maximum level of in is with , then (and also ).

Induction step: Assume that is an argument that is strongly acceptable/deniable with respect to and the maximum level in is . We have to show that (and ). Since is strongly acceptable/deniable with respect to and and the maximum level of in is , there exists a non-empty set such that . Since is a parents of , by Definition 3.1, is also strongly acceptable/deniable with respect to and . Thus, by Definition 3.6 the maximum level of each is strictly less than the maximum level of i.e. the maximum level of in is at most . Then, by the induction hypothesis, , for each . Therefore, . Further, because and is a monotonic function. Therefore, (and also ). That is, there exists an , such that .

Further, we have to show that the natural number assumed in the beginning of the proof is the least natural number that satisfies the condition of the theorem. Toward a contradiction assume that there exists an such that . By our assumption the greatest maximum level of an argument of , namely is and is a least set that satisfies the conditions of Definition 3.1 for all arguments the truth values of them appear in . It is easy to check that . Thus, . That is, is the least natural number that satisfies the condition of the theorem.

Theorem 3.13 ().

Let be an ADF and let be an interpretation of . If argument is strongly acceptable/deniable with respect to and a least set , then each is also strongly acceptable/deniable with respect to and a .

Proof.

Toward a contradiction assume that there exists that is not strongly acceptable/deniable with respect to and any . By Definition 3.1, any argument in set is an ancestor of that is strongly acceptable/deniable. Thus, is not any of the ancestors of that appears in set , otherwise it is strongly acceptable/deniable. Therefore, is also strongly acceptable/deniable with respect to and . Then, is not a least set that satisfies the conditions of Definition 3.1 for . This is a contradiction by the assumption of the theorem that is a least set. Thus, the assumption that there exists an argument in that is not strongly acceptable/deniable with respect to and a subset of is wrong. ∎

To show that the set of strongly admissible interpretations of a given ADF make a lattice, first, in Theorem 3.17 we show that every two strongly admissible interpretations of have a unique supremum. To this end, we first introduce the notion of join of two strongly admissible interpretations in Definition 3.14.

Definition 3.14 ().

Let be an ADF and let and be two strongly admissible interpretations of . The join is defined as

Proposition 3.15 ().

The join of two strongly admissible interpretations of is a well-defined function.

Proof.

Let be an ADF and let and be two strongly admissible interpretations of . We show that the join operator is a well-defined function. That is, we have to show that there is no that has two different values in . Toward a contradiction assume that there is a that has two different outputs in . That is, is assigned to in one of the interpretations and to in another one. For instance, and . By Theorem 3.12, there exists the least natural numbers and such that and , respectively. Since and , . Further, since and , . That is, and . This is a contradiction by Lemma 3.9, that says either or , because and are elements of the sequence of interpretations presented in Lemma 3.9. Thus, the assumption that there exists that is acceptable in a strongly admissible interpretation of but that is deniable in another strongly admissible of is wrong. Thus, is a well-defined function. ∎

Lemma 3.16, presents that the join of two strongly admissible interpretations of a given ADF is also a strongly admissible interpretation of that ADF.

Lemma 3.16 ().

Let be an ADF and let and be strongly admissible interpretations of . Then is also a strongly admissible interpretation of .

Proof.

Toward a contradiction assume that is not a strongly admissible interpretation of . Thus, there exists an such that but it is not strongly acceptable/deniable with respect to and any set. By Definition 3.14, either or . Since and are strongly admissible interpretations, is strongly acceptable/deniable with respect to or . Since and , by Lemma 3.8, is strongly acceptable/deniable with respect to . This is a contradiction with the assumption that is not strongly acceptable/deniable with respect to . Thus, the assumption that is not a strongly admissible interpretation was wrong. That is, the join of two strongly admissible interpretations of is a strongly admissible interpretation of . ∎

Theorem 3.17 ().

Let be an ADF. Every two strongly admissible interpretations of have a unique supremum.

Proof.

Let be an ADF and let and be two strongly admissible interpretations of . We show that is a supremum of and . By Definition 3.14, is an upper bound of and . By Lemma 3.16, is a strongly admissible interpretation of . It remains to show that is a least upper bound of and . Toward a contradiction, assume that is not the least upper bound of and . That is, there exists a strongly admissible interpretation of such that , and . Thus there exists with and . Thus, either or . That is, either or . This is a contradiction by the assumption that is the least upper bound of and . Thus, the assumption that is not the least upper bound of and was wrong. ∎

Further, to show that the set of strongly admissible interpretations of ADF make a lattice, in Theorem 3.20 we show that every two strongly admissible interpretations of have an infimum. To this end, in Definition 3.18, we present the concept of the maximum strongly admissible interpretation contained in an interpretation of .

Definition 3.18 ().

Let be an ADF and let be an interpretation of . Interpretation is called a unique maximum strongly admissible interpretation that is less than or equal to , with respect to ordering if the following conditions hold:

  • is a strongly admissible interpretation of s.t. ,

  • there is no strongly admissible interpretation of such that .

Lemma 3.19 ().

Let be an ADF and let be an interpretation of . Then, there exists a unique maximum strongly admissible interpretation that is less than or equal to , with respect to ordering.

Proof.

Each interpretation of has at least as much information as the trivial interpretation. Thus, each of has at least as much information as , which is a strongly admissible interpretation. Since the number of arguments of is finite, there exists at least one maximal strongly admissible interpretation of , namely for the given interpretation . We show that this is unique. Toward a contradiction assume that there are two maximal strongly admissible interpretations that satisfy the condition of the lemma, namely and . By Lemma 3.16, is a strongly admissible interpretation of s.t. . However, and together with the assumption that and are maximal strongly admissible interpretations lead to and . That is, . Thus, the maximum strongly admissible interpretation which is contained in is unique. ∎

Theorem 3.20 ().

Let be an ADF. Every two strongly admissible interpretations of have a unique infimum.

Proof.

Let be an ADF and let and be two strongly admissible interpretations of . Let . By Lemma 3.19, there exists a unique maximum strongly admissible interpretation that is less than or equal to , i.e. . That is is a lower bound of and . It remains to show that is the greatest lower bound of and . Toward a contradiction assume that there exists that is the greatest lower bound of and . That is, and . Then by the definition . By the assumption is the maximum strong admissible that is less or equal to , thus, . Thus, is an infimum of and . ∎

Theorem 3.21 ().

Let be an ADF. The strongly admissible interpretations of form a lattice with respect to the -ordering, with the least element and the top element .

Proof.

We have to show that every two strongly admissible interpretations of have a supremum and an infimum. Theorem 3.17 shows the former one and Theorem 3.20 indicates the latter one. Thus, the strongly admissible interpretations of make a lattice with respect to the -ordering. In Lemma 3.10, it is shown that is the least strongly admissible interpretation and is the largest strongly admissible interpretation of the sequence of the interpretations presented in Lemma 3.9. This fact together with Theorem 3.12, shows that is the greatest element of this lattice. It is trivial that is the least element of this lattice. ∎

The set of strongly admissible interpretations of ADF , given in Example 3.3 form a lattice, depicted in Figure 3. The top element of this lattice is .

Figure 3. Complete lattice of the strongly admissible interpretations of the ADF of Example 3.3

4. Conclusion

In this work we have defined strongly admissible semantics for ADFs, based on the concept of strongly acceptable/deniable arguments. From a theoretical perspective, we have observed that the strongly admissible interpretations of a given ADF form a lattice with the trivial interpretation as the unique minimal element and the grounded interpretation as the unique maximal element.

The concept of strong admissibility is related to grounded semantics in a similar way as the concept of admissibility is related to preferred semantics. That is, to answer the credulous decision problem of an ADF under the grounded semantics, there is no need of constructing the full grounded interpretation of the given ADF. Instead, it is enough to construct a strongly admissible interpretation of the given ADF that satisfies the decision problem. Similarly, to answer the credulous decision problem of ADFs under preferred semantics, it is enough to investigate whether there exists an admissible interpretation in order to solve the decision problem.We used this method in preferred discussion games in (Keshavarzi Zafarghandi et al., 2019) to answer the credulous decision problem of ADFs under preferred semantics.

Possible future research questions include whether the concept of strongly admissible semantics for ADFs, presented in this work, is a proper generalization of the concept of strongly admissible semantics for AFs (Baroni and Giacomin, 2007; Caminada, 2014).

Further, it is interesting to investigate how the concept of strong admissibility of ADFs relates to the grounded discussion game presented in (Keshavarzi Zafarghandi et al., 2020). In other words, investigation is required of the question whether the discussion game presented in (Keshavarzi Zafarghandi et al., 2020) to answer the credulous decision problem of ADFs under the grounded semantics is equivalent to answer the same decision problem under strong admissibility interpretation. The grounded discussion game was defined over ADFs without any redundant links, however, the concept of strongly admissible semantics is presented for all kinds of ADFs. Thus, we will investigate whether the concept of strongly admissible semantics is at the basis of the proof procedures of the grounded discussion games for ADFs without any redundant links.

Further, we would like to investigate whether the grounded discussion game presents the shortest discussion/explanation that answers the credulous decision problems under strongly admissible/ grounded semantics for the given argument of ADFs.

Computational complexity classes of semantics of AFs and ADFs are presented in (Dvořák and Dunne, 2017). Computational complexity of strongly admissible semantics of AFs is studied in (Dvořák and Wallner, 2020). Further, in (Caminada and Dunne, 2020), the computational complexity of identifying strongly admissible labellings with bounded or minimal size was studied. As a future work, it would be interesting to clarify the computational complexity of investigating of the truth value of an argument in a strongly admissible interpretation of a given ADF.

Acknowledgements.

The authors would like to thank Dr. M. Caminada and Prof. dr. S. Woltran for their recommendations for presenting the notion of strongly admissible semantics for ADFs. The work is supported by the Center of Data Science

Systems Complexity (DSSC) Doctoral Programme, at the University of Groningen.

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