Abstract Interpretation in Formal Argumentation: with a Galois Connection for Abstract Dialectical Frameworks and May-Must Argumentation (First Report)
aa r X i v : . [ c s . L O ] J u l Preprint Abstract Interpretation in Formal Argumentation: witha Galois Connection for Abstract DialecticalFrameworks and May-Must Argumentation(First Report)
Ryuta Arisaka and Takayuki Ito
Nagoya Institute of Technology, Nagoya, Japan ( e-mail: [email protected], [email protected] ) submitted -; revised -; accepted - Abstract
Labelling-based formal argumentation relies on labelling functions that typically assign one of 3 labels toindicate either acceptance, rejection, or else undecided-to-be-either, to each argument. While a classicallabelling-based approach applies globally uniform conditions as to how an argument is to be labelled, theycan be determined more locally per argument. Abstract dialectical frameworks (
ADF ) is a well-known ar-gumentation formalism that belongs to this category, offering a greater labelling flexibility. As the size ofan argumentation increases in the numbers of arguments and argument-to-argument relations, however, itbecomes increasingly more costly to check whether a labelling function satisfies those local conditions oreven whether the conditions are as per the intention of those who had specified them. Some compromiseis thus required for reasoning about a larger argumentation. In this context, there is a more recently pro-posed formalism of may-must argumentation (
MMA ) that enforces still local but more abstract labellingconditions. We identify how they link to each other in this work. We prove that there is a Galois connec-tion between them, in which
ADF is a concretisation of
MMA and
MMA is an abstraction of
ADF . Weexplore the consequence of abstract interpretation at play in formal argumentation, demonstrating a soundreasoning about the judgement of acceptability/rejectability in
ADF from within
MMA . As far as we areaware, there is seldom any work that incorporates abstract interpretation into formal argumentation in theliterature, and, in the stated context, this work is the first to demonstrate its use and relevance.
KEYWORDS : abstract interpretation, formal argumentation, abstract dialectical frameworks, may-must ar-gumentation, galois connection
Abstract argumentation (Dung 1995) represents an argumentation as a directed graph of: nodesfor arguments; and edges for attacks from the source arguments to the target arguments, with anintent to infer acceptance statuses of arguments. Classically (Dung 1995; Jakobovits and Vermeir 1999;Caminada 2006), conditions for acceptance and rejection are defined globally uniformly. How-ever, it is also possible to localise the conditions to a sub-argumentation or even to each argu-ment. Abstract Dialectical Frameworks (
ADF ) (Brewka et al. 2013) and May-Must Argumenta-tion (
MMA ) (Arisaka and Ito 2020a) both belong to the last category.In
ADF , for each argument, one of 3 acceptance statuses: in to mean accepted; out to meanrejected; and undec to mean undecided-to-be-either, is chosen based on acceptance statuses of R. Arisaka and T. Ito the arguments attacking. In effect, to each argument a attacked by n arguments a , . . . , a n , ADF attaches instructions: “If acceptance statuses of ( a , . . . , a n ) are ([ x ] , . . . , [ x n ]) , then choose [ x ] for a ’s acceptance status”, where every [ · ] is either of in , out , and undec , with [ x ] , . . . , [ x n ] covering all combinations of the 3 statuses (and thus there can be up to 3 n instructions). As a smallexample, suppose an argumentation graph of: a p a r a q with 3 arguments, where a p isgiven: “If (acceptance status of) a r is in , then choose [ x ] for a p (’s acceptance status)”; “If a r is out , then choose [ x ] for a p ”; and “If a r is undec , then choose [ x ] for a p ”. Similarly 3 cases areconsidered for a q . Meanwhile, instructions for a r have to cover 9 cases; e.g. “If ( a p , a q ) is: • ( undec , undec ) or ( undec , out ) , then choose undec for a r . • ( out , undec ) or ( out , out ) , then choose in for a r . • ( in , undec ) or ( undec , in ) or ( in , in ) , then choose out for a r . • ( in , out ) or ( out , in ) , then choose undec for a r .” With
ADF , a user can specify argument’s acceptability status independently for each combi-nation. The freedom, however, comes with an associated cost. Given the complexity results(Brewka et al. 2013), increases in the numbers of arguments and argument-to-argument relationscan make it exponentially more costly to check whether an acceptance status of an argumentsatisfies the instructions attached to it, or even whether the instructions are as per the intentionof those who had specified them. For scalability of reasoning about a larger argumentation, tech-niques of search space reduction ought to be explored.
We identify in this paper that
MMA (Arisaka and Ito 2020a) serves to abstract
ADF ’s instruc-tions. We prove that there is in fact a Galois connection for the two formalisms, in which, onone hand, ADF is a concretisation of
MMA and, on the other hand,
MMA is an abstraction of
ADF . The consequence, as we will show, is abstract interpretation (Cousot and Cousot 1977;Cousot and Cousot 1979) enabling reasonings about acceptance statuses of arguments in
ADF from within
MMA .To give contexts to our idea, in
MMA , every argument a in an argumentation graph is assignedtwo pairs of natural numbers. One pair, say ( n , n ) , states that at least n (resp. n ) arguments at-tacking a need to be rejected for a to be judged possibly accepted (resp. accepted). Another pair,say ( m , m ) , states that at least m (resp. m ) arguments attacking a need to be accepted for a tobe judged possibly rejected (resp. rejected). As we see, these conditions only specify the cardinal-ity of rejected or accepted attacking arguments and not specifically which ones. Moreover, while n , n , , m , m specify the minimum numbers for the respective conditions, any numbers that ex-ceed them satisfy the same conditions, that is, these conditions are monotonic. Along with any ADF actually keeps the argument-to-argument relation to be of an unspecified nature, allowing in particular the supportrelation to be expressed; however, in the present paper, we will abstract any non-attack relations. Also, it uses t for in and f for out ; for this, however, we use more widely-used notations (Jakobovits and Vermeir 1999; Caminada 2006). A Galois connection for two systems is a pair of mappings betwen them; formal detail is in Section 2. bstract Interpretation in Formal Argumentation
MMA than
ADF . Suppose the fol-lowing for the above example a p (( n ap , n ap ) , ( m ap , m ap )) a r (( n ar , n ar ) , ( m ar , m ar )) a q (( n aq , n aq ) , ( m aq , m aq )) , with associated pairs, then (( n a r , n a r ) , ( m a r , m a r )) = (( , ) , ( , )) signifies the following: • a r is not judged accepted or rejected in any degree if neither a p nor a q is in or out . Accord-ing to (Arisaka and Ito 2020a) (Section 2 of this paper for more detail), this case results inthe choice of undec for a r . • a r is judged only possibly accepted, and not judged rejected in any degree if one of a p and a q is out and the other is undec . This case results in the choice of either in or undec for a r on a non-deterministic basis. • a r is judged accepted but not rejected in any degree, if both a p and a q are out . This caseresults in the choice of in for a r . • a r is not judged accepted in any degree, but judged rejected if both a p and a q are in . Thiscase results in the choice of out for a r . • a r is judged only possibly accepted and judged rejected if one of a p and a q is out andthe other is in . This case results in the choice of either out or undec for a r on a non-deterministic basis. • (The other cases do not happen with the chosen n a r , n a r , m a r , m a r .)There is the following correspondence between ADF and
MMA for this example, for any accep-tance statuses of ( a p , a q ) . • When there is only one acceptance status to choose for a r in this MMA for the acceptancestatuses of ( a p , a q ) , then the same acceptance status is chosen for a r in ADF . • When there are more than one acceptance status to choose for a r in this MMA for theacceptance statuses of ( a p , a q ) , then one of them is chosen for a r in ADF .In other words, this
MMA soundly over-approximates the
ADF instructions.This kind of a technique to reason about a system from within its abstraction, in a mannerensuring that some properties of the abstracted system be sound over-approximations of someproperties of the concrete system, is known as abstract interpretation (Cousot and Cousot 1977;Cousot and Cousot 1979), which is popular in static program analysis. It is almost not studied informal argumentation, however. Perhaps, to the static analysis community, it is a question justwhat of formal argumentation may require abstract interpretation; and, for the formal argumenta-tion community, its focus having been more on making the prediction of abstract argumentation(Dung 1995) increasingly more precise may explain why the concept of abstract interpretationin formal argumentation has been rather elusive. Nonetheless, we contend that a stronger movetowards reasoning about a larger argumentation is bound to gather force, especially with anincreasing interest in argumentation mining technology to automatically extract large-scale ar-gumentations. It is a reasonable projection that abstract interpretation will play just as importanta part in formal argumentation as it does in static analysis. We take an initiating step towards thedevelopment.
As far as we are aware, there is one preprint for loop abstraction (Arisaka and Dauphin 2018)that takes an inspiration from abstract interpretation. However, it gives more weights to learn-ing about some otherwise unlearnable acceptance statuses in an original argumentation than to
R. Arisaka and T. Ito making generally sound reasoning about the properties of the original argumentation. For itsapplication in the stated context, there is, to the best of our knowledge, none existing in the lit-erature. We make clear to what extent a Galois connection for
ADF and
MMA permits us theabove-mentioned sound reasoning about
ADF within
MMA .In the rest, we will go through technical preliminaries (in Section 2), and establish a Galoisconnection between
MMA and
ADF , showing how it can be utilised for soundly reasoning about
ADF properties in a corresponding abstract space, within
MMA (in Section 3).
Let A denote a class of abstract entities that we understand as arguments, and let R denote aclass of all binary relations over A . We refer to a member of A (resp. R ) by A (resp. R ) with orwithout a subscript. By R A for A ⊆ A , we denote a subclass of R which contains all and onlymembers R of R over A , i.e. for every R ∈ R A and every ( a , a ) ∈ R , it holds that a , a ∈ A . A(finite) abstract argumentation is then a tuple ( A , R ) with A ⊆ fin A and R ∈ R A .For characterisation of acceptance statuses, we will make use of labellings (Jakobovits and Vermeir 1999;Caminada 2006) uniformly for a compatibility with ADF (Brewka et al. 2013) and
MMA (Arisaka and Ito 2020a),both of which are to be introduced below. Readers interested in non-labelling-based approachesare referred to (Dung 1995; Baroni and Giacomin 2007).Let L denote { in , out , undec } , and let Λ denote the class of all partial functions A → L . Let Λ A for A ⊆ A denote a subclass of Λ that includes all (but nothing else) λ ∈ Λ that are definedfor all members (but nothing else) of A . For the order among members of Λ , let (cid:22) be a binaryrelation over Λ such that λ (cid:22) λ for λ , λ ∈ Λ iff all the following conditions hold. (1) Thereis some A ⊆ fin A such that λ , λ ∈ Λ A . (2) For every a ∈ A , λ ( a ) = in (resp. λ ( a ) = out )materially implies λ ( a ) = in (resp. λ ( a ) = out ). We may write λ ≺ λ when λ (cid:22) λ but not λ (cid:22) λ . MMA and labelling instructions
Let N be the class of natural numbers including 0, and for any tuple T of n -components, let ( T ) i for 1 ≤ i ≤ n refer to T ’s i -th component. A may-must scale is some XXX ∈ N × N with ( XXX ) ≤ ( XXX ) . ( XXX ) (resp. ( XXX ) ) is called may- condition (resp. must- condition ) of XXX . A nuance tuple is apair ( XXX , XXX ) of may-must scales, the first one XXX for acceptance judgement and the secondone XXX for rejection judgement. Cf. Section 1. With Q ≡ (( n , n ) , ( m , m )) , ( Q ) = ( n , n ) , ( Q ) = ( m , m ) , (( Q ) ) i = n i , and (( Q ) ) i = m i ( i ∈ { , } ). We denote the class of all nuancetuples by Q and refer to its member by Q with or without a subscript. For any Q ∈ Q , we call ( Q ) its may-must acceptance scale and ( Q ) its may-must rejection scale. A MMA is then atuple ( A , R , f Q ) with A ⊆ fin A ; R ∈ R A ; and f Q : A → Q . We denote the class of all MMA tuplesby F MMA , and refer to its member by F MMA with or without a subscript.In
MMA , acceptance and rejection of an argument are independently considered, which are Google Scholar Search as of 26th May 2020 had produced only 7 results out of 4,300+ papers that cited (Dung 1995)and that included “abstract interpretation”, of which only (Arisaka and Dauphin 2018) refers to a Galois connectionand concrete/abstract spaces. bstract Interpretation in Formal Argumentation F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , any a ∈ A and any λ ∈ Λ , let pred F MMA ( a ) be the set of all a x ∈ A with ( a x , a ) ∈ R , let pred F MMA λ , out ( a ) be the set of all a x ∈ pred F MMA ( a ) such that λ is defined for a x and that: λ ( a x ) = in if a x attacks , and let pred F MMA λ , in ( a ) be the set of all a x ∈ pred F MMA ( a ) such that λ is defined for a x and that λ ( a x ) = out , then a is said to satisfy: • may-a(cceptance condition) (resp. may-r(ejection condition)) under λ in F MMA iff (( f Q ( a )) ) ≤ | pred F MMA λ , out ( a ) | (resp. (( f Q ( a )) ) ≤ | pred F MMA λ , in ( a ) | ). • must-a(cceptance condition) (resp. must-r(ejection condition)) under λ in F MMA iff (( f Q ( a )) ) ≤ | pred F MMA λ , out ( a ) | (resp. (( f Q ( a )) ) ≤ | pred F MMA λ , in ( a ) | ). • may s -a(cceptance condition) (resp. may s -r(ejection condition)) under λ in F MMA iff (( f Q ( a )) ) ≤ | pred F MMA λ , out ( a ) | < (( f Q ( a )) ) (resp. (( f Q ( a )) ) ≤ | pred F MMA λ , in ( a ) | < (( f Q ( a )) ) ). • not-a(cceptance condition) (resp. not-r(ejection condition)) under λ in F MMA iff | pred F MMA λ , out ( a ) | < (( f Q ( a )) ) (resp. | pred F MMA λ , in ( a ) | < (( f Q ( a )) ) ).Clearly, the may-must conditions are monotonic over the increase in the number of rejected/acceptedattacking arguments. If obvious from the context, “under λ in F MMA ” may be omitted.
MMA labelling instructions that we saw in Section 1 are technically termed label designations in (Arisaka and Ito 2020a). They are derived from combining these independent judgements.Specifically, for any F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , any a ∈ A , and any λ ∈ Λ , λ is said todesignate l ∈ L for a iff all the following conditions hold. (Cf. Fig. 1 for which label(s) may bedesignated for each combination.) must-r may s -r not-rmust-a undec in ? in may s -a out ? any in ?not-a out out ? undec Fig. 1: Label designation for each com-bination of satisfied may-must conditionsunder a given λ ∈ Λ . any is any of in , out , undec . in ? is any of in , undec . out ? is anyof out , undec . λ is defined for every member of pred F MMA ( a ) .2. If l = in , a satisfies may-a but not must-r (under λ ).3. If l = out , a satisfies may-r but not must-a.4. If l = undec , then either of the following holds. • a satisfies must-a and must-r. • a satisfies at least either may s -a or may s -r. • a satisfies not-a and not-r.While we refer a reader to (Arisaka and Ito 2020a) fora slower explanation, these label designations are as theresult of MMA ’s interpretation of the satisfaction condi-tions under a possible-world perspective. In the classic (i.e. non-intuitionistic) interpretation ofmodalities (see for example (Garson 2018)), a necessary (resp. possible) proposition is true iff itis true in every (resp. some) possible world accessible from the current world, and a not possi-ble proposition is false in every accessible possible world.
MMA transposes these to the must-may s - conditions by taking acceptance for truth and rejection for falsehood, obtainining for each a ∈ A that its satisfaction of: must-a (resp. must-r) implies acceptance (resp. rejection) of a inevery accessible possible world; may s -a (resp. may s -r) implies acceptance (resp. rejection) of a in some accessible possible world; and not-a (resp. not-r) implies rejection (resp. acceptance) of a in every accessible possible world. (Note the use of “may s ” instead of “may” here, once both“may” and “must” are satisfied, it suffices to simply consider “must”.) Any possible world im-plying only acceptance (resp. rejection) of a is implying in (resp. out ) for a . Any possible world R. Arisaka and T. Ito implying both acceptance and rejection of a is implying an inconsistent acceptance status, i.e. undec , for a in the possible world. In the above definition of label designation, λ designates any l ∈ L for a ∈ A so long as there is some structure of possible worlds one of which is the currentworld and an accessility relation such that l is implied for a in at least one accessible possibleworld. a ’s label is said to be proper under λ iff (1) λ is defined for a , and (2) λ designates λ ( a ) for a . If every a ∈ A ’s label is proper under λ ∈ Λ A , then we call λ an exact labelling of F MMA .Suppose the following F MMA a p (( , ) , ( , )) a r (( , ) , ( , )) a q (( , ) , ( , )) with associated may-must scales. Let [ a : l , . . . , a n : l n ] λ for a , . . . , a n ∈ A and l , . . . , l n ∈ L denote some mem-ber of Λ { a ,..., a n } with [ a : l , . . . , a n : l n ] λ ( a i ) = l i , then both [ a p : out , a r : out , a q : in ] λ and [ a p : out , a r : undec , a q : in ] λ are all exact labellings of F MMA . ( a p satisfies must-r and not-a, and a q must-a and not-r, irrespective of attackers’ labels, and therefore a r satisfies must-r and may s -a.There are no other cases.) ADF and labelling instructions
While
ADF uses its own set of symbols and terminology different from those used in abstractargumentation, in this paper we keep them consistent with
MMA notations. A finite
ADF isa tuple ( A , R , C ) with: A ⊆ fin A ; R ∈ R A ; and C = S a ∈ A { C a } where, for each a ∈ A , C a is afunction: Λ pred ( A , R , C ) ( a ) → L . Here and elsewhere, pred ( A , R , C ) ( a ) = { a x ∈ A | ( a x , a ) ∈ R } for anyfinite ADF tuple ( A , R , C ) . We denote the class of all finite ADF tuples by F ADF , and refer to itsmember by F ADF with or without a subscript.Moreover, to ease the juxtaposition with
MMA , we define the notion of label designationfor
ADF as well. For any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) , any a ∈ A and any λ ∈ Λ defined atleast for each member of pred F ADF ( a ) , let λ ↓ pred F ADF ( a ) denote a member of Λ pred F ADF ( a ) with λ ( a x ) = λ ↓ pred F ADF ( a ) ( a x ) for any a x ∈ pred F ADF ( a ) . Then, for any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) ,any a ∈ A and any λ ∈ Λ , we say that λ designates l ∈ L for a ∈ A iff (1) λ is defined at leastfor each member of pred F ADF ( a ) , and (2) C a ( λ ↓ pred F ADF ( a ) )( a ) = l . We say that a ∈ A ’s label isproper under λ ∈ Λ in F ADF iff (1) λ is defined for a , and (2) λ designates λ ( a ) for a .Since we defined exact labellings of MMA , again to ease the juxtaposition, we define it for
ADF here. If every a ∈ A ’s label is designated under λ ∈ Λ A , then we say λ is an exact labelling of F ADF . Suppose the following a pC p a qC q with associated conditions. Assume C p ([ a q : in ] λ ) = undec and C p ([ a q : l ] λ ) = in for l ∈ { undec , out } . Assume C q ([ a p : in ] λ ) = undec and C q ([ a p : l ] λ ) = in for l ∈ { undec , out } . Then [ a p : in , a q : undec ] λ and [ a p : undec , a q : in ] λ are all the exactlabellings of F ADF . One natural semantics for both
MMA and
ADF is the exact semantics as the set of all exactlabellings. However, it is in general not possible to guarantee existence of an exact labelling; see In (Arisaka and Ito 2020a), “designated” instead of “proper” is used. bstract Interpretation in Formal Argumentation
7a counter-example in (Arisaka and Ito 2020a). While the non-existence is not in itself a problem,
ADF and
MMA both propose some approximation, the former with a consensus operator andthe latter with maximisation of the number of arguments whose labels are designated (with theremaining labelled undec ), for gaining the existence property.Since our objective is to consider properties of
ADF from within
MMA , it makes sense totouch upon
ADF ’s consensus operator here, and define it also for
MMA (which is incidentallynew; however, being straightforward, we do not have to claim any novelty in this formulation for
MMA ). Let twoVal : A × Λ → Λ , which we alternatively state twoVal A : Λ → Λ , be such that,for any F ≡ ( A , R , X ) ( ∈ F ADF ∪ F MMA ) , any A ⊆ fin A and any λ ∈ Λ A , twoVal A ( λ ) = { λ x ∈ Λ A | λ (cid:22) λ x and λ x is maximal in ( Λ A , (cid:22) ) } .Every member λ x of twoVal A ( λ ) is such that λ x ( a ) ∈ { in , out } for every a ∈ A . Now, let Θ : ( F ADF ∪ F MMA ) × Λ → Λ , which we alternatively state Θ ( F ADF ∪ F MMA ) : Λ → Λ , be such that,for any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) , any F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) and any λ ∈ Λ A , all thefollowing hold, with Y denoting either of F ADF and F MMA .1. Θ Y ( λ ) ∈ Λ A .2. For every a ∈ A and every l ∈ { in , out } , Θ Y ( λ )( a ) = l iff, for every λ x ∈ twoVal A ( λ ) , λ x designates only l for a in Y .In a nutshell (Brewka et al. 2013), Θ Y ( λ ) gets a consensus of every λ x ∈ twoVal A ( λ ) on the labelof each a ∈ A : if each one of them says only in for a , then Θ Y ( λ )( a ) = in , if each one of themsays only out for a , then Θ Y ( λ )( a ) = out , and for the other cases Θ Y ( λ )( a ) = undec .Then the ADF -grounded semantics of Y ∈ ( F ADF ∪ F MMA ) contains just the least fixpoint of Θ Y (the order is (cid:22) ). Readers are referred to (Brewka et al. 2013) for any other ADF -semantics.
Abstract interpretation (Cousot and Cousot 1977; Cousot and Cousot 1979) is a popular tech-nique in static analysis, useful for reasoning about properties of a large-scale program throughabstraction. It abstracts a concrete program into an abstract program while ensuring that the ab-straction be a sound over-approximation of the concrete program for some property. The sound-ness is in the sense that if an abstracted program satisfies some property, then some property isguaranteed to hold in the concrete program.Important to abstract interpretation is the notion of Galois connection (see any standard text,e.g. (Davey and Priestley 2002)). Briefly, let S and S each be an ordered set, partially orderedin ≤ and respectively in ≤ . Let f → : S → S be a function that maps each element of S onto an element of S , and let f → : S → S be a function that maps each element of S ontoan element of S . If f → ( s ) ≤ s materially implies s ≤ f → ( s ) and vice versa, then thepair of f → and f → is said to be a Galois connection. The following properties hold good. AGalois connection is: contractive, i.e. ( f → ◦ f → )( s ) ≤ s for every s ∈ S ; extensive, i.e. s ≤ ( f → ◦ f → )( s ) for every s ∈ S ; and monotone for both f → and f → (to follow fromthe contractiveness and the extensiveness). Further, it holds that f → ◦ f → ◦ f → = f → andthat f → ◦ f → ◦ f → = f → . R. Arisaka and T. Ito
MMA and
ADF and abstract interpretation
In this section, we firstly establish a Galois connection for
MMA and
ADF . F MMA F MMA F MMA into F ADF F ADF F ADF . Let us begin by defining mappings of F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) onto F ADF ≡ ( A , R , C ) ( ∈ F ADF ) . For every a ∈ A , λ ∈ Λ designates at most one member of L for a in F ADF whereas it may designate more than one member of L for a in F MMA . This differencehas to be taken into account. Example 3.1 shows a concrete mapping example.
Definition 3.1 ( Concretisation: from F
MMA to F
ADF )Let Γ be a class of all functions γ : F MMA → F ADF , each of which is such that, for any F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , γ ( F MMA ) is some ( A , R , C ) ∈ F ADF with C satisfying the following. Forany a ∈ A and any λ ∈ Λ pred F MMA ( a ) , if L ⊆ L is such that λ designates each l ∈ L but does notdesignate any l ∈ ( L \ L ) for a in F MMA , then C a ( λ ) ∈ L .We say that F ADF ∈ F ADF is a concretisation of F MMA ∈ F MMA iff there is some γ ∈ Γ with F ADF = γ ( F MMA ) . By ΓΓΓ [ F MMA ] we denote the set of all concretisations of F MMA ∈ F MMA . Example 3.1 ( Concretisation )Consider the F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) in the diagram below. We show some of its con-cretisations. Since γ ∈ Γ does not modify A and R , the problem at hand is identification of C a x tocorrespond to f Q ( a x ) for each x ∈ { , . . . , } . F MMA : a (( , ) , ( , )) a (( , ) , ( , )) a (( , ) , ( , )) a (( , ) , ( , )) a (( , ) , ( , )) F ADF : a C a a C a a C a a C a a C a γ For both a and a , every λ ∈ Λ designates just in . As per Definition 3.1, for any λ ∈ Λ /0 , C a ( λ ) = C a ( λ ) = in irrespective of which member of Γ is referred to by γ .For a , there are 3 distinct labellings [ a : in ] λ , [ a : out ] λ , [ a : undec ] λ ∈ Λ pred F MMA ( a ) . Wehave: [ a : in ] λ (satisfying not-a and may s -r) designates out and undec for a ; [ a : out ] λ (satis-fying may s -a and not-r) designates in and undec for a ; and [ a : undec ] λ (satisfying not-a andnot-r) designates undec for a , in F MMA . Thus, C a is any one of the following.1. C a ([ a : in ] λ ) = out , C a ([ a : out ] λ ) = in , C a ([ a : undec ] λ ) = undec .2. C a ([ a : in ] λ ) = out , C a ([ a : out ] λ ) = undec , C a ([ a : undec ] λ ) = undec .3. C a ([ a : in ] λ ) = undec , C a ([ a : out ] λ ) = in , C a ([ a : undec ] λ ) = undec .4. C a ([ a : in ] λ ) = undec , C a ([ a : out ] λ ) = undec , C a ([ a : undec ] λ ) = undec .Hence, some γ ∈ Γ has the first C a , some others have the second, third, or the fourth C a . Anal-ogously for a and a . ♣ When we either decrease a may- condition ( ∈ N ) or increase a must- condition of a may-mustscale in F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , we obtain at least as large a set of concretisations asbefore the change (Theorem 3.1). For the proof, we first read the following subsumption relationoff Fig. 1. bstract Interpretation in Formal Argumentation Lemma 3.1 ( Label designation subsumption )Let x , y be a member of { must , may s , not } . For any F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , any a ∈ A and any λ , λ , λ ∈ Λ , if a satisfies: x -a and y -r under λ ; may s -a and y -r under λ ; and x -a andmay s -r under λ , and if λ designates l ∈ L for a , then both λ and λ designate l for a .Also, for convenience, we define the following order. Definition 3.2 ( Abstract order )Let ✂ ⊆ Q × Q be such that ( Q , Q ) ∈ ✂ , alternatively Q ✂ Q , holds iff, for any i ∈ { , } , (( Q ) i ) ≤ (( Q ) i ) and (( Q ) i ) ≤ (( Q ) i ) both hold. We define ⊑ ⊆ F MMA × F MMA to besuch that, for any F MMA ≡ ( A , R , f Q ) and any F MMA ≡ ( A , R , f ′ Q ) , ( F MMA , F MMA ) ∈ ⊑ , alterna-tively F MMA ⊑ F MMA , holds iff, for any a ∈ A , f Q ( a ) ✂ f ′ Q ( a ) holds.We also extend ⊑ for 2 F MMA in the following manner. For F MMA x , F MMA y ⊆ F MMA , wedefine: F MMA x ⊑ F MMA y iff, for any F MMA x ∈ F MMA x , there exists some F MMA y ∈ F MMA y suchthat F MMA x ⊑ F MMA y . Theorem 3.1 ( Monotonicity )For any F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) and any F MMA ′ ≡ ( A , R , f ′ Q ) ( ∈ F MMA ) , if F MMA ⊑ F MMA ′ holds, then ΓΓΓ [ F MMA ] ⊆ ΓΓΓ [ F MMA ′ ] holds. Proof
By Definition 3.1, it suffices to show that, for any a ∈ A and any λ ∈ Λ pred F MMA ( a ) , if λ designates l ∈ L for a in F MMA , then λ designates l for a in F MMA ′ . Now, the differences be-tween f Q and f ′ Q are such that, for any λ ∈ Λ , firstly, if a satisfies must-a (resp. must-r) under λ in F MMA , a satisfies either must-a or may s -a (resp. must-r or may s -r) under λ in F MMA ′ , and,secondly, if a satisfies not-a (resp. not-r) under λ in F MMA , a satisfies either not-a or may s -a(resp. not-r or may s -r) under λ in F MMA ′ . Apply Lemma 3.1. ✷ F ADF F ADF F ADF into F MMA F MMA F MMA . Into the other direction of mapping F ADF ≡ ( A , R , C ) ( ∈ F ADF ) onto F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , recall that F MMA only requires a ( ∈ A ) ’s may-must scales, | pred F MMA λ , in ( a ) | and | pred F MMA λ , out ( a ) | (for λ ∈ Λ ) for label designation. For any λ , λ ∈ Λ , as long as a satisfies x -aand y -r ( x , y ∈ { must , may s , not } ) under both λ , λ , it holds that λ and λ designate the samelabel(s) for a .On the other hand (see also Example 3.1), F ADF ’s C a determines label designation inde-pendently for each λ ∈ Λ pred F ADF ( a ) . For distinct λ , λ ∈ Λ pred F ADF ( a ) with | pred F ADF λ , in ( a ) | = | pred F ADF λ , in ( a ) | and | pred F ADF λ , out ( a ) | = | pred F ADF λ , out ( a ) | , it can happen that C a ( λ ) = C a ( λ ) .As such, we need to abstract the specificity of F ADF ’s C for the mapping. Formally, we considerthe following class of functions. See Example 3.2 to follow for a concrete example. Definition 3.3 ( Abstraction: from F
ADF to F
MMA )Let ∆ be a class of all functions α : F ADF → F MMA , each of which is such that, for any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) , α ( F ADF ) is some ( A , R , f Q ) ∈ F MMA where, for every a ∈ A and every λ ∈ Λ pred F ADF ( a ) , f Q ( a ) ≡ (( n a , n a ) , ( m a , m a )) satisfies all the following conditions.1. 0 ≤ n a ≤ n a ≤ | pred F ADF ( a ) | +
1. Also 0 ≤ m a ≤ m a ≤ | pred F ADF ( a ) | + R. Arisaka and T. Ito
2. If | pred F ADF λ , out ( a ) | < n a and | pred F ADF λ , in ( a ) | < m a , then C a ( λ ) = undec .(This corresponds to not-a, not-r satisfaction.)3. If n a ≤ | pred F ADF λ , out ( a ) | < n a and | pred F ADF λ , in ( a ) | < m a , then C a ( λ ) ∈ { in , undec } .( may s -a, not-r )4. If n a ≤ | pred F ADF λ , out ( a ) | and | pred F ADF λ , in ( a ) | < m a , then C a ( λ ) = in .( must-a, not-r )5. If | pred F ADF λ , out ( a ) | < n a and m a ≤ | pred F ADF λ , in ( a ) | < m a , then C a ( λ ) ∈ { out , undec } .( not-a, may s -r )6. If n a ≤ | pred F ADF λ , out ( a ) | < n a and m a ≤ | pred F ADF λ , in ( a ) | < m a , then C a ( λ ) ∈ { in , out , undec } .( may s -a, may s -r )7. If n a ≤ | pred F ADF λ , out ( a ) | and m a ≤ | pred F ADF λ , in ( a ) | < m a , then C a ( λ ) ∈ { in , undec } .( must-a, may s -r )8. If | pred F ADF λ , out ( a ) | < n a and m a ≤ | pred F ADF λ , in ( a ) | , then C a ( λ ) = out .( not-a, must-r )9. If n a ≤ | pred F ADF λ , out ( a ) | < n a and m a ≤ | pred F ADF λ , in ( a ) | , then C a ( λ ) ∈ { out , undec } .( may s -a, must-r )10. If n a ≤ | pred F ADF λ , out ( a ) | and m a ≤ | pred F ADF λ , in ( a ) | , then C a ( λ ) = undec .( must-a, must-r )For any F ADF ∈ F ADF , we say that F MMA ∈ F MMA is an abstraction of F ADF iff there is some α ∈ ∆ with F MMA = α ( F ADF ) . We denote the set of all abstractions of F ADF by ∆∆∆ [ F ADF ] . Example 3.2 ( Abstraction )Suppose F ADF : a C a a C a a C a . Then Λ pred F ADF ( a ) = S l , l ∈ L { [ a : l , a : l ] λ } . Assume:1. C a ([ a : undec , a : undec ] λ ) = undec . 2. C a ([ a : l , a : l ] λ ) = out if in ∈ { l } ∪ { l } .3. C a ([ a : undec , a : out ] λ ) = in . 4. C a ([ a : out , a : undec ] λ ) = undec .5. C a ([ a : out , a : out ] λ ) = undec .Let us consider which f Q ( a ) ≡ (( n a , n a ) , ( m a , m a )) can be in an abstraction of F ADF . Firstly,by 1. and 2., for any λ ∈ Λ defined for a and a , we see that λ does not designate out for a solong as | pred F ADF λ , in ( a ) | =
0; however, as soon as | pred F ADF λ , in ( a ) | >
0, we have that λ designatesonly out . As the result, we can set ( m a , m a ) to ( , ) . On the other hand, for ( n a , n a ) , wehave 3. and 4., and it cannot be that n a ≤
1, but also we have 5., and n a =
2. Thus, we musthave n a =
3. However, because of 3., n a cannot be greater than or equal to 2. Hence we canset n a to be 1, resulting in ( n a , n a ) = ( , ) . It is trivial to see that any F MMA ∈ ∆∆∆ [ F ADF ] withthis f Q ( a ) least abstracts C a of F ADF among all possible f Q ( a ) in members of ∆∆∆ [ F ADF ] . Other f Q ( a ) also appear in other members of ∆∆∆ [ F ADF ] . Specifically, due to Lemma 3.1 and Definition bstract Interpretation in Formal Argumentation F MMA ∈ ∆∆∆ [ F ADF ] can come with any f Q ( a ) = (( n a ′ , ) , ( m a ′ , m a ′ )) with 0 ≤ n a ′ ≤ ≤ m a ′ ≤
1, 1 ≤ m a ′ ≤
3. Analogously for f Q ( a ) and f Q ( a ) . ♣ Until now, we have left the cardinality of
ΓΓΓ [ F MMA ] and ∆∆∆ [ F ADF ] all up to intuition. The followingtheorem establishes the bounds, with an implication of existence of the two sets (Corollary 3.1). Theorem 3.2 ( Cardinality of the maps )For any F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , we have 1 ≤ ΓΓΓ [ F MMA ] ≤ | L | | A | ( | L | | A | ) , and for any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) , we have 1 ≤ ∆∆∆ [ F ADF ] ≤ ( ( | A | + )( | A | + ) ) | A | . It holds that ( ( | A | + )( | A | + ) ) | A | ≪ | L | | A | ( | L | | A | ) for any 2 ≤ | A | and | L | = Proof.
For the former, the lower bound is due to the fact that any λ ∈ Λ pred F MMA ( a ) for a ∈ A designates at least one l ∈ L for a in F MMA . For the upper bound, | pred F MMA ( a ) | ≤ | A | , and thus | Λ pred F MMA ( a ) | ≤ | L | | A | . Any λ ∈ Λ pred F MMA ( a ) may designate as many labels as there are in L for a , hence there may be up to | L | ( | L | | A | ) alternatives for the third component of F ADF ∈ ΓΓΓ ( F MMA ) .There are | A | arguments. Put together, we obtain the result. For the latter, the lower bound isdue to the fact that f Q ( a ) = (( , | pred F MMA ( a ) | + ) , ( , | pred F MMA ( a ) | + )) designates each of in , out and undec . For the upper bound, for each a ∈ A , C a may map to member(s) of X ≡{ (( n , n ) , ( m , m )) | ≤ n , n , m , m ≤ | pred F ( a ) | + } . Since we have | pred F MMA ( a ) | + ≤| A | +
1, it holds that | X | ≤ ( ( | A | + )( | A | + ) ) . There are | A | arguments. ✷ Corollary 3.1 ( Existence )For any F MMA ∈ F MMA , there exists some concretisation of F MMA , and for any F ADF ∈ F ADF ,there exists some abstraction of F ADF .While ∆∆∆ [ F ADF ] is still rather large, note that it covers every possible abstraction of F ADF . Inpractice, we adopt only a single set of criteria for abstraction e.g. minimal abstraction (see f α in Theorem 3.3) in ⊑ , which leaves only a handful of the set, or just one in case the abstractionminimum in ⊑ exists.Any abstraction of F ADF ∈ F ADF correctly overapproximates C a ’s label designation for every a ∈ A , following trivially from the definition of label designation (Section 2) and Definition 3.3. Proposition 3.1 ( Abstraction soundness )For any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) and any F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , if F MMA is anabstraction of F ADF , then for any a ∈ A and any λ ∈ Λ pred F MMA ( a ) all the following hold.1. λ designates C a ( λ ) for a in F MMA .2. if λ designates at most one l ∈ L for a in F MMA , then λ designates l for a in F ADF . When there are two systems related in concrete-abstract relation, it is of interest to establishGalois connection (Cf. Section 2). Galois connection is used for abstraction interpretation instatic analysis for verification of properties of large-scale programs as the verification in concretespace is often undecidable or very costly. In our view, it is no different with formal argumentation;reasoning about a large-scale argumentation will benefit from utilising the technique. Here, weidentify a Galois connection between F MMA and F ADF based on Γ and ∆ that we introduced.2 R. Arisaka and T. ItoTheorem 3.3 ( Galois connection )Let 2 F MMA ( A , R ) be a subclass of 2 F MMA which contains every F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) for some f Q but nothing else. Let 2 F ADF ( A , R ) be a subclass of 2 F ADF which contains every F ADF ≡ ( A , R , C ) ( ∈ F ADF ) for some C but nothing else.Let f γ : 2 F MMA → F ADF and f α : 2 F ADF → F MMA be the following. • for any F MMA x ∈ F MMA ( A , R ) , f γ ( F MMA x ) = S F MMA x ∈ F MMA x ΓΓΓ [ F MMA x ] . • for any F ADF x ∈ F ADF ( A , R ) , f α ( F ADF x ) = S F ADF x ∈ F ADF x { F MMA ∈ ∆∆∆ [ F ADF x ] | ∀ F MMA ′ ∈ ∆∆∆ [ F ADF x ] . F MMA ′ ⊏ F MMA } .Then ( f α , f γ ) is a Galois connection for ( F MMA ( A , R ) , ⊑ ) and ( F ADF ( A , R ) , ⊆ ) . Proof.
Suppose F MMA x ∈ F MMA ( A , R ) and F ADF x ∈ F ADF ( A , R ) . Suppose f α ( F ADF x ) ⊑ F MMA x , then wehave to show that F ADF x ⊆ f γ ( F MMA x ) . By Theorem 3.1, we have S F MMA y ∈ f α ( F ADF x ) ΓΓΓ [ F MMA y ] ⊆ S F MMA x ∈ F MMA x ΓΓΓ [ F MMA x ] . By the definition of f γ , we have S F MMA x ∈ F MMA x ΓΓΓ [ F MMA x ] ⊆ f γ ( F MMA x ) .By Proposition 3.1 and Definition 3.1, we have F ADF x ⊆ S F MMA y ∈ f α ( F ADF x ) ΓΓΓ [ F MMA y ] . Hence F ADF x ⊆ f γ ( F MMA x ) , as required.Suppose on the other hand that F ADF x ⊆ f γ ( F MMA x ) , then we have to show that f α ( F ADF x ) ⊑ F MMA x . First, trivially, we have f α ( F ADF x ) ⊑ f α ( f γ ( F MMA x )) , since every F ADF ∈ F ADF x is con-tained in f γ ( F MMA x ) by the present assumption. By Lemma 3.1, we have f α ( f γ ( F MMA x )) ⊑ F MMA x , thus we have f α ( F ADF x ) ⊑ F MMA x , as required. ✷ We close this section by presenting that the Galois connection allows us to infer semantic prop-erties of F ADF ∈ F ADF from within F MMA , for both the exact semantics and the
ADF -groundedsemantics (Cf. Section 2). We assume L : { exact , adfg } × ( F ADF ∪ F MMA ) → Λ to be suchthat, for any Y ∈ ( F ADF ∪ F MMA ) , L ( exact , Y ) is the exact semantics of Y and L ( adfg , Y ) is the ADF -grounded semantics of Y . Theorem 3.4 ( Abstract interpretation (exact semantics) )For any F MMA ≡ ( A , R , f Q ) ( ∈ F MMA ) , let L one ( exact , F MMA ) denote the largest subset of L ( exact , F MMA ) satisfying: if λ ∈ L one ( exact , F MMA ) , then, for every a ∈ A , λ designates atmost one l ∈ L for a in F MMA .Then, for any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) and any F MMA ∈ f α ( { F ADF } ) , it holds that L one ( exact , F MMA ) ⊆ L ( exact , F ADF ) . Proof.
Follows from 2. of Proposition 3.1. ✷ Corollary 3.2
For any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) and any F MMA ∈ f α ( { F ADF } ) , if L one ( exact , F MMA ) = /0,then L ( exact , F ADF ) = /0. bstract Interpretation in Formal Argumentation Theorem 3.5 ( Abstract interpretation (
ADF -grounded semantics) )For any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) and any F MMA ∈ f α ( { F ADF } ) , it holds for any λ ∈ L ( adfg , F MMA ) that there exists some λ ′ ∈ L ( adfg , F ADF ) with λ (cid:22) λ ′ . Proof.
Follows from 2. of Proposition 3.1. ✷ In the literature of formal argumentation, two types of acceptance (and rejection) of an ar-gument are popularly referred to. In the context of labelling-based argumentations, for any ( A , R , C ) ∈ F ADF or ( A , R , f Q ) ∈ F MMA , a ∈ A is called: credulously accepted (resp. rejected)with respect to a semantics iff there exists at least one member λ ∈ Λ A of the semantics with λ ( a ) = in (resp. λ ( a ) = out ); and skeptically accepted (resp. rejected) with respect to a seman-tics iff a is credulously accepted and λ ( a ) = in (resp. λ ( a ) = out ) for every member λ ∈ Λ A ofthe semantics. Theorem 3.6 ( Acceptance and rejection )With respect to both exact and
ADF -grounded semantics, for any F ADF ≡ ( A , R , C ) ( ∈ F ADF ) ,any F MMA ∈ f α ( { F ADF } ) and any a ∈ A , all the following hold good.1. if a is credulously accepted (resp. rejected) in F MMA with respect to the exact or the
ADF -grounded semantics, then a is credulously accepted (resp. rejected) in F ADF with respectto the same semantics.2. if a is skeptically accepted (resp. rejected) in F MMA with respect to the exact semantics andif | L one ( exact , F MMA ) | = | L ( exact , F ADF ) | , then a is skeptically accepted (resp. rejected)in F ADF with respect to the exact semantics.3. if a is skeptically accepted (resp. rejected) in F MMA with respect to the
ADF -groundedsemantics, then a is skeptically accepted (resp. rejected) in F ADF with respect to the
ADF -grounded semantics.
Proof.
Follows from Theorem 3.4 and Theorem 3.5. ✷ We identified a Galois connection for abstract dialectical frameworks and may-must argumen-tation, demonstrating abstract interpretation at play in formal argumentation. The technique ofabstract interpretation or its significance has almost not been known to the argumentation com-munity. As far as we are aware, there is a preprint (Arisaka and Dauphin 2018) that contemplatesits application to abstraction of loops in an argumentation graph to sharpen acceptability statusesof undec -labelled arguments. However, while, to an extent, it carries an underlying motivation ofabstract interpretation to know better what could not be otherwise known, it is not entirely clearwhether the abstraction of loops intends a sound over-approximation. In comparison, we identi-fied a Galois connection between abstract dialectical frameworks and may-must argumentationswith results stipulating what the semantic predictions in
MMA space mean in
ADF space.For future work, it is possible to make the situation more complex with concurrency in multi-agent argumentation (Arisaka and Satoh 2018; Arisaka and Ito 2019). Complication is unbounded.In static analysis, abstract interpretation generally involves consideration for widening and nar-rowing operations. Studies on more algorithmic approaches with them should be also interestingand practically worthwhile.4
R. Arisaka and T. Ito
References A RISAKA , R.
AND D AUPHIN , J. 2018. Abstractly Interpreting Argumentation Frameworks for SharpeningExtensions.
ArXiv e-prints:1802.01526 .A RISAKA , R.
AND I TO , T. 2019. Numerical Abstract Persuasion Argumentation for Expressing Con-current Multi-Agent Negotiations. In IJCAI Best of Workshops 2019 (to appear, found also at https: // arxiv. org/ abs/ 2001. 08335 ) .A RISAKA , R.
AND I TO , T. 2020a. Broadening Label-based Argumentation Semantics with May-MustScales. In CLAR . Springer, Hangzhou, China, 22–41.A
RISAKA , R.
AND I TO , T. 2020b. Broadening Label-based Argumentation Semantics with May-MustScales (May-Must Argumentation. ArXiv e-prints:2001.05730 .A RISAKA , R.
AND S ATOH , K. 2018. Abstract Argumentation / Persuasion / Dynamics. In
PRIMA .Springer, Tokyo, Japan, 331–343.B
ARONI , P.
AND G IACOMIN , M. 2007. On principle-based evaluation of extension-based argumentationsemantics.
Artificial Intelligence 171,
REWKA , G., S
TRASS , H., E
LLMAUTHALER , S., W
ALLNER , J.,
AND W OLTRAN , S. 2013. AbstractDialectical Frameworks Revisited. In
IJCAI . Morgan Kaufmann Publishers, Beijing, China.C
AMINADA , M. 2006. On the Issue of Reinstatement in Argumentation. In
JELIA . Springer, Liverpool,UK, 111–123.C
OUSOT , P.
AND C OUSOT , R. 1977. Abstract interpretation: a unified lattice model for static analysis ofprograms by construction or approximation of fixpoints. In
POPL . ACM Press, Los Angeles, California,238–252.C
OUSOT , P.
AND C OUSOT , R. 1979. Systematic design of program analysis frameworks. In
POPL . ACMPress, New York, NY, USA, 269–282.D
AVEY , B. A.
AND P RIESTLEY , H. A. 2002.
Introduction to Lattices and Order . Cambridge UniversityPress.D
UNG , P. M. 1995. On the Acceptability of Arguments and Its Fundamental Role in NonmonotonicReasoning, Logic Programming, and n-Person Games.
Artificial Intelligence 77,
2, 321–357.G
ARSON , J. 2018. Modal Logic. In
The Stanford Encyclopedia of Philosophy , E. N. Zalta, Ed.J
AKOBOVITS , H.
AND V ERMEIR , D. 1999. Robust Semantics for Argumentation Frameworks.