# Weighted Abstract Dialectical Frameworks: Extended and Revised Report

Abstract Dialectical Frameworks (ADFs) generalize Dung's argumentation frameworks allowing various relationships among arguments to be expressed in a systematic way. We further generalize ADFs so as to accommodate arbitrary acceptance degrees for the arguments. This makes ADFs applicable in domains where both the initial status of arguments and their relationship are only insufficiently specified by Boolean functions. We define all standard ADF semantics for the weighted case, including grounded, preferred and stable semantics. We illustrate our approach using acceptance degrees from the unit interval and show how other valuation structures can be integrated. In each case it is sufficient to specify how the generalized acceptance conditions are represented by formulas, and to specify the information ordering underlying the characteristic ADF operator. We also present complexity results for problems related to weighted ADFs.

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## 1 Introduction

Computational models of argumentation are a highly active area of current research. The field has two main subareas, namely logic-based argumentation and abstract argumentation. The former studies the structure of arguments, how they can be constructed from a given formal knowledge base, and how they logically interact with each other. The latter, in contrast, assumes a given set of abstract arguments together with specific relations among them. The focus is on evaluating the arguments based on their interactions with one another. This evaluation typically uses a specific semantics, thus identifying subsets of the available arguments satisfying intended properties so that the chosen set arguably can be viewed as representing a coherent world view.

In the abstract approach, Dung’s argumentation frameworks (AFs) [23] and their associated semantics are widely used. In a nutshell, an AF is a directed graph with each vertex being an abstract argument and each directed edge corresponding to an attack from one argument to another. These attacks are then resolved using appropriate semantics. The semantics are typically based on two important concepts, namely conflict-freeness and admissibility. The former states that if there is a conflict between two arguments, i.e. one argument attacks the other, then the two cannot be jointly accepted. The latter specifies that every set of accepted arguments must defend itself against attacks. A variety of semantics has been defined, ranging from Dung’s original complete, preferred, stable, and grounded semantics to the more recent ideal and cf2 semantics. The different semantics reflect different intuitions about what “coherent world view” means in this context, see e.g. [7] for an overview.

Despite their popularity, there have been various attempts to generalize AFs as many researchers felt a need to cover additional relevant relationships among arguments (see e.g. the work of [16]). One of the most systematic and flexible outcomes of this research are abstract dialectical frameworks (ADFs) [13, 11, 12]. ADFs allow for arbitrary relationships among arguments. In particular, arguments can not only attack each other, they also may provide support for other arguments and interact in various complex ways. This is achieved by adding explicit acceptance conditions to the arguments which are most naturally expressed in terms of a propositional formula (with atoms referring to parent arguments). This way, it is possible to specify individually for a particular argument, say, under what conditions the available supporting arguments outweigh the counterarguments. Meanwhile various applications of ADFs have been presented, for instance in legal reasoning [1, 2] and text exploration [15]. A mobile argumentation app based on ADF techniques was developed by [38].

The operator-based semantics of ADFs can be traced back to work on approximation fixpoint theory (AFT) [19, 18, 20], an algebraic framework for studying semantics of knowledge representation formalisms. We refer to the work of [39] for a detailed analysis of the relationship between ADFs and AFT. The presentation of our approach in this paper does not assume specific background knowledge in AFT.

The motivation for the work presented here is as follows. The definition of the various ADF semantics is based on an analysis in terms of partial two-valued (or, equivalently, three-valued) interpretations. The output provided by ADFs (and AFs, for that matter) is thus restricted to three options: an argument either is true (accepted) in an intended interpretation, or it is false (rejected), or its value is unknown. However, many situations in argumentation call for more fine-grained distinctions (see, e.g. [3] for an application of weighted argumentation in the Twitter domain). For instance, it is sometimes natural to assume numerical acceptance degrees, say, taken from the unit interval, and to explore the effect of these degrees on other arguments. The availability of such acceptance degrees allows for new, interesting types of queries to be asked. For instance, under a given semantics (stable, preferred, complete, …), we may want to know whether the value of a particular argument is above/below a certain threshold in some or all interpretations of the required type. It also may be useful to be able to distinguish among a finite number of acceptance degrees, say strong accept, weak accept, neutral, weak reject and strong reject. Or it may even be useful to operate on intervals of acceptance degrees.

The goal of this paper is to show how the ADF approach (and thus AFs) can accommodate such acceptance degrees. To put it differently, we aim to bridge two rich research areas, multi-valued logics on the one hand and computational models of argumentation on the other.

We start with the necessary ADF background in Section 2. We then introduce our general framework for weighted ADFs in Section 3. Section 4 focuses on ADFs with acceptance degrees in the unit interval. Section 5 applies the same idea to three other valuation structures. Complexity results for various problems related to weighted ADFs are presented in Section 6. Section 7 discusses related work and concludes.

## 2 Background

An ADF is a directed node-labelled graph whose nodes represent statements. The links in represent dependencies: the status of a node only depends on the status of its parents (denoted ), that is, the nodes with a direct link to . In addition, each node is labelled by an associated acceptance condition specifying the conditions under which is acceptable, whence . Formally, the acceptance condition of node with parents is a function . It is convenient to represent the acceptance conditions as a collection of propositional formulas (using atoms from and connectives , , ). Then, for any interpretation , we have , that is, the acceptance condition evaluates just like evaluates . This leads to the logical representation of ADFs we will frequently use, where an ADF is a pair with the set of links implicitly given as iff appears in .

Semantics assign to ADFs a collection of partial two-valued interpretations, i.e. mappings of the statements to values where indicates that the value is undefined. Mathematically such interpretations are equivalent to 3-valued interpretations, but for the purposes of this paper it is beneficial to view them (interchangeably) also as partial interpretations. The three values are partially ordered by according to their information content: is the -least partial order containing and . As usual we write whenever and not . The information ordering extends in a straightforward way to partial interpretations over in that if and only if for all .

A partial interpretation is total if all statements are mapped to or . For interpretations and , we say that extends iff . We denote by the set of all completions of , i.e. total interpretations that extend .

For an ADF , statement and a partial interpretation , the characteristic operator is given by

 ΓD(v)(s)=⎧⎨⎩\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0tif Cs(w)=\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0t for all w∈[v]2,\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0fif Cs(w)=\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0f for % all w∈[v]2,uotherwise.

That is, the operator returns an interpretation mapping a statement to (resp. ) if and only if all two-valued interpretations extending evaluate to true (resp. false). Intuitively, checks which truth values can be justified based on the information in and the acceptance conditions.

Given an ADF , a partial interpretation is grounded with respect to if it is the least fixpoint of ; it is admissible with respect to if ; it is complete with respect to if ; it is a model of if it is complete and total; it is preferred with respect to if is maximally admissible with respect to . As shown in [11] these semantics generalize the corresponding notions defined for AFs. For , denotes the set of all admissible (resp. complete, preferred) interpretations with respect to .

###### Example 2.1

Given ADF over with , , and interpretations

 v1 = {a↦u,b↦u}, v2 = {a↦\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0t,b↦u}, v3 = {a↦\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0t,b↦\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0f}, v4 = {a↦\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0f,b↦\definecolor[named]pgfstrokecolorrgb0,0,0\pgfsys@color@gray@stroke0\pgfsys@color@gray@fill0t}.

We get , (note that , and thus ), and

## 3 The General Framework

In this section we introduce weighted ADFs (wADFs). More precisely, we introduce a general framework which allows us to define wADFs over a chosen set of values (acceptance degrees) based on an information ordering on .111Slightly abusing notation we write for both the specific ADF ordering and the generic ordering used here.

###### Definition 3.1

A weighted ADF (wADF) over is a tuple , where

• is a set (of nodes, statements, arguments; anything one might accept or not),

• is a set of links,

• is a set of truth values with ,

• is a collection of acceptance conditions over , that is, functions ,

• – where – forms a complete partial order with least element .

As for standard ADFs, the special value represents an undefined truth value. As usual, forms a complete partial order (CPO) iff: (1) it has a least element, here , (2) each non-empty subset has a greatest lower bound , and (3) each ascending chain over has a least upper bound .

ADFs are a special case of wADFs with and the information ordering as defined in the background section. We provide a formal result in Theorem 3.19 below.

###### Example 3.2

In Figure 1a a simple wADF with three arguments is shown that are intended to decide the acceptance status of a paper based on that paper’s significance and scientific methodology. On the right side of that figure (b), a value ordering is shown, with the intended meaning that denotes no knowledge w.r.t. the arguments, tendencies denote a certain leaning towards acceptance or rejection, and the information maximal values denote acceptance, borderline acceptance, or rejection.

As for ADFs, we will use propositional formulas interpreted over to specify acceptance conditions. The understanding is that a formula specifies a function such that for each interpretation , is obtained by considering , the evaluation of the formula under the interpretation . Unlike in classical propositional logic, there is no single standard way of interpreting formulas in the multi-valued case. Thus the user (specifying the wADF) should state how formulas are to be evaluated under interpretations of atoms by values from .

###### Example 3.3

Continuing Example 3.2, let us define acceptance conditions for each argument. Say argument “paper’s significance” (shortened to ) shall be set to “accept” and that the paper’s “scientific methodology” (shortened to ) shall be set to “borderline” (e.g. because of peer reviewing). This can be expressed simply by stating: accept, borderline. The third argument, shortened to , shall depend on the status of the two other arguments, namely by taking the most informative value “compatible” (w.r.t. information ordering) to the values given to the other arguments. That is, take the usual greatest lower bound for two values w.r.t. the information ordering shown in Figure 1b. Say we formalize this by , defining the conjunction as the meet.

In case the truth values in are -incomparable, the information ordering on the truth values can be defined analogously to the ordering for standard ADFs (where with and ).

###### Definition 3.4

Let be a set of truth values with . A relation is flat iff for all :

 x≤iyiffx=u or x=y

Likewise, a wADF is flat iff is flat.

As mentioned above, clearly all standard ADFs are flat. For flat orderings, the greatest lower bound of a subset is obtained thus:

 ⨅iX={xif X={x}uotherwise

We now define the semantics. A semantics takes a wADF over and produces a collection of partial interpretations from to , that is, functions with where represents the fact that the value of a certain node is undefined. Given that for standard ADFs the interpretations of interest are partial functions from to (or, equivalently, functions from to ), this is the obvious generalization we need.222This differs from approaches like [4] which consider weight assignments as part of the input and is more in line with research in multi-valued logics. Let be a wADF over . As for standard ADFs, the characteristic operator for takes a partial interpretation and produces a new interpretation, . The new partial interpretation collects information from and mediates between all completions of . As in the standard case a completion of is any total interpretation that extends with respect to the information ordering:333The notion of completion in the original AAAI paper considered total interpretations obtained by replacing with arbitrary values from only. This turns out to be insufficient for some of our results to hold.

###### Definition 3.5

Let be a wADF over , a partial interpretation. A total interpretation is a completion of if .

The set of all completions of is denoted by .

###### Example 3.6

Continuing Example 3.2, say we have interpretation “tendency accept”, “accept”, “borderline” . Then there are three completions in , , : “accept”, “accept”, “borderline” and “borderline”, “accept”, “borderline” . The reason is that both arguments and already have information maximal values associated with them (any completion assigns thus the same value), but argument has a non-maximal value, thus any completion of assigns to any value s.t. . In the example, this means that is then assigned “tendency accept” (same as for ), “accept” (), and “borderline” ().

Formally, the operator is defined as follows: for each , the truth value is the greatest lower bound with respect to (the consensus) of the set . With these specifications the rest is entirely analogous to the definitions for standard ADFs.

###### Definition 3.7

Let be a wADF and . Applying to yields a new interpretation (the consensus over ) defined as

 ΓD(v):S→Vuwiths↦⨅i{Cs(w) ∣∣ w∈[v]c}.

As usual, we can now define the semantics via fixpoints.

###### Definition 3.8

An interpretation of a wADF is

• grounded for iff , i. e., is the least fixpoint of .
Intuition: collects all the information which is beyond any doubt.

• admissible for iff .
Intuition: does not contain unjustifiable information.

• preferred for iff it is -maximal admissible for .
Intuition: has maximal information content without giving up admissibility.

• complete for iff .
Intuition: contains exactly the justifiable information.

• a model of iff for all and .
Intuition: contains exactly the information that is justifiable when each statement has a defined truth value.

Again we use , and to denote the set of all admissible, complete and preferred interpretations for , respectively. Moreover, gives the set of all models of .

###### Example 3.9

Continuing Example 3.2, let us consider some interpretations. As usual, is admissible. Since “accept” and “borderline”, for any interpretation (due to definition of their acceptance conditions), it holds that it is admissible to assign these two arguments any value lower or equal to these values (and argument “tendency accept”, “no tendency”, or ). There is only one complete interpretation: “tendency accept”, “accept”, “borderline” . This implies that is the (unique in this case) model of this wADF, and also the only preferred interpretation, as well as the grounded interpretation.

Independently of the previous example, we want to emphasize that we have to show existence of the least fixpoint of , otherwise the grounded interpretation is not well-defined in general. The simplest way to do this is to show monotonicity of the operator .

To this end, we lift the information ordering on point-wise to interpretations over . For , we set

 v≤iwiff∀s∈S:v(s)≤iw(s)

The pair then forms a CPO in which the characteristic operator of wADFs is monotone. The least element of this CPO is the interpretation that maps every statement to and we will also use to denote the least upper bound for subset of .

###### Proposition 3.10

The operator is -monotone, that is: for all interpretations we have that implies .

• Proof. Let be two interpretations such that . By definition of , we find that . Thus also for any . It follows that , that is, for any .

Existence of the least fixpoint of then follows via the fixpoint theorem for monotone operators in complete partial orders (see, e.g., [17], Theorem 8.22).

The following result is a generalization of Theorem 25 of [23] and Theorem 1 by [11].

###### Theorem 3.11

Let be a weighted adf with an information ordering .

1. Each preferred interpretation for is complete, but not vice versa.

2. The grounded interpretation for is the -least complete interpretation.

3. The complete interpretations for form a complete meet-semilattice with respect to .

• Proof. The first item is shown analogous to a previous result in [39, Theorem 3.10]. More concretely, the first statement in the theorem holds for every approximating operator, such as . The second item follows directly from definition (the grounded interpretation is a fixed point, as are all the complete interpretations; the grounded interpretation is the least one). The proof of the third item follows the same line of reasoning as the proof of the third item of [11, Theorem 1].

Next, we show that the well-known relationships between Dung semantics carry over to our generalizations.

###### Theorem 3.12

Let be a weighted adf. It holds that

If is flat, then additionally .

• Proof. The “non-flat” case follows from definitions and from Theorem 3.11: every preferred interpretation is complete, every complete interpretation is admissible. Every model is a complete interpretation by definition. For the flat case, every value assigned to a statement by a model is, directly, information maximal (due to the flat ordering). Thus, a model is an information maximal complete interpretation, which directly implies that a model is a preferred interpretation under the assumption that the information ordering is flat.

The proviso that be flat is necessary for the inclusion : consider with and given by (that is, ); now if there are with , then we find that is a model that is not preferred.

The flatness property is also crucial in the following result that lets us compute the grounded semantics by iterative application of the characteristic operator.

###### Proposition 3.13

Let be a wADF such that every ascending chain in is finite. Then, there is some such that is the grounded interpretation of . Note that denotes iterative applications of .

• Proof. As every ascending sequence of interpretations is finite and since is monotone, it follows from a known result ([17], Theorem 8.8(2)) that is Scott-continuous. Then, it follows from the Kleene Fixed-Point Theorem that is the grounded interpretation of . As the chain is finite, we get that for some .

Note that this result applies to important families of wADFs such as all flat wADFs where is finite or all wADFs with finite .

In general, one cannot guarantee that we reach the grounded interpretation by iterative application of the characteristic operator.

###### Example 3.14

Consider wADF , where the acceptance condition of the single statement is defined as

 Cs(v)={v(s)+0,5−v(s)2for v(s)<0.5,1otherwise.

We then have

 ΓD(vu)(s)=0.25,Γ2D(vu)(s)=0.375,Γ3D(vu)(s)=0.4375,Γ4D(vu)(s)=0.46875,…,

that is, for iterative application of on , the values converge towards . However, the interpretation is not the grounded interpretation of as it is no fixed point of . In fact, . Note that every wADF is guaranteed to have a grounded interpretation, in this case it is .

Despite we do not always get the grounded interpretation by iterative application of the characteristic operator, if we do reach a fixed point this way then it is the grounded interpretation.

###### Proposition 3.15

Let be a wADF. If is a fixed point of , then is the grounded interpretation of .

• Proof. The proposition follows from a known result ([17], Theorem 8.15(i)).

Another result concerns acyclic wADFs, i.e. ADFs where the pair forms an acyclic directed graph and generalizes a recent result of [33].

###### Theorem 3.16

For any acyclic wADF with countable , with the grounded interpretation of .

• Proof. We show that an acyclic wADF possesses exactly one complete interpretation. The result then follows from Theorem 3.11. Given any acyclic wADF , we can (partly) order the statements according to their “depth” in the wADF, starting from statements without parents. A path from a statement to a statement , in , is defined as with each and for all , () we have . The length of (denoted as ) is . We define the depth of a statement in as . Towards a contradiction assume such that . Let be a statement with such that is minimal (i.e., for all with ).

Consider the case , where is a leaf. As , maps to the same value for every interpretation . Therefore, we have

 ⨅i{Cs(w) ∣∣ w∈[v1]c}=⨅i{Cs(w) ∣∣ w∈[v2]c}

which is a contradiction as this is the same as which amounts to . Now assume . For every parent we have and thus, by choice of , it holds that . When we restrict the completions of and to , being the domain relevant for evaluating , we get two identical sets of functions , where

 [v1]′c={s↦w(s) ∣∣ s∈par(s),w∈[v1]c}

and

 [v2]′c={s↦w(s) ∣∣ s∈par(s),w∈[v2]c}.

One can see that

 ⨅i{Cs(w) ∣∣ w∈[v1]c}=⨅i{Cs(w) ∣∣ w∈[v1]′c}

and

 ⨅i{Cs(w) ∣∣ w∈[v2]c}=⨅i{Cs(w) ∣∣ w∈[v2]′c}.

We end up in the contradiction like in the previous case.

In the rest of this section we show how stable semantics can be generalized to weighted ADFs. The basic idea underlying stable semantics is to treat truth values asymmetrically. For standard ADFs where only and can appear in models, (false) can be assumed to hold (by default), whereas (true) needs to be justified by a derivation. Technically this is achieved by building the reduct of an ADF and then checking whether the grounded interpretation of the reduct coincides with the original model on the nodes which “survive” in the reduct.

Moving from the two-valued to the multi-valued case offers an additional degree of freedom: it is not clear a priori what the assumed, respectively derived truth values are. The stable semantics we introduce here will thus be parameterized by a subset

of the set of values over which the weighted ADF is defined.

###### Definition 3.17

Let be a wADF. Let be a model of (that is, is total). Let be the set of assumed truth values. The -reduct of is the wADF where

• ,

• ,

• where is obtained from by fixing the value of each parent of in such that to .

Note that whenever the acceptance function is represented using propositional formulas, the new acceptance function is simply obtained by replacing atoms not in by their -values. Now stable models can be defined as usual:

###### Definition 3.18

Let be a wADF and let be a model of . Let be the grounded interpretation of the -reduct of . is a -stable model of iff for each .

This clearly generalizes stable semantics for standard ADFs: just let and . We conclude this section by showing the exact relationship between ADFs and wADFs.

###### Theorem 3.19

Let be an ADF. The wADF associated to is with as defined in the background section. An interpretation is a model/admissible/complete/preferred/grounded for iff it is a model/ admissible/complete/preferred/grounded for . Moreover, is stable for iff it is -stable for .

• Proof. It suffices to show that , i.e., that the characteristic operator for the weighted ADF is the same as the characteristic operator for the non-weighted (standard) ADF . To see this, note that recovers all ingredients of standard ADFs: is a flat ordering, the same as the “pre-defined” one for on , evaluation of acceptance conditions (or formulas if the representation is different) is the same, and truth values are the same, as well. For the stable semantics, note that the same notion of reduct is recovered for -stable semantics.

## 4 Weighted ADFs Over the Unit Interval

In this section we focus on weighted ADFs over the unit interval, that is, wADFs over .

We first assume a flat information ordering . As discussed in Section 3, we will use propositional formulas to specify acceptance conditions over . In the subsequent examples, we employ a formula evaluation that is defined via structural induction as follows:

 w(φ∧ψ) =min{w(φ),w(ψ)},