Enforcing Sets of Formulas in Structured Argumentation
AnneMarie Borg1 , Floris Bex1,2
1
Department of Information and Computing Sciences, Utrecht University
2
Tilburg Institute for Law, Technology, and Society, Tilburg University{A.Borg, F.J.Bex}@UU.nl

Abstract
Enforcement, adjusting an argumentation framework suchthat a certain set of arguments becomes acceptable, is an important research topic within the study of dynamic argumentation, but one that has been little studied for structured argumentation. In this paper we study enforcement in a general
structured argumentation setting. In particular, we study conditions on the argumentation setting and the knowledge basethat ensure (or prevent) the acceptability of sets of formulas
for structured argumentation frameworks.

1

Introduction

Argumentation is a dynamic process, in which derived conclusions have to be revised in view of new information.
However, basic abstract Argumentation Frameworks (AFs)– arguments and the attacks between them (Dung 1995)– and their conclusions – set(s) of arguments (extensions)that can collectively be accepted – are static. Dynamic approaches to abstract argumentation have been studied, suchas enforcement (Baumann and Brewka 2010): given a set ofarguments that we want to accept, how to change the argumentation framework in such a way that this set becomes(part of) an extension in the resulting framework?
Although enforcement has been extensively studied for
abstract argumentation (see (Doutre and Mailly 2018) foran overview), not much research has been conducted on enforcement for structured argumentation, where arguments
can be constructed by inferring conclusions (formulas) froma knowledge base (a set of formulas). In abstract argumentation, dynamics is often modeled by adding arguments/attacks, but in structured argumentation it is not possible to simply add specific arguments/attacks without directly
influencing other arguments/attacks in the framework. The
structure of individual arguments means that there is a subargument relation, for example, when an argument is basedon a subset of the premises (formulas) of another argument.
Thus, adding an argument means that its sub-arguments haveto be added as well. Similarly, because attacks are dependent
on the formulas in the arguments, adding an attack could
mean that we have to add an argument as well. Hence, in a
To appear in the Proceedings of the 18th International Conference on Principles of Knowledge Representation and Reasoning(KR’21).

structured setting we should define enforcement in terms offormulas instead of in terms of arguments: given a set of formulas that you want to accept, how to change the framework
in such a way that this set becomes (part of) the conclusionsof an extension in the resulting framework?
For our study of enforcement of sets of formulas, we takea general approach to structured argumentation (Borg and
Straßer 2018) so the results hold for many of the well-known
approaches (Besnard et al. 2014). We allow formulas to be
added to the knowledge base, and show under which conditions on the attack relation and on the new information in
the knowledge base a set of formulas can be enforced. We
see that a set of formulas can always be enforced if it can bederived from a consistent set of formulas and either (i) that
set is maximally consistent; or (ii) the acceptance relation
is monotonic; or (iii) the acceptance relation satisfies noninterference and the set does not share any information withthe already existing information. To the best of our knowledge, this is the first study on enforcement of sets of formulas in a general structured argumentation setting.
Our study of enforcement in a structured argumentation
setting is not only interesting from a theoretical perspective. There are many real-world applications of structured
argumentation, including those in legal reasoning (Prakken2020), crime and forensics (Bex, Testerink, and Peters 2016;
Odekerken and Bex 2020), medicine (Čyras et al. 2020;
Zeng et al. 2020), technologies for behaviour change (Chalaguine et al. 2018) and debunking fake news (Visser,
Lawrence, and Reed 2020). In these application domains,
dynamics and specifically enforcement for structured argumentation is imperative. For example, when reasoning inlegal or medical cases we want to know whether new observations can enforce a new (legal/medical) conclusion orwhether we can stop looking for more evidence. Similarly,
in persuasive dialogues for behaviour change or about fakenews, it makes sense to enforce one’s conclusion in as shorta dialogue as possible, since in the case of long dialogues
the user will simply disengage.
We start our paper by further illustrating the need for enforcement in a real-life application of structured argumentation (Section 2). The setting from (Borg and Straßer 2018)
and enforcement for abstract argumentation (Baumann andBrewka 2010) are recalled in Section 3. Then, in Section 4,
enforcement in a structured setting are introduced. Finally,we discuss the implications of some of the assumptions andrestrictions made in the paper in Section 5 and related workin Section 6. We conclude in Section 7.

2

Enforcement in Real-world Applications

At the Dutch National Police several argumentation-based
applications are being deployed (Bex, Testerink, and Peters
2016). These applications are aimed at assisting the police
at working through high volume tasks, leaving more time fortasks that require human attention. For example, there is anapplication that helps Dutch citizens to file a complaint on
online trade fraud and can then determine whether it is a caseof fraud (Odekerken, Borg, and Bex 2020), and an application helps the police to identify malafide webshops (Odekerken and Bex 2020). Each of these applications is basedon an inquiry process in which the system gathers information until the argumentation component (which is based ona variation of ASPIC+ (Prakken 2010)) determines that thederived conclusion will not change if more information isadded (i.e., the conclusion is stable).
Take, for example, the application for online trade fraud.
This system tries to determine whether, given the information that the citizen has provided, the conclusion for fraud
can be stably accepted (i.e., it is currently accepted and willremain so with any future expansion of the knowledge base),or not (e.g., no (accepted) argument for fraud is possible
given the current knowledge base, or the acceptance of fraud
might still change when new information is obtained). Essentially, the question is whether the conclusion fraud can be
enforced for all possible expansions of the knowledge base,and if so, what information is necessary for this enforcement. The application contains a rule-base (based on Dutch
fraud law) and a set of queryables, formulas that can be
added to the knowledge base and from which (given the rulebase) it might be possible to build arguments for the conclusion fraud. These queryables are observations that can begathered by, for example, asking the citizen more questions(e.g., ‘how much did you pay to the suspected fraudster?’) or
querying a database (e.g., ‘is the suspected fraudster known
to the police?’). The question of enforcement becomes then:which of these queryables should be added to the knowledge
base to enforce the conclusion fraud? Since the application(and the other applications mentioned here and in Section 1)
are based on structured argumentation frameworks (in thecase of the trade fraud application a variation of ASPIC+ ),the existing literature on enforcement does not help to answer this question. In this paper we therefore study enforcement for structured argumentation in a general setting, theresults of which are general enough to be implemented in
the police applications as well.

3

Preliminaries

In this paper it is assumed that there is a formal language L.
Sets of formulas are denoted by S, T, finite sets of formulas
are denoted by Γ, ∆, Θ, formulas are denoted by γ, δ, φ, ψand atoms are denoted by p, q, r, all of which can be primed
or indexed if needed. Furthermore, it is assumed that there isa deducability relation `⊆ ℘fin (L) × L (where ℘fin denotes

the set of finite subsets).
For the results in this paper it is not necessary to assume
all the Tarskian conditions on ` (i.e., reflexivity, transitivity
and monotonicity). Given a language L, a set of L-formulas
S and a deducability relation `, we denote the closure of S
w.r.t. ` by CN` (S) = {φ | ∃Γ ⊆ S s.t. Γ ` φ}. When ` isclear from the context it will be omitted.

3.1

General Argumentation Setting

Abstract argumentation frameworks (Dung 1995) are pairs
AF = hArgs, Ai, where Args is a set of arguments andA ⊆ Args × Args is an attack relation on those arguments –in such frameworks, the arguments are abstract entities withno internal content or structure. In structured argumentation the arguments in an argumentation framework are constructed from a knowledge base and a set of rules and theattack relation is based on the structure of these arguments.
Rather than choosing one of the well-known approaches
to structured argumentation (see e.g., (Besnard et al. 2014))
as the framework for this paper, the general argumentation
setting from (Borg and Straßer 2018) is taken as the argumentation approach. This allows us to keep the resultsof the current paper as general as possible and does not
require to introduce all the specifics of one particular approach. Moreover, the results apply to some (e.g., ASPIC+ ,ABA and logic-based argumentation) of the approaches discussed in (Besnard et al. 2014), since it was shown in (Borg
and Straßer 2018) that these can be translated into the general setting. We recall here the most important definitions
from (Borg and Straßer 2018).
Definition 1 (Argument). Given a set of L-formulas S, theset of S-based arguments is denoted by Args` (S), such that(Γ, γ) ∈ Args` (S) iff Γ ` γ for some Γ ⊆ S. Given anargument a = (Γ, γ) ∈ Args` (S), Conc(a) = γ denotes theconclusion of a and Supp(a) = Γ denotes the support
set ofS
a. Where S is a set of arguments, Supps(S) = {Supp(a) |a ∈ S} and Concs(S) = {Conc(a) | a ∈ S}.
To accommodate argumentative attacks two types of functions are introduced: a contrariness function · : L → ℘(L)
that associates each formula with a set of conflicting formulas and a target function b· : ℘fin (L) \ ∅ → ℘fin (L) thatassociates the support set of each argument with the sets offormulas in which the argument can be attacked.
Example 1. Let L be a formal language and `CL the consequence relation from classical logic (CL). Some examplesof the contrariness and target function are:negation (¬): φ = {¬φ} conflict: φ = {ψ | ψ, φ ` ⊥}b = Γ conjunction: Γb = {V Γ0 | Γ0 ⊆ Γ}.
identity (id): ΓAn argumentation setting is based on the deducability relation, the contrariness function and the target function.
Definition 2 (Setting). An (argumentation) setting is a tripleAS` = (`, ·,b·). A setting based on S ⊆ L is given byAS` (S) = (S, `, ·,b·).
Attacks between arguments in a setting are based on thecontrariness function and the target function:

a
b

f

e

d

c

Figure 1: Graphical representation of part of the argumentation
framework from Example 2.

Definition 3 (Attacks). Let AS` (S) be a setting and let a =(Γ, γ) ∈ Args` (S) and b = (∆, δ) ∈ Args` (S). Then ab such that γ ∈ φ.
attacks b (in φ) iff there is a φ ∈ ∆Given a setting the corresponding argumentation framework can be defined.
Definition 4 (Argumentation framework). Given a settingAS` (S), the corresponding argumentation framework is thepair AF(AS` (S)) = hArgs` (S), Ai, where Args` (S) is theset of S-based arguments and (a, b) ∈ A iff a, b, ∈ Args` (S)and a attacks b.
Example 2. Let ASCL (S) = (S, `CL , ¬, id) be an argumentation setting, based on classical logic, negation (¬) as
the contrariness function and id as the target function, for
S = {p, q, ¬p ∨ ¬q}. Then some of the arguments are:a = ({p}, p)d = ({p, q}, ¬(¬p ∨ ¬q))b = ({q}, q)e = ({p, ¬p ∨ ¬q}, ¬q)c = ({¬p ∨ ¬q}, ¬p ∨ ¬q) f = ({q, ¬p ∨ ¬q}, ¬p)Here, for example, f attacks a, d and e since p ∈\\ ∩ Supp(e)\ and ¬p ∈ p. Also, d attacks
Supp(a) ∩ Supp(d)c since Supp(c) = {¬p ∨ ¬q} and ¬(¬p ∨ ¬q) ∈ ¬p ∨ ¬q.
See Figure 1 for a graphical representation of part of theinduced argumentation framework.

{Adm, Cmp, Prf, Grd, Stb}.1
In this paper two types of acceptance relations are considered. These relations are functions with which sets ofaccepted formulas can be derived from an argumentation
framework, given a semantics.
Definition 6 (Acceptance relations). Let AF(AS` (S)) be anargumentation framework and let Sem ∈ {Grd, Cmp, Prf,Stb}. A non-empty set T of L-formulas is:• skeptically accepted iff for all E ∈ Sem(AF(AS` (S))),
T ⊆ Concs(E);• credulously accepted iff there is some E∈
Sem(AF(AS` (S))), T ⊆ Concs(E).
Example 3. In the argumentation framework for ASCL (S),with S = {p, q, ¬p ∨ ¬q} from Example 2, it can be shownthat Args({p, q}), Args({p, ¬p ∨ ¬q}) and Args({q, ¬p ∨¬q}) are the extensions for Sem ∈ {Prf, Stb}.
It follows that a formula φ is only skeptically accepted for
Sem ∈ {Grd, Cmp} if it is a tautology in classical logic, p∨q
is skeptically accepted for Sem ∈ {Prf, Stb}, since at leastp or q is part of each extension and from either p ∨ q can bederived in classical logic and all formulas ψ ∈ {p, q, ¬p ∨¬q} are credulously accepted for Sem ∈ {Prf, Stb}.

3.2

Enforcement for Abstract Argumentation• S is preferred (Prf) iff it is ⊆-maximal complete;• S is grounded (Grd) iff it is ⊆-minimal complete;• S is stable (Stb) iff it is admissible and for all a ∈Args` (S) \ S there is a b ∈ S that attacks a.

In this section, we recall three types of expansions (addingarguments/attacks) and enforcement as introduced in (Baumann and Brewka 2010).
Definition 7. (Expansion, (Baumann and Brewka 2010,
Definition 5)) An argumentation framework AF? is an expansion of the framework AF = hArgs, Ai iff AF? =hArgs ∪ Args? , A ∪ A? i, such that Args? 6= ∅ and Args ∩Args? = ∅. Such an expansion is:• normal iff for all a and b, if (a, b) ∈ A? then a ∈ Args? orb ∈ Args? ;• strong iff it is normal and for all a and b, if (a, b) ∈ A?
then it is not the case that a ∈ Args and b ∈ Args? ;• weak iff it is normal and for all a and b if (a, b) ∈ A? thenit is not the case that a ∈ Args? and b ∈ Args.
Definition 8. (Enforcement, (Baumann and Brewka 2010,
Definition 6)) Let AF = hArgs, Ai be an argumentation
framework, Sem a semantics and let S be a set of arguments,
such that S ∈/ Sem(AF). An enforcement of S is a framework AF? such that (1) AF? is AF or a normal expansion ofit and (2) S ∈ Sem(AF? ).
Example 4. Consider the argumentation framework AF =hArgs, Ai, where Args = {a, b, c} and A = {(a, b), (b, a)}shown as the solid part in Figure 2. Now suppose that AF isexpanded: AF0 = Args0 , A0 , where Args0 = Args ∪ {d, e}
and A0 = A ∪ {(a, d), (b, d), (d, c), (e, d)}. The resulting
graphical representation is the full graph of Figure 2. AF0 is
a normal expansion of AF, since no new attacks are added
between the arguments of Args. It would be a strong expansion if (a, d) and (b, d) would not be part of A0 and it would
be a weak expansion if (d, c) would not be part of A0 .

Extensions will be denoted by E, the set of all Semextensions will be denoted by Sem(AF(AS` (S))) for Sem ∈1
In what follows we will sometimes identify Grd(AF(AS` (S)))
with its single element.

To determine which arguments can collectively be considered as accepted, Dung-style argumentation semantics (Dung 1995) can be applied to an argumentation framework.
Definition 5 (Argumentation semantics). Let AF(AS` (S))
be an argumentation framework for a setting AS` (S) and
let S ⊆ Args` (S) be a set of arguments. Then:• S is conflict-free iff there are no a, b ∈ S such that a attacks b;• S defends a ∈ Args` (S) iff for each attacker b ∈Args` (S) of a there is a c ∈ S that attacks b;• S is admissible (Adm) iff it is conflict-free and it defends
every a ∈ S; and• S is complete (Cmp) iff it is admissible and it contains
every a ∈ Args` (S) it defends.
Some specific complete extensions are:

a
c

b
d

e

Figure 2: Graphical representation of the argumentation frameworkfrom Example 4.

4

Enforcement for Structured
Argumentation

Enforcement in structured argumentation differs from enforcement for abstract argumentation. While in abstract
argumentation enforcement is often based on the additionof arguments or attacks, in structured argumentation it isusually not possible to add one specific argument without
adding others as well, viz.
Example 5 (Example 2 continued). Consider again theargumentation setting ASCL (S), with S = {p, q, ¬p ∨¬q}. If one would want to add the argument ({r}, r) toAF(ASCL (S)), r has to be added to S: S0 = S ∪ {r}. However, then arguments such as ({p, r}, p ∧ r) and ({r}, p ∨ r)
are added to Args` (S0 ) as well.
Therefore, the notion of an expansion has to be redefinedfor a structured setting: rather than changing the argumentation framework directly, the argumentation setting on which
the framework is based is expanded.
Definition 9 (Expansion, structured argumentation). Let
AF(AS` (S)) be an argumentation framework, for an argumentation setting AS` (S) and some set of formulas S ⊆ L.
An expansion for AF(AS` (S)) is a set of formulas S0 ⊆ L,
such that S0 6= ∅ and S0 ∩ S = ∅, and the resulting framework is AF(AS` (S ∪ S0 )) = hArgs` (S ∪ S0 ), A0 i, where(a, b) ∈ A0 iff a, b ∈ Args` (S∪S0 ) and a attacks b as definedin Definition 3.
Now enforcement is not about (sets of) arguments, but
about (sets of) formulas. Before defining enforcement interms of sets of formulas, first some useful notions.
Definition 10 ((Strong) (in)consistency). Given an argumentation setting AS` = h`, ·,b·i a set of formulas S anda set of formulas Θ ⊆ S. Then:• Θ is AS` -inconsistent iff there is a γ ∈ Θ for whichΘ \ {γ} ` γ 0 where γ 0 ∈ γ. Θ is strongly AS` (S)inconsistent if Θ ⊆ Θ0 ⊆ S implies that Θ0 is inconsistent.
• Θ is AS` -consistent iff it is not AS` -inconsistent and Θis strongly AS` -consistent if Θ0 ⊆ Θ implies that Θ0 isAS` -consistent.
• Θ is maximal AS` -consistent iff there is no AS` consistent Θ0 ⊆ S such that Θ ⊂ Θ0 and Θ is maximal
strongly AS` (S)-consistent if Θ ( Θ0 ⊆ S implies that
Θ0 is not strongly AS` (S)-consistent.
We denote CS(AS` (S)) [MCS(AS` (S))] for all [maximal]
strongly AS` (S)-consistent sets of S. If the argumentation
setting is clear from the context we will write CS(S) [resp.
MCS(S)].

Remark 1. Since ` is not assumed to be monotonic, we use
the definition of strong inconsistency from (Brewka, Thimm,
and Ulbricht 2019). Note that, when ` is monotonic, thisdefinition coincides with the definition of (in)consistencyfrom (Borg and Straßer 2018).
Example 6 (Example 3 continued). In the argumentation
framework from the previous subsection, where ASCL =h`CL , ¬, idi and S = {p, q, ¬p ∨ ¬q}, the set S is ASCL inconsistent. However, every S0 ⊂ S is ASCL -consistent.
In particular, there are three maximally consistent subsets:
MCS(S) = {{p, q}, {p, ¬p ∨ ¬q}, {q, ¬p ∨ ¬q}}.
In what follows we will assume that the set of formulas
to be enforced is strongly AS` (S)-consistent, since otherwise the consistency postulate from (Caminada and Amgoud2007) could be violated.
Next, we define enforcement for structured argumentation. Enforcing a set of formulas can be done strictly, where
the conclusions are exactly that set of formulas, or nonstrictly, where the set of formulas has to be part of the conclusions (cf. (Baumann and Brewka 2010)).
Definition 11 (Enforcement, structured argumentation). Let
AF(AS` (S)) be an argumentation framework and let T ⊆ L
be the set of formulas to be enforced. Then:• T is [strictly] credulously enforced by the expansion S0 ⊆L if there is some extension E ∈ Sem(AF(AS` (S ∪ S0 ))such that [Concs(E) = CN(T)] Concs(E) ⊇ T.
• T is [strictly] skeptically enforced by the expansion S0 ⊆L if for all extensions E ∈ Sem(AF(AS` (S ∪ S0 )) it holds
that [Concs(E) = CN(T)] Concs(E) ⊇ T.
Enforcement of a single formula φ means that T = {φ}in the definition above. The definition is kept general inthat S0 ⊆ L. Depending on the application S0 might be restricted. For example, in the online trade fraud example from
Section 2 S0 would be restricted to the set of queryables,
since these are the only formulas that can be added to theknowledge base.
Example 7 (Example 2 continued). Consider again theframework AF(ASCL (S)) for S = {p, q, ¬p ∨ ¬q}. Suppose that S0 = S ∪ {r} as in Example 5. For this expanded
set it holds that Prf(AF(AS` (S0 ))) = {ArgsCL ({p, q, r}),ArgsCL ({p, ¬p ∨ ¬q, r}), ArgsCL ({q, ¬p ∨ ¬q, r})}. Therefore r is both credulously and skeptically enforced by {r}.
In the next sections formal results on whether enforcing a
set of formulas is possible are shown. The question of how
such a set of formulas should be enforced is left for future
work. We start by introducing some useful properties.

4.1

Relating Sets of Formulas and Sets of
Arguments

Since enforcement is about extensions (sets of arguments)
and in the structured setting defined for sets of formulas,sets of formulas and sets of arguments have to be related. In
this section results for this relation for static argumentation
are presented.2
2
Due to space restrictions we cannot provide proofs in this section. For full proofs we refer to Appendix A.

Most of the results in the remainder of the paper will beshown for argumentation settings with the identity (id) targetfunction.3 In such settings, when the set of formulas is AS` consistent, there is only one extension and all conclusions ofthat extension are skeptically accepted.
Lemma 1. Let AS` = h`, ·, idi be an argumentation setting and let S be a strongly AS` (S)-consistent set of Lformulas. Then: Sem(AF(AS` (S))) = {Args` (S)} forSem ∈ {Grd, Cmp, Prf, Stb}, and CN(S) is skeptically accepted in AF(AS` (S)).
The next proposition shows how extensions of argumentation frameworks and sets of formulas are related. While therelation between formal argumentation and reasoning withconsistent subsets is well-known (Arieli, Borg, and Heyninck 2019), our result here (i.e., Proposition 1) is more general than the existing results. This is the case since we employ the general setting from (Borg and Straßer 2018), but
we restrict the setting less than is done in that paper (e.g., it
is not assumed that the support set of an argument is consistent, nor that ` is transitive). Moreover, we apply the moregeneral notion of inconsistency from (Brewka, Thimm, andUlbricht 2019), which is also applicable to nonmonotonic`. It is, however, assumed that the setting is contrapositable(see also Footnote 3):
Definition 12. (Contrapositable settings (Borg and Straßer
2018, Definition 18)) A setting (`, ·) is contrapositable iff forall Θ ∈ ℘fin (L), if Θ ` γ 0 where γ 0 ∈ γ then for all δ ∈ Θ,(Θ ∪ {γ}) \ {δ} ` δ 0 for some δ 0 ∈ δ. By extension we callAS` = h`, ·,b·i contrapositable if (`, ·) is contrapositable.
Proposition 1. Let AF(AS` (S)) be such that AS` (S) =h`, ·, idi is contrapositable and let Sem ∈ {Cmp, Prf, Stb}:
1. if T ∈ CS(S), then there is some T0 ⊇ T such thatArgs` (T0 ) ∈ Sem(AF(AS` (S)));
2. if T ∈ MCS(S), then Args` (T) ∈ Sem(AF(AS` (S)));
3. if E ∈ Sem(AF(AS` (S))) then Supps(E) ∈ CS(S);4. if E ∈ Prf(AF(AS` (S))) then Supps(E) ∈ MCS(S) .
The first item states that for any consistent subset there is asuperset from which an extension can be constructed and thesecond item shows that arguments constructed from a maximally consistent subset form an extension. The third item
shows that the union of the support set of all arguments inan extension is always consistent and the fourth shows thatthe support set of the arguments of a preferred extension isa maximally consistent subset. Since enforcement in structured argumentation is defined for sets of formulas, theseresults show how sets of formulas that should be enforced
and extensions can be related.

4.2

Enforcing Formulas

Throughout this section we will consider several conditions
that make enforcement of sets of formulas (im)possible. In
addition to general conditions on, for example, the set of formulas that should be enforced and the acceptance relation,
we will discuss the expansion types and the notion of monotonicity from (Baumann and Brewka 2010).
3

See Section 5 for a discussion on this assumption.

Skeptical enforcement. The type of acceptance (i.e.,
skeptical or credulous) determines, partially, the possibility of enforcing a set of formulas. Intuitively, it is easierto enforce a set of formulas credulously, since then only one
extension has to be considered, while for skeptical enforcement all extensions have to be considered. Indeed, when theset of formulas that should be enforced can only be derived
from sets of formulas that are conflicting with the current
knowledge base, the set cannot be skeptically enforced. Before we show this, we introduce the next lemma:4
Lemma 2. Let AS` = h`, ·, idi be an argumentation setting that is contrapositable and let S be a set of L-formulas.
Then: Grd(AF(AS` (S))) ⊆ Args` (Free(S)).5
Proposition 2. Let AS` = h`, ·, idi such that (`, ·) is
contrapositable, let S be a set of L-formulas and supposethat T ⊆ L has to be skeptically enforced for Sem ∈{Grd, Cmp}. Then T cannot be enforced if there is no minimal strongly AS` (S)-consistent S0 ⊆ L such that T ⊆CN(S? ), where S? ⊆ S0 ∪ S and S0 ∪ S00 is strongly AS` (S)consistent for each AS` -consistent S00 ⊆ S.
Proof. Let AS` = h`, ·, idi such that (`, ·) is contrapositable, let S ⊆ L and suppose that T ⊆ L has to be
skeptically enforced. Moreover, suppose that for each minimal S0 ⊆ L such that there is some minimal S? ⊆ S ∪ S0 for
which T ⊆ CN(S? ), there is some AS` -consistent S00 ⊆ Ssuch that S0 ∪S00 is AS` -inconsistent. It follows that for eachS0 , there is some γ ∈ S0 such that γ ∈/ Free(S ∪ S0 ). By as0
sumption S is minimal, from which it follows that γ ∈ S?
as well. Since Grd(AF(AS` (S ∪ S0 ))) ⊆ Args` (Free(S ∪S0 )), it follows that γ ∈/ Supps(Grd(AF(AS` (S ∪ S0 )))).
?
Since S is minimal to obtain T, it follows that T 6⊆
Concs(Grd(AF(AS` (S ∪ S0 )))).
The next example illustrates what happens when T is inconsistent with the current set of formulas S.
Example 8. Consider the setting ASCL = h`CL , ¬, idi, let
S = {p} and suppose that T = {¬p} has to be enforced. Note that S ∪ T is ASCL -inconsistent. We have that
Grd(AF(ASCL (S ∪ T))) = ∅ and that Sem(AF(ASCL (S ∪T))) = {{ArgsCL (S)}, {ArgsCL (T)}} for Sem ∈ {Prf, Stb}.
Therefore, while T is credulously enforced, it is not skeptically enforced.
Consistent sets. Now we consider the relation between
extensions and sets of formulas as shown in Proposition 1.
In particular, any set of formulas T can be skeptically [resp.
credulously] enforced if there is some set of L-formulas S0such that T is part of the consequences of all [resp. some]maximal strongly AS` (S)-consistent subsets of S ∪ S0 :Theorem 1. Let AF(AS` (S)) be such that AS` = h`, ·, idi
is contrapositable and let T ⊆ L be the set that should be
enforced. Then:
4

In this section, where space allows we provide proof sketchesof (parts of) the proofs. Full proofs can be found in Appendix B.
5
Free(S) is the set of L-formulasfrom S that is part of everyT
T ∈ MCS(S): Free(S) = T∈MCS(S) T.

1. If T can be skeptically enforced then there is some S0 ⊆ Lsuch that T ⊆ CN(S? ) and S? ⊆ Free(S ∪ S0 ) for Sem ∈{Grd, Cmp};
2. T can be skeptically enforced iff there is some S0 ⊆ Lsuch that for every S? ∈ MCS(S ∪ S0 ), T ⊆ CN(S? ) for
Sem ∈ {Prf, Stb};
3. T can be strictly skeptically enforced iff there is someS0 ⊆ L such that for every S? ∈ MCS(S ∪ S0 ), CN(T) =CN(S? ) for Sem ∈ {Prf, Stb};
4. T can be credulously enforced iff there is some S0 ⊆ L
and some S? ∈ CS(S ∪ S0 ) such that T ⊆ CN(S? ) for
Sem ∈ {Cmp, Prf, Stb};
5. T can be strictly credulously enforced iff there is someS0 ⊆ L such that there is some S? ∈ MCS(S ∪ S0 ) suchthat CN(T) = CN(S? ) for Sem ∈ {Prf, Stb}.
Proof. Let AF(AS` (S)) be such that AS` = h`, ·, idi is contrapositable and let T ⊆ L be a set of formulas. We show
Items 2 and 4:
2. (⇒) Suppose that T can be skeptically enforced byexpanding S with S0 .
Then, for each extension
E ∈ Sem(AF(AS` (S ∪ S0 ))), T ⊆ Concs(E).
By Proposition 1.2 it follows immediately that for
any S? ∈ MCS(S ∪ S0 ), T ⊆ CN(S? ), since
Args` (S? ) ∈ Sem(AF(AS` (S ∪ S0 ))) and Concs(E) ⊆CN(Args` (S? )).
(⇐) Suppose that T cannot be skeptically enforced, thenfor each S0 ⊆ L there is some E ∈ Sem(AF(AS` (S∪S0 )))such that for some φ ∈ T, φ ∈/ Concs(E). By Proposition 1.4 Supps(E) ∈ MCS(S ∪ S0 ). Hence, for any
S0 ⊆ L, there is some S? ∈ MCS(S ∪ S0 ) such that forsome φ ∈ T, φ ∈/ CN(S? ).
4. (⇒) Suppose that T can be credulously enforced. Then
there is some S0 ⊆ L, such that there is an extension E ∈Sem(AF(AS` (S ∪ S0 ))) for which T ⊆ Concs(E). Notethat T ⊆ CN(Supps(E)). By Proposition 1.3 it follows
that Supps(E) is consistent.
(⇐) Suppose that there is some S0 ⊆ L and someS? ∈ CS(S ∪ S0 ) such that T ⊆ CN(S? ). By Proposition 1.1 there is some S? ⊆ S# such that Args` (S# ) ∈Sem(AF(AS` (S ∪ S0 ))). Since T ⊆ CN(S? ), for eachφ ∈ T there is some Γ ⊆ S? ⊆ S# such that Γ `φ. Therefore (Γ, φ) ∈ Args` (S# ) and hence T ⊆Concs(Args` (S# )).
Example 9. (Example 7 continued) With the results from
Theorem 1, the skepticalT enforcement of r in Example 7 is
not surprising: r ∈ MCS({p, q, ¬p ∨ ¬q, r}). This also
explains why r cannot be strictly enforced: there is no maximally consistent subset of S∪{r} with r as the only member.
Remark 2. In the proof for skeptical enforcement, the results from Items 2 and 4 of Proposition 1 were used, while
for credulous enforcement, the results from Items 1 and 3 of
Proposition 1 were used. Note that in Proposition 1, Items 1
and 3 are implied by Items 2 and 4 respectively. This meansthat the results for credulous enforcement are more generally applicable, but also that these results hold when only
Items 2 and 4 of Proposition 1 are known for a setting.

Remark 3. In view of this theorem, one might suspect thatenforcement in structured argumentation comes down tofinding maximally consistent subsets of which the desired set
is a consequence. However, some remarks are in order.
• As was shown in (Vesic and van der Torre 2012), it ispossible to instantiate abstract argumentation with classical logic and obtain a meaningful setting (i.e., the rationality postulates from (Caminada and Amgoud 2007)
are satisfied) of which the conclusions do not correspondto maximally consistent subsets. Therefore, argumentation is a richer formalism than reasoning with maximallyconsistent subsets and the result from Theorem 1 only illustrates the possibility of enforcement under these conditions. See (Arieli, Borg, and Heyninck 2019) for a moredetailed discussion on and (Arieli, Borg, and Straßer
2021) for a characterization of the relation between structured argumentation and maximally consistent subsets.
• In this paper the results are about the (im)possibility ofenforcement. When it comes to how a set of formulas
should be enforced the implementation might yield a different result. For example, new information might be obtained over time in a dialogue setting (recall the online
trade fraud example from Section 2). In such a case, theresult of a question might be different from what was necessary to enforce a set of formulas and with that one answer the possibility of enforcement changes.
• Further possibilities to enforce sets of formulas are discussed in the remainder of the paper. For example, theresults from Proposition 3 and Theorem 2 do not rely on
the results from Proposition 1 nor is it assumed that theargumentation setup is contrapositable.
Acceptance relation. Next we consider monotonic acceptance relations: when new information is added the acceptance of the previously derived information does not change.
While argumentation is mainly employed to model nonmonotonic reasoning, as Lemma 3 shows, not all acceptance
relations are nonmonotonic. Proposition 3 is therefore relevant in our setting as well.
Definition 13 (Monotonic acceptance relation). Let
AF(AS` (S)) be an argumentation framework. An acceptance relation is called monotonic iff for any set of formulas
T, if T is accepted in the framework, then T is accepted inany expansion of that framework.
The next proposition shows that a (set of) formula(s) thatcan be derived from a consistent set of formulas can be enforced in an argumentation framework in which the acceptance relation is monotonic. We give an example of such a
relation in Lemma 3.
Proposition 3. Let AF(AS` (S)) be an argumentation
framework for AS` = h`, ·, idi and suppose that the considered acceptance relation is monotonic. Then:• Any formula φ ∈ L, such that Γ ` φ for an AS` consistent Γ ⊆ L, can be enforced by expanding S with
Γ; and• Let S0 be an AS` -consistent set of L-formulas, such that
T ⊆ CN(S0 ). Then T can be enforced by expanding theframework with S0 .
Proof. Let AF(AS` (S)) be an argumentation framework forAS` = h`, ·, idi with a monotonic acceptance relation. We
show the first item, the proof of the second item is similar.
Suppose that Γ ` φ for some strongly AS` -consistent Γ ⊆ L
and that φ should be enforced. By Lemma 1 it follows thatφ is accepted in AF(AS` (Γ)). Since acceptance is assumed
to be monotonic, φ is accepted in AF(AS` (S ∪ Γ)).
The above results hold for contrapositable frameworks
and credulous acceptance:
Lemma 3. If AF(AS` (S)) is an argumentation framework
for a contrapositable setting AS` = h`, ·, idi then credulous
acceptance is monotonic for Sem ∈ {Cmp, Prf, Stb}.
Proof. Let AF(AS` (S)) be such that AS` = h`, ·, idi is contrapositable, let Sem ∈ {Cmp, Prf, Stb} and suppose that
T is credulously accepted in AF(AS` (S)). Moreover, let
S0 ⊆ L be arbitrary. Since T is credulously accepted inAF(AS` (S)), there is some E ∈ Sem(AF(AS` (S))) suchthat T ⊆ Concs(E). By Proposition 1.3 Supps(E) is stronglyAS` (S)-consistent (i.e., Supps(E) ∈ CS(S)) and thus, by
Proposition 1.1 there is some Supps(E) ⊆ T0 ⊆ S ∪ S0such that Args` (T0 ) ∈ Sem(AF(AS` (S ∪ S0 ))). Note that
T ⊆ Concs(Args` (T0 )). Therefore T is credulously accepted in AF(AS` (S ∪ S0 ))), for any S0 .
The next examples are counterexamples of acceptance relations that are not monotonic. The first shows that skepticalacceptance is non-monotonic.
Example 10. Consider the setting ASCL = h`CL , ¬, idi and
let S = {p}. Note that p is a credulous consequence inAF(ASCL (S)) and since credulous acceptance is monotonic(Lemma 3) p will remain a credulous consequence. Let ¬pbe the formula that should be skeptically enforced. Note
that, by Proposition 3, ¬p is credulously enforced by expanding the framework with S0 = {¬p}. However, then bothp and ¬p will be credulously accepted in AF(ASCL (S ∪ S0 )).
Therefore, ¬p will never be skeptically accepted.
The second example considers argumentation frameworks
in which a preference relation is defined on arguments.
Adding a preference relation is a common tool in the argumentation literature to refine conclusions, see e.g., (Amgoud
and Cayrol 2002; Modgil and Prakken 2013). However, as
the example will show, the credulous acceptance relation is
no longer monotonic when preferences are added. In thispaper we use the following notion of preferences among formulas and sets of formulas:
Definition 14 (Preference relation). Let L be a formal language, v : L → N a preference relation on formulas in L
and let φ, ψ ∈ L. We denote that φ is [strictly] preferredover ψ w.r.t. v by [ψ ≺v φ] ψ v φ.
The preference relation can be lifted to sets of formulas,such that S ⊆ L is [strictly] preferred over T ⊆ L w.r.t. v isdenoted by [T ≺v S] T v S. The formal definition of thelifting of the preference relation to sets of formulas is left abstract in this paper. Examples can be found in, e.g., (Modgil
and Prakken 2013). An argument a is preferred over an argument b if Supp(a) is preferred over Supp(b), given some
preference relation v. We accommodate preferences in theattack relation in the usual way (Amgoud and Cayrol 2002;
Modgil and Prakken 2013): for a, b, ∈ Args` (S), a attacks b iff a attacks b and Supp(b) is not strictly preferredover Supp(a).
Example 11. Let ASCL = h`CL , ¬, idi and S = {p} asin the previous example and suppose that there is a preference relation v defined such that p ≺v ¬p. In the flat case,both ¬p and p are credulously accepted in the framework expanded with {¬p}, since there was an argument ({¬p}, ¬p)which attacks and is attacked by ({p}, p). However, in theprioritized case ({p}, p) can no longer attack ({¬p}, ¬p):there is no defense for the -attack from ({¬p}, ¬p). Hencep is no longer accepted given this v.
Relevance. The results in (Borg and Straßer 2018), on
non-interference (Caminada, Carnielli, and Dunne 2011),
can also be used to say something about the possibility to
enforce a (set of) formula(s). Intuitively, non-interference isa helpful property to have, since it indicates that adding irrelevant information does not change the acceptability of thealready derived conclusions. In what follows let Atoms(S)denote the set of atoms that occur in the formulas of S. It issaid that two sets of formulas S1 and S2 are syntactically disjoint (denoted by S1 | S2 ) if Atoms(S1 ) ∩ Atoms(S2 ) = ∅.
Definition 15. (Non-interference, (Caminada, Carnielli,and Dunne 2011)) An acceptance relation satisfies noninterference iff for all S1 ∪ {φ} ∪ S2 ⊆ L such that(S1 ∪ {φ}) | S2 it holds that: φ is a consequence of S1
iff φ is a consequence of S1 ∪ S2 .
In (Borg and Straßer 2018) an extensive discussion canbe found on the requirements on the elements of an argumentation setting in order for the acceptability relations tosatisfy non-interference (e.g., the deducability relation is assumed to satisfy the basic relevance criterion from relevancelogic (Avron 2014) and the target function is assumed to be
monotonic (id is monotonic)). In a dynamic setting, we havethe following result:
Theorem 2. Let AF(AS` (S)) be such that AS` = h`, ·, idi
and that the acceptance relation satisfies non-interference.
A set of formulas T can be credulously and skeptically enforced if there is some AS` -consistent S0 ⊆ L such that
T ⊆ CN(S0 ) and S0 | S.
Proof. Let AF(AS` (S)) be an argumentation framework
such that the considered acceptance relation satisfies noninterference. Moreover, let S0 be a strongly AS` -consistent
set of L-formulas, such that S0 | S and let T ⊆CN(S0 ). Since S0 is strongly AS` -consistent, it follows by
Lemma 1 that T is credulously and skeptically accepted inAF(AS` (S0 )). Since S0 | S and the acceptance relation satisfies non-interference, it follows that T is credulously andskeptically accepted in AF(AS` (S ∪ S0 )) as well.

Expansion types. Although enforcement in structured argumentation is different from enforcement in abstract argumentation, the expansion types from (Baumann and Brewka2010) (recall Definition 7) are available in structured argumentation as well. Intuitively, the next proposition follows
since the structure of the existing arguments does not change
when an argumentation framework is expanded, hence there
are no new attacks between the existing arguments after the
expansion.
Proposition 4. Any expansion of an argumentation framework is normal.
Proof. Let AF(AS` (S)) = hArgs` (S), Ai be an argumentation framework and suppose that S is expanded by S0 .
Let the result be AF(AS` (S ∪ S0 )) = hArgs? , A? i, whereArgs? = Args` (S ∪ S0 ) and A? = {(a, b) ∈ Args? × Args? |a attacks b}. Now suppose that S0 is such that the expansion is not normal. Then there are a, b ∈ Args` (S) such that(a, b) ∈ A? \ A. However, since a, b ∈ Args` (S), by the definition of attack (Definition 3) it follows immediately that a
attacks b in AF(AS` (S)) as well. A contradiction.
The next two remarks discuss conditions on the argumentation setting and framework which ensure that an expansion
is not only normal, but also weak or strong.
Remark 4. In some structured argumentation approaches(e.g., ABA (Bondarenko et al. 1997) and ASPIC+ (Modgil
and Prakken 2013)) it is possible to construct arguments thatcannot be attacked since these are derived from strict rules
and/or premises. On the one hand, if the framework before
the expansion consists of such arguments, the expansion is
weak: the existing arguments are not attacked. On the other
hand, if the new arguments that can be created with the expansion cannot be attacked, the expansion is strong: only
the existing arguments can be attacked.
Remark 5. Consider a language L and a preference relation v for it. If an argumentation framework AF(AS` (S)) is
expanded in such a way that the new arguments are strictly
stronger than the existing ones, the expansion is strong. If
the framework is expanded such that the existing arguments
are strictly stronger, the expansion is weak.
Monotonicity. In (Baumann and Brewka 2010) conditions
on the semantics and the expansion type are studied to ensure that the expansion is monotonic, i.e., that the numberof extensions does not decrease and the acceptance statusof the already existing arguments does not change when aframework is expanded. In view of Proposition 1 and Theorem 1 we can show the structured counterpart of (Baumann
and Brewka 2010, Theorem 5).
Proposition 5. Let AF(AS` (S)) be an argumentation
framework for a contrapositable AS` = h`, ·, idi and let
S0 be an expansion. Then, for Sem ∈ {Prf, Stb}:• |Sem(AF(AS` (S)))| ≤ |Sem(AF(AS` (S ∪ S0 )))|;• ∀E ∈ Sem(AF(AS` (S))), ∃E 0 ∈ Sem(AF(AS` (S ∪ S0 )))such that E ⊆ E 0 .

Proof sketch. Let AF(AS` (S)) be an argumentation framework for a contrapositable AS` and let S0 be an expansion.
Note that MCS is monotonic: for every T ∈ MCS(S), there
is some T ⊆ T0 ∈ MCS(S ∪ S0 ).
• Thisfollowssince|Stb(AF(AS` (T)))|≤|Prf(AF(AS` (T)))| for any set of formulas T (i.e.,every stable extension is a preferred extension (Dung1995)).
Since, |MCS(S)| ≤ |MCS(S ∪ S0 )|,|Prf(AF(AS` (S)))| ≤ |MCS(S)| (Proposition 1.4)
and |MCS(S ∪ S0 )| ≤ |Stb(AF(AS` (S ∪ S0 )))| (Proposition 1.2).
• Let E ∈ Sem(AF(AS` (S))). Then, by Proposition 1.4 itfollows that Supps(E) ∈ MCS(S). Hence, there is someT ∈ MCS(S ∪ S0 ) such that Supps(E) ⊆ T. By Proposition 1.2 it follows that Args` (T) ∈ Sem(AF(AS` (S ∪S0 ))) (recall that every stable extension is also preferred).
Since Supps(E) ⊆ T, E ⊆ Args` (T). Because E wasarbitrary, this holds for all Prf- and Stb-extensions.

Remark 6. (Baumann and Brewka 2010, Theorem 5) contains a third item (which serves as the converse of the second item) that in a structured setting does not hold without further restrictions. Consider again Example 10, we
have that Prf(AF(ASCL ({p}))) = {Args` ({p})}, while
Prf(AF(ASCL ({p, ¬p}))) = {Args` ({p}), Args` ({¬p})}.
Note that the extension Args` ({¬p}) does not extend an extension in the framework before the expansion.

5

Discussion

Throughout the paper some assumptions were made to show
the results. We will discuss some of these in this section, toillustrate that such restrictions are common in the literature.
In many of the results it was assumed that the target function is id. When looking at well-known approaches to structured argumentation (Besnard et al. 2014) and the translationof these approaches as provided by (Borg and Straßer 2018)it can be seen that the target function is often id. In ABA andASPIC+ this is always the case, because of the translation,and in logic-based argumentation attack rules such as direct
defeat and direct undercut are covered by id.
The first formal results required that the argumentation
setting was contrapositive. In, e.g., (Heyninck and Arieli
2018; Modgil and Prakken 2013) the same assumption ismade to show, e.g., the rationality postulates from (Caminada and Amgoud 2007) and the relation to reasoning withmaximally consistent subsets. As we consider it essentialthat these rationality postulates are satisfied, the argumentation setting would be contrapositive in most settings anyway.
Like in (Borg and Straßer 2018), we do not consider apreference ordering on the arguments. However, throughout
the paper some remarks were already made about preference
orderings; we plan to formally investigate the use of preferences in future work.

6

Related Work

Enforcement is an important research direction in the studyof dynamics in formal argumentation (Doutre and Mailly

2018). In (Baumann and Brewka 2010) (im)possibility results are shown for the enforcement of extensions, based on
abstract principles of semantics, rather than specific Dungstyle semantics. Moreover, conditions under which the addition of new arguments and attacks between them does
not change the acceptance status of already existing arguments are investigated as well. Other, related questions
have been investigated such as extension removal (Baumann and Brewka 2019), enforcement by changing the semantics (Doutre and Mailly 2017) and the minimal change
necessary to enforce an extension (Baumann 2012; CosteMarquis et al. 2014; Wallner, Niskanen, and Järvisalo 2017),
all for abstract argumentation frameworks (Dung 1995).
An enforcement operator for structured argumentation is
introduced in (Wallner 2020). However, the question of enforcement is still at the argument level: how to adjust theknowledge base such that a certain (set of) argument(s) becomes a part of some extension. In our paper we are interested in enforcement on the formula level. This might leadto different conclusion, since, for example, there might bemore than one argument for a certain formula.
In a structured argumentation setting we are not awareof any further research on enforcement. There is, however, some work on other aspects of dynamics. In (Alfano et al. 2018), an incremental approach to efficiently
recalculating the acceptability status of literals in updated
DeLP programs is introduced, in (Modgil and Prakken2012) resolutions of attacks are studied in the context of
ASPIC+ , and in (Testerink, Odekerken, and Bex 2019;
Odekerken, Borg, and Bex 2020) the stability of the acceptability status of a certain formula is investigated. Finally,(Booth et al. 2014) study concepts related to enforcementin (abstract) abductive argumentation frameworks and showthat these frameworks can be instantiated by abductive logic
programming.
By definition, enforcement is related to belief revision (Alchourrón, Gärdenfors, and Makinson 1985; Katsuno and Mendelzon 1991). This relation has been surveyed in, e.g., (Falappa, Kern-Isberner, and Simari 2009;
Falappa et al. 2011) where it is argued that formal argumentation and belief revision model two complementary aspectsof human commonsense reasoning. In particular, where belief revision concerns how an agent’s beliefs can be updated
consistently, argumentation is concerned with the process ofinferring beliefs, and the reasons for and against these beliefs (Falappa, Kern-Isberner, and Simari 2009, p. 355).
In the dynamic argumentation literature, belief revision (like) postulates have been employed to handle changes informal argumentation. We mention some general trends.
Many dynamic argumentation approaches that are combined with belief revision postulates are based on a translation of the argumentation framework and notions suchas acceptance and Dung-style semantics into logical formulas. For example, (de Saint-Cyr et al. 2016) introduce a new logical language called YALLA, (Doutre,
Herzig, and Perrussel 2014; Doutre, Maffre, and McBurney 2017) use the Dynamic Logic of Propositional Assignments and (Baumann and Brewka 2015) introduce so-called
Dung-logics (see (Doutre and Mailly 2018) for a more de-tailed overview). Belief revision has also been combinedwith existing structured argumentation approaches. For
example, (Snaith and Reed 2016) adjust the belief revision postulates to account for the structure of ASPIC+ elements, thus modelling dynamic ASPIC+ . Furthermore,
non-prioritized belief revision is modelled in (Shakarian etal. 2016) for a probabilistic variation of DeLP, and revisionin logic-based argumentation is modeled in (Krümpelmann
et al. 2012). Each of these examples studies the question
how to revise the argumentation framework to obtain good
conclusions, which is the central question in belief revision. In contrast, enforcement as studied in this paper isuseful for structured argumentation applications based on
dialogues, which are exactly the kind of applications thathave been implemented at the Dutch National Police and inother application domains, see e.g., (Chalaguine et al. 2018;
Visser, Lawrence, and Reed 2020).

7

Conclusion and Future WorkIn this paper we have formulated enforcement for a structured setting: given a set of formulas T, is it possible toexpand the knowledge base such that T becomes credulously or skeptically accepted? We discussed a real-life application where this question is relevant (Section 2) and argued why enforcement in structured argumentation shouldbe about formulas (Example 5). For the general approachto structured argumentation from (Borg and Straßer 2018),
we have studied a variety of conditions under which a setof formulas T can be (strictly) credulously/skeptically enforced: (maximal) consistency of the expansion, relevance,and monotonicity of the acceptance relation. Moreover, wehave shown that, although the setting and the notion of enforcement is different from (Baumann and Brewka 2010),
any expansion is normal and satisfies some monotonicity
properties. While some of our results build on existing literature, we have shown that the existing studies on static
argumentation settings can be employed in dynamic settings
as well. This suggests that the extensive research on static
structured argumentation can be generalized to dynamic setting, which makes the research better applicable.
As far as we know, this is the first study on enforcementof sets of formulas in a general structured argumentation setting. This opens up the possibility to further investigate dynamics for specific structured settings (Besnard et al. 2014;
Bondarenko et al. 1997; Modgil and Prakken 2013), and toextend the investigation of other notions of dynamics fromknowledge representation in formal argumentation.
In addition to extending the current setting with preferences, in future work we will also look into other approaches to structured argumentation that are not coveredby the general setting from (Borg and Straßer 2018) (e.g.,
DeLP (Garcı́a and Simari 2004) and claim-augmented argumentation frameworks (Dvořák and Woltran 2020)). Moreover, we will study how a set of formulas can be enforced.
To this end, we will study different aspects of enforcement
for structured argumentation, such as the minimal change
necessary to enforce a set of formulas, efficient enforcement,
removal of formulas for enforcement, and enforcing the removal of sets of formulas from the conclusion. While someof these more computational aspects of enforcement can bestudied in the general setting, for others (e.g., efficient algorithms or complexity studies), we will have to look into
specific approaches to structured argumentation.

Acknowledgements
This research has been partly funded by the Dutch Ministryof Justice and the Dutch National Police.

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A

Proofs Relating Sets of Formulas and Sets
of Arguments

We start the appendix by providing full proofs of the resultsfrom Section 4.1.
Lemma 1. Let AS` = h`, ·, idi be an argumentation setting and let S be a strongly AS` (S)-consistent set of Lformulas. Then: Sem(AF(AS` (S))) = {Args` (S)} forSem ∈ {Grd, Cmp, Prf, Stb}, and CN(S) is skeptically accepted in AF(AS` (S)).
Proof. Let AS` = h`, ·, idi be an argumentation setting andlet S be a strongly AS` -consistent set of L-formulas. We
show that Args` (S) is not attacked and is therefore the single extension for Sem ∈ {Grd, Cmp, Prf, Stb}. Suppose
towards a contradiction, that there are a, b ∈ Args` (S) suchthat a = (Γ, φ), b = (∆, ψ) and (a, b) ∈ A. Thus there issome δ ∈ ∆ such that φ ∈ δ. By Definition 10, Γ ∪ {δ}is AS` -inconsistent. A contradiction, since Γ ∪ δ ⊆ S isstrongly AS` -consistent. Hence no argument in Args` (S) is
attacked. By the definition of Args` (S):Concs(Args` (S)) = {Conc(a) | a ∈ Args` (S)}= {φ | (Γ, φ) ∈ Args` (S)}= {φ | Γ ` φ for some Γ ⊆ S}
= {φ | ∃Γ ⊆ S s.t. Γ ` φ} = CN(S).

It follows that CN(S) is skeptically accepted.
When the target function is id the argumentation framework satisfies support closure: given an extension, any argument that can be constructed from formulas already usedin the support set of arguments in that extension will be partof that extension. This result will be used in the proof of
Proposition 1.
Lemma 4. AF(AS` (S)) for AS` = h`, ·, idi satisfies support closure: if Supp(a) ⊆ Supps(E) then a ∈ E, for
Sem ∈ {Grd, Cmp, Prf, Stb}.
Proof. Let AF(AS` (S)) be such that AS` = h`, ·, idi, let
E ∈ Sem(AF(AS` (S))) for Sem ∈ {Grd, Cmp, Prf, Stb}and let a ∈ Args` (S) such that Supp(a) ⊆ Supps(E). If
a is not attacked, by the completeness of E it follows thata ∈ E. Now suppose that there is some b ∈ Args` (S) such
\ = Supp(a)that b attacks a. Then there is some γ ∈ Supp(a)such that Conc(b) ∈ γ. By assumption, there is some c ∈ Esuch that γ ∈ Supp(c) as well. Hence b attacks c. Sincec ∈ E, there is some d ∈ E such that d defends c against
the attack from b and therefore defends a against the attackfrom b as well. Again by the completeness of E, a ∈ E.
In addition to the above lemma, we will also use the following remark, sometimes implicitly.
Remark 7. Let AS` = h`, ·, idi be an argumentation setting, S a set of L-formulas and let S0 ∈ CS(S). Then there
is some T ∈ MCS(S) such that S0 ⊆ T.
Proposition 1. Let AF(AS` (S)) be such that AS` (S) =h`, ·, idi is contrapositable and let Sem ∈ {Cmp, Prf, Stb}:
1. if T ∈ CS(S), then there is some T0 ⊇ T such thatArgs` (T0 ) ∈ Sem(AF(AS` (S)));
2. if T ∈ MCS(S), then Args` (T) ∈ Sem(AF(AS` (S)));
3. if E ∈ Sem(AF(AS` (S))) then Supps(E) ∈ CS(S);4. if E ∈ Prf(AF(AS` (S))) then Supps(E) ∈ MCS(S) .
Proof. Let AF(AS` (S)) be such that AS` (S) = h`, ·, idi iscontrapositable and let Sem ∈ {Cmp, Prf, Stb}.
1. Let T0 ∈ CS(S). Then, by Remark 7, there is some T0 ⊆T such that T ∈ MCS(S). By Lemma 1 it follows thatArgs` (T) is conflict-free. It can be shown that Args` (T)is stable (and therefore preferred and complete as well).
Suppose that there is some b = (∆, δ) ∈ Args` (S) \Args` (T). Then there is some φ ∈ ∆ \ T. By assumption, T is maximal strongly AS` (S)-consistent. Therefore T ∪ {φ} is not strongly AS` (S)-consistent. From
which it follows that there is some Γ ⊆ T ∪ {φ} andsome γ ∈ Γ such that Γ \ {γ} ` γ 0 for some γ 0 ∈ γ.
If γ = φ, then γ 0 ∈ φ and a = (Γ \ {φ}, γ 0 ) ∈ Args` (T).
If γ 6= φ, then φ ∈ Γ \ {γ} (since Γ \ {φ} ⊆ T), by
contraposition Γ \ {φ} ` φ0 for φ0 ∈ φ. Then also a =(Γ\{φ}, φ0 ) ∈ Args` (T). In both cases a attacks b. It follows that Args` (T) ∈ Stb(AF(AS` (S))). Since any stable extension is also preferred and complete Args` (T) ∈Sem(AF(AS` (S))) for Sem ∈ {Cmp, Prf, Stb}.
2. This follows immediately by the proof of Item 1.

3. Now suppose that E ∈ Sem(AF(AS` (S))). Let T =Supps(E) and suppose that T ∈/ CS(S). Then there issome Γ ⊆ T and some γ ∈ Γ such that Γ\{γ} ` γ 0 whereγ 0 ∈ γ. By Lemma 4, b = (Γ \ {γ}, γ 0 ) ∈ E. However,since γ ∈ T, there is some a ∈ E such that γ ∈ Supp(a).
Hence b attacks a. A contradiction to the conflict-freenessof E. Thus T is strongly AS` (S)-consistent.
4. Let E ∈ Prf(AF(AS` (S))). By Item 3 Supps(E) isstrongly AS` -consistent. Suppose that Supps(E) is not
maximal strongly AS` (S)-consistent. Then, by Remark 7, there is some S ⊇ T ⊃ Supps(E) such that
T is maximal strongly AS` (S)-consistent. By Item 2,
Args` (T) ∈ Prf(AF(AS` (S))). However, E ( Args` (T).
A contradiction. Hence Supps(E) = T.

B

Proofs Enforcing Formulas

In this appendix full proofs of the results from Section 4.2
are collected.
While Free(S) is not necessarily strongly consistent (Brewka, Thimm, and Ulbricht 2019), from Proposition 1 it is known that the support set of the grounded extension is strongly consistent.
Lemma 2. Let AS` = h`, ·, idi be an argumentation setting that is contrapositable and let S be a set of L-formulas.
Then: Grd(AF(AS` (S))) ⊆ Args` (Free(S)).6
Proof. Let AS` = h`, ·, idi be an argumentation system thatis contrapositable and let S be a set of L-formulas. Moreover, let Egrd ∈ Grd(AF(AS` (S))) be the grounded extension and a ∈ Egrd . Suppose, towards a contradiction, that
there is some γ ∈ Supp(a) such that γ ∈/ Free(S). Then
there is some T ∈ MCS(S) such that γ ∈/ T. By Proposition 1.2, Args` (T) ∈ Stb(AF(AS` (S))). Since γ ∈/ T,a ∈/ Args` (T). However, since every stable extension iscomplete, a ∈/ Egrd . A contradiction.
Theorem 1. Let AF(AS` (S)) be such that AS` = h`, ·, idi
is contrapositable and let T ⊆ L be the set that should be
enforced. Then:
1. If T can be skeptically enforced then there is some S0 ⊆ Lsuch that T ⊆ CN(S? ) and S? ⊆ Free(S ∪ S0 ) for Sem ∈{Grd, Cmp};
2. T can be skeptically enforced iff there is some S0 ⊆ Lsuch that for every S? ∈ MCS(S ∪ S0 ), T ⊆ CN(S? ) for
Sem ∈ {Prf, Stb};
3. T can be strictly skeptically enforced iff there is someS0 ⊆ L such that for every S? ∈ MCS(S ∪ S0 ), CN(T) =CN(S? ) for Sem ∈ {Prf, Stb};
4. T can be credulously enforced iff there is some S0 ⊆ L
and some S? ∈ CS(S ∪ S0 ) such that T ⊆ CN(S? ) for
Sem ∈ {Cmp, Prf, Stb};
5. T can be strictly credulously enforced iff there is someS0 ⊆ L such that there is some S? ∈ MCS(S ∪ S0 ) suchthat CN(T) = CN(S? ) for Sem ∈ {Prf, Stb}.
6
Free(S) is the set of L-formulasfrom S that is part of everyT
T ∈ MCS(S): Free(S) = T∈MCS(S) T.

Proof. Let AF(AS` (S)) be such that AS` = h`, ·, idi andlet T ⊆ L be a set of formulas. Consider each item, in both
directions:
1. This follows immediately from Proposition 2.
2. (⇒) Suppose that T can be skeptically enforced byexpanding S with S0 .
Then, for each extension
E ∈ Sem(AF(AS` (S ∪ S0 ))), T ⊆ Concs(E).
By Proposition 1.2 it follows immediately that for
any S? ∈ MCS(S ∪ S0 ), T ⊆ CN(S? ), since
Args` (S? ) ∈ Sem(AF(AS` (S ∪ S0 ))) and Concs(E) ⊆CN(Args` (S? )).
(⇐) Suppose that T cannot be skeptically enforced, thenfor each S0 ⊆ L there is some E ∈ Sem(AF(AS` (S∪S0 )))such that for some φ ∈ T, φ ∈/ Concs(E). By Proposition 1.4 Supps(E) ∈ MCS(S ∪ S0 ). Hence, for any
S0 ⊆ L, there is some S? ∈ MCS(S ∪ S0 ) such that forsome φ ∈ T, φ ∈/ CN(S? ).
3. (⇒) Suppose that T can be strictly skeptically enforced byexpanding the framework with S0 . Then for any extension
E ∈ Sem(AF(AS` (S ∪ S0 ))), CN(T) = Concs(E). Likefor Item 2, by Proposition 1.2 it follows that for all S? ∈MCS(S ∪ S0 ), CN(T) = CN(S? ).
(⇐) Now assume that T cannot be strictly skeptically enforced. If T cannot be skeptically enforced either, it follows by the previous item that for each S0 ⊆ L, there issome S? ∈ MCS(S ∪ S0 ) such that for some φ ∈ T, φ ∈/CN(S? ). If T can be skeptically enforced, then for eachS0 ⊆ L such that there is some S? ∈ MCS(S ∪ S? ) withT ⊆ CN(S? ) (note that also CN(T) ⊆ CN(S? )), there issome φ ∈ CN(S? ) \ CN(T). Hence CN(T) 6= CN(S? ) for
any S0 ⊆ L and any S? ∈ MCS(S ∪ S0 ).
4. (⇒) Suppose that T can be credulously enforced. Then
there is some S0 ⊆ L, such that there is an extension E ∈Sem(AF(AS` (S ∪ S0 ))) for which T ⊆ Concs(E). Notethat T ⊆ CN(Supps(E)). By Proposition 1.3 it follows
that Supps(E) is consistent.
(⇐) Suppose that there is some S0 ⊆ L and someS? ∈ CS(S ∪ S0 ) such that T ⊆ CN(S? ). By Proposition 1.1 there is some S? ⊆ S# such that Args` (S# ) ∈Sem(AF(AS` (S∪S0 ))). Since T ⊆ CN(S? ), for each φ ∈T there is some Γ ⊆ S? ⊆ S# such that Γ ` φ. Therefore(Γ, φ) ∈ Args` (S# ) and hence T ⊆ Concs(Args` (S# )).
5. (⇒) Assume that T can be strictly credulously enforced by some S0 ⊆ L. Then there is some E ∈Sem(AF(AS` (S∪S0 ))) such that Concs(E) = CN(T). By
Proposition 1.4 it follows that Supps(E) ∈ MCS(S ∪ S0 ).
Since CN(Supps(E)) = Concs(E) = CN(T), it follows that there is some S? ∈ MCS(S ∪ S0 ) such that
CN(T) = CN(S? ).
(⇐) Now assume that there is some S0 ⊆ L and someS? ∈ MCS(S ∪ S0 ) such that CN(T) = CN(S? ). By
Proposition 1.2, Args` (S? ) ∈ Stb(AF(AS` (S ∪ S0 ))).
Since Concs(Args` (S? )) = CN(S? ) = CN(T) it follows
that T can be strictly credulously enforced by expanding
the framework with S0 .

Proposition 3. Let AF(AS` (S)) be an argumentation
framework for AS` = h`, ·, idi and suppose that the considered acceptance relation is monotonic. Then:• Any formula φ ∈ L, such that Γ ` φ for an AS` consistent Γ ⊆ L, can be enforced by expanding S with
Γ; and• Let S0 be an AS` -consistent set of L-formulas, such that
T ⊆ CN(S0 ). Then T can be enforced by expanding theframework with S0 .
Proof. Let AF(AS` (S)) be an argumentation framework forAS` = h`, ·, idi with a monotonic acceptance relation. Consider both items:• Suppose that Γ ` φ for some strongly AS` -consistent Γ ⊆L and that φ should be enforced. By Lemma 1 it follows
that φ is accepted in AF(AS` (Γ)). Since acceptance isassumed to be monotonic, φ is accepted in AF(AS` (S ∪Γ)) as well.
• Let S0 be a strongly AS` -consistent set of of L formulas such that T ⊆ CN(S0 ) and suppose that T has to
be enforced. Since S0 is strongly AS` (S)-consistent, byLemma 1 CN(S0 ) is accepted in AF(AS` (S0 )). Since acceptance is assumed to be monotonic it follows that T is
accepted in AF(AS` (S ∪ S0 )). Hence T is enforced by the
expansion of S0 .
Proposition 5. Let AF(AS` (S)) be an argumentation
framework for a contrapositable AS` = h`, ·, idi and let
S0 be an expansion. Then, for Sem ∈ {Prf, Stb}:• |Sem(AF(AS` (S)))| ≤ |Sem(AF(AS` (S ∪ S0 )))|;• ∀E ∈ Sem(AF(AS` (S))), ∃E 0 ∈ Sem(AF(AS` (S ∪ S0 )))such that E ⊆ E 0 .
Proof. Let AF(AS` (S)) be an argumentation framework forAS` = h`, ·, idi and let S0 be an expansions. Consider eachof the items:• Note that, given the definition of strong consistency (Definition 10), |MCS(S)| ≤ |MCS(S ∪ S0 )|. By Proposition 1.4 it is known that |Prf(AF(AS` (S)))| ≤ |MCS(S)|.
By Proposition 1.2 it is known that |MCS(S ∪ S0 )| ≤|Stb(AF(AS` (S ∪ S0 )))|. Since any stable extension
is also a preferred extension (Dung 1995) and therefore |Stb(AF(AS` (T)))| ≤ |Prf(AF(AS` (T)))| for any
set of formulas T, it follows that |Stb(AF(AS` (S)))| ≤|MCS(S)| and |MCS(S ∪ S0 )| ≤ |Prf(AF(AS` (S ∪ S0 )))|
as well. It therefore follows that |Sem(AF(AS` (S)))| ≤|Sem(AF(AS` (S ∪ S0 )))|, for Sem ∈ {Prf, Stb}.
• Let E ∈ Sem(AF(AS` (S))). Then, by Proposition 1.4 itfollows that Supps(E) ∈ MCS(S). Hence, there is someT ∈ MCS(S ∪ S0 ) such that Supps(E) ⊆ T. By Proposition 1.2 it follows that Args` (T) ∈ Sem(AF(AS` (S ∪S0 ))) (recall that every stable extension is also preferred).
Since Supps(E) ⊆ T, E ⊆ Args` (T). Because E wasarbitrary, this holds for all Prf- and Stb-extensions.