A Logical Account of Formal Argumentation

Martin Caminada Dov Gabbay A Logical Account of Formal Argumentation Paper 347, May 2009 Abstract. In the current paper, we re-examine how abstract argumentation can be formulated in terms of labellings, and how the resulting theory can be applied in the field of modal logic. In particular, we are able to express the (complete) extensions of an argumentation framework as models of a set of modal logic formulas that represents the argumentation framework. Using this approach, it becomes possible to define the grounded extension in terms of modal logic entailment. Keywords: abstract argumentation, argument labellings, modal logic, grounded semantics 1. Introduction Formal argumentation has become a popular approach for purposes varying from nonmonotonic reasoning [2, 7], multi-agent communication [1] and reasoning in the semantic web [23]. Although some research on formal argumentation can be traced back to the early 1990s (like for instance the work of Vreeswijk [27] and of Simari and Loui [24]) the topic really started to take off with Dung’s theory of abstract argumentation [12]. Here, arguments are seen as abstract entities (although they can be instantiated using approaches like [7] and [22]) among which an attack relationship is defined. The thus formed argumentation framework can be represented as a directed graph in which the arguments serve as nodes and the attack relation as the arrows. Given such a graph, an interesting question is which sets of nodes can reasonably be accepted. Several criteria of acceptance have been stated, including grounded, complete, preferred and stable semantics [12], as well as more recent approaches like semi-stable semantics [10] and ideal semantics [13]. Despite its relative popularity, formal argumentation has been criticized for its lack of meta-theory [6]. Although quite extensive work has been done describing the properties of the various argument-based semantics [4, 3], what is lacking is a comprehensive way of expressing properties of argumentation using existing logical approaches. Presented by ; Received June 1, 2009 Studia Logica (2009) 0: 1–38 c Springer 2009 2 Caminada and Gabbay In this paper, we describe several alternative ways to express argumentation (and in particular argumentation semantics) other then the traditional extensions approach. In particular, we focus on argument labellings, classical logic and various forms of modal logic. The remaining part of this paper is structured as follows. First, in Section 2, we treat some basic concepts of abstract argumentation, including the traditional extensions approach. Then, in Section 3, we describe the labellings approach and show how this can be used an alternative way to describe the standard admissibility based argumentation semantics. Section 4 then describes how argumentation can be expressed in modal logic. In section 5 argumentation is described using classical logic, and in Section 6 we introduce an alternative approach for using modal logic to describe argumentation. We then round off with a discussion of related work in Section 7. 2. Argumentation Preliminaries An argumentation framework [12] consists of a set of arguments and an attack relation on these arguments. In order to simplify the discussion, we consider only finite argumentation frameworks. Definition 1. Let U be the universe of all possible arguments. An argumentation framework is a pair (Ar , att ) where Ar is a finite subset of U and att ⊆ Ar × Ar. We say that an argument A attacks an argument B iff (A, B) ∈ att . An argumentation framework can be depicted as a directed graph in which the arguments are represented as nodes and the attack relation is represented as arrows. For instance, argumentation framework (Ar , att) where Ar = {A, B, C, D, E} and att = {(A, B), (B, A), (B, C), (C, D), (D, E), (E, C)} is represented in Figure 1. D A B C E Figure 1. An argumentation framework represented as a directed graph. A Logical Account of Formal Argumentation 3 The shorthand notation A+ and A− stands for, respectively, the set of arguments attacked by argument A and the set of arguments that attack argument A. Likewise, if Args is a set of arguments, then we write Args + for the set of arguments that are attacked by at least one argument in Args, and Args − for the set of arguments that attack at least one argument in Args. In the definition below, F (Args) stands for the set of arguments that are acceptable in the sense of [12]. Definition 2 (defense / conflict-free). Let (Ar , att ) be an argumentation framework, A ∈ Ar and Args ⊆ Ar. We define A+ as {B | A att B} and Args + as {B | A att B for some A ∈ Args}. We define A− as {B | B att A} and Args − as {B | B att A for some A ∈ Args}. Args is conflict-free iff Args ∩ Args + = ∅. Args defends an argument A iff A− ⊆ Args + . We define the function F : 2Ar → 2Ar as F (Args) = {A | A is defended by Args}. In the definition below, definitions of grounded, preferred and stable semantics are described in terms of complete semantics, which has the advantage of making the proofs in the remainder of this paper more straightforward. These descriptions are not literally the same as the ones provided by Dung [12], but as was first stated in [8], these are in fact equivalent to Dung’s original versions of grounded, preferred and stable semantics. Definition 3 (acceptability semantics). Let (Ar , att ) be an argumentation framework and let Args ⊆ Ar be a conflict-free set of arguments. - Args is admissible iff Args ⊆ F (Args). - Args is a complete extension iff Args = F (Args). - Args is a grounded extension iff Args is the minimal (w.r.t. set-inclusion) complete extension. - Args is a preferred extension iff Args is a maximal (w.r.t. set-inclusion) complete extension. - Args is a stable extension iff Args is a complete extension that attacks every argument in Ar \Args. - Args is a semi-stable extension iff Args is a complete extension where Args ∪ Args + is maximal (w.r.t. set-inclusion). 4 Caminada and Gabbay As an example, in the argumentation framework of Figure 1 {B, D} is a stable extension, {A} is a preferred extension which is neither stable nor semi-stable, ∅ is the grounded extension, and {B} is an admissible set which is not a complete extension. It is known that for every argumentation framework, there exists at least one admissible set (the empty set), exactly one grounded extension, one or more complete extensions, one or more preferred extensions and zero or more stable extensions. Moreover, when the set of arguments in the argumentation framework is finite, as is assumed in the current paper, there also exist one or more semi-stable extensions. An overview of how the various extensions are related to each other is provided in Figure 2. The fact that every stable extension is also a semistable extension, and that every semi-stable extension is also a preferred extension was first stated in [10]. All other relations shown in Figure 2 have originally been stated in [12]. stable extension is a semi−stable extension is a preferred extension grounded extension is a is a complete extension is a admissible set is a conflict−free set Figure 2. An overview of argumentation semantics (extension based). 3. Argument Labellings The concepts of admissibility, as well as that of complete, grounded, preferred, stable or semi-stable semantics were originally stated in terms of sets of arguments. It is equally well possible, however, to express these concepts using argument labellings. This approach was pioneered by Pollock [21] and Jakobovits and Vermeir [20], and has more recently been extended by Caminada [8, 11], Vreeswijk [28] and Verheij [26]. The idea of a labelling is to associate with each argument exactly one label, which can either be in, out A Logical Account of Formal Argumentation 5 or undec. The label in indicates that the argument is explicitly accepted, the label out indicates that the argument is explicitly rejected, and the label undec indicates that the status of the argument is undecided, meaning that one abstains from an explicit judgment whether the argument is in or out. Definition 4. Let (Ar , att ) be an argumentation framework. A labelling is a total function L : Ar −→ {in, out, undec}. We write in(L) for {A | L(A) = in}, out(L) for {A | L(A) = out} and undec(L) for {A | L(A) = undec}. Sometimes, we write a labelling L as a triple (Args 1 , Args 2 , Args 3 ) where Args 1 = in(L), Args 2 = out(L) and Args 3 = undec(L). We distinguish three special kinds of labellings. The all-in labelling is a labelling that labels every argument in. The all-out labelling is a labelling that labels every argument out. The all-undec labelling is a labelling that labels every argument undec. We say that a labelling is conflict-free if no in-labelled argument attacks an (other or the same) in-labelled argument. 3.1. Complete Labellings We will now define the concept of a complete labelling and show its relationship with Dung’s concept of a complete extension. Definition 5. Let (Ar , att ) be an argumentation framework. A complete labelling is a labelling such that for every A ∈ Ar it holds that: 1. if A is labelled in then all attackers of A are labelled out 2. if all attackers of A are labelled out then A is labelled in 3. if A is labelled out then A has a attacker that is labelled in, and 4. if A has a attacker that is labelled in then A is labelled out Conditions 1 and 2 essentially form a bi-implication (“A is labelled in iff all its attacker are labelled out”), just like conditions 3 and 4 (“A is labelled out iff it has a attacker that is labelled in”). It is also possible to characterize a complete labelling as a labelling without arguments that are illegally in, illegally out, or illegally undec. Definition 6. Let L be a labelling of argumentation framework (Ar, att ) and let A ∈ Ar . We say that: 1. A is illegally in iff A is labelled in but not all its attackers are labelled out 6 Caminada and Gabbay 2. A is illegally out iff A is labelled out but it does not have an attacker that is labelled in 3. A is illegally undec iff A is labelled undec but either all its attackers are labelled out or it has a attacker that is labelled in. We say that an argument is legally in iff it is labelled in and is not illegally in. We say that an argument is legally out iff it is labelled out and is not illegally out. We say that an argument is legally undec iff it is labelled undec and is not illegally undec. Theorem 1. Let L be a labelling of argumentation framework (Ar, att ). It holds that L is a complete labelling iff • every in-labelled argument is legally in, • every out-labelled argument is legally out, and • every undec-labelled argument is legally undec. Proof. “=⇒”: The fact that each in-labelled argument is legally in follows from point 1 of Definition 5. The fact that each out-labelled argument is legally out follows from point 3 of Definition 5. We will now prove that each undeclabelled argument is legally undec. Let A be an argument that is labelled undec. Then from point 2 of Definition 5 it follows that not all attackers of A are labelled out and from point 4 of Defintion 5 it follows that A does not have a attacker that is labelled in. Hence, A is legally undec. “⇐=”: Point 1 of Definition 5 follows from the fact that each in-labelled argument is legally in. Point 3 of Definition 5 follows from the fact that each out-labelled argument is legally out. Point 2 of Definition 5 can be proved as follows. Let A be an argument of which all attackers are labelled out. Then A cannot be labelled out (otherwise A would be illegally out) and A cannot be labelled undec (otherwise A would be illegally undec). Therefore, A can only be labelled in. Point 4 of Definition 5 can be proved as follows. Let A be an argument that has an attacker that is labelled in. Then A cannot be labelled in (otherwise A would be illegally in) and A cannot be labelled undec (otherwise A would be illegally undec). Hence, A can only be labelled out. Using the results of Theorem 1 we can restate the concept of a complete labelling as follows. Proposition 1. Let L be a labelling of argumentation framework (Ar , att). It holds that L is a complete labelling iff for each argument A it holds that: A Logical Account of Formal Argumentation 7 • if A is labelled in then all its attackers are labelled out, • if A is labelled out then it has at least one attacker that is labelled in, and • if A is labelled undec then it has at least one attacker that is labelled undec and it does not have an attacker that is labelled in. Lemma 1. Let L1 and L2 be complete labellings of argumentation framework AF = (Ar , att ). It holds that: • in(L1 ) ⊆ in(L2 ) iff out(L1 ) ⊆ out(L2 ), and • in(L1 ) ( in(L2 ) iff out(L1 ) ( out(L2 ) Proof. We first prove that in(L1 ) ⊆ in(L2 ) iff out(L1 ) ⊆ out(L2 ). “=⇒”: Suppose in(L1 ) ⊆ in(L2 ). Let A ∈ out(L1 ). From point 3 of Definition 5 it then follows that A has an attacker (say B) that is labelled in by L1 . That is, B ∈ in(L1 ). From the fact that in(L1 ) ⊆ in(L2 ) it then follows that B ∈ in(L2 ). From point 4 of Definition 5 it then follows that A is labelled out by L2 . That is, A ∈ out(L2 ). “⇐=”: Suppose out(L1 ) ⊆ out(L2 ). Let A ∈ in(L1 ). From point 1 of Definition 5 it then follows that each attacker of A must be labelled out by L1 . From the fact that out(L1 ) ⊆ out(L2 ) it then follows that each attacker of A is also labelled out by L2 . From point 2 of Definition 5 it then follows that A is labelled in by L2 . That is, A ∈ in(L2 ). We now prove that in(L1 ) ( in(L2 ) out(L1 ) ( out(L2 ). “=⇒”: Suppose in(L1 ) ( in(L2 ). This means that in(L1 ) ⊆ in(L2 ) and not in(L2 ) ⊆ in(L1 ). It then follows that out(L1 ) ⊆ out(L2 ) and not out(L2 ) ⊆ out(L1 ). This means that out(L1 ) ( out(L2 ). “⇐=”: Suppose out(L1 ) ( out(L2 ). This means that out(L1 ) ⊆ out(L2 ) and not out(L2 ) ⊆ out(L1 ). It then follows that in(L1 ) ⊆ in(L2 ) and not in(L2 ) ⊆ in(L1 ). This means that in(L1 ) ( in(L2 ). Lemma 1 implies that a complete labelling is uniquely defined by the in-labelled part, as well as by the out-labelled part. Lemma 2. Let L1 and L2 be complete labellings of argumentation framework (Ar, att ). 1. if in(L1 ) = in(L2 ) then L1 = L2 2. if out(L1 ) = out(L2 ) then L1 = L2 8 Caminada and Gabbay Proof. 1. Suppose in(L1 ) = in(L2 ). Then in(L1 ) ⊆ in(L2 ) and in(L2 ) ⊆ in(L1 ), so from Lemma 1 it follows that out(L1 ) ⊆ out(L2 ) and out(L2 ) ⊆ out(L1 ), so out(L1 ) = out(L2 ). From the fact that in(L1 ) = in(L2 ) and out(L1 ) = out(L2 ) it then follows that undec(L1 ) = undec(L2 ) so L1 = L2 . 2. Suppose out(L1 ) = out(L2 ). Then out(L1 ) ⊆ out(L2 ) and out(L2 ) ⊆ out(L1 ), so from Lemma 1 it follows that in(L1 ) ⊆ in(L2 ) and in(L2 ) ⊆ in(L1 ), so in(L1 ) = in(L2 ). From the fact that in(L1 ) = in(L2 ) and out(L1 ) = out(L2 ) it then follows that undec(L1 ) = undec(L2 ) so L1 = L2 . It turns out that there is a one-to-one relationship between complete labellings and complete extensions, and that it is relatively straightforward to convert a complete labelling to a complete extension and vice versa. Definition 7. Let AF = (Ar , att ) be an argumentation framework, clabellings be its set of all conflict-free labellings and csets be its set of all conflict-free sets. We define a function Ext2LabAF : csets → clabellings such that Ext2LabAF (Args) = {(A, in) | A ∈ Args} ∪ {(A, out | A ∈ Args + } ∪ {(A, undec | A 6∈ Args and A 6∈ Args + }. We define a function Lab2ExtAF : clabellings → csets such that Lab2ExtAF (L) = in(L). Sometimes, when the argumentation framework is either clear or not relevant, we write Ext2Lab and Lab2Ext instead of Ext2LabAF and Lab2ExtAF . Theorem 2. Let AF = (Ar , att ) be an argumentation framework and let L be a complete labelling. Then Lab2ExtAF (L) is a complete extension. Proof. Let Args = Lab2ExtAF (L). We now prove that Args is a fixpoint of F . Args ⊆ F (Args): Let A ∈ Args. Then L(A) = in. The fact that L is a complete labelling implies (point 1 of Definition 5) that each attacker B of A is labelled out. Point 3 of Definition 5 then implies that each such B has an attacker (say C) that is labelled in. From the definition A Logical Account of Formal Argumentation 9 of Lab2ExtAF it then follows that C ∈ Args. This means that for each attacker B of A, there is a C ∈ Args that attacks B. Therefore, A ∈ F (Args). F (Args) ⊆ Args: Let A ∈ F (Args). Then each B that attacks A is attacked by some C ∈ Args. From the definition of Lab2ExtAF it follows that C is labelled in. The fact that L is a complete labelling then implies (point 4 of Definition 5) that each such B is labelled out, which then implies (point 2 of Definition 5) that A is labelled in. Therefore, by definition of Lab2ExtAF , it holds that A ∈ Args. The fact that Args is a fixpoint of F , together with the fact that Args is conflict-free (which follows from the fact that each complete labelling is also a conflict-free labelling) implies that Args is a complete extension. Theorem 3. Let AF = (Ar, att ) be an argumentation framework and let Args be a complete extension. Then Ext2LabAF (Args) is a complete labelling. Proof. We first observe that Ext2LabAF (Args) is well-defined because the fact that Args is a complete extension implies that it is conflict-free. We now prove the four properties of Definition 5. 1. “if A is labelled in then all attackers of A are labelled out” Let A be an argument that is labelled in. From the definition of Ext2LabAF it then follows that A ∈ Args. The fact that Args is a complete extension implies that it is an admissible set. That is, Args attacks every attacker of A. From the definition of Ext2LabAF it then follows that every attacker of A is labelled out. 2. “if all attackers of A are labelled out then A is labelled in” Let A be an argument such that every attacker of A is labelled out. From the definition of Ext2LabAF it then follows that every attacker of A is an element of Args + . This means that A is defended by Args (A ∈ F (Args)). From the fact that Args is a complete extension it then follows that A ∈ Args. From the definition of Ext2LabAF it then follows that A is labelled in. 3. “if A is labelled out then A has an attacker that is labelled in” Let A be an argument that is labelled out. From the definition of Ext2LabAF it then follows that A ∈ Args + , so there is an argument B ∈ Args that attacks A. From the definition of Ext2LabAF it follows that B is labelled in. 10 Caminada and Gabbay 4. “if A has an attacker that is labelled in then A is labelled out” Let A be an argument that has an attacker (say B) that is labelled in. From the definition of Ext2LabAF it follows that B ∈ Args, so A ∈ Args + . From the definition of Ext2LabAF it then follows that A is labelled out. When the domain and range of Lab2ExtAF are restricted to complete labellings and complete extensions, and the domain and range of Ext2LabAF are restricted to complete extensions and complete labellings, then the resulting functions (call them Lab2ExtcAF and Ext2LabcAF ) are bijective and each other’s inverse. Theorem 4. Let AF = (Ar , att ) be an argumentation framework, cextensions its set of complete extensions and clabellings its set of complete labellings. Let Ext2LabcAF : cextensions → clabellings be a function such that Ext2LabcAF (Args) = Ext2LabAF (Args) and Lab2ExtcAF : clabellings → cextensions be a function such that Lab2ExtcAF (L) = Lab2ExtAF (L). The functions Ext2LabcAF and Lab2ExtcAF are bijective and each other’s inverse. Proof. As every function that has an inverse is bijective, we only need to prove that Lab2ExtcAF and Ext2LabcAF are each other’s inverses. That is (Lab2ExtcAF )−1 = Ext2LabcAF and (Ext2LabcAF )−1 = Lab2ExtcAF . For this, we prove the following two things: 1. For every complete labelling L it holds that Ext2LabcAF (Lab2ExtcAF (L)) = L. Let L be a complete labelling of AF and let A ∈ Ar . If L(A) = in then A ∈ Lab2ExtcAF (L), so Ext2LabcAF (Lab2ExtcAF (L)) (A) = in. If L(A) = out then A is attacked by Lab2ExtcAF (L), so Ext2LabcAF (Lab2ExtcAF (L))(A) = out. If L(A) = undec then A 6∈ Lab2ExtcAF (L) and A is not attacked by Lab2ExtcAF (L), so Ext2LabcAF (Lab2ExtcAF (L))(A) = undec. 2. For every complete extension Args it holds that Lab2ExtcAF (Ext2LabcAF (Args)) = Args. Let Args be a complete extension of AF . We now prove two things: (a) Lab2ExtcAF (Ext2LabcAF (Args)) ⊆ Args Let A ∈ Lab2ExtcAF (Ext2LabcAF (Args)). Then A is labelled in by Ext2LabcAF (Args). Therefore A ∈ Args. A Logical Account of Formal Argumentation 11 (b) Args ⊆ Lab2ExtcAF (Ext2LabcAF (Args)) Let A ∈ Args. Then A is labelled in by Ext2LabcAF (Args). Therefore A ∈ Lab2ExtcAF (Ext2LabcAF (Args)). From Theorem 4 it follows that complete labellings and complete extensions stand in a one-to-one relationship to each other. In essence, complete labellings and complete extensions are different ways of describing the same thing. Given the one-to-one relationship between complete extensions and complete labellings, the next step would be to examine what kind of labellings are associated with stable, grounded, preferred and semi-stable extensions, all of which are essentially special forms of complete extensions. What would be the properties of the labellings associated with these types of extensions? This question is studied in the following sections. 3.2. Stable Labellings We start the discussion with examining the labellings that are associated with stable extensions. These labellings will be called stable labellings. Definition 8. Let AF = (Ar , att ) be an argumentation framework. A stable labelling is a complete labelling L such that Lab2ExtAF (L) is a stable extension. The fact that for complete labellings and complete extensions, Lab2Ext and Ext2Lab are inverse functions implies that if Args is a stable extension, then Ext2Lab(Args) is a stable labelling. Stable labellings can also be characterized as complete labellings without undec. Theorem 5. Let AF = (Ar , att) be an argumentation framework. The following statements are equivalent: 1. L is a complete labelling such that undec(L) = ∅ 2. L is a stable labelling Proof. from 1 to 2: Suppose L is a complete labelling such that undec(L) = ∅. Let Args be Lab2ExtAF (L). We now prove that Args is a stable extension 12 Caminada and Gabbay (Definition 3). Let A ∈ Ar \Args. From the fact that A 6∈ Args it follows that A is not labelled in by L. From the fact that undec(L) = ∅ it follows that A is not labelled undec either. Therefore, A is labelled out by L. From the fact that L is a complete labelling, it follows that A is attacked by an argument (say B) that is labelled in. From the fact that B is labelled in it follows that B ∈ Args. This means that A is attacked by Args. from 2 to 1: Suppose L is a stable labelling, meaning that Args = Lab2ExtAF (L) is a stable extension. We now prove that undec(L) = ∅. Let A ∈ Ar. We distinguish two cases: 1. A ∈ Args. Then A is labelled in by L. 2. A 6∈ Args. Then from the fact that Args is a stable extension, it follows that A is attacked by Args, so A is labelled out by L. In both cases, A 6∈ undec(L). Since this folds for an arbitrary A ∈ Ar, it follows that undec(L) = ∅. As an aside, one can also raise the question whether there exist labellings with empty in or empty out, and if so, what would be the meaning of these labellings. As for complete labellings with empty in it follows that these also have empty out, so each argument has to be labelled undec. Thus, a complete labelling with empty in is essentially the grounded labelling (see Definition 9 and Theorem 6) in an argumentation framework where each argument has at least one attacker. As for complete labellings with empty out, it follows that each argument has to be labelled either in or undec. This means that no argument that is labelled in attacks any (other) argument. In essence, a complete labelling with empty out is the grounded labelling in an argumentation framework where each argument without attackers also does not attack any argument itself. 3.3. The Grounded Labelling The next thing to be examined are the properties of the labelling associated with the grounded extension. Definition 9. Let AF = (Ar , att ) be an argumentation framework. A grounded labelling is a complete labelling L such that Lab2ExtAF (L) is the grounded extension. A Logical Account of Formal Argumentation 13 The fact that the grounded extension is unique, and that for complete labellings and complete extensions, Lab2Ext and Ext2Lab are inverse functions, implies that the grounded labelling is unique, and that if Args is the grounded labelling, then Ext2Lab(Args) is the grounded extension. The grounded labelling can be characterized as the complete labelling with minimal in, as the complete labelling with minimal out, or as the complete labelling maximal undec. Theorem 6. Let AF = (Ar , att) be an argumentation framework. The following statements are equivalent: 1. L is a complete labelling where in(L) is minimal (w.r.t. set inclusion) among all complete labellings of AF 2. L is a complete labelling where out(L) is minimal (w.r.t. set inclusion) among all complete labellings of AF 3. L is a complete labelling where undec(L) is maximal (w.r.t. set inclusion) among all complete labellings of AF 4. L is the grounded labelling Proof. from 1 to 2: Let L be a complete labelling where out(L) is not minimal. Then there exists a complete labelling L′ with out(L′ ) ( out(L). From Lemma 1 it then follows that in(L′ ) ( in(L), so L is a labelling where in(L) is not minimal. from 2 to 1: Let L be a complete labelling where in(L) is not minimal. Then there exists a complete labelling L′ with in(L′ ) ( in(L). From Lemma 1 it then follows that out(L′ ) ( out(L), so L is a labelling where out(L) is not minimal. from 1 to 4: Let L be a complete labelling where in(L) is minimal. Then Lab2ExtAF (L) is a minimal complete extension, which implies it is the grounded extension, so L is the grounded labelling. from 4 to 1: Let L be the grounded labelling. Then Lab2ExtAF (L) is the grounded extension, which means it is the minimal complete extension. It then follows that in(L) is minimal. from 1 to 3: Let L be a complete labelling where in(L) is minimal. Then Lab2ExtAF (L) is the grounded extension. Now suppose that undec(L) is not maximal. Then there exists a complete labelling L′ with undec(L) ( 14 Caminada and Gabbay undec(L′ ). It holds that Lab2ExtAF (L′ ) is a complete extension, and from the fact that the grounded extension is a subset of each complete extension, it follows that Lab2ExtAF (L) ⊆ Lab2ExtAF (L′ ), so in(L) ⊆ in(L′ ). From Lemma 1 it then follows that out(L) ⊆ out(L′ ). From the fact that in(L) ⊆ in(L′ ) and out(L) ⊆ out(L′ ) it then follows that undec(L′ ) ⊆ undec(L). Contradiction. from 3 to 1: Let L be a complete labelling where in(L) is not minimal. Then there exists a complete labelling L′ with in(L′ ) ( in(L). It then also follows (Lemma 1) that out(L′ ) ( out(L), so undec(L) ( undec(L′ ). Contradiction. 3.4. Preferred Labellings We now examine the properties of the labellings associated with preferred extensions. Definition 10. Let AF = (Ar , att) be an argumentation framework. A preferred labelling is a complete labelling L such that Lab2ExtAF (L) is a preferred extension. The fact that for complete labellings and complete extensions, Lab2Ext and Ext2Lab are inverse functions implies that if Args is a preferred extension, then Ext2Lab(Args) is a preferred labelling. Preferred labellings can be characterized as complete labellings with maximal in, or as complete labellings with maximal out. Theorem 7. Let AF = (Ar , att) be an argumentation framework. The following statements are equivalent: 1. L is a complete labelling where in(L) is maximal (w.r.t. set inclusion) among all complete labellings of AF 2. L is a complete labelling where out(L) is maximal (w.r.t. set inclusion) among all complete labellings of AF 3. L is a preferred labelling Proof. from 1 to 2: Let L be a complete labelling where out(L) is not maximal. Then there exists a complete labelling L′ with out(L) ( out(L′ ). From Lemma 1 it then follows that in(L) ( in(L′ ), so L is a labelling where in(L) is not maximal. A Logical Account of Formal Argumentation 15 from 2 to 1: Let L be a complete labelling where in(L) is not maximal. Then there exists a complete labelling L′ with in(L) ( in(L′ ). From Lemma 1 it then follows that out(L) ( out(L′ ), so L is a labelling where out(L) is not maximal. from 1 to 3: Let L be a complete labelling where in(L) is maximal. Then Lab2ExtAF (L) is a maximal complete extension, which implies it is a preferred extension, so L is a preferred labelling. from 3 to 1: Let L be a preferred labelling. Then Lab2ExtAF (L) is a preferred extension, which means it is a maximal complete extension. It then follows that in(L) is maximal. 3.5. Semi-Stable Labellings We now examine the properties of the labellings associated with semi-stable extensions. Definition 11. Let AF = (Ar , att) be an argumentation framework. A semi-stable labelling is a complete labelling L such that Lab2ExtAF (L) is a semi-stable extension. The fact that for complete labellings and complete extensions, Lab2Ext and Ext2Lab are inverse functions implies that if Args is a semi-stable extension, then Ext2Lab(Args) is a semi-stable labelling. Semi-stable labellings can be characterized as complete labellings with minimal undec. Theorem 8. Let AF = (Ar , att) be an argumentation framework. The following statements are equivalent: 1. L is a complete labelling where undec(L) is minimal (w.r.t. set inclusion) among all complete labellings of AF 2. L is a semi-stable labelling Proof. from 1 to 2: Suppose L is a complete labelling, but not a semi-stable labelling. Then Lab2ExtAF (L) is a complete extension but not a semistable extension. This implies that Lab2ExtAF (L) ∪ Lab2ExtAF (L)+ is not maximal. So there exists a complete extension Args ′ such that 16 Caminada and Gabbay Lab2ExtAF (L) ∪ Lab2ExtAF (L)+ ( Args ′ ∪ Args ′+ . Let L′ be the labelling associated with Args ′ (that is: L′ = Ext2LabAF (Args ′ ) and Args ′ = Lab2ExtAF (L′ )). It then follows that Lab2ExtAF (L)∪Lab2ExtAF (L)+ ( Lab2ExtAF (L′ ) ∪ Lab2ExtAF (L′ )+ , so in(L) ∪ out(L) ( in(L′ ) ∪ out(L′ ), so undec(L′ ) ( undec(L). This means that L is not a labelling where undec(L) is minimal. from 2 to 1: Suppose that L is a complete labelling where undec(L) is not minimal. This implies that there exists a complete labelling L′ such that undec(L′ ) ( undec(L). It then follows that in(L) ∪ out(L) ( in(L′ ) ∪ out(L′ ). Let Args = Lab2ExtAF (L) and Args ′ = Lab2ExtAF (L′ ). It then follows that Args ∪ Args + ( Args ′ ∪ Args ′+ , which means that Args is not a semi-stable extension, which implies that L is not a semistable labelling. 3.6. Roundup The main results of the discussion until so far are summarized in Table 1. Notice that we have covered all combinations of minimal or maximal in, out or undec. Almost all combinations turn out to correspond to the traditional Dung-style semantics. The only exception are the labellings with minimal undec, which corresponds with a semantics not introduced in [12] but in [10]. restriction complete labellings no restrictions empty undec maximal in maximal out maximal undec minimal in minimal out minimal undec Dung-style semantics complete semantics stable semantics preferred semantics preferred semantics grounded semantics grounded semantics grounded semantics semi-stable semantics linked by def. and th. Def. 5 and Th. 1 Def. 8 and Th. 5 Def. 10 and Th. 7 Def. 10 and Th. 7 Def. 9 and Th. 6 Def. 9 and Th. 6 Def. 9 and Th. 6 Def. 11 and Th. 8 Table 1. Argument labellings and Dung-style semantics. Overall, one can see labellings as an alternative way to specify argumentation semantics. The essential rule is that an argument has to be accepted A Logical Account of Formal Argumentation 17 iff all its attackers are rejected, and an argument has to be rejected iff it has at least one attacker that is accepted. Thus, argumentation can be explained without referring to things like admissibility or fixpoints of Dung’s characteristic function. In a gunfight, one stays alive iff all attackers are dead, and one dies iff at least one attacker is still alive. Those who can understand this have basically understood what abstract argumentation is all about. 3.7. Admissible Labellings Until so far, we have modelled the concepts of complete, stable, grounded, preferred and semi-stable semantics in terms of labellings. This was done by strengthening the concept of complete labellings. Another route would be to weaken the concept of a complete labelling. An obvious way to do this would be to take only a subset of the four properties of Definition 5. However, these four properties are not completely independent. If one requires property 1 (“if A is labelled in then all attackers of A are labelled out”) then it makes sense also to require property 3 (“if A is labelled out then A has an attacker that is labelled in”), and vice versa. If one requires each in label to make sense, and defines this in terms of out labels, then one should also require each out label to make sense (and vice versa). Together, properties 1 and 3 stand for admissibility. Definition 12. Let (Ar, att ) be an argumentation framework. An admissible labelling is a labelling such that for every A ∈ Ar it holds that: 1. if A is labelled in then all attackers of A are labelled out 2. if A is labelled out then A has an attacker that is labelled in Admissible labellings correspond to admissible sets, but the relationship is no longer one-to-one. Theorem 9. Let AF = (Ar , att ) be an argumentation framework. It holds that: 1. if Args is an admissible set of AF then Ext2LabAF (Args) is an admissible labelling of AF , and 2. if L is an admissible labelling of AF then Lab2ExtAF (L) is an admissible set of AF . Proof. 18 Caminada and Gabbay 1. Let Args be an admisssible set of AF , and let L = Ext2LabAF (Args). We now prove that L is an admissible labelling. (a) Let A be an argument that is labelled in by L. Let B be an arbitrary attacker of A. The fact that A is labelled in by L implies that A ∈ Args. The fact that Args is an admissible set implies that Args attacks B. That is, B ∈ Args + , so B is labelled out by L. Since this holds for any attacker B of A it follows that every attacker of A is labelled out by L (b) Let A be an argument that is labelled out by L. It then follows that A ∈ Args + , so there exists a B ∈ Args that attacks A. This B is labelled in by L. This means that A has a attacker that is labelled in by L. 2. Let L be an admissible labelling of AF , and let Args = Lab2ExtAF (L). We now prove that Args is an admissible set. (a) We first prove that Args is conflict-free. Suppose this is not the case. Then there exist A, B ∈ Args such that A attacks B. It then follows that both A and B are labelled in by L. But this cannot be the case because the fact that L is an admissible labelling implies that all attackers of A (including B) are labelled out by L. (b) We now prove that Args defends all its elements. Let A ∈ Args. Then A is labelled in by L. Let B be an arbitrary argument that attacks A. From the fact that L is an admissible labelling, it follows that B is labelled out by L. From the fact that L is an admissible labelling it then also follows that B has an attacker C that is labelled in by L, so C ∈ Args. Therefore Args is self-defending. Admissible labellings and admissible sets have a many-to-one relationship. That is, each admissible labelling is associated with exactly one admissible set, but each admissible set is associated with one or more admissible labellings. As an example, consider again the argumentation framework of Figure 1. Here, ({B}, {A, C}, {D, E}) and ({B}, {A}, {C, D, E}) are two admissible labellings associated with the same admissible set {B}. For admissible labellings (say L1 and L2 ) it does not hold that if in(L1 ) ⊆ in(L2 ) then out(L1 ) ⊆ out(L2 ). As a counter example, consider again the argumentation framework of Figure 1, with L1 = ({B}, {A, C}, {D, E}) and L2 = ({B}, {A}, {C, D, E}). Similarly, it also does not hold that if out(L1 ) ⊆ out(L2 ) then in(L1 ) ⊆ in(L2 ). As a counter example, consider A Logical Account of Formal Argumentation 19 the argumentation framework AF = ({A, B, C, D}, {(A, C), (B, C), (C, D)}) with L1 = ({A, B, D}, {C}, ∅) and L2 = ({A, D}, {C}, {B}). Between the concept of an admissible labelling and a complete labelling, one can distinguish two intermediate forms. Definition 13. Let AF = (Ar , att ) be an argumentation framework. A JV-labelling is a labelling that satisfies: 1. if A is labelled in then all attackers of A are labelled out 2. if A is labelled out then A has an attacker that is labelled in 3. if A has an attacker that is labelled in then A is labelled out A VJ-labelling is a labelling that satisfies: 1. if A is labelled in then all attackers of A are labelled out 2. if A is labelled out then A has an attacker that is labelled in 3. if all attackers of A are labelled out then A is labelled in In essence, an admissible labelling satisfies point 1 and 3 of a complete labelling (Definition 5), a JV-labelling satisfies point 1, 3 and 4, and a VJ-labelling satisfies point 1, 3 and 2. It immediately follows that every complete labelling is also a JV-labelling and a VJ-labelling, and that every JV-labelling or VJ-labelling is also an admissible labelling. We use the term JV-labelling, because these are quite close to a proposal of Jakobovits and Vermeir [20]. An important difference, however, is that in our approach each argument gets exactly one out of three possible labels (in, out or undec) whereas in Jakobovits and Vermeir’s original proposal, there are four possibilities (either no label, single in, single out, or both in and out).1 It turns out that JV-labellings are uniquely identified by their in labelled part, whereas VJ-labellings are uniquely identified by their out labelled part. Lemma 3. Let AF = (Ar , att ) be an argumentation framework. Let L1 and L2 be JV-labellings of AF and let L3 and L4 be VJ-labellings of AF . It holds that: 1. if in(L1 ) ⊆ in(L2 ) then out(L1 ) ⊆ out(L2 ) 2. if in(L1 ) ( in(L2 ) then out(L1 ) ( out(L2 ) 3. if out(L3 ) ⊆ out(L4 ) then in(L3 ) ⊆ in(L4 ) 1 Another difference is that Jakobovits and Vermeir use the symbol “+” instead of in and the symbol “−” instead of out. 20 Caminada and Gabbay 4. if out(L3 ) ( out(L4 ) then in(L3 ) ( in(L4 ) Proof. Similar to the proof of Lemma 1. Lemma 4. Let AF = (Ar , att ) be an argumentation framework. Let L1 and L2 be JV-labellings of AF and let L3 and L4 be VJ-labellings of AF . It holds that: 1. if in(L1 ) = in(L2 ) then L1 = L2 2. if out(L3 ) = out(L4 ) then L3 = L4 Proof. Similar to the proof of Lemma 2. Since JV-labellings are uniquely identified by their in labelled part, and the fact that an admissible set essentially specifies a set of in labelled arguments, it does not come as a surprise that there exists a one-to-one relation between admissible sets and JV-labellings. Theorem 10. Let AF = (Ar , att ) be an argumentation framework, asets be its set of admissible sets and JV−labellings be its set of JV-labellings. Let Ext2LabJV : asets → JV−labellings be a function such that AF JV Ext2LabAF (Args) = Ext2LabAF (Args) and Lab2ExtJV AF : JV−labellings → asets be a function such that Lab2ExtJV (L) = Lab2Ext AF (L). The funcAF JV are bijective and each other’s inverse. tions Ext2LabJV and Lab2Ext AF AF Proof. Similar to the proof of Theorem 4. A global overview of the relations between the various forms of labellings is provide in Figure 3. 4. Argumentation and Modal Logic There are three main methods of introducing modal logic into argumentation theory: the metalevel approach, the object level approach and the mixed approach. We view an argumentation system as a combination of an argumentation framework AF = (Ar , att ) (where Ar is a set of atomic arguments and att ⊆ Ar × Ar is the attack relation) and a complete labelling L of AF . To us, the thus described argumentation system serves as the object level. The metalevel approach talks about the argumentation framework from “above”, using another language and logic. The metalevel language can be classical logic or modal logic. These are traditional metalevel languages 21 A Logical Account of Formal Argumentation stable labelling is a semi−stable labelling is a preferred labelling grounded labelling is a is a complete labelling is a is a JV−admissible labelling VJ−admissible labelling is a is a admissible labelling is a conflict−free labelling Figure 3. An overview of argumentation semantics (labelling based). and are used in this way in many areas. Classical logic can be full classical logic or the computational Horn clause logic programming language. When classical logic is used, one can think of the process as translation. There is a traditional translation of modal logic into classical logic and through this translation the metalevel approaches are related. For the sake of clarity, we shall present all three metalevel versions in the appropriate sections: the classical logic one, the Horn clause logic programming one, and the modal logic one. They are related but are not the same; each one has its advantages and limitations. 4.1. Modal Logic Preliminaries This subsection introduces some modal logic background needed for introducing our approaches. The modal logic K is a propositional system with the modal operator  and the atomic propositions Q = {q1 , q2 , q3 . . .} and ¬, ∧, ∨, →, ⊤ and ⊥. Models for K have the form M = (S, R, h) where S is the set of possible worlds, R ⊆ S × S, S 6= ∅ and h is the assignment function, giving to each atomic letter q a subset h(q) ⊆ S. Satisfaction is defined as follows for t ∈ S: • t  q iff t ∈ h(q), for atomic q • t  A ∧ B, A ∨ B, ¬A, A → B are defined as usual 22 Caminada and Gabbay • t  A iff for all s such that tRs we have s  A • A holds in (S, R, h) iff for all t ∈ S we have t  A. K can be axiomatised as follows: 1. all substitution instances of truth functional tautologies 2. (A → B) → (A → B) ⊢A 3. ⊢ A It is complete for the class of all finite frames of the form (S, R) where S 6= ∅, S is finite, and R ⊆ S × S. 4.2. Products of Modal Logics We need the concept of an H-product of a fixed frame modal logic. To define this, we need to introduce the concept of a fixed frame modal logic as well as the concept of a product. Definition 14. A modal logic L is said to be a fixed frame modal logic (FF modal logic) if it can be characterised by a family of models of the form (S0 , R0 , h), h ∈ H, where (S0 , R0 ) is a fixed frame and H is a set of assignments to this frame. We have L ⊢ A iff A holds in all FF-models (S0 , R0 , h), h ∈ H. We can now define the cross product of the two modal logics. Let 1 be a modality of modal logic E1 that is completely chacterised by a class of models of the form M1 = {(Si1 , Ri1 , h1i )}, i ∈ I1 and similarly for 2 , E2 , M2 = {(Sj2 , Rj2 , h2j )}, j ∈ I2 . We define the flat-cross product of these two logics semantically, through their models. The language contains the two modalities {1 , 2 }. The models have the form (S 1 × S 2 , R1 ∪ R2 , h1,2 ) where (S 1 , R1 , h1 ) and (S 2 , R2 , h2 ) are any models of 1 and 2 respectively and S 1 × S 2 is the product of the sets S 1 and S 2 . We define R1 and R2 as follows. (x1 , x2 )R1 (y1 , x2 ) iff x1 R1 y1 (x1 , x2 )R2 (x1 , y2 ) iff x2 R2 y2 We define h1,2 by some boolean function of h1 and h2 , for example (x1 , x2 ) ∈ h1,2 (q) iff x1 ∈ h1 (q) or x2 ∈ h2 (q) or another definition (x1 , x2 ) ∈ h1,2 (q) iff x1 ∈ h1 (q). We have (x1 , x2 )  1 A iff for all y1 , x1 R1 y1 we have (y1 , x2 )  A. 23 A Logical Account of Formal Argumentation (x1 , x2 )  2 A iff for all y2 , x2 R2 y2 we have (x1 , y2 )  A. The above definition defines a general flat product. When we have a single FF-logic, with  characterised by a fixed frame (S0 , R0 ) and a family of assignments H, we can form the universal product of the logic with  along the axis of H. We need a new modality for the H axis, which we denote by ⊡. Definition 15. Let (S0 , R0 ) be a frame for a fixed frame modal logic L with . Let H be a family of assignments h ∈ H such that (S0 , R0 , h) is a model of L. We form the universal H product of the frame as follows. We form the set M = {(S0 , R0 , h) | h ∈ H}. We use a modality ⊡ to move around M. We then have two modalities:  and ⊡. Satisfaction in M is defined as follows. (t, h)  q iff t ∈ h(q), for t ∈ S, h ∈ H (t, h)  A iff for all s, tRs implies (s, h)  A (t, h)  ⊡A iff for all h′ 6= h, (t, h′ )  A A holds in the model iff for all t, h, (t, h)  A. Figure 4 shows this is indeed a product. R0 axis s t model (S0,R 0,t,h) H axis h h´ Figure 4.  moves from t to s on the h vertical.⊡ moves from h to h′ on the t horizontal. See the book [18] for a wealth of material products. 4.3. Modal Provability Logic Löb’s logic for one modality  has the following axioms and rules. 24 Caminada and Gabbay 1. axioms and rules of modal logic K 2. ♦A → ♦(A ∧ ¬A) 3. A → A Axiom 2 says that if A is possible then it is possible for the last time. Axiom 3 says R is transitive. It is complete for the class of all frames which are finite and acyclic (we can take all finite tree frames). The following holds. Theorem 11 (fixed point theorem). Let Ψ(x) be a modal formula with the propositional variable x such that x is in the scope of a modality . Then there is a formula φ which is a solution to the fixed point equation x ↔ Ψ(x). Namely, we have ⊢ φ ↔ Ψ(x/φ). Proof. See Theorem 3.4 (page 464) of [25]. We shall use an extension of this logic in our object level modal approach. The logic we use we call LN 1; it is characterised by linear chains. The axiom we add for it is: 4. ♦A ∧ ♦B → ♦(A ∧ B) ∨ ♦(A ∧ ♦B) ∨ ♦(B ∧ ♦A) If we then also add the following axiom 5. ⊥ we get the logic of chains of length 3 (3-chains) of the form of Figure 5. 3 2 1 Figure 5. A 3-chain. The next step will be to connect these chains to labellings. A Logical Account of Formal Argumentation 4.4. 25 The Metalevel Approach We use the modal logic K to describe the system of Definition 4. We can view (Ar, att ) as a modal frame and view L as an assignment to three propositional atomic letters: q1 , q0 and q? , which we regard as constants. We have a  q1 iff L(a) = in, a  q0 iff L(a) = out, and a  q? iff L(a) = undec. The modality goes by the direction of “being attacked from”, namely a  A iff for all y such that y att a (that is, (y, a) ∈ att ), we have y  A. We therefore must adopt the following axioms, written as ∆(q1 , q0 , q? ). 1. ⊥ ∨ q0 → q1 (if a is not attacked by any argument, i.e. ⊥ holds, or all attackets of a are out then a is in) 2. ♦q1 → q0 (if a is attacked by an argument which is in then a is out) 3. (q0 ∨ q? ) ∧ ♦q? → q? (if all the attackers of a are either out or undec and at least one attacker of a is undec then a is undec) 4. ⊢m (q0 ∨ q1 ∨ q? ), m ≥ 0 (each argument has at least one label out, in or undec) 5. ⊢m (¬(qi ∧ qj )), i 6= j, i, j ∈ {0, 1, ?}, m ≥ 0 (no argument has more than one label) Let ∆ be the above theory and let AF = (Ar , att ) be an argumentation framework and L be a labelling of AF . Then for any a ∈ Ar it holds that a  ∆(q1 , q0 , q? ), provided that a  q1 iff L(a) = in a  q0 iff L(a) = out a  q? iff L(a) = undec Conversely if (S, R, h) is a modal model of ∆ (i.e. for all t ∈ S, t  ∆) then it is an argumentation system with L(a) = in iff a  q1 L(a) = out iff a  q0 L(a) = undec iff a  q? This turns g into a complete labelling (see Theorem 1 and Proposition 1). The above is may be a nice model but there is still not much we can do with it. Since the extensions are assignments to q0 , q1 , q? satisfying ∆ we cannot directly talk about them, except for the grounded labelling. We shall see later how to deal with the other labellings, using circumscription. We shall see that to be able to fully understand our metalevel modal model 26 Caminada and Gabbay and its options we should also introduce and compare with the classical logic metamodel. This we shall do in section 5. Using the modality  as above and results from [5], we can define extensions. Let E be a propositional letter. Then E defines a set of points in Ar. So according to our notation, t  E iff for all s such that sRt we have s  E. So if E denotes a set of arguments, the ♦E is the set of arguments attacked by E and ♦E is the set of arguments defended by E. We have: 1. E is stable iff E = ¬♦E = ¬E 2. (a) E is conflict-free iff E → ¬♦E (b) E is admissible iff E → (♦E) ∧ (¬E) iff E → (¬E ∧ ♦E) 3. E is a complete extension iff E = (¬E ∧ ♦E) Using fixed points methods of Section 4.5 and reference [16], we can find the fixed point solutions for (1) (stable extensions) and (3) (complete extensions) for finite frames. Altogether, the metalevel approach can be summarized as follows. Given AF = (Ar , att ) 1. We view elements a ∈ Ar as possible worlds and so AF becomes a model for modal logic K (with the modality ). 2. Labellings become assignments in the modal logic. 3. The frame of our modal logic is fixed, it is (Ar , att ). What changes are the assignments defined by the labellings. This means we have what we call a fixed frame modal logic FF modality. 4. Properties of labellings are studied in a circumscription logic based on modal logic K (to be introduced in Section 5. 4.5. The object level approach The previous view used modal logic to talk about argumentation. The now to be introduced object level approach will model argumentation from within. To introduce our point of view, let us ask what does an argumentation framework of the form AF = (Ar, att ) say to us? Here we view the arguments a ∈ Args as atoms in some logic. So let X be the logical content of the argumentation framework AF and labelling L. Hence if a is in then X ⊢ a in some logic. If a is out then X ⊢ ¬a and if a is undec then we have neither. We might take the simple minded view and let XL = {a | L(a) = in} ∪ {¬a | L(a) = out} and propose this as a solution. The problem, however, is that this is too simplistic because 27 A Logical Account of Formal Argumentation 1. It does not take into account the internal structure of AF (i.e. the attack relation) 2. It has an explicit dependence on L Our aim is therefore to introduce a more sophisticated approach. We borrow the Löb modal provability logic and use a suitable extension of it, which we call LN 1, and view the logical content of (AF, L) as a formula M(AF, L) of LN 1. LN 1 is a fixed frame modal logic, the frame being a chain of 3 elements. We should therefore have (using ⊟ as the modal provability operator) 1. M(AF, L) ⊢ ⊟a if a is in 2. M(AF, L) ⊢ ⊟¬a if a is out 3. neither, if a is undec We read ⊟ as provability. Figure 6 shows how we find M(AF, L). y 1 x V y a n not attacked by any y Figure 6. Finding M(AF, L). Let x ∈ Ar and let y1 , . . . , yn denote all arguments attacking it (i.e. (yi , x) ∈ att). The labelling conditions basically say that L(x) = in iff for each V attacker yi it holds that L(yi ) is out. So if in means provable, we get x ↔ yi ¬ ⊟ ¬(M(AF, L) ∧ ¬yi ) so we have something like M(AF, L) ↔ ∧a not atttacked by any V y ⊟ a ∧ [ a∈Ar ,x attacked by y1 ,...,yn V (x ↔ yi ¬ ⊟ ¬(M(AF, L) ∧ ¬yi ))]. The exact solution formula for M(AF, L) is given in the beginning of Section 6. Thus, M(AF, L) is obtained as a fixed point solution in a suitable modal provability logic. The fixpoint equation is generated by AF . All models of M(AF ) should give us all labellings of AF . A model of M(AF ) gives the 28 Caminada and Gabbay atoms, the elements of Ar assignments and from the assignments we can tell which a is in, out or undec. This we will see in later sections. Altogether, the object level approach can be summarised as follows. Let AF = (Ar , att ). • We regard elements of Ar as atomic propositions in some modal provability logic LN 1. The graph (Ar , att ) generates a fixed point equation in the modal logic and the unique solution M(AF ) of this equation is a formula of modal logic representing the logical content of AF . • The possible world models of M(AF ) are in one to one correspondence with all labellings L of AF . • The logic we use, LN 1 is a fixed frame modal logic. 4.6. The Mixed Approach We saw that the metalevel approach regards arguments as possible worlds, while the object level approach regards them as propositions in a logic. The problem in both approaches is how to characterise extensions. We will now introduce a mixed approach. Consider the previous two approaches. In each of them, the fixed argumentation framework gives rise to a fixed frame modal logic. On the metalevel approach, the fixed frame is (Ar , att ), and the set of assignments H is what all the possible labellings give us. On the object level approach the fixed frame is a chain of three elements (see Figure 5) and the assignments are all those assignments L that give us models of M(AF ). We shall see later that there is a one-to-one correspondance between the assignments which are models of M(AF ) and the labellings of AF . In either case we get a fixed frame modal logic, one with  and the other with ⊟. The mixed approach adds the modality ⊡ to the existing modality and forms the universal products with {, ⊡} and with {⊟, ⊡} respectively. To show how the mixed approach works, consider the semi-stable semantics of Table 1. For this we need to say that our labelling has a minimal undec. Let us explain the approach for the metalevel mixed case. Let gs−s be the assignment arising from the semi-stable labelling. Then there is no other h for a model such that h(q2 ) $ gs−s (q? ). We can express this in the universal model. The grounded extension for example is characterised by minimal in, so we obtain  q1 ∧ ⊡q1 . A Logical Account of Formal Argumentation 5. 29 The metalevel approach We saw that the metalevel approach uses modal logic K to talk about argumentation frameworks of the form AF = (Ar , att ). The arguments become the possible worlds of a Kripke model, the attack relation becomes the accessibility relation and an associated labelling L gives rise to and assignment. By varying L we get a fixed frame modal logic based on K. To fully appreciate the advantages and limitations of this approach we need to compare it to two other approaches, the Logic Programming and the classical logic ones. 5.1. The Classical Logic Metalevel Approach We start with classical logic with equality (“=”) and three modal predicates Q1 , Q0 and Q? and a binary relation R. Given an argumentation framework AF = (Ar , att ) and a labelling L we construct the associated model of classical logic. We take Ar as the domain, att as the relation R and use L to get the extensions of tree predicates Q0 , Q1 and Q? as follows. 1. a ∈ Q1 iff L labels a as in 2. a ∈ Q0 iff L labels a as out 3. a ∈ Q? iff L labels a as undec Consider the following classical theory ∆(R, Q0 , Q1 , Q? ). 1. ∀x(Q0 (x) ∨ Q1 (x) ∨ Q? (x)) 2. ¬∃x(Qi (x) ∧ Qj (x)) for i 6= j, i, j ∈ {0, 1, ?} 3. ∀y(∀x(xRy → Q0 (x)) → Q1 (y) 4. ∀y(∃x(xRy ∧ Q1 (x)) → Q0 (y)) 5. ∀y(∀x(xRy → (Q0 (x) ∨ Q? (x))) ∧ ∃x(xRy ∧ Q? (x)) → Q? (y)) Any model of ∆ with domain D defines an argumentation framework with Ar = D, att = R and L is what we obtain from the elements satisfying the respective predicates Q0 , Q1 and Q? . Notice that we are not using “=”. If we want to characterise any specific argumentation framework AF = (Ar, att ) we need equality and we need constant names for every element of Ar. We write the following additional axioms θ(AF ). W 6. ∀x( a∈Ar x = a) V 7. a,b∈Ar ,a6=b a 6= b 30 8. Caminada and Gabbay V a,b∈att aRb We want to see how to characterise the different extensions of the argumentation frameworks obtained in classical logic as models of ∆. This means for example that we want to say that the predicate Q1 is minimal in the model (in order to obtain the grounded extension) or for example that the predicate Q? is minimal to get the semi-stable extension, or that the predicate Q1 is maximal to get the preferred extension. The concept “Q is maximal” or “Q is minimal” is not first order. We therefore need second order formula to express this, an approach that is know as predicate circumscription of John McCarthy; the subject has a very well developed theory. So the theory we want is for example ∆semi−stable (R, Q0 , Q1 , Q? ) = ∆(R, Q0 , Q1 , Q? ) ∧ “ Q? is minimal”. Any model of ∆semi−stable yields an argumentation framework based on the domain of the model, with L (derived from Q1 , Q1 , Q? ) which is semi-stable. Using circumscription we write ∆semi−stable = ∆(AF, Q0 , Q1 , Q? ) ∧ ∀Q0 ∀Q1 ∀Q∗? ((Q∗? ( Q? ) → ¬∆(AF, Q0 , Q1 , Q∗? )) where X ( Y is defined as ∀y(X(y) → Y (y)) ∧ ∃z(Y (z) ∧ ¬X(z)). Similarly to maximise we use ) in the circumscription formulas for Q1 and Q∗1 . According to our book [14] it is possible to eliminate such second order quantifiers under certain circumstance. We need to check whether this can be done in our case. When we deal with a specific finite argumentation framework AF , our set of axioms is ∆(AF ) = ∆ ∪ θ(AF ). In this case characterising some of the extensions is easier. For one thing we can use provability to characterise some extensions and for others circumscription becomes first order. The above axioms ∆(AF ) = ∆ ∪ θ(AF ) can immediately characterise the grounded extension of AF as the set of all x such that ∆(AF ) ⊢ Q1 (x). The preferred semantics is characterised by maximal in, which implies min that Q1 is maximal, so ¬Q1 is minimal. So let Q1 = {x | ∆(L) ⊢ ¬Q1 } min and add the axiom ∀(Q1 (x) ↔ Q1 (x)). The problem with the above min definition is that Qmin and Q1 are defined using provability, which is ? outside the logic itself. To bring this in, we need to use circumscription as we did before. However, since for a given AF , Ar is finite, we can turn the second order qualtifier ∀X into a big conjunction by enumerating all possible 31 A Logical Account of Formal Argumentation subsets B ⊆ Ar. We get ∀XΨ[y ∈ X] is replaced by This makes circumscription first order for AF fixed. 5.2. V B⊆Ar W Ψ[ a∈B y = a]. The Logic Programming Metalevel Approach The logic programming metalevel approach has been worked out in [29]. The approach is similar to the classical logic approach except that we use logic programming to represent the argumentation framework. The extensions correspond to the models of the corresponding logic program. For each argument x we write the clause x ← ¬y1 , . . . , ¬yn (n ≥ 0), where {y1 , . . . , yn } is the set of all arguments attacking x. We get a resulting logic program satisfying 1. each atom is the head of exactly one clause 2. the bodies of clauses consist of weakly negated atoms only For more information, we refer to [29] and [17]. 6. The Modal Provability Object Level Approach Let AF = (Ar , att ) be an argumentation framework. Let x be an argument whose set of attackers is y1 , . . . , yn , as was illustrated in Figure 6. We construct the following modal formula in the modal logic LN 3 of provability. V V M(AF ) = G(⊥ ∨ y∈Ar and y has attackers yi y ↔ i ♦¬yi )∧ V y∈Ar and y is not attacked Gx Here, GA stands for A ∧ A. The logic LN 3 has the following axioms. 1. all substitution instances of classical tautologies 2. all K4 axioms • (A → B) → (A → B) • A → A ⊢A • ⊢ A 3. all substitution instances of Löb’s axioms ♦A → ♦(A ∧ ¬A) 4. linearity axiom ♦A ∧ ♦B → ♦(A ∧ B) ∨ ♦(A ∧ ♦B) ∨ ♦(B ∧ ♦A) 32 Caminada and Gabbay 5. 3-chain axiom ⊥ 6. axioms for atoms q only • q → (¬q → q) • (⊥ ∨ q) ↔ q • (⊥ ∨ ¬q) ↔ ¬q Lemma 5. 1. The logic LN 3 is complete for all 3-chain models whose assignment to atoms has one of the types of Table 2. 2. Any 3-chain model of LN 1 allows for the atoms to have one of types 1, 0 or 2 assignments. Proof. The axioms of LN1 force the following for atomic q in the chain 1 < 2 < 3: • 1  q and 2  q ⇒ 3  q • 1 ∼ q and 2 ∼ q ⇒ 3 ∼ q • 1  q and 2 ∼ q ⇒ 3  q. We need to prove that the option “1  q and 2 ∼ q and 3 ∼ q” cannot arise. Consider a point y such as y is being attacked. (If V not then Gy is in M (AF ) and so y is of type 1.) We have G(⊥ ∨ (y ↔ ♦ ∼ yi )) holds in V the model. Hence y ↔ i ♦ ∼ yi holds at points 1 and 2. We distinguish several cases: 1. If for any j, if yj = ⊥ at 3 then ♦ ∼ yj is true at 1 and 2 and it plays no role in the conjunction. If for all yj , 3 ∼ yj then y = ⊤ at 1 and 2 and y is of type 1. 2. Some yj are ⊤ at 3. Then ♦ ∼ yj is ⊥ at 2 and y = ⊥ at 2. 3. We now check whether yj is true or false at 2. If for some j, yj = ⊤ at 2 then ♦ ∼ yj = ⊥ at 1 and y = ⊥ at 1 and hence y = ⊥ everywhere, and hence y is of type 0. If for all j, yj = ⊥ at 2 then y = ⊤ at 1 and we have that y = ⊤ at 1 and y = ⊥ at 2 and we must have by the axiom 4.1 that 3  y and y is of type 3. 33 A Logical Account of Formal Argumentation world 3 2 1 type 1 ⊤ ⊤ ⊤ type 0 ⊥ ⊥ ⊥ type 2 ⊤ ⊥ ⊤ Table 2. 3-chain models. Theorem 12. Let AF = (Ar, att ) be an argumentation framework and let M(AF ) be the modal sentence as defined. Then: 1. Any complete labelling L for AF gives rise to a model of M(AF ), where for any q, the correspondence is as in Table 2. • q is assigned type 1 assignment iff L(q) = 1 • q is assigned type 0 assignment iff L(q) = 0 • q is assigned type 2 assignment iff L(q) =? 2. Let M1 , M2 , M3 , . . . be all the 3-chain models of M(AF ). Then {Mi } are all the complete labellings for AF , where these complete labellings are obtained as in Table 2. Proof. Direction 1. Let h be an assignment model satisfying M (AF ). By Lemma 5 every atom node x in P gets assigned values h(x) of types 0, 1, or ?. Define a Caminada candidate function L according to the types, namely L(x) = i iff h(x) is of type i, i ∈ {0, 1, ?}. We now show that L satisfies the conditions of a Caminada function of Definition 5. 1 If x is an initial point then Gx is a conjunct of M (AF ), therefore h(x) is of type 1 and hence L(x) = 1. 2 If y1 , . . . , yn are allV nodes with arrows to y, then we have the conjunct y ↔ i ♦ ∼ yi in M (AF ) in the clause ^ ^ G(⊥ ∨ (y ↔ ♦ ∼ yi )) y i V This means that at nodes 1 and 2 of the chain y ↔ i ♦ ∼ yi must hold. Assume all yi are of type 0, then y = ⊤ at 2 34 Caminada and Gabbay and 1. Hence by axiom 6.2, y = ⊤ also at node 3 and y is of type 1. This means that if all yi are ‘out’ then y is ‘in’. Assume one of yi is of type 1. Then for this yi , ♦ ∼ yi is ⊥ at nodes 1 and 2. Hence y is false at 1 and 2 and by axiom 4.3, y = ⊥ at node 3. Hence y is of type 0. Now assume all of yi are either of type 0 or of type ?, and at least one of yi say y1 is of type?. The yi of type 0 have no influence on y because ♦ ∼ yi is ⊤ at nodes 1 and 2. The crucial nodes are the nodes like y1 which are of type ?. This means that 1  y1 , 2 ∼ y1 , 3  y1 . Thus ♦ ∼ y1 = ⊥ at 2 and ♦ ∼ y1 = ⊤ at 1. This holds for any type yi of type ?. Thus y = ⊤ at 1 and y = ⊥ at 2. Hence by axiom 6.1, y = ⊤ at 3 and hence y is of type ?. Direction 2. Let L be a Caminada function. Define a model by assigning values to the propositions according to the code of Table 2. We claim M (AF ) holds in this model. First all nodes y without arrows into them are assigned type 1 and hence Gy holds. For the other nodes we must check the formula ^ ^ G(⊥ ∨ (y ↔ ♦ ∼ yi )) y y has arrows leading to it i and show it holds in the model. We distinguish several cases. Assume all yi are of type 0. This means L(yi ) = 0 for all yi . So ♦ ∼ yi is ⊤ at nodes 1 and 2. But V since all L(yi ) = 0 we get that L(y) = 1 and hence y ↔ i ♦ ∼ yi holds at 1 and 2 as required. If one V of yi is of type 1, this means ♦ ∼ yi is ⊥ at 1 and 2. Hence i ♦ ∼ yi is ⊥ at 1 and 2. But also since L(yi ) = 1, we get L(y) = 0 and so y is of type 0 and y = ⊥ at 1 and 2. V Hence y ↔ i ♦ ∼ yi holds at 1 and 2. Now assume that all yi are either of type 0 or type ?. This means L(yi ) is either 0 or ?. Assume that at least one yi is of type ? (i.e. L(yi ) =?). Then L(y) =? and V we have y = ⊤ at 1 and y = ⊥ at 2. Let us check whether i ♦ ∼ yi is ⊥ at and ⊤ at 2. Since all yi are of type ? or ⊥ with at least one yi of type 35 A Logical Account of Formal Argumentation ?, the type ? yi will be ⊤ at node 2 and hence ♦ ∼ yi = ⊥ at node 1 and hence y = ⊥ at node 1. V On the other hand all yi are ⊥ at nodeV3 and hence i ♦ ∼ yi is ⊤ at node 2. This shows that y ↔ i ♦ ∼ yi is ⊤ at nodes 1 and 2. Thus the above argument shows that ^ ^ G(⊥ ∨ (y ↔ ♦ ∼ yi )) u i holds at all nodes 1, 2, and 3. This completes the proof of the theorem. Corollary 1. M(AF ) characterises all the extensions of AF through the modal logic LN 3. In particular, we obtain the following theorem. Theorem 13. x is in the grounded extension of AF iff M(AF ) ⊢LN 3 Gx. Proof. This holds because the grounded extension is the minimal complete extension [12]. Since Gx is in every model of M(AF ) we cannot characterise other extensions, e.g. preferred extensions, in a similar way. We need the mixed approach with additional modalities. 7. Discussion Grossi in [19] uses modal logic to represent argumentation frameworks. His approach is metalevel. We use (in Section 5.1) classical logic to talk about argumentation frameworks and use circumscription to define the various extensions. Grossi uses modal logic to describe the argumentation frameworks. He needs two modalities, one to go with the attack relation and one to go with the converse of the attack relation (like two temporal logic modalities). He also uses a universal modality and to get the extensions he needs µ-calculus on top. Our approach, on the other hand, is to use classical logic with circumscription to do the job. We would not be surprised if the approach of Grossi could simulate the relevant classical logic with circumscription, since all the ingredients are present. Note that our use of modal logic in section 6 is object level and is completely different from its use as a metalevel tool. 36 Caminada and Gabbay As was mentioned in Section 3 the approach of argument labellings can be traced back to Pollock, who in his 1995 book [21] described his oscar system in terms of labellings. As explained in [20], Pollock’s approach essentially boils down to preferred semantics. The labelling approach of Jakobovits and Vermeir [20] is aimed not so much at describing Dung’s original semantics but rather to defining additional semantics driven by what they perceive to be problems in Dung’s original semantics. Caminada first described complete semantics in terms of labellings in [9] and showed how this can be used to provide labelling based descriptions of other semantics as well. This approach was then applied in [11] and [28] to provide labelling-based algorithms for computing particular argumentation sets and extensions. Besnard and Doutre [5] examine how complete and preferred semantics can be expressed in terms of set theoretical equations, but do not provide a logical account of these semantics. A connection between the current work and the topic of linear programming equations can be found in [15]. We did use some of the equations of [15] in Section 4.4. References [1] L. Amgoud, N. Maudet, and S. Parsons. Modelling dialogues using argumentation. In Proceedings of the Fourth International Conference on MultiAgent Systems (ICMAS00), pages 31–38, Boston, MA, 2000. [2] ASPIC-consortium. Deliverable D2.5: Draft formal semantics for ASPIC system, June 2005. [3] Pietro Baroni and Massimiliano Giacomin. Comparing argumentation semantics with respect to skepticism. In Proc. ECSQARU 2007, pages 210–221, 2007. [4] Pietro Baroni and Massimiliano Giacomin. 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An algorithm to compute minimally grounded and admissible defence sets in argument systems. In P.E. Dunne and TJ.M. Bench-Capon, editors, Computational Models of Argument; Proceedings of COMMA 2006, pages 109–120. 38 Caminada and Gabbay IOS, 2006. [29] Yining Wu, Martin Caminada, and Dov Gabbay. Complete extensions in argumentation coincide with 3-valued stable models in logic programming. submitted to Studia Logica, 2009. Martin W.A. Caminada Interdisciplinary Lab for Intelligent and Adaptive Systems University of Luxembourg [email protected] Dov M. Gabbay King’s College, London Bar-Alan University, Israel University of Luxembourg [email protected]