On the Logic of Argumentation Theory

Davide Grossi Institute for Logic, Language and Computation University of Amsterdam Science Park 904, 1098 XH Amsterdam, The Netherlands [email protected]

ABSTRACT So, the intuitive reading of a  b is that argument a at- b The paper applies modal logic to formalize fragments of ar- tacks argument . This paper investigates the simple but gumentation theory. Such formalization allows to import, yet unexplored idea which consists in viewing abstract ar- S, R for free, a wealth of new notions (e.g., argument equiva- gumentation frameworks as Kripke frames ( ) [1] where S A lence), new techniques (e.g., calculi, model-checking games, = , that is, the set of states is the set of arguments, and R −1 bisimulation games), and results (e.g., completeness of cal- = , that is, the accessibility relation is the inverse of culi, adequacy of games, complexity of model-checking) from the attack relation or, intuitively, the ‘being attacked’ rela- logic to argumentation. tion. The entire paper hinges on this simple assumption. For space reasons the paper cannot introduce argumen- tation theory in an extensive way but, to make it as most Categories and Subject Descriptors self-contained as possible, the main argumentation-theoretic I.2.4 [Knowledge Representation Formalisms and Meth- notions from [6] have been recapitulated in Table 1. As such ods]: Modal logic notions are formalized along the paper, their intuitive read- ing will also be provided. This said, the paper is organized General Terms as follows. Section 2 starts off by applying a well-known modal logic to study a first set of notions of argumenta- Theory tion theory. This enables the possibility of using calculi to derive argumentation-theoretic results such as the Funda- Keywords mental Lemma [6], and import complexity results concern- Argumentation theory, modal logic ing, for instance, checking whether a given set is a stable extension. Along the same line, Section 3 tackles the for- malization of the notion of grounded extension within the 1. INTRODUCTION modal μ-calculus. In Section 4 semantic games are studied The paper analyzes argumentation from the point of view for the logic introduced in Section 2 which provide a ver- of formal logic. It shows how standard results in argu- sion of dialogue games as model-checking games. Section 5 mentation theory obtain elegant reformulations within well- tackles the question—not yet addressed in the literature— investigated modal logics. This allows to import—for free— of when two arguments, or two argumentation frameworks, a number of techniques (e.g., calculi, logical games) as well are “indistinguishable” from the point of view of argumen- as results (e.g. completeness, adequacy, complexity) from tation theory. For this purpose the model-theoretic notion modal logic to argumentation theory. Also, as is often the of bisimulation is introduced and bisimulation games are case in the cross-fertilization of different formalisms, this presented as a procedural method to check the “behavioral perspective opens up new lines of research which were thus equivalence” of two argumentation frameworks. Section 6 far hidden to the attention of argumentation theorists. addresses the problem of the representation of preferred ex- Although the results presented are theoretical, they set tensions, briefly discusses related work and concludes. the stage for the development of logic-based techniques for argumentation in multi-agent systems such as, eminently, the formal verification (via model-checking) of argumenta- 2. ARGUMENTS IN MODAL DISGUISE tion systems, and the design of multi-agent argumentation protocols via logic games. 2.1 Argumentation models Let us start with the basic structure of argumentation If an argumentation framework can be viewed as a Kripke theory. An abstract argumentation framework is a rela- frame, then an argumentation framework plus a function A A,  A tional structure =( ) where is a non-empty set assigning names from a set P to sets of arguments can be ⊆ A2 A of arguments, and is an ‘attack’ relation on [6]. viewed as a Kripke model [1].

Cite as: On the Logic of Argumentation Theory, Davide Grossi, Proc. Definition 1 (Argumentation models). Let P be a of 9th Int. Conf. on Autonomous Agents and Multiagent Sys- set of propositional atoms. An argumentation model M = tems (AAMAS 2010), van der Hoek, Kaminka, Lespérance, Luck and A, I A A,   ( ) is a structure such that: =( ) is an argumen- Sen (eds.), May, 10–14, 2010, Toronto, Canada, pp. 409-416 I −→ A c tation framework; : P 2 is an assignment from P to Copyright 2010, International Foundation for Autonomous Agents and A Multiagent Systems (www.ifaamas.org). All rights reserved. subsets of . The set of all argumentation models is called

409 cA characteristic function of A iff ∀X, cA(X)={a |∀b :[b  a ⇒∃c ∈ X : c  b]}

X is acceptable w.r.t. Y in A iff X ⊆ cA(Y )

X conflict-free in A iff  ∃a, b ∈ X s.t. a  b

X admissible set of A iff X is a pre-fixpoint of cA (i.e. X ⊆ cA(X))

X complete extension of A iff X is conflict-free and is a fixpoint of cA (i.e., X = cA(X))

X stable extension of A iff X is a complete extension of A and ∀b ∈ X, ∃a ∈ X : a  b

X grounded extension of A iff X is the minimal complete extension of A

X preferred extension of A iff X is a maximal complete extension of A

Table 1: Basic notions of argumentation theory (X denotes a set of arguments).

A. A pointed argumentation model is a pair (M,a) where 2.2.2 Semantics. M a A is an argumentation model and an argument from . ∀ Definition 2 (Satisfaction). Let ϕ ∈LK . The sat- Argumentation models are nothing but argumentation frames isfaction of ϕ by a pointed argumentation model (M,a) is together with a way of “naming” sets of arguments or, to put inductively defined as follows (Boolean clauses are omitted): it otherwise, of “labeling” arguments. The fact that an ar- −1 gument a belongs to I(p) in a given model M, which in M,a|= ϕ iff ∃b ∈ A :(a, b) ∈  and M,b|= ϕ logical notation reads (A, I),a |= p, can be interpreted as M,a|= ∀ϕ iff ∃b ∈ A : M,b|= ϕ stating that “argument a has property p” , or that “p is true ϕ M of a”. By substituting p with a Boolean compound ϕ (e.g., As usual, is valid in an argumentation model iff it is M M| ϕ ϕ ϕ := p∧q) we can say that “a belongs to both the sets called satisfiedinallpointedmodelsof , i.e., = ; is valid M p and q”, and the same can be done for all other Boolean in a class of argumentation models iff it is valid in all M | ϕ ϕ connectives. The following example applies this insight to its models, i.e., = . The truth-set of a formula is |ϕ| argumentation labeling functions [3]. denoted M. K∀ Example 1. (Argument labelings as argumentation mod- Logic is therefore endowed with modal operators of the els) In argumentation theory, a labeling function [3] is a type “there exists an argument attacking the current one  function l : { , , ?}−→A from the set of three labels { , such that”, i.e., , and “there exists an argument such 1 0 1 ∀ , }—intuitively in, out, undecided—to the set of argu- that”, i.e., , together with their duals. Given an argumen- 0 ? M ments A. From a logical point of view, such a function is tation model we can thereby express statements such as a equivalent to a valuation function I : P −→ 2A with the the ones adverted to above: “ is attacked by an argument in ϕ ϕ further constraint that each argument can get at most one a set called ” corresponds to being true in the pointed M,a a ϕ label which, in propositional logic, amounts to the following model ( ) and “ is defended by the set ” corresponds to ϕ being true in the pointed model (M,a). formula Label := (1∧¬0∧¬?)∨(¬1∧0∧¬?)∨(¬1∧¬0∧?). Hence, a framework A with a labeling function is nothing but 2.2.3 Axiomatics. an argumentation model M =(A, I) s.t. M|= Label. Logic K∀ is axiomatized as follows, where i ∈{, ∀}: Formula Label in the example is just a propositional formula (Prop) propositional tautologies but what is typically interesting in argumentation theory i ϕ → ϕ → i ϕ → i ϕ are statements of the sort: “argument a is attacked by an (K)[]( 1 2) ([ ] 1 [ ] 2) argument in a set ϕ”; “argument a is defended by the set (T)[∀]ϕ → ϕ ϕ”, or, “ϕ attacks an attacker of argument a”. These are (4)[∀]ϕ → [∀][∀]ϕ modal statements, and in order to express them, it suffices (5) ¬[∀]ϕ → [∀]¬[∀]ϕ to introduce a dedicated modal operator  whose intuitive reading is “there exists an attacking argument such that”. (Incl)[∀]ϕ → [i]ϕ (Dual) iϕ ↔¬[i]¬ϕ 2.2 Logic K∀ This section introduces logic K∀, an extension of the min- 2.2.4 Meta-theoretical results. imal modal logic K with universal modality. We list the following known relevant results. ∀ 2.2.1 Language. • Logic K is sound and strongly complete for the class ∀ A of argumentation frames [1, Ch. 7]. The language of K is a standard modal language with  ∀ ∀ two modalities: and , i.e., the universal modality. It • The complexity of checking whether a formula of LK is built on the set of atoms P by the following BNF: is satisfied by a pointed model M is P-complete [10]. K∀ L : ϕ ::= p |⊥|¬ϕ | ϕ ∧ ϕ |ϕ | ∀ϕ 2.3 Doing argumentation in K∀ where p ranges over P. The other standard boolean { , ∨, → Perhaps surprisingly, logic K∀ is already expressive enough } and modal {[], [∀]} connectives are defined as usual. to capture several basic notions of argumentation theory

410 α → γ ∧ β → γ → α ∨ β → γ Prop such as: conflict freeness, acceptability, admissibility, com- (( ) ( )) ( ) plete extensions, stable extensions. ([∀](α → γ) ∧ [∀](β → γ)) → [∀](α ∨ β → γ)2, N, K, MP Acc(ϕ, ψ):=[∀](ϕ → []ψ) (1) ([∀](ϕ → []  ϕ) ∧ [∀](ψ → []  ϕ)) → CFree(ϕ):=[∀](ϕ → []¬ϕ) (2) [∀](ϕ ∨ ψ → []  ϕ)Instanceof3 ϕ ∀ ϕ →  ¬ϕ ∧  ϕ Adm ( ):=[]( ([ ] [ ] )) (3) []  ϕ → []  (ϕ ∨ ψ) Prop, K, N Compl(ϕ):=[∀]((ϕ → []¬ϕ) ∧ (ϕ ↔ []ϕ)) (4) ([∀](ϕ → []  ϕ) ∧ [∀](ψ → []  ϕ)) → Stable(ϕ):=[∀](ϕ ↔ []¬ϕ) (5) [∀](ϕ ∨ ψ → []  ϕ ∨ ψ)4, Prop, K, N Intuitively, a set of arguments ϕ is acceptable with respect Acc(ϕ, ϕ) ∧ Acc(ψ, ϕ) → Acc(ϕ ∨ ψ, ϕ ∨ ψ)5, definition to the set of arguments ψ if and only all ϕ-arguments are such that for all their attackers there exists a defender in ψ Figure 1: Example of a derivation in K∀. (Formula 1). A set of arguments ϕ is conflict free if and only if all ϕ-arguments are such that none of their attackers is in ϕ (Formula 2). A set of arguments ϕ is admissible if and 1 the derivation of a sub-result of Formula 8: The proof only if it is conflict free and acceptable with respect to itself is completed by proving that: Adm(ϕ) ∧ Acc(ψ ∨ ξ,ϕ) → (Formula 3). A set ϕ is a complete extension if and only if Acc(ϕ, ϕ) ∧ Acc(ψ, ϕ), Adm (ϕ) ∧ Acc(ψ ∨ ξ,ϕ) → CFree(ϕ ∨ it is conflict free and it is equivalent to the set of arguments ψ), and Adm (ϕ) ∧ Acc(ψ ∨ ξ,ϕ) → Acc(ξ,ϕ ∨ ψ). all the attackers of which are attacked by some ϕ-argument (Formula 4). Finally, a set ϕ is a stable extension if and only Formula 8 is a generalized version of the so-called Funda- if it is equivalent to the set of arguments whose attackers are mental Lemma proven in [6]. It states that if ϕ is admissible not in ϕ (Formula 5). The adequacy of these definitions with and both ψ and ξ are acceptable with respect to it then also respect to the standard ones in Table 1 is easily checked. ψ ∨ ξ is admissible and ξ is acceptable with respect to ϕ ∨ ψ. The following examples applies K∀ to Example 1. Formulae 9 and 10 state well-known facts about the relative strength of admissible, complete and stable extensions. ∀ Example 2. (Argumentation labelings in K ) According Other results can be formalized along the same lines. What to [3], a labeling function is a complete labeling if and only the section has shown is that, already within a rather stan- if the following holds for each argument: a) an argument dard modal systems such as K∀, quite many notions and is labeled 1, i.e., in, iff all its attackers are labeled 0, i.e., results of abstract argumentation can be accommodated. out. b) an argument is labeled 0, i.e., out, iff there exists at Besides, by the results reported in Section 2.2.4 it follows least one attacker labeled 1. The reformulation of a)-b) in that model-checking whether a given formula is conflict free, ∀ K goes as follows: admissible, acceptable (with respect to another formula), complete or stable can be done in polynomial time: e.g., [∀]((1 ↔ []0) ∧ (0 ↔1) ∧ Label) (6) “M,a |= Stable(ϕ)?”. Similarly, it follows that it can be where Label is the propositional formula described in Exam- checked in polynomial time whether an argument belongs ple 1. So, a valuation I on an alphabet containing 1, 0 and to the truth-set of a formula which is conflict free, admissi- ? is a complete labeling for an argumentation framework A ble, acceptable (with respect to another formula), complete iff the model (A, I) satisfies Formula 6. Also, it is a matter or stable: e.g., “M,a|= ϕ ∧ Stable(ϕ)?”. of propositional reasoning to see that Formula 6 is equivalent to the following formula: 3. MODAL FIXPOINTS Compl(1) ∧ [∀]((0 ↔1) ∧ Label)(7)The present section shows what kind of modal machinery is needed to capture the notion of grounded extension left I In words, this means that a function on an alphabet con- aside in Section 2. In [6], the grounded extension is defined A taining 1, 0 and ? is a complete labeling of if and only if as the smallest fixpoint of the characteristic function of an A, I the model ( ) makes 1 to be a complete extension (For- argumentation framework (see Table 1). mula 4) and evaluates the labels 0 and ? accordingly. We obtain therefore a direct correspondence between complete 3.1 Characteristic functions in K labelings and complete extensions. The same could be done Each argumentation framework A =(A, ) determines a for stable extensions. A A characteristic function cA :2 −→ 2 such that for any set We can now prove results of argumentation theory, such as of arguments X, cA(X) yields the set of arguments in A which are acceptable with respect to X, i.e., {a ∈ A |∀b ∈ the ones proven in [6], which concern the notions formalized ∀ in Formulae 1-5 as theorems of K∀. A :[b  a ⇒∃c ∈ X : c  b]}. Does logic K have a syntactic counterpart of the characteristic function? The Theorem 1 ([6] formalized). The following formulae answer turns out to be yes. [] are theorems of K∀: Let L be the language defined by the following BNF: [] Adm (ϕ) ∧ Acc(ψ ∨ ξ,ϕ) → Adm(ϕ ∨ ψ) ∧ Acc(ξ,ϕ ∨ ψ)(8 ) L : ϕ ::= p |⊥|¬ϕ | ϕ ∧ ϕ | []ϕ ϕ → ϕ Stable( ) Adm( ) (9) where p belongs to the set of atoms P. Language L[] is ϕ → ϕ K∀ Stable( ) Compl( ) (10) the fragment of L containing only the compounded modal K Proof (Sketch). The theorem is easily proven semanti- operator [] or, also, simply the fragment of L (i.e., f ∀ cally by then calling in completeness. However, as an exam- LK without universal modality) containing only the []- + A ple of the application of the calculus, we provide in Figure operator. Let A =(2 , ∩, −, ∅,cA) be the power set alge-

411 A braon2 extended with operator cA, and consider the term where |ϕ|M[p:=X] denotes the truth-set of ϕ once I(p) is set [] algebra terL[] =(L , ∧, ¬, ⊥, []). Finally, let to be X. As usual, we say that: ϕ isvalidinanargumen- I∗ : L[] −→ 2A be the inductive extension of a valua- tation model M iff it is satisfied in all pointed models of tion function I : P −→ 2A according to the semantics given M, i.e., M|= ϕ; ϕ is valid in a class M of argumentation in Definition 2. We can prove the following result. models iff it is valid in all its models, i.e., M |= ϕ.

Theorem 2 (cA vs. []). Let M =(A, I) be an We have now all the logical machinery in place to express argumentation model. Function I∗ is a homomorphism from the notion of grounded extension. Set ϕ(p):=[]p, + ϕ p   terL[] to A . that is, take ( ) to be the modal version [ ] of the characteristic function, and apply it to formula p. What we Proof. The case of Boolean connectives is trivial. It re- obtain is a modal formula expressing the least fixpoint of a ϕ |  ϕ| c |ϕ| . mains to be proven that for any : [ ] M = A( M) characteristic function, that is, the grounded extension: It suffices to spell out the semantics of []: μp.  p −1 −1 Grounded := [ ] (11) |[]ϕ|M = {a |∀b : a  b, ∃c : b  c & c ∈|ϕ|M} Notice that, unlike the notions formalized in Formulae 1-5, = {a |∀b : b  a, ∃c : c  b & c ∈|ϕ|M} the grounded extension of a framework is always unique and c |ϕ| . = A( M) does not depend on the particular labeling of a given model. This completes the proof. 3.2.3 Axiomatics. μ In other words, Theorem 2 shows that the complex modal Logic K is axiomatized by the following rules and axiom operator [], under the semantics provided in Defini- schemata. tion 2, behaves exactly like the characteristic function of (Prop) propositional schemata the argumentation frameworks on which the argumentation  ϕ → ϕ →  ϕ →  ϕ models are built. To put it yet otherwise, formulae of the (K)[]( 1 2) ([ ] 1 [ ] 2) form []ϕ denote the value of the characteristic function (Fixpoint) ϕ(μp.ϕ(p)) ↔ μp.ϕ(p) ϕ applied to the set of arguments. Notice also that from (MP) if  ϕ1 → ϕ2 and  ϕ1 then ϕ2 Theorem 2 the adequacy of Formulae 1-5 with respect to ( ) if  ϕ then  []ϕ the definitions in Table 1 follows straightforwardly. N Characteristic functions are known to be monotonic [6] (Least) if  ϕ1(ϕ2) → ϕ2 then  μp.ϕ1(p) → ϕ2   hence, by Theorem 2, we get that [ ] denotes a mono- So, the axiomatics of Kμ consists of the axiom system K tonic function and therefore, by the Knaster-Tarski theo- 1 axiomatizing  plus schema Fixpoint and rule Least. rem we have that there always exist a greatest and a least Axiom states that μp.ϕ(p) is indeed a fixpoint   Fixpoint [ ] -fixpoint. From a logical point of view this means since a further application of ϕ still yields μp.ϕ(p) and vice that, in order to be able to express the grounded extension, versa. Instead, rule Least guarantees that μp.ϕ(p)isinfact it suffices to add to the K fragment of K∀ a least fixpoint the least fixpoint by imposing that if ϕ2 is provably a pre- operator. This takes us to the realm of μ-calculi. fixpoint of ϕ1,thenμp.ϕ1(p) provably implies ϕ2. μ 3.2 -calculus for argumentation 3.2.4 Meta-theoretical results. 3.2.1 Language. We list two relevant known results. To add the least fixpoint operator μ to logic K we first • Kμ A μ Logic is sound and complete for the class of all define language LK via the following BNF: argumentation models under the semantics given in ∀ Kμ Definition 3 [14]. Notice however that, unlike K , the L ϕ p |⊥|¬ϕ | ϕ ∧ ϕ |ϕ | μp.ϕ p μ : ::= ( ) given axiomatics of K is not strongly complete since where p ranges over P and ϕ(p) indicates that p occurs free it is obviously not compact. ϕ in (i.e., it is not bounded by fixpoint operators) and under • The complexity of the model-checking problem for a 2 ϕ ψ an even number of negations. In general, the notation ( ) formula of size m and alternation depth d on a system ψ ϕ stands for occurs in . The usual definitions for Boolean of size n is O(m·nd+1) [8] where the alternation depth and modal operators can be applied. Intuitively, μp.ϕ(p) μ of a formula of LK is the maximum number of μ/¬μ¬ denotes the smallest formula p such that p ↔ ϕ(p). This μ in a chain of nested fixpoints. intuition is made precise in the semantics of LK . 3.3 Doing argumentation in Kμ 3.2.2 Semantics. Like in Section 2.3 we give a couple of examples of the μ Kμ kind of argumentation-theoretic results formalizable in K . Definition 3 (Satisfaction). Let ϕ ∈L . The sat- isfaction of ϕ by a pointed model (M,a),withM =(A, I), Theorem 3 (Grounded extension is conflict-free). is inductively defined as follows (Boolean clauses, as well as The following formula is a validity of Kμ: the clause for , are as in Definition 2): \ Grounded →¬[]Grounded (12) M,a| μp.ϕ p a ∈ {X ∈ A ||ϕ| ⊆ X} = ( ) iff 2 M[p:=X] Proof. We proceed per absurdum applying the defini- 1We refer the interested reader to [5]. tion in Formula 11. Take an argumentation model satis- 2This syntactic restriction guarantees that every formula fying Formula 12 and assume that there exist arguments ϕ(p) defines a monotonic set transformation. a, b such that a −1 b and M,b |= μp.[]p while also

412 M,a |= μp.[]p. We distinguish two cases: 1) there Position Turn Available moves a −1 b −1 b −1 ... −1 b (ϕ1 ∨ ϕ2,a) ∃{(ϕ1,a), (ϕ2,a)} exists a finite chain ( 1 n)of ϕ ∧ ϕ ,a ∀{ϕ ,a , ϕ ,a } a ( 1 2 ) ( 1 ) ( 2 ) successors starting from ; 2) there exists an infinite such (  ϕ, a) ∃{(ϕ, b) | (a, b) ∈−1} chain. If 1) is the case, then M,bn |=[]ϕ for any ϕ. ([]ϕ, a) ∀{(ϕ, b) | (a, b) ∈−1} Since both M,a|= μp.[]p and M,b|= μp.[]p, then ( ∀ ϕ, a) ∃{(ϕ, b) | b ∈ A} M,bn−1 |= μp.[]p which, by Definition 3, means that ([∀]ϕ, a) ∀{(ϕ, b) | b ∈ A} ⊥,a ∃∅ for any p such that |[]p|M ⊆|p|M, M,bn−1 |=[]p, ( ) (,a) ∀∅ which is impossible given that for any ϕ M,bn |=[]ϕ and M,b |   ¬p (p, a)&a ∈I(p) ∃∅ hence that n−1 = [ ] . If 2) is the case, then p, a a ∈I p ∀∅ |μp.  p| ∅ ( )& ( ) we show that [ ] M = . This is the case since (¬p, a)&a ∈I(p) ∃∅ the two following sets are both pre-fixpoints but they have (¬p, a)&a ∈I(p) ∀∅ −1 −1 empty intersection: {c ∈ A |a 2m c} and {c ∈ A |b 2m c} −1 −1 K∀ where 2m denotes reachability via  in an even number Table 2: Rules of the model-checking game for . of steps. We thus obtain a contradiction.

Like Theorem 2, Theorem 3 provides a modal logic formula- The important thing to notice is that positions of the game tion of an argumentation-theoretic result. Let us now look are pairs of a formula and an argument, and that the type at the complexity. of formula in the position determines which player has to play: ∃ if the formula is a disjunction, a box, a false atom 3 Theorem 4 (Model-checking grounded). Let M be or ⊥,and∀ in the remaining cases. an argumentation model. It can be decided in polynomial Definition 5 (Winning strategies and positions). time whether an argument a belongs to the grounded exten- A strategy for player P in C(ϕ, M)@(ϕ, a) is a function sion of M, that is, whether M,a|= Grounded. telling P what to do in any match played from position Proof. Since μp.[]p has alternation depth 0 it fol- (ϕ, a). Such a strategy is winning for P if and only if, in lows, by the result reported in Section 3.2.4, that model- any match played according to the strategy, P wins. A posi- checking μp.[]p can be done in O(m · n) where m is the tion (ϕ, a) in C(ϕ, M) is winning for P if and only if P has size of μp.[]p and n the size of M. a winning strategy in C(ϕ, M)@(ϕ, a). The set of winning positions of C(ϕ, M) is denoted WinP (C(ϕ, M)).

4. DIALOGUE GAMES & LOGIC GAMES By Definitions 4 and 5 it follows that the model-checking The proof-theory of abstract argumentation is commonly game is a two-players zero-sum game. It is known that such given in terms of dialogue games [12]. The present section games are determined, that is, each match has a winner [15]. introduces a new game-theoretic proof procedure for argu- It remains to be proven that the game is adequate with mentation theory based on model-checking games. In model- respect to the semantics of K∀. To put it otherwise, we have checking games, a proponent or verifier (∃ve) tries to prove to prove that if ∃ always wins then the formula defining that a given formula ϕ holds in a point a of a model M, the game is true at the point of instantiation, and that if a while an opponent or falsifier (∀dam) tries to disprove it. formula is true at a point in a model, then ∃ always wins The present section deals with the model-checking game for the corresponding game instantiated at that point. K∀ Kμ .For -variant we refer the reader to [13]. ∀ Theorem 5 (Adequacy). Let ϕ ∈LK , and let M = ∀ 4.1 Model-checking game for K (A, I) be an argumentation model. Then, for all a ∈ A: A model-checking game is a graph game, that is, a game (ϕ, a) ∈ Win∃(C(ϕ, M)) ⇐⇒ M ,a|= ϕ. played by two agents on a directed graph, where each node— called position—is labelled by the player that is supposed to Proof (Sketch). The proof is by induction on the length move next. The structure of the graph determines which l of ϕ. A proof for K without the universal modality can are the admissible moves at any given position. If a player be found in [13]. It suffices to extend the inductive case has to move in a certain position but there are no available to cover formulae with the universal modality. The base moves, then it loses and its opponent wins. In general, graph case l = 0 is straightforward. For the step l>0wepro- games might have infinite paths, but this is not the case in vide a proof of the modal case ϕ = ∀ψ. From left to the game we are going to introduce. A match of a graph right. Assume (ϕ, a) ∈ Win∃(C(ϕ, M)). It is ∃’s turn to game is then just the set of positions visited during play, move. It follows that there exists a position (ψ, b) s.t. it that is, a complete path through the graph. is a winning position for ∃. By induction hypothesis we conclude that M,b |= ψ and hence M,a |= ∀ψ. From Definition 4 (K∀-model-checking game). Let ϕ ∈ right to left. Assume M,a |= ϕ. It follows that there ex- K∀ b M,b | ψ L ,andM be an argumentation model. The model-checking ists s.t. = . By induction hypothesis we have that ψ, b ∈ C ψ, M ∃ game C(ϕ, M) is defined by the following items. Players: ( ) Win∃( ( )). But it is ’s turn to move, hence ϕ, a ∈ C ϕ, M The set of players is {∃, ∀}. An element from {∃, ∀} will be we conclude ( ) Win∃( ( )). denoted P and its opponent P . Game form: The game form of C(ϕ, M) is defined by the board game in Table 2. 4.2 Games for model-checking extensions Winning conditions: Player P wins if and only if P has The next example illustrates a model-checking game for to play in a position with no available moves. Instantia- stable extensions run on the so-called Nixon diamond [12]. tion: The instance of C(ϕ, M) with starting point (ϕ, a) is denoted C(ϕ, M)@(ϕ, a). 3 Notice that positions use formulae in positive normal form.

413 (1 ∧ [∀](1 ↔¬  1),a) ∀ Admis. : C(ϕ ∧ Adm(ϕ), M)@(ϕ ∧ Adm(ϕ),a) a Compl. : C(ϕ ∧ Compl(ϕ), M)@(ϕ ∧ Compl(ϕ),a) 1 Stable : C(ϕ ∧ Stable(ϕ), M)@(ϕ ∧ Stable(ϕ),a) ([∀](1 ↔¬  1),a) ∀ ∃ (1,a) wins!wins! Grounded : C(Grounded, M)@(Grounded,a)

(1 ↔¬  1,a) ∀ (1 ↔¬  1,b) ∀ Table 3: Games for model-checking extensions.

(¬ ∨¬  ,a) ∃ ∨  ,a ∃ 1 1 (1 1 ) “A|= ∀p1,...,pn¬STa(ϕ)?” where p1,...,pn are the atoms 0 occurring in ϕ and STa(ϕ) is the standard translation of ϕ b 5 (¬  1,a) ∀ (¬1,a) ∀ wwins!ins! realized in state a. To conclude, we might say that the games defined in Sec- Nixon tion 4.1 provide a proof procedure for a reasoning task which (1,b) ∃ wins! Diamond is computationally simpler than the one tackled by standard Figure 2: Game for stable extensions in the 2-cycle. dialogue games. It should be noted, however, that this is no intrinsic limitation to the logic-based approach advocated in the present paper. Model-checking games for monadic Example 3 (Model-checking the Nixon diamond). second-order logic (or rather for appropriate fragments of it) would accommodate dialogue games in their entirety, lifting Let A =({a, b}, {(a, b), (b, a)}) be an argumentation frame- the sort of systematization they enable—in the form exem- work consisting of two arguments a and b attacking each plified by Table 3—to dialogue games. other (i.e., the Nixon diamond), and consider the labeling I assigning 1 to a and 0 to b (top right corner of Figure 2). We now want to run an evaluation game for checking whether a 5. INDISTINGUISHABLE ARGUMENTS belongs to a stable extension corresponding to the truth-set Since abstract argumentation neglects the internal struc- of 1. Such game is the game C(1 ∧ Stable(1), (A, I)) initial- ture of arguments, the natural question arises of when two ized at position (1 ∧ Stable(1),a). That is, spelling out the arguments can be said to be the same from the point of view definition of Stable(1): C(1 ∧ [∀](1 ↔¬1))@(1 ∧ [∀](1 ↔ of argumentation theory. Studying such notion of “same- ¬1),a). Such a game, played according to the rules in ness” or “equivalence” of arguments is not just a mathemat- Definitions 4 and 5, gives rise to the tree in Figure 2. ical diversion. A simple example where this issue appears is in legal reasoning, and in particular within common-law In general, model-checking games provide a proof procedure systems. Often, in such systems the so-called principle of for checking whether an argument belongs to a certain ex- stare decisis [11] holds. According to such a principle, a tension given an argumentation model, which we have seen judge should rule cases that are “substantially the same” in in Sections 2.3 and 3.3 to be a polynomial problem. The the same way. Now, an essential aspect of a judicial case structure of such proof procedure is invariant, and the dif- is its argumentation framework, so being the same in this ferent games are obtained simply by changing the formula 4 respect seems to mean something like exhibiting the “same to be checked (Table 3). This feature confers a high sys- argumentative structure”. In the present section we present tematic flavor to this sort of games for argumentation. a formal study of this simple intuition based on K∀ and Kμ. Now the natural question arises of what the precise rela- tionship is between model-checking games and the sort of 5.1 Bisimilar arguments games studied in argumentation, called dialogue games [12]. The logical analysis of abstract argumentation enables us 4.3 Model-checking games vs. dialogue games directly with a well-investigated formal notion of“behavioral equivalence” between arguments/points in a model: bisimu- The best way to highlight the difference between model- lation [1, 9]. It is well-known that logic Kμ is invariant under checking games and dialogue games is by pointing consid- bisimulation [13]. In the present section we will focus on the erations of a complexity-theoretic kind. We have seen, in specific notion of bisimulation which is tailored to K∀, also Sections 2.3 and 3.3, that checking whether an argument be- called total bisimulation. longs to a specific admissible set, or an extension (complete, stable or grounded) can be done in polynomial time. How- Definition 6 (Bisimulation). Let M =(A, , I) and ever, it is well-known that checking whether an argument M =(A, , I) be two argumentation models. A bisimula- belongs to an extension can be harder (e.g. NP-complete tion between M and M is a non-empty relation Z ⊆ A×A for stable extensions [7]). So where is the trick? such that for any a, a s.t. aZa: Atom: a and a are propo- In model-checking games you are given a model M = sitionally equivalent; Zig: if a −1 b for some b ∈ A,then (A, I), a formula ϕ and an argument a, and ∃ve is asked a −1 lb for some b ∈ A and bZb; Zag: if a −1 b to prove that M,a |= ϕ. In dialogue games, the check ap- for some b ∈ A then a −1 b for some inA and aZa.A ¯  pointed to ∃ve is inherently more complex since the input total bisimulation is a bisimulation Z ⊆ A × A such that consists there of only an argumentation framework A, a for- its left projection covers A and its right projection covers mula ϕ and an argument a. ∃ve is then asked to prove A. When a total bisimulation exists between M and M we that there exists a labeling I such that (A, I),a |= ϕ. This write (M,a)  (M,a). is not a model-checking problem but a satisfiability prob- ∀ lem in a pointed frame [1] which, in turn, is essentially Now, since logic K is invariant under total bisimulation μ a model-checking problem in monadic second-order logic: [1] and logic K under bisimulation [9], we obtain a natural notion of “sameness” of arguments, which is weaker than the 4Note that the game for checking grounded extensions is, obviously, the model-checking game for Kμ [13]. 5For the standard translation we refer the reader to [1].

414 Position Available moves From Definition 7 we obtain the following notions of win- ning strategies and winning positions. ((M,a)(M,a)) {((M,a)(M,b)) |∃b ∈ A : a −1 b} Definition 8 (Winning strategies and positions). ∪{ M,b M,a |∃b ∈ A a −1 b} (( )( )) : A strategy for player P in B(M, M)@(a, a) is a function ∪{((M,a)(M,b)) |∃b ∈ A} telling P what to do in any match played from position (a, a). Suchastrategyiswinning for P if and only if, ∪{((M,b)(M ,a)) |∃b ∈ A} in any match played according to the strategy, P wins. A position ((M,a)(M,a)) in B(M, M) is winning for P if Table 4: Rules of the bisimulation game and only if P has a winning strategy in B(M, M)@(a, a). The set of all winning positions of game B(M, M) for P is  notion of isomorphism of argumentation frameworks. If two denoted by WinP (B(M, M )). arguments are “the same” in this perspective, then they are We have the following adequacy theorem. The proof is stan- equivalent from the point of view of argumentation theory, dard and the reader is referred to [9]. as far as the notions expressible in those logics are concerned. In particular, we obtain the following simple theorem. Theorem 7 (Adequacy). Take (M,a) and (M,a) Theorem 6 (Bisimilar arguments). Let (M,a) and to be two argumentation models. It holds that: M,a Z      ( ) be two pointed models, and let be a total bisimu- ((M,a)(M ,a)) ∈ WinD(B(M, M )) ⇐⇒ (M,a)  (M ,a). lation between M and M. It holds that: In words, D has a winning strategy in the total bisimulation M,a| ϕ ∧ ϕ ⇐⇒ M ,a | ϕ ∧ ϕ = Adm ( ) = Adm( ) game B(M, M)@(a, a) if and only if M,a and M,a are where Adm(ϕ) can be substituted by CFree(ϕ), Compl(ϕ) or totally bisimilar. An example of such a game follows. Stable(ϕ) and Adm(ϕ) ∧ ϕ can be substituted by Grounded. Example 4 (A total bisimulation game). Consider Proof. Follows directly from the fact that bisimulation two simple legal cases concerning the innocence or guiltiness Kμ implies -equivalence [9], and total bisimulation implies of two defendants in two different trials. In the first one, K∀ -equivalence [1]. two arguments a and b claiming the defendant to be guilty a In other words, Theorem 6 states that if two arguments are attack an argument claiming his/her innocence. In the x totally bisimilar, then they are indistinguishable from the second one, only one argument claiming the defendant’s y point of view of abstract argumentation in the sense that guiltiness attacks an argument for his/her innocence. The M M the first belongs to a given conflict-free, or admissible set two argumentation models, and , are depicted at the c y ϕ if and only if also the second does, and the first belongs top of Figure 3. A total bisimulation connects with ,and a b x to a given stable, complete extension ϕ, or to the grounded and with . Part of the extensive bisimulation game B M, M c, y extension, if and only if also the second does. ( )@( ) is depicted in Figure 3. Notice that D wins on those infinite paths where it can always duplicate 5.2 Total bisimulation games S’s moves. On the other hand, it looses for instance when M,b M,y We can associate a game to Definition 6. Such game it replies to one of S’s moves (( )( )) by moving in   a y checks whether two given pointed models (M,a) and (M ,a) the first model to state which is labelled guilty while is are bisimular or not. The game is played by two players: labelled innocent. Spoiler, which tries to show that the two given pointed mod- Pushing the legal analogy further, bisimulation games are els are not bisimilar, and Duplicator which pursues the op- an idealized version of the sort of dialogues in which lawyers posite goal. A match is started by S,thenD responds, and compare old cases with new ones. The lawyer arguing for dif- so on. If and only if D moves to a position where the two ference proceeds like the Spoiler, while the lawyer claiming pointed models are not propositionally equivalent, or if it the equivalence of the cases, proceeds like the Duplicator. cannot move, S wins. Definition 7 (Total bisimulation game). Take two 6. RELATED AND FUTURE WORK pointed models M and M.Thetotal bisimulation game  B(M, M ) is defined by the following items. Players: The 6.1 Related work set of players is {D, S}. An element from {D, S} will be To the best of our knowledge, only two papers have dealt denoted P and its opponent P . Game form: The game  with the application of logic to the formalization of abstract form of B(M, M ) is defined by Table 4. Turn function: argumentation theory. The first one is [2] which presents If the round is even S plays, if it is odd D plays. Winning preliminary work aimed at generalizing abstract argumen- conditions: S wins if and only if either D has moved to    tation within a logical language. There are two main differ- a position ((M,a)(M ,a)) where a and a do not satisfy ences with our approach: first, propositional atoms denote the same labels, or D has no available moves. Otherwise D  arguments instead of sets of arguments; second, the var- wins. Instantiation: The instance of B(M, M ) with start-     ious extensions, instead of being defined in the logic, are ing position ((M,a)(M ,a)) is denoted B(M, M )@(a, a ). taken to be primitives. The resulting logic is non-standard So, as we might expect, positions in a (total) bisimulation and no proof procedures (e.g., calculi or games) nor meta- game are pairs of pointed models, that is, the pointed models theoretical results are studied. that D tries to show are bisimilar. It might also be instruc- The second one [4] is closer in purpose to our work. It aims tive to notice that such a game can have infinite matches, at defining several notions of extensions within modal logic. which, according to Definition 7, are won by D. However, while our approach is eminently model-theoretical,

415 axb guilty guilty guilty 6.3 Conclusions The paper has shown how well-known modal logics—the M M extensions of K with universal modality and least fixpoint innocent innocent operator—can be fruitfully applied to argumentation theory c y in an almost direct way. Future work will aim at filling the (M,c)(M,y) gaps in the analysis presented—eminently the formalization of preferred extensions and the study of monadic second- (M,c)(M,y) (M,a)(M,y) (M,b)(M,y) (M,c)(M,x) order games—as well as pursuing some of the research lines that this perspective opens up, such as pushing further the (M,a)(M,x) (M,b)(M,x) (M,c)(M,y) application of bisimulation in argumentation, and applying dynamic logic techniques to study argumentation dynamics. (M,b)(M,x) (M,a)(M,y)

(M,b)(M,x) (M,a)(M,y) (M,a)(M,x) Acknowledgments. D wwins!ins! S winss!! D wwins!ins! This work has been funded by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) under the VENI Figure 3: A total bisimulation game. grant 639.021.816. The author wishes to thank Sanjay Mod- gil for the inspiring conversation that sparked this study.

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