The abstract argumentation frameworks (AFs) introduced by Dung (1995) have garnered increasing attention in the recent past. In his seminal paper, Dung showed how an abstract notion of argument (seen as an atomic entity) and the notion of individual attacks between arguments together could reconstruct several established KR formalisms in argumentative terms. Despite the generality of those and many more results in the field that was sparked by that paper, researchers also noticed that the restriction to individual attacks is often overly limiting, and devised extensions and generalizations of Dung’s frameworks: directions included generalizing individual attacks to collective attacks (Nielsen and Parsons, 2006), leading to so-called SETAFs; others started offering a support relation between arguments (Cayrol and Lagasquie-Schiex, 2005), preferences among arguments (Amgoud and Cayrol, 2002; Modgil, 2009), or attacks on attacks into arbitrary depth (Baroni et al., 2011). This is only the tip of an iceberg, for a more comprehensive overview we refer to the work of Brewka, Polberg, and Woltran (2014).
One of the most recent and most comprehensive generalizations of AFs has been presented by Brewka and Woltran (2010) (and later continued by Brewka et al., 2013) in the form of abstract dialectical frameworks (ADFs). These ADFs offer any type of link between arguments: individual attacks (as in AFs), collective attacks (as in SETAFs), and individual and collective support, to name only a few. This generality is achieved through so-called acceptance conditions associated to each statement. Roughly, the meaning of relationships between arguments is not fixed in ADFs, but is specified by the user for each argument in the form of Boolean functions (acceptance functions) on the argument’s parents. However, this generality comes with a price: Strass and Wallner (2015) found that the complexity of the associated reasoning problems of ADFs is in general higher than in AFs (one level up in the polynomial hierarchy). Fortunately, the subclass of bipolar ADFs (defined by Brewka and Woltran, 2010) is as complex as AFs (for all considered semantics) while still offering a wide range of modeling capacities (Strass and Wallner, 2015). However, there has only been little concerted effort so far to exactly analyze and compare the expressiveness of the abovementioned languages.
This paper is about exactly analyzing means of expression for argumentation formalisms. Instead of motivating expressiveness in natural language and showing examples that some formalisms seem to be able to express but others do not, we tackle the problem in a formal way. We use a precise mathematical definition of expressiveness: a set of interpretations is realizable by a formalism under a semantics if and only if there exists a knowledge base of the formalism whose semantics is exactly the given set of interpretations. Studying realizability in AFs has been started by Dunne et al. (2013, 2015), who analyzed realizability for extension-based semantics, that is, interpretations represented by sets where arguments are either accepted (in the extension set) or not accepted (not in the extension set). While their initial work disregarded arguments that are never accepted, there have been continuations where the existence of such “invisible” arguments is ruled out (Baumann et al., 2014; Linsbichler, Spanring, and Woltran, 2015). Dyrkolbotn (2014) began to analyze realizability for labeling-based semantics of AFs, that is, three-valued semantics where arguments can be accepted (mapped to true), rejected (mapped to false) or neither (mapped to unknown). Strass (2015) started to analyze the relative expressiveness of two-valued semantics for ADFs (relative with respect to related formalisms). Most recently, Pührer (2015) presented precise characterizations of realizability for ADFs under several three-valued semantics, namely admissible, grounded, complete, and preferred. The term “precise characterizations” means that he gave necessary and sufficient conditions for an interpretation set to be ADF-realizable under a semantics.
The present paper continues this line of work by lifting it to a much more general setting. We combine the works of Dunne et al. (2015), Pührer (2015), and Strass (2015) into a unifying framework, and at the same time extend them to formalisms and semantics not considered in the respective papers: we treat several formalisms, namely AFs, SETAFs, and (B)ADFs, while the previous works all used different approaches and techniques. This is possible because all of these formalisms can be seen as subclasses of ADFs that are obtained by suitably restricting the acceptance conditions.
Another important feature of our framework is that we uniformly use three-valued interpretations as the underlying model theory. In particular, this means that arguments cannot be “invisible” any more since the underlying vocabulary of arguments is always implicit in each interpretation. Technically, we always assume a fixed underlying vocabulary and consider our results parametric in that vocabulary. In contrast, for example, Dyrkolbotn (2014) presents a construction for realizability that introduces new arguments into the realizing knowledge base; we do not allow that. While sometimes the introduction of new arguments can make sense, for example if new information becomes available about a domain or a debate, it is not sensible in general, as these new arguments would be purely technical with an unclear dialectical meaning. Moreover, it would lead to a different notion of realizability, where most of the realizability problems would be significantly easier, if not trivial.
The paper proceeds as follows. We begin with recalling and introducing the basis and basics of our work – the formalisms we analyze and the methodology with which we analyze them. Next we introduce our general framework for realizability; the major novelty is our consistent use of so-called characterization functions, firstly introduced by Pührer (2015), which we adapt to further semantics. The main workhorse of our approach will be a parametric propagate-and-guess algorithm for deciding whether a given interpretation set is realizable in a formalism under a semantics. We then analyze the relative expressiveness of the considered formalisms, presenting several new results that we obtained using an implementation of our framework. We conclude with a discussion.
We make use of standard mathematical concepts like functions and partially ordered sets. For a function we denote the update of with a pair by with if , and otherwise. For a function and , its preimage is . A partially ordered set is a pair with a partial order on . A partially ordered set is a complete lattice if and only if every has both a greatest lower bound (glb) and a least upper bound (lub) . A partially ordered set is a complete meet-semilattice iff every non-empty subset has a greatest lower bound (the meet) and every ascending chain has a least upper bound .
Let be a fixed finite set of statements. An interpretation is a mapping that assigns one of the truth values true (), false () or unknown () to each statement. An interpretation is two-valued if , that is, the truth value is not assigned. Two-valued interpretations can be extended to assign truth values to propositional formulas as usual.
The three truth values are partially ordered according to their information content: we have and and no other pair in , which intuitively means that the classical truth values contain more information than the truth value unknown. As usual, we denote by the partial order associated to the strict partial order . The pair forms a complete meet-semilattice with the information meet operation . This meet can intuitively be interpreted as consensus and assigns , , and returns otherwise.
The information ordering extends in a straightforward way to interpretations over in that iff for all . We say for two interpretations that extends iff . The set of all interpretations over forms a complete meet-semilattice with respect to the information ordering . The consensus meet operation of this semilattice is given by for all . The least element of is the valuation mapping all statements to unknown – the least informative interpretation. By we denote the set of two-valued interpretations; they are the -maximal elements of the meet-semilattice . We denote by the set of all two-valued interpretations that extend . The elements of form an -antichain with greatest lower bound .
Abstract Argumentation Formalisms
An abstract dialectical framework (ADF) is a tuple where is a set of statements (representing positions one can take or not take in a debate), is a set of links (representing dependencies between the positions), is a collection of functions , one for each statement . The function is the acceptance condition of and expresses whether can be accepted, given the acceptance status of its parents . We usually represent each by a propositional formula over . To specify an acceptance condition, then, we take to hold iff is a model for .
Brewka and Woltran (2010) introduced a useful subclass of ADFs: an ADF is bipolar iff all links in are supporting or attacking (or both). A link is supporting in iff for all , we have that implies . Symmetrically, a link is attacking in iff for all , we have that implies . If a link is both supporting and attacking then has no actual influence on . (But the link does not violate bipolarity.) We write BADFs as and mean that contains all supporting links and all attacking links.
The semantics of ADFs can be defined using an operator over three-valued interpretations (Brewka and Woltran, 2010; Brewka et al., 2013). For an ADF and a three-valued interpretation , the interpretation is given by
That is, for each statement , the operator returns the consensus truth value for its acceptance formula , where the consensus takes into account all possible two-valued interpretations that extend the input valuation . If this is two-valued, we get and thus .
The standard semantics of ADFs are now defined as follows. For ADF , an interpretation is
admissible iff ;
complete iff ;
preferred iff it is -maximal admissible;
a two-valued model iff it is two-valued and .
We denote the sets of interpretations that are admissible, complete, preferred, and two-valued models by , , and , respectively. These definitions are proper generalizations of Dung’s notions for AFs: For an AF , where is the attack relation, the ADF associated to is with and for all . AFs inherit their semantics from the definitions for ADFs (Brewka et al., 2013, Theorems 2 and 4). In particular, an interpretation is stable for an AF if and only if it is a two-valued model of .
A SETAF is a pair where is the (set) attack relation. We define three-valued counterparts of the semantics introduced by Nielsen and Parsons (2006), following the same conventions as in three-valued semantics of AFs (Caminada and Gabbay, 2009) and argumentation formalisms in general. Given a statement and an interpretation we say that is acceptable wrt. if and is unacceptable wrt. if . For an interpretation it holds that
iff for all , is acceptable wrt. if and is unacceptable wrt. if ;
iff for all , is acceptable wrt. iff and is unacceptable wrt. iff ;
iff is -maximal admissible; and
iff and .
For a SETAF the corresponding ADF has acceptance formula for each statement . (Polberg, 2016)
For any SETAF it holds that , where . Given interpretation and statement , it holds that iff iff iff is acceptable wrt. and iff iff iff is unacceptable wrt. . Hence for .
A set of interpretations is realizable in a formalism under a semantics if and only if there exists a knowledge base having exactly . Pührer (2015) characterized realizability for ADFs under various three-valued semantics. We will reuse the central notions for capturing the complete semantics in this work.
Definition 1 (Pührer 2015).
Let be a set of interpretations. A function is a -characterization of iff: for each we have iff for each :
implies for all and
implies and for some .
From a function of this kind we can build a corresponding ADF by the following construction. For a function , we define as the ADF where the acceptance formula for each statement is given by
Observe that we have iff by definition. Intuitively, the acceptance condition is constructed such that is a model of if and only if we find .
Proposition 2 (Pührer 2015).
Let be a set of interpretations. (1) For each ADF with , there is a -characterization for ; (2) for each -characterization for we have .
The result shows that can be realized under complete semantics if and only if there is a -characterization for .
3 A General Framework for Realizability
The main underlying idea of our framework is that all abstract argumentation formalisms introduced in the previous section can be viewed as subclasses of abstract dialectical frameworks. This is clear for ADFs themselves and for BADFs by definition; for AFs and SETAFs it is fairly easy to see. However, knowing that these formalisms can be recast as ADFs is not everything. To employ this knowledge for realizability, we must be able to precisely characterize the corresponding subclasses in terms of restricting the ADFs’ acceptance functions. Alas, this is also possible and paves the way for the framework we present in this section. Most importantly, we will make use of the fact that different formalisms and different semantics can be characterized modularly, that is, independently of each other.
Towards a uniform account of realizability for ADFs under different semantics, we start with a new characterization of realizability for ADFs under admissible semantics that is based on a notion similar in spirit to -characterizations.
Let be a set of interpretations. A function is an -characterization of iff: for each we have iff for every :
implies for all .
Note that the only difference to Definition 1 is dropping the second condition related to statements with truth value .
Let be a set of interpretations. (1) For each ADF such that , there is an -characterization for ; (2) for each adm-characterization for we have . (1) We define the function as for every and where is the acceptance formula of in . We will show that is an -characterization for . Let be an interpretation. Consider the case and for some and some . From we get . By definition of is follows that . Now assume and consequently . There must be some such that and . Hence, there is some with and by definition of . Thus, is an -characterization
(2) Observe that for every two-valued interpretation and every we have . : Let be an interpretation and a statement such that . Let be a two-valued interpretation with . Since we have . Therefore, by our observation it must also hold that . Thus, by Definition 2, . : Consider an interpretation such that . We show that . From we get . There must be some such that and . Hence, there is some with and consequently . Thus, by Definition 2 we have .
When listing sets of interpretations in examples, for the sake of readability we represent three-valued interpretations by sequences of truth values, tacitly assuming that the underlying vocabulary is given and has an associated total ordering. For example, for the vocabulary we represent the interpretation by the sequence .
Consider the sets and of interpretations over . The mapping is an -characterization for . Thus, the ADF has as its admissible interpretations. Indeed, the realizing ADF has the following acceptance conditions:
For no -characterization exists because but the implication of Definition 2 trivially holds for , , and .
We have seen that the construction for realizing under complete semantics can also be used for realizing a set of interpretations under admissible semantics. The only difference is that we here require to be an -characterization instead of a -characterization for . Note that admissible semantics can be characterized by properties that are easier to check than existence of an -characterization (see the work of Pührer, 2015). However, using the same type of characterizations for different semantics allows for a unified approach for checking realizability and constructing a realizing ADF in case one exists.
For realizing under the model semantics, we can likewise present an adjusted version of -characterizations.
Let be a set of interpretations. A function is a -characterization of if and only if: (1) is defined on (that is, ) and (2) for each , we have iff .
As we can show, there is a one-to-one correspondence between -characterizations and ADF realizations.
Let be a set of interpretations. (1) For each ADF such that , there is a -characterization for ; (2) vice versa, for each -characterization for we find . (1) Let be an ADF with . It immediately follows that . To define we can use the construction in the proof of Proposition 3. It follows directly that for any , we find iff . Thus is a -characterization for .
(2) Let and be a -characterization of . For any we have:
A related result was given by Strass (2015, Proposition 10). The characterization we presented here fits into the general framework of this paper and is directly usable for our realizability algorithm. Wrapping up, the next result summarizes how ADF realizability can be captured by different types of characterizations for the semantics we considered so far.
Let be a set of interpretations and consider . There is an ADF such that if and only if there is a -characterization for .
The preferred semantics of an ADF is closely related to its admissible semantics as, by definition, the preferred interpretations of are its -maximal admissible interpretations. As a consequence we can also describe preferred realizability in terms of -characterizations. We use the lattice-theoretic standard notation to select the -maximal elements of a given set of interpretations.
Let be a set of interpretations. There is an ADF with iff there is an -characterization for some with and .
Finally, we give a result on the complexity of deciding realizability for the mentioned formalisms and semantics.
Let be a formalism and be a semantics. The decision problem “Given a vocabulary and a set of interpretations over , is there a such that ?” can be decided in nondeterministic time that is polynomial in the size of .111We assume here that the representation of any over has size . There might be specific with smaller representations, but we cannot assume any better for the general case.
For all considered and , computing all -interpretations of a given witness can be done in time that is linear in the size of . Comparing the result to can also be done in linear time.
3.1 Deciding Realizability: Algorithm 1
Our main algorithm for deciding realizability is a propagate-and-guess algorithm in the spirit of the DPLL algorithm for deciding propositional satisfiability (Gomes et al., 2008). It is generic with respect to (1) the formalism and (2) the semantics for which should be realized. To this end, the propagation part of the algorithm is kept exchangeable and will vary depending on formalism and semantics. Roughly, in the propagation step the algorithm uses the desired set of interpretations to derive certain necessary properties of the realizing knowledge base (line 2). This is the essential part of the algorithm: the derivation rules (propagators) used there are based on characterizations of realizability with respect to formalism and semantics. Once propagation of properties has reached a fixed point (line 7), the algorithm checks whether the derived information is sufficient to construct a knowledge base. If so, the knowledge base can be constructed and returned (line 9). Otherwise (no more information can be obtained through propagation and there is not enough information to construct a knowledge base yet), the algorithm guesses another assignment for the characterization (line 11) and calls itself recursively.
The main data structure that Algorithm 1 operates on is a set of triples consisting of a two-valued interpretation , an atom and a truth value . This data structure is intended to represent the -characterizations introduced in Definitions 3, 2 and 1. There, a -characterization is a function from two-valued interpretations to two-valued interpretations. However, as the algorithm builds the -characterization step by step and there might not even be a -characterization in the end (because is not realizable), we use a set of triples to be able to represent both partial and incoherent states of affairs. The -characterization candidate induced by is partial if we have that for some and , neither nor ; likewise, the candidate is incoherent if for some and , both and . If is neither partial nor incoherent, it gives rise to a unique -characterization that can be used to construct the knowledge base realizing the desired set of interpretations. The correspondence to the characterization-function is then such that iff .
In our presentation of the algorithm we focused on its main features, therefore the guessing step (line 11) is completely “blind”. It is possible to use common CSP techniques, such as shaving (removing guessing possibilities that directly lead to inconsistency). Finally, we remark that the algorithm can be extended to enumerate all possible realizations of a given interpretation set – by keeping all choice points in the guessing step and thus exhaustively exploring the whole search space.
In the case where the constructed relation becomes functional at some point, the algorithm returns a realizing knowledge base . For ADFs, this just means that we denote by the -characterization represented by and set . For the remaining formalisms we will introduce the respective constructions in later subsections.
The algorithm is parametric in two dimensions, namely with respect to the formalism and with respect to the semantics . These two aspects come into the algorithm via so-called propagators. A propagator is a formalism-specific or semantics-specific set of derivation rules. Given a set of desired interpretations and a partial -characterization , a propagator derives new triples that must necessarily be part of any total -characterization for such that extends . In the following, we present semantics propagators for admissible, complete and two-valued model (in (SET)AF terms stable) semantics, and formalism propagators for BADFs, AFs, and SETAFs.
3.2 Semantics Propagators
These propagators (cf. Figure 1) are directly derived from the properties of -characterizations presented in Definitions 3, 2 and 1. While the definitions provide exact conditions to check whether a given function is a -characterization, the propagators allow us to derive definite values of partial characterizations that are necessary to fulfill the conditions for being a -characterization.
For admissible semantics, the condition for a function to be an -characterization of a desired set of interpretations (cf. Definition 2) can be split into a condition for desired interpretations and two conditions for undesired interpretations . Propagator derives new triples by considering interpretations . Here, for all two-valued interpretations that extend , the value has to be in accordance with on ’s Boolean part, that is, the algorithm adds whenever . On the other hand, derives new triples for in order to ensure that there is a two-valued interpretation extending where differs from on a Boolean value of . Note that while immediately allows us to derive information about for each desired interpretation , propagator is much weaker in the sense that it only derives a triple of if there is no other way to meet the conditions for an undesired interpretation. Special treatment is required for the interpretation that maps all statements to and is admissible for every ADF. This is not captured by and as these deal only with interpretations that have Boolean mappings. Thus, propagator serves to check whether . If this is not the case, the propagator immediately makes the relation incoherent and the algorithm correctly answers “no”.
For complete semantics and interpretations , propagator derives triples just like in the admissible case. Propagator deals with statements having for which there have to be at least two having and . Hence derives triple if for all other we find a triple . For interpretations it must hold that there is some such that (i) and for some or (ii) but for all , assigns the same Boolean truth value to . Now if neither (i) nor (ii) can be fulfilled by any statement due to the current contents of , propagators and derive triple for if needed for to fulfill (i) and for if needed for to fulfill (ii), respectively.
Consider the set . First, we consider a run of . In the first iteration, propagator ensures that in line 2 contains , , , and . Based on the latter three tuples and , propagator derives in the second iteration which together with causes the algorithm to return “no”. Consequently, is not -realizable. A run of on the other hand returns -characterization for that maps to , to , and to and all other to . Hence, ADF , given by the acceptance conditions
has as its complete semantics.
Finally, for two-valued model semantics, propagator derives new triples by looking at interpretations . For those, we must find in each -characterization by definition. Thus the algorithm adds for each to the partial characterization . Propagator looks at interpretations , for which it must hold that . Thus there must be a statement with , which is exactly what this propagator derives whenever it is clear that there is only one statement candidate left. This, in turn, is the case whenever all with the opposite truth value and all with cannot coherently become the necessary witness any more. The propagator checks whether , that is, the desired set of interpretations consists entirely of two-valued interpretations. In that case this propagator makes the relation incoherent, following a similar strategy as .
Realizing a given set of interpretations under preferred semantics requires special treatment. We do not have a -characterization function for at hand to directly check realizability of but have to find some such that is realizable under admissible semantics (cf. Corollary 6). Algorithm 2 implements this idea by guessing such a (line 7) and then using Algorithm 1 to try to realize under admissible semantics (line 11). If returns a knowledge base realizing under we can directly use as solution of since it holds that , given that is an -antichain (line 2).