# weighted_abstract_dialectical_frameworks__96ab7bd4.pdf Weighted Abstract Dialectical Frameworks Gerhard Brewka, Hannes Strass Leipzig University, Computer Science Institute Leipzig, Germany Johannes P. Wallner, Stefan Woltran TU Wien, Institute of Information Systems Vienna, Austria 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. 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) (Dung 1995) 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 Copyright c 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. 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. (Baroni, Caminada, and Giacomin 2011) 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 Cayrol and Lagasquie-Schiex, 2009). One of the most systematic and flexible outcomes of this research are abstract dialectical frameworks (ADFs) (Brewka and Woltran 2010; Brewka et al. 2013). 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 (Al-Abdulkarim, Atkinson, and Bench-Capon 2014; 2016) and text exploration (Cabrio and Villata 2016). A mobile argumentation app based on ADF techniques was developed by P uhrer (2017). The operator-based semantics of ADFs can be traced back to the work of Denecker, Marek, and Truszczy nski (2000; 2003; 2004) on approximation fixpoint theory (AFT), an algebraic framework for studying semantics of knowledge representation formalisms. We refer to the work of Strass (2013) 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: The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18) 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. (Alsinet et al. 2017) 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 s 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 (S, L, C) whose nodes represent statements. The links in L represent dependencies: the status of a node s only depends on the status of its parents (denoted par(s)), that is, the nodes with a direct link to s. In addition, each node s is labelled by an associated acceptance condition Cs specifying the conditions under which s is acceptable, whence C = {Cs}s S. Formally, the acceptance condition Cs of node s with parents par(s) is a function Cs : (par(s) {t, f}) {t, f}. It is convenient to represent the acceptance conditions as a collection Φ = {ϕs}s S of propositional formulas (using atoms from par(s) and connectives , , ). Then, for any interpretation w : par(s) {t, f}, we have Cs(w) = w(ϕs), that is, the acceptance condition Cs evaluates w just like w evaluates ϕs. This leads to the logical representation of ADFs we will frequently use, where an ADF is a pair (S, Φ) with the set of links L implicitly given as (a, b) L iff a appears in ϕb. Semantics assign to ADFs a collection of partial twovalued interpretations, i.e. mappings of the statements to values {t, f, u} where u 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 i according to their information content: i is the -least partial order containing u i t and u i f. As usual we write v1 0.5 and y > 0.5; x y = 0 otherwise x y = 1 if x > 0.5 or y > 0.5; x y = 0 otherwise However, this approach appears to throw out the baby with the bath water: the different behavior of w ADFs in such examples stems from the fact that they make more fine-grained distinctions. It is thus not unintended. Note also that there is an easy alternative option to specify tautological acceptance conditions for w ADFs: simply replace a a with 1. So far we have used a flat information ordering with least element u and different elements in [0, 1] incomparable. Of course, nothing prevents us from choosing a more refined ordering, for instance the ordering i given by x i y iff x i y or y < x 0.5 or 0.5 x < y That is, 0.5 is immediately above u, and a value smaller than 0.5 is more informative if it is closer to 0, a value greater than 0.5 is more informative if it is closer to 1. The pair ([0, 1] {u} , i) again forms a complete partial order; for any non-empty X [0, 1] {u}, its greatest lower bound is given by u if u X inf X if X [0.5, 1] sup X if X [0, 0.5] 0.5 otherwise where inf and sup are greatest lower bound and least upper bound in the complete lattice ([0, 1], ). The CPO property extends to the pointwise extension of i to valuations. Thus for any D = (S, L, C, [0, 1], i), its characteristic w ADF operator ΓD is well-defined, in particular its least fixpoint lfp(ΓD) exists and is uniquely determined. Example 4. Consider again Example 2, but this time with information ordering i. Again we iterate ΓD on the interpretation (u, u, u, u). With the extended information ordering we obtain the least fixpoint v1 = (0.8, 0.5, 0.5, 0.6). The more refined information ordering thus leads to more informative results, as expected. Due to space constraints determining the other semantics is left to the reader. In general, arbitrary CPO-preserving refinements of the information ordering have the following effects: (1) The model semantics is unaffected, since it is independent of the information ordering, but only depends on the truth values V ; (2 4) grounded, admissible, preferred, and complete semantics may increase in information content, but never decrease. Proposition 7. Let D = (S, L, C, V, i) be a w ADF with information-ordering CPO (Vu, i). Furthermore let i Vu Vu with i i be such that (Vu, i) is a CPO, and define D = (S, L, C, V, i). Then for each v : S Vu and each w : S V , we have: 1. ΓD(w) = w if and only if ΓD (w) = w; 2. lfp(ΓD) i lfp(ΓD ); 3. adm(ΓD) adm(ΓD ); and 4. if v prf(ΓD) (v com(ΓD)) then there is a v prf(ΓD ) (v com(ΓD )) with v i v . 5 Alternative Valuation Structures In the last section we considered values in [0, 1]. Of course, there are many more options which have been studied intensively in the area of multi-valued logics (an excellent overview was given by Gottwald, 2015), e.g. Wm = { k m 1 | 0 k m 1},3 or Belnap s 4-valued system with { , { }, { }, { , }}. Given D = (S, L, C, V, i), for any chosen set of truth degrees V , an interpretation assigns a value from V to nodes in S, and the characteristic operator ΓD works on partial V -interpretations, that is, on functions of the type S Vu where u / V . Acceptance conditions are of the form Cs : (par(s) V ) V . As before we will represent acceptance conditions as propositional formulas, this time interpreted over V . To make ADF techniques work we need to define the evaluation of propositional connectives over V , thus specifying which acceptance condition a formula actually represents, and the information ordering, as done for [0, 1] in Section 4. The literature on multi-valued logics provides a rich source of alternative valuation structures with different benefits and properties. It also offers a wide range of options regarding different evaluations of propositional formulas (e. g., G odel, Łukasiewicz, etc.). The only general constraint is that the information ordering i on V {u} must form a complete partial order (CPO) with least element u. In the following we illustrate the use of alternative valuation structures using three different examples. 3As an example consider W5 = {0, 1 W3: Let us start with W3 = {0, 0.5, 1} We define formula evaluation as before, that is, 0, 0.5, and 1 evaluate to themselves, to min, to max, and y to 1 y. We choose as the information ordering the smallest reflexive relation containing u i x for all x W3, where the remaining values are incomparable if they are different. This fully specifies w ADFs based on W3. Note that 0.5 = u. This makes perfect sense as saying the acceptance degree is 0.5 is different from saying the acceptance degree is unknown . Belnap: We show how the truth degrees in Belnap s fourvalued logic can be used in w ADFs. The truth degrees are B = { , { }, { }, { , }}. As to formula evaluation we use the standard definitions for Belnap s logic: conjunction/disjunction are the infimum/supremum under the truth ordering t; negation swaps { } and { }, and leaves the other two values unchanged. The truth ordering is the reflexive closure of: { } t t { } { } t { , } t { } The information ordering is the reflexive closure of: u i i { } i { , } u i i { } i { , } With these definitions, the operator ΓD and thus the 4-valued w ADF system is fully specified. Note again that u = ; treating them as identical would yield a different system. Intervals: Our approach can also handle intervals. Let us illustrate this idea using intervals from within the unit interval as truth degrees, that is we consider truth values in INT = {[a, b] | 0 a b 1}. Formula evaluation can be defined as follows: [a, b] [c, d] = [min(a, c), min(b, d)] [a, b] [c, d] = [max(a, c), max(b, d)] [a, b] = [1 b, 1 a] The information ordering is the -least relation satisfying (1) u i v for all v INT, and (2) [a, b] i [c, d] whenever [c, d] [a, b], which fully defines the characteristic operator ΓD. 6 Computational Complexity In this section we give a preliminary complexity analysis of weighted ADFs. For this we assume that all acceptance conditions are specified via propositional formulas, which, additionally, may contain constants from a pre-specified and fixed set of values V . Further, we assume that evaluating an acceptance condition under an interpretation that assigns no statement to u can be done in polynomial time, and that comparing two values under the, again pre-specified and fixed, information ordering can, likewise, be computed in polynomial time with respect to the size of the given w ADF. That is, both V and i are fixed (|V | is a constant) and not part of the input for the problems we study in this section. We first show that, under fixed and finite valuation structures V , complexity upper bounds of w ADFs remain the same as for classical ADFs (Strass and Wallner 2015) for several central computational tasks. As for ADFs, a cornerstone auxiliary complexity result is the following for checking whether a given interpretation is admissible in a given w ADF. Proposition 8. Verifying whether a given interpretation is admissible in w ADFs with fixed and finite valuation structures is in co NP. Based on the previous result, checking whether there exists an admissible interpretation for a given w ADF that assigns a given value, different to u, to a given statement, has the same complexity upper bound as the analogous task of credulous acceptance on ADFs. The complexity class Σp 2 contains all problems solvable via a non-deterministic polynomial-time algorithm that has access to an NP-oracle (which can solve problems in NP in constant time). Proposition 9. Checking whether there is an admissible interpretation assigning a given statement a given value for w ADFs with fixed and finite valuation structures is in Σp 2. Checking whether all preferred interpretations for a given w ADF assign a given value, different to u, to a given statement, has the same complexity upper bound as the analogous task of skeptical acceptance on ADFs. The class Πp 3 is the complement class of Σp 3 (contains the complements of all problems in Σp 3), which in turn contains all problems solvable via a non-deterministic polytime algorithm with access to a Σp 2-oracle. Proposition 10. Checking whether all preferred interpretations assign a given statement a given value for w ADFs with fixed and finite valuation structures is in Πp 3. Analogously to ADFs, the same complexity upper bound for existence of stable models can be derived for w ADFs with fixed and finite valuation structures. The assumed values can be arbitrarily chosen among the fixed V . Proposition 11. Checking existence of stable models is in Σp 2 for w ADFs with fixed and finite valuation structures. Complexity lower bounds depend on the fixed V , information ordering, and formula evaluation. Non-weighted ADFs (i. e., V = {t, f}) are an example where, for the corresponding fixed components, the complexity lower bounds match the previously shown upper bounds. Finally, we show a result for w ADFs over the unit interval. As before, we make the same assumptions on the acceptance conditions, but let V be the unit interval and assume a flat information ordering (u is strictly lower than all other elements, and all other elements are incomparable), and assume formula evaluation as defined in Section 4. Although ADFs are not a special case, with respect to all semantics, of such w ADFs, co NP-hardness of verifying whether a given interpretation is admissible follows from a similar argument as for classical (non-weighted) ADFs. Proposition 12. Verifying whether a given interpretation is admissible in w ADFs over the unit interval with flat information ordering is co NP-hard. 7 Related Work and Conclusions In this paper we introduced a framework for defining weighted ADFs, a generalization of ADFs allowing to assign acceptance degrees to arguments. The framework is fully flexible regarding the choice of acceptance degrees and their associated information ordering. We have provided definitions of the main semantics and showed a number of properties together with a preliminary complexity analysis. There is quite some work on weighted argumentation frameworks, and even a section entitled Weighted ADFs in (Brewka and Woltran 2010). In many cases weights of some sort are added to the links in the argument graph, not to the nodes. For instance, Brewka and Woltran use weights on links to simplify the definition of acceptance conditions, an idea that has later been extended to the GRAPPA framework (Brewka and Woltran 2014). In (Dunne et al. 2011) weights on links are used as an inconsistency budget : attacks may be disregarded as long as the sum of the weights of disregarded attacks remains under some threshold. Here we focus on papers assigning acceptance degrees to argument nodes. One such approach is Gabbay s equational theory of argumentation frameworks (Gabbay 2012). He allows for values in the unit interval and represents AFs in an equational form, where the equations specify certain constraints value assignments need to satisfy. There are also probabilistic extensions of AFs, e.g. (Dung and Thang 2010; Li, Oren, and Norman 2011; Hunter 2013; Hunter and Thimm 2014a; 2014b) (for a complexity analysis for probabilistic AFs see (Fazzinga, Flesca, and Parisi 2015)), and even of ADFs (Polberg and Doder 2014). The main idea is to generate several standard AFs (resp. ADFs) which represent the possible situations induced by the probabilities. The latter can be assigned to arguments, attacks, and in case of ADFs, to acceptance conditions. The evaluation of frameworks generated this way follows the standard approach, and the results of these evaluations are aggregated accordingly. The behavior of the semantics is thus triggered via all relevant subgraphs. A related approach in a multivalued setting is (Dondio 2014; 2017). Social AFs (Leite and Martins 2011) extend standard AFs by adding to each argument associated numbers of positive and negative votes. The semantics describe how the votes propagate through the network, yielding a non-linear system of equations. Recently, several properties and semantics for weighted AFs have been proposed in (Amgoud, Ben Naim, and Vesic 2017; Amgoud et al. 2017). In those works, weights for arguments are also given from the unit interval, interpreted in the sense that the greater the value the more acceptable the argument. The focus is on the definition of new semantics dedicated for weighted AFs, rather than on generalizing standard semantics. However, the properties proposed in (Amgoud et al. 2017) adapted to our setting are of interest and thus are on our agenda for future work. Finally in (Besnard and Hunter 2001) acceptance grades of arguments are derived from the structure of the argument tree. The authors attempt to provide an abstraction of an argument tree in the form of a single number . In a similar vein Grossi and Modgil (2015) derive acceptance grades from the structure of the underlying AF, e.g. the number of attacks against which a particular argument is defended. These approaches are orthogonal to ours. Our generalization of ADFs differs significantly from the mentioned papers in at least the following respects: 1. We are more general than existing work (with the exception of the work of Polberg and Doder, 2014) in taking ADFs rather than AFs as starting point. 2. Rather than focusing on a single set of acceptance values, we provide a framework where the values can be freely selected based on the needs of a particular application. 3. Our semantics are a direct generalization of the operatorbased ADF semantics and does not require the computation and aggregation of results for various subgraphs. Moreover, we obtain reasonable results also in cases where equational approaches do not have solutions. 4. Finally, the choice of an adequate information ordering allows us to do some fine tuning which is not possible in any approach we are aware of. As to future work, we first want to explore restricted subclasses of weighted ADFs. In particular, we would like to exploit the known concept to express Dung AFs as ADFs (see (Brewka et al. 2013), Theorem 2) in order to investigate a new form of weighted AFs as a subclass of weighted ADFs. Our general definitions of the standard semantics for weighted ADFs will readily deliver natural definitions of semantics for such weighted AFs. Furthermore, the subclass of bipolar ADFs has been recognised as a useful class, as they are strictly more expressive than AFs while of equal computational complexity (Strass and Wallner 2015; Strass 2015; Linsbichler, P uhrer, and Strass 2016), so we intend to investigate weighted bipolar ADFs. A first step would be the generalization of supporting and attacking links to the multi-valued setting, for example via regarding of acceptance functions monotonicity and antimonotonicity in single (function) arguments (Baumann and Strass 2017). We also would like to explore an idea that goes back to Bogaerts, Vennekens, and Denecker (2016). In our approach, as in standard AFT, interpretations v : S V of atoms S with truth values V are approximated by functions v : S Vu with Vu = V {u} for u / V . Such threevalued/partial interpretations consequently represent the set of their completions. However, not all sets of total interpretations can be represented as completions of a partial interpretation. This is due to the fact that partial interpretations either assign a specific value, or leave the value completely undefined. This suggests the following: A generalized partial interpretation (gpi) of S in V is a total function v : S 2V \ { }, that is, a gpi assigns to each element of S a non-empty subset of the values in V . In this new setting, the total function w : S V is a completion of v if and only if for all s S, we have w(s) v(s). Based on the notion of a gpi we can generalize the characteristic ADF operator ΓD to operate on gpis rather than partial interpretations. For each node s, the revised gpi ΓD(g) returns the set of values that are obtained by evaluating the acceptance condition of s under any completion of the input gpi g. A further investigation of this topic is on our agenda. Acknowledgements This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) under grant BR 1817/7-2 and by the Austrian Science Fund (FWF): I2854 and P30168-N31. Al-Abdulkarim, L.; Atkinson, K.; and Bench-Capon, T. J. M. 2014. Abstract dialectical frameworks for legal reasoning. In Proc. 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