Argumentation in Artificial Intelligence: 20 years after Dung's work. Right margin for notes

Computing Science
Argumentation in Artiﬁcial Intelligence:
20 Years after Dung’s Work
Federico Cerutti
Department of Computing Science July 2015
University of Aberdeen
King’s College
Aberdeen AB24 3UE
Copyright © 2015, The University of Aberdeen

Argumentation in Artiﬁcial Intelligence: 20
Years after Dung’s Work
Federico Cerutti
Department of Computing Science
University of Aberdeen
July 2015
Abstract: Handouts for the IJCAI 2015 tutorial on Argumentation.
This document is a collection of technical deﬁnitions as well as ex-
amples of various topics addressed in the tutorial. It is not supposed
to be an exhaustive compendium of twenty years of research in ar-
gumentation theory.
This material is derived from a variety of publications from many
researchers who hold the copyright and any other intellectual prop-
erty of their work. Original publications are thoroughly cited and
reported in the bibliography at the end of the document. Errors and
misunderstandings rest with the author of this tutorial: please send
an email to federico.cerutti@acm.org for reporting any.
Keywords: argumentation; tutorial; IJCAI2015
University of Aberdeen, 2015 Page 1

1 Dung’s Argumentation
Framework
Acknowledgement
This handout include material from a number of collaborators including
Pietro Baroni, Massimiliano Giacomin, and Stefan Woltran.5
Definition 1 ([Dun95]). A Dung argumentation framework AF is a pair
〈A ,→ 〉
where A is a set of arguments, and → is a binary relation on A i.e. →⊆
A ×A . ♠
An argumentation framework has an obvious representation as a di-
rected graph where the nodes are arguments and the edges are drawn10
from attacking to attacked arguments.
The set of attackers of an argument a1 will be denoted as a−
1 {a2 :
a2 → a1}, the set of arguments attacked by a1 will be denoted as a+
1 {a2 :
a1 → a2}. We also extend these notations to sets of arguments, i.e. given
E ⊆ A , E−
{a2 | ∃a1 ∈ E,a2 → a1} and E+
{a2 | ∃a1 ∈ E,a1 → a2}.15
With a little abuse of notation we define S → a ≡ ∃a ∈ S : a → b. Simi-
larly, b → S ≡ ∃a ∈ S : b → a.
1.1 Principles for Extension-based Semantics:
[BG07]
Definition 2.

Given an argumentation framework AF = 〈A ,→ 〉, a set20
S ⊆ A is D-conflict-free, denoted as D-cf(S), if and only if a,b ∈ S such
that a → b. A semantics σ satisfies the D-conflict-free principle if and only
if ∀AF,∀E ∈ Eσ(AF) E is D-conflict-free . ♠
Definition 3. Given an argumentation framework AF = 〈A ,→ 〉, an ar-
gument a ∈ A is D-acceptable w.r.t. a set S ⊆ A if and only if ∀b ∈ A25
b → a ⇒ S → b.
The function FAF : 2A
→ 2A
which, given a set S ⊆ A , returns the
set of the D-acceptable arguments w.r.t. S, is called the D-characteristic
function of AF. ♠
Definition 4. Given an argumentation framework AF = 〈A ,→ 〉, a set30
S ⊆ A is D-admissible (S ∈ AS (AF)) if and only if D-cf(S) and ∀a ∈ S

Dung’s AF • Acceptability of Arguments [PV02; BG09a]
a is D-acceptable w.r.t. S. The set of all the D-admissible sets of AF is
denoted as AS (AF). ♠
Dσ = {AF|Eσ(AF) = }
Deﬁnition 5.

A semantics σ satisfies the D-admissibility principle if and
only if ∀AF ∈ Dσ Eσ(AF) ⊆ AS (AF), namely ∀E ∈ Eσ(AF) it holds that:
a ∈ E ⇒ (∀b ∈ A ,b → a ⇒ E → b). ♠
Definition 6. Given an argumentation framework AF = 〈A ,→ 〉, a ∈
A and S ⊆ A , we say that a is D-strongly-defended by S (denoted as5
D-sd(a,S)) iff ∀b ∈ A , b → a, ∃c ∈ S {a} : c → b and D-sd(c,S {a}). ♠
Definition 7.

A semantics σ satisﬁes the D-strongly admissibility prin-
ciple if and only if ∀AF ∈ Dσ, ∀E ∈ Eσ(AF) it holds that
a ∈ E ⊃ D-sd(a,E) ♠
Deﬁnition 8.

A semantics σ satisﬁes the D-reinstatement principle if and
only if ∀AF ∈ Dσ, ∀E ∈ Eσ(AF) it holds that:
(∀b ∈ A ,b → a ⇒ E → b) ⇒ a ∈ E. ♠
Deﬁnition 9.

A set of extensions E is D-I-maximal if and only if ∀E1,E2 ∈
E , if E1 ⊆ E2 then E1 = E2. A semantics σ satisfies the D-I-maximality
principle if and only if ∀AF ∈ Dσ Eσ(AF) is D-I-maximal. ♠
Definition 10. Given an argumentation framework AF = 〈A ,→ 〉, a non-10
empty set S ⊆ A is D-unattacked if and only if ∃a ∈ (A S) : a → S. The
set of D-unattacked sets of AF is denoted as US (AF). ♠
Definition 11. Let AF = 〈A ,→ 〉 be an argumentation framework. The
restriction of AF to S ⊆ A is the argumentation framework AF↓S = 〈S,→
∩(S × S)〉. ♠15
Definition 12.

A semantics σ satisfies the D-directionality principle if
and only if ∀AF = 〈A ,→ 〉,∀S ∈ US (AF),AE σ(AF,S) = Eσ(AF↓S), where
AE σ(AF,S) {(E ∩ S) | E ∈ Eσ(AF)} ⊆ 2S
. ♠
1.2 Acceptability of Arguments [PV02; BG09a]
Definition 13. Given a semantics σ and an argumentation framework20
〈A ,→ 〉, an argument AF ∈ Dσ is:
• skeptically justified iff ∀E ∈ Eσ(AF), a ∈ S;
• credulously justified iff ∃E ∈ Eσ(AF), a ∈ S. ♠

Dung’s AF • (Some) Semantics [Dun95]
Definition 14. Given a semantics σ and an argumentation framework
〈A ,→ 〉, an argument AF ∈ Dσ is:
• justified iff it is skeptically justified;
• defensible iff it is credulously justified but not skeptically justified;
• overruled iff it is not credulously justified. ♠5
1.3 (Some) Semantics [Dun95]
Lemma 1 (Dung’s Fundamental Lemma, [Dun95, Lemma 10]). Given an
argumentation framework AF = 〈A ,→ 〉, let S ⊆ A be a D-admissible set
of arguments, and a,b be arguments which are acceptable with respect to
S. Then:10
1. S = S ∪{a} is D-admissible; and
2. b is D-acceptable with respect to S . ♣
Theorem 1 ([Dun95, Theorem 11]). Given an argumentation framework
AF = 〈A ,→ 〉, the set of all D-admissible sets of 〈A ,→ 〉 form a complete
partial order with respect to set inclusion. ♣15
Definition 15 (Complete Extension).

Given an argumentation frame-
work AF = 〈A ,→ 〉, S ⊆ A is a D-complete extension iff S is D-conﬂict-free
and S = FAF(S). C O denotes the complete semantics. ♠
Deﬁnition 16 (Grounded Extension).

work AF = 〈A ,→ 〉. The grounded extension of AF is the least complete20
extension of AF. GR denotes the grounded semantics. ♠
Deﬁnition 17 (Preferred Extension).

work AF = 〈A ,→ 〉. A preferred extension of AF is a maximal (w.r.t. set
inclusion) complete extension of AF. P R denotes the preferred seman-
tics. ♠25
Deﬁnition 18. Given an argumentation framework AF = 〈A ,→ 〉 and
S ⊆ A , S+
{a ∈ A | ∃b ∈ S ∧ b → a}. ♠
Deﬁnition 19 (Stable Extension).

Given an argumentation framework
AF = 〈A ,→ 〉. S ⊆ A is a stable extension of AF iff S is a preferred exten-
sion and S+
= A S. S T denotes the stable semantics. ♠30

Dung’s AF • Labelling-Based Semantics Representation
[Cam06]
C O GR P R S T
D-conflict-free Yes Yes Yes Yes
D-admissibility Yes Yes Yes Yes
D-strongly admissibility No Yes No No
D-reinstatement Yes Yes Yes Yes
D-I-maximality No Yes Yes Yes
D-directionality Yes Yes Yes No
Table 1.1: Satisfaction of general properties by argumentation semantics
[BG07; BCG11]
S T
P R
C O GR
Figure 1.1: Relationships among argumentation semantics
1.4 Labelling-Based Semantics Representation
[Cam06]
Definition 20. Let ∆ = Γ be an argumentation framework. A labelling
L ab ∈ L(∆) is a complete labelling of ∆ iff it satisfies the following condi-
tions for any a1 ∈ A :5
• L ab(a1) = in ⇔ ∀a2 ∈ a−
1 L ab(a2) = out;
• L ab(a1) = out ⇔ ∃a2 ∈ a−
1 : L ab(a2) = in. ♠
The grounded and preferred labelling can then be defined on the basis
of complete labellings.
Definition 21. Let ∆ = Γ be an argumentation framework. A labelling10
L ab ∈ L(∆) is the grounded labelling of ∆ if it is the complete labelling
of ∆ minimizing the set of arguments labelled in, and it is a preferred
labelling of ∆ if it is a complete labelling of ∆ maximizing the set of argu-
ments labelled in. ♠
In order to show the connection between extensions and labellings, let15
us recall the definition of the function Ext2Lab, returning the labelling
corresponding to a D-conflict-free set of arguments S.
Definition 22. Given an AF ∆ = Γ and a D-conflict-free set S ⊆ A , the
corresponding labelling Ext2Lab(S) is defined as Ext2Lab(S) ≡ L ab, where
• L ab(a1) = in ⇔ a1 ∈ S20
• L ab(a1) = out ⇔ ∃ a2 ∈ S s.t. a2 → a1

[Cam06]
σ = C O σ = GR σ = P R σ = S T
EXISTSσ trivial trivial trivial NP-c
CAσ NP-c polynomial NP-c NP-c
SAσ polynomial polynomial Π
p
2 -c coNP-c
VERσ polynomial polynomial coNP-c polynomial
NEσ NP-c polynomial NP-c NP-c
Table 1.2: Complexity of decision problems by argumentation semantics
[DW09]
• L ab(a1) = undec ⇔ a1 ∉ S ∧ a2 ∈ S s.t. a2 → a1 ♠
[Cam06] shows that there is a bijective correspondence between the
complete, grounded, preferred extensions and the complete, grounded,
preferred labellings, respectively.
Proposition 1. Given an an AF ∆ = Γ, L ab is a complete (grounded,5
preferred) labelling of ∆ if and only if there is a complete (grounded, pre-
ferred) extension S of ∆ such that L ab = Ext2Lab(S). ♣
The set of complete labellings of ∆ is denoted as LC O (∆), the set of
preferred labellings as LP R(∆), while LGR(∆) denotes the set including
the grounded labelling.10

[Cam06]

Dung’s AF • Skepticism Relationships [BG09b]
GR
C O
P R
GR
C O
P RS T
Figure 1.2: S
⊕ relation for any argumentation framework (left) and for
argumentation framework where stable extensions exist (right).
1.5 Skepticism Relationships [BG09b]
E1
E
E2 denotes that E1 is at least as skeptical as E2.
Definition 23. Let E
be a skepticism relation between sets of exten-
sions. The skepticism relation between argumentation semantics S
is
such that for any argumentation semantics σ1 and σ2, σ1
S
σ2 iff ∀AF ∈5
Dσ1 ∩Dσ2 , EAF(σ1) E
EAF(σ2). ♠
Definition 24. Given two sets of extensions E1 and E2 of an argumenta-
tion framework AF:
• E1
E
∩+ E2 iff ∀E2 ∈ E2, ∃E1 ∈ E1: E1 ⊆ E2;
• E1
E
∪+ E2 iff ∀E1 ∈ E1, ∃E2 ∈ E2: E1 ⊆ E2. ♠10
Lemma 2. Given two argumentation semantics σ1 and σ2, if for any
argumentation framework AF EAF(σ1) ⊆ EAF(σ2), then σ1
E
∩+ σ2 and
σ1
E
∪+ σ2 (σ1
E
⊕ σ2). ♣
1.6 Signatures [Dun+14]
Let A be a countably infinite domain of arguments, and15
AFA = {〈A ,→〉 | A ⊆ A,→⊆ A ×A }.
Definition 25. The signature Σσ of a semantics σ is defined as
Σσ = {σ(F) | F ∈ AFA}
(i.e. the collection of all possible sets of extensions an AF can possess
under a semantics). ♠20
Given S ⊆ 2A
, ArgsS = S∈S S, PairsS = {〈a,b〉 | ∃S ∈ S s.t. {a,b} ⊆ S}. S
is called an extension-set if ArgsS is finite.
Definition 26. Let S ⊆ 2A
. S is incomparable if ∀S,S ∈ S, S ⊆ S implies
S = S . ♠

Dung’s AF • Signatures [Dun+14]
Deﬁnition 27. An extension-set S ⊆ 2A
is tight if ∀S ∈ S and a ∈ ArgsS
it holds that if S ∪ {a} ∈ S then there exists an b ∈ S such that 〈a,b〉 ∈
PairsS. ♠
Deﬁnition 28. S ⊆⊆ 2A
is adm-closed if for each A,B ∈ S the following
holds: if 〈a,b〉 ∈ PairsS for each a,b ∈ A ∪B, then also A ∪B ∈ S. ♠5
Proposition 2. For each F ∈ AFA:
• S T (F) is incomparable and tight;
• P R(F) is non-empty, incomparable and adm-closed. ♣
Theorem 2. The signatures for S T and P R are:
• ΣS T = {S | S is incomparable and tight};10
• ΣP R = {S = | S is incomparable and adm-closed}. ♣

Dung’s AF • Signatures [Dun+14]
Consider
S = { { a,d, e },
{ b, c, e },
{ a,b,d } }

Dung’s AF • Decomposability and Transparancy [Bar+14]
1.7 Decomposability and Transparancy [Bar+14]
Definition 29. Given an argumentation framework AF = (A ,→),
a labelling-based semantics σ associates with AF a subset of L(AF), de-
noted as Lσ(AF). ♠
Definition 30. Given AF = (A ,→) and a set Args ⊆ A , the input of Args,5
denoted as Argsinp, is the set {B ∈ A Args | ∃A ∈ Args,(B, A) ∈→}, the con-
ditioning relation of Args, denoted as ArgsR
, is defined as → ∩(Argsinp ×
Args). ♠
Definition 31. An argumentation framework with input is a tuple
(AF,I ,LI ,RI ), including an argumentation framework AF = (A ,→), a10
set of arguments I such that I ∩A = , a labelling LI ∈ LI and a rela-
tion RI ⊆ I × A . A local function assigns to any argumentation frame-
work with input a (possibly empty) set of labellings of AF, i.e.
F(AF,I ,LI ,RI ) ∈ 2L(AF)
. ♠
Definition 32. Given an argumentation framework with input15
(AF,I ,LI ,RI ), the standard argumentation framework w.r.t.
(AF,I ,LI ,RI ) is defined as AF = (A ∪ I ,→ ∪R I ), where I = I ∪
{A | A ∈ out(LI )} and R I = RI ∪ {(A , A) | A ∈ out(LI )} ∪ {(A, A) | A ∈
undec(LI )}. ♠
Definition 33. Given a semantics σ, the canonical local function of σ20
(also called local function of σ) is defined as Fσ(AF,I ,LI ,RI ) = {Lab↓A |
Lab ∈ Lσ(AF )}, where AF = (A ,→) and AF is the standard argumenta-
tion framework w.r.t. (AF,I ,LI ,RI ). ♠
Definition 34. A semantics σ is complete-compatible iff the following
conditions hold:25
1. For any argumentation framework AF = (A ,→), every labelling L ∈
Lσ(AF) satisfies the following conditions:
• if A ∈ A is initial, then L(A) = in
• if B ∈ A and there is an initial argument A which attacks B,
then L(B) = out30
• if C ∈ A is self-defeating, and there are no attackers of C be-
sides C itself, then L(C) = undec
2. for any set of arguments I and any labelling LI ∈ LI , the ar-
gumentation framework AF = (I ,→ ), where I = I ∪ {A | A ∈
out(LI )} and → = {(A , A) | A ∈ out(LI )}∪{(A, A) | A ∈ undec(LI )},35
admits a (unique) labelling, i.e. |Lσ(AF )| = 1. ♠

Dung’s AF • Decomposability and Transparancy [Bar+14]
Definition 35. A semantics σ is fully decomposable (or simply decom-
posable) iff there is a local function F such that for every argumenta-
tion framework AF = (A ,→) and every partition P = {P1,...Pn} of A ,
Lσ(AF) = U (P , AF,F) where U (P , AF,F) {LP1 ∪ ... ∪ LPn |
LPi
∈ F(AF↓Pi
,Pi
inp,( j=1···n,j=i LPj
)↓
Pi
inp,Pi
R
)}. ♠5
Definition 36. A complete-compatible semantics σ is top-down decom-
posable iff for any argumentation framework AF = (A ,→) and any parti-
tion P = {P1,...Pn} of A , it holds that Lσ(AF) ⊆ U (P , AF,Fσ). ♠
Definition 37. A complete-compatible semantics σ is bottom-up decom-
posable iff for any argumentation framework AF = (A ,→) and any parti-10
tion P = {P1,...Pn} of A , it holds that Lσ(AF) ⊇ U (P , AF,Fσ). ♠
C O S T GR P R
Full decomposability Yes Yes No No
Top-down decomposability Yes Yes Yes Yes
Bottom-up decomposability Yes Yes No No
Table 1.3: Decomposability properties of argumentation semantics.

2 Argumentation Schemes
Argumentation schemes [WRM08] are reasoning patterns which generate
arguments:
• deductive/inductive inferences that represent forms of common types
of arguments used in everyday discourse, and in special contexts5
(e.g. legal argumentation);
• neither deductive nor inductive, but defeasible, presumptive, or ab-
ductive.
Moreover, an argument satisfying a pattern may not be very strong by
itself, but may be strong enough to provide evidence to warrant rational10
acceptance of its conclusion, given that it premises are acceptable.
According to Toulmin [Tou58] such an argument can be plausible and
thus accepted after a balance of considerations in an investigation or dis-
cussion moved forward as new evidence is being collected. The investiga-
tion can then move ahead, even under conditions of uncertainty and lack15
of knowledge, using the conclusions tentatively accepted.
2.1 An example: Walton et al. ’s Argumentation
Schemes for Practical Reasoning
Suppose I am deliberating with my spouse on what to do
with our pension investment fund — whether to buy stocks,20
bonds or some other type of investments. We consult with a
ﬁnancial adviser, and expert source of information who can
tell us what is happening in the stock market, and so forth at
the present time [Wal97].
Premises for practical inference:25
1. states that an agent (“I” or “my”) has a particular goal;
2. states that an agent has a particular goal.
〈S0,S1,...,Sn〉 represents a sequence of states of affairs that can be
ordered temporally from earlier to latter. A state of affairs is meant to be
like a statement, but one describing some event or occurrence that can30
be brought about by an agent. It may be a human action, or it may be a
natural event.

Argumentation Schemes • AS and Dialogues
Practical Inference
Premises:
Goal Premise Bringing about Sn is my goal
Means Premise In order to bring about Sn, I need to bring
about Si
Conclusions:
Therefore, I need to bring about Si.
Critical questions:
Other-Means
Question
Are there alternative possible actions to
bring about Si that could also lead to the
goal?
Best-Means
Question
Is Si the best (or most favourable) of the
alternatives?
Other-Goals
Question
Do I have goals other than Si whose
achievement is preferable and that
should have priority?
Possibility
Question
Is it possible to bring about Si in the
given circumstances?
Side Effects
Question
Would bringing about Si have known bad
consequences that ought to be taken into
account?
2.2 AS and Dialogues
Dialogue for practical reasoning: all moves (propose, prefer, justify) are co-
ordinated in a formal deliberation dialogue that has eight stages [HMP01].
1. Opening of the deliberation dialogue, and the raising of a governing5
question about what is to be done.
2. Discussion of: (a) the governing question; (b) desirable goals; (c)
any constraints on the possible actions which may be considered;
(d) perspectives by which proposals may be evaluated; and (e) any
premises (facts) relevant to this evaluation.10
3. Suggesting of possible action-options appropriate to the governing
question.
4. Commenting on proposals from various perspectives.
5. Revising of: (a) the governing question, (b) goals, (c) constraints, (d)
perspectives, and/or (e) action-options in the light of the comments15

presented; and the undertaking of any information-gathering or
fact-checking required for resolution.
6. Recommending an option for action, and acceptance or non-accept-
ance of this recommendation by each participant.
7. Confirming acceptance of a recommended option by each partici-5
pant.
8. Closing of the deliberation dialogue.
Proposals are initially made at stage 3, and then evaluated at stages
4, 5 and 6.
Especially at stage 5, much argumentation taking the form of practi-10
cal reasoning would seem to be involved.
As discussed in [Wal06], there are three dialectical adequacy condi-
tions for defining the speech act of making a proposal.
The Proponent’s Requirement (Condition 1). The proponent
puts forward a statement that describes an action and says that15
both proponent and respondent (or the respondent group) should
carry out this action.
The proponent is committed to carrying out that action: the state-
ment has the logical form of the conclusion of a practical inference,
and also expresses an attitude toward that statement.20
The Respondent’s Requirement (Condition 2). The statement
is put forward with the aim of offering reasons of a kind that will
lead the respondent to become committed to it.
The Governing Question Requirement (Condition 3). The job
of the proponent is to overcame doubts or conflicts of opinions, while25
the job of the respondent is to express them. Thus the role of the
respondent is to ask questions that cast the prudential reasonable-
ness of the action in the statement into doubt, and to mount attacks
(counter-arguments and rebuttals) against it.
Condition 3 relates to the global structure of the dialogue, whereas30
conditions 1 and 2 are more localised to the part where the proposal was
made. Condition 3 relates to the global burden of proof [Wal14] and the
roles of the two parties in the dialogue as a whole.
Speech acts [MP02], like making a proposal, are seen as types of
moves in a dialogue that are governed by rules. Three basic character-35
istics of any type of move that have to be defined:
1. pre-conditions of the move;

2. the conditions defining the move itself;
3. the post-conditions that state the result of the move.
Preconditions
• At least two agents (proponent and opponent);
• A governing question;5
• Set of statements (propositions);
• The proponent proposes the proposition to the respondent if and
only if:
1. there is a set of premises that the proponent is committed to,
and fit the premises of the argumentation scheme for practical10
reasoning;
2. the proponent is advocating these premises, that is, he is mak-
ing a claim that they are true or applicable in the case at issue;
3. there is an inference from these premises fitting the argumen-
tation scheme for practical reasoning; and15
4. the proposition is the conclusion of the inference.
The Defining Conditions
The central defining condition sets out the conditions defining the struc-
ture of the move of making a proposal.
The Goal Statement: We have a goal G.20
The Means Statement: Bringing about p is necessary (or suffi-
cient) for us to bring about G.
Then the inference follows.
The Proposal Statement: We should (practically ought to) bring
about p.25

Proposal Statement in form of AS
Premises:
Goal Statement We have a goal G.
The Means
Statement
Bringing about p is necessary (or suffi-
cient) for us to bring about G.
Conclusions:
We should (practically ought to) bring
about p.
The Post-Conditions
The central post-condition is the response condition.
The proposal must be open to critical questioning by opponent. The5
proponent should be open to answering doubts and objections correspond-
ing to any one of the five critical questions for practical reasoning; as well
as to counter-proposals, and is in charge of giving reasons why her pro-
posal is better than the alternatives.
The response condition set by these critical questions helps to explain10
how and why the maker of a proposal needs to be open to questioning and
to requests for justification.

3 A Semantic-Web View of
Argumentation
Acknowledgement
Chris Reed. An overview can also be find at [Bex+13].5
3.1 The Argument Interchange Format [Rah+11]
Node Graph
(argument
network)
has-a
Information
Node
(I-Node)
is-a
Scheme Node
S-Node
has-a
Edge
is-a
Rule of inference
application node
(RA-Node)
Conflict application
node (CA-Node)
Preference
application node
(PA-Node)
Derived concept
application node (e.g.
defeat)
is-a
...
ContextScheme
Conflict
scheme
contained-in
Rule of inference
scheme
Logical inference
scheme
Presumptive
inference scheme
...
is-a
Logical conflict
scheme
is-a
...
Preference
scheme
Logical preference
scheme
is-a
...
Presumptive
preference scheme
is-a
uses uses uses
Figure 3.1: Original AIF Ontology [Che+06; Rah+11]
3.2 An Ontology of Arguments [Rah+11]
Please download Protégé from http://protege.stanford.edu/ and the
AIF OWL version from http://www.arg.dundee.ac.uk/wp-content/
uploads/AIF.owl10
Representation of the argument described in Figure 3.2
___jobArg : PracticalReasoning_Inference
fulfils(___jobArg, PracticalReasoning_Scheme)
hasGoalPlan_Premise(___jobArg, ___jobArgGoalPlan)
hasConclusion(___jobArg, ___jobArgConclusion)15
hasGoal_Premise(___jobArg, ___jobArgGoal)
___jobArgConclusion : EncouragedAction_Statement
fulfils(___jobArgConclusion, EncouragedAction_Desc)

Semantic Web Argumentation • AIF-OWL
Practical
Inference
Bringing about
is my goal
Sn
Si
In order to bring about
I need to bring about
Sn
Therefore I need
to bring about Si
hasConcDeschasPremiseDesc
hasPremiseDesc
Bringing about being rich
is my goal
In order to bring about being rich
I need to bring about having a job
fulfilsPremiseDesc
fulfilsPremiseDesc
fulfilsScheme
supports
supports
Therefore I need
to bring about
having a job
hasConclusion
fulfils
Figure 3.2: An argument network linking instances of argument and
scheme components
Symmetric attack
r → p
r pMP2
A1
A2
p → q
p
qMP1
neg1
Undercut attack
r MP2
A3
A2 s → v
s
vMP1
cut1
p
r → p
Figure 3.3: Examples of conflicts [Rah+11, Fig. 2]
claimText (___jobArgConclusion "Therefore I need to bring about hav-
ing a job")
___jobArgGoal : Goal_Statement
fulfils(___jobArgGoal, Goal_Desc)
claimText (___jobArgGoal "Bringing about being rich is my goal")5
___jobArgGoalPlan : GoalPlan_Statement
fulfils(___jobArgGoalPlan, GoalPlan_Desc)
claimText (___jobArgGoalPlan "In order to bring about being rich I
need to bring about having a job")

Relevant portion of the AIF ontology
EncouragedAction_Statement
EncouragedAction_Statement Statement
GoalPlan_Statement
GoalPlan_Statement Statement5
Goal_Statement
Goal_Statement Statement
I-node
I-node ≡ Statement
I-node Node10
I-node ¬ S-node
Inference
Inference ≡ RA-node
Inference ∃ fulﬁls Inference_Scheme
Inference ≥ 1 hasPremise Statement15
Inference Scheme_Application
Inference = hasConclusion (Scheme_Application Statement)
Inference_Scheme
Inference_Scheme Scheme ≥
1 hasPremise_Desc Statement_Description = hasConclusion_Desc20
(Scheme Statement_Description)
PracticalReasoning_Inference
PracticalReasoning_Inference ≡ Presumptive_Inference ∃ hasCon-
clusion EncouragedAction_Statement ∃ hasGoalPlan_Premise Goal-
Plan_Statement ∃ hasGoal_Premise Goal_Statement25
RA-node
RA-node ≡ Inference
RA-node S-node
S-node
S-node ≡ Scheme_Application30
S-node Node
S-node ¬ I-node

Scheme
Scheme Form
Scheme ¬ Statement_Description
Scheme_Application
Scheme_Application ≡ S-node5
Scheme_Application ∃ fulfils Scheme
Scheme_Application Thing
Scheme_Application ¬ Statement
Statement
Statement ≡ NegStatement10
Statement ≡ I-node
Statement Thing
Statement ∃ fulfils Statement_Description
Statement ¬ Scheme_Application
Statement_Description15
Statement_Description Form
Statement_Description ¬ Scheme
fulfils
∃ fulfils Thing Node
hasConclusion_Desc20
∃ hasConclusion_Desc Thing Inference_Scheme
hasGoalPlan_Premise
hasPremise
hasGoal_Premise
hasPremise25
claimText
∃ claimText DatatypeLiteral Statement
∀ claimText DatatypeString
Individuals of EncouragedAction_Desc
EncouragedAction_Desc : Statement_Description30
formDescription (EncouragedAction_Desc "A should be brought about")

Individuals of GoalPlan_Desc
GoalPlan_Desc : Statement_Description
formDescription (GoalPlan_Desc "Bringing about B is the way to bring
about A")
Individuals of Goal_Desc5
Goal_Desc : Statement_Description
formDescription (Goal_Desc "The goal is to bring about A")
Individuals of PracticalReasoning_Scheme
PracticalReasoning_Scheme : PresumptiveInference_Scheme
hasPremise_Desc(PracticalReasoning_Scheme, Goal_Desc)10
hasConclusion_Desc(PracticalReasoning_Scheme, EncouragedAction_Desc)
hasPremise_Desc(PracticalReasoning_Scheme, GoalPlan_Desc)

4 Argumentation Frameworks:
Graphs and Models
Acknowledgement
(in alphabetic order):5
• Pietro Baroni;
• Trevor J. M. Bench-Capon;
• Claudette Cayrol;
• Paul E. Dunne;
• Anthony Hunter;10
• Hengfei Li;
• Sanjay Modgil;
• Nir Oren;
• Guillermo R. Simari.
4.1 Graphs15
Value-Based Argumentation Framework [BA09]
Example 1 ([AB08], derived from [Col92; Chr00]). The situation involves
two agents, called Hal and Carla, both of whom are diabetic. Hal, through
no fault of his own, has lost his supply of insulin and urgently needs to
take some to stay alive. Hal is aware that Carla has some insulin kept in20
her house, but Hal does not have permission to enter Carla’s house. The
question is whether Hal is justiﬁed in breaking into Carla’s house and
taking her insulin in order to save his life. Note that by taking Carla’s in-
sulin, Hal may be putting her life in jeopardy, since she will come to need
that insulin herself. One possible response is that if Hal has money, he25
can compensate Carla so that her insulin can be replaced before she needs
it. Alternatively if Hal has no money but Carla does, she can replace her
insulin herself, since her need is not immediately life threatening. There
is, however, a serious problem if neither of them have money, since in that
case Carla’s life is really under threat.30
Partial formalisation:

Frameworks • Graphs
a2
LC, FC
a3
LC, FH
a1
LC
Figure 4.1: Graphical representation of Ex. 1.
• a1 suggests that Hal should not take insulin, thus allowing Carla
to be alive (which promotes the value of Life for Carla LC);
• a2 suggests that Hal should take insulin and compensate Carla,
thus both of them stay alive (which promotes the value of Life for
Carla, and the Freedom — of using money — for Carla FC);5
• a3 suggests that Hal should take insulin and that Carla should buy
insulin, thus both of them stay alive (which promotes the value of
Life for Carla, and the Freedom — of using money — for Hal FH).
a2 defeats a1, a3 defeats a1, a3 and a2 defeat each other. ♥
Extended Argumentation Framework [Mod09]10
Example 2 (From [Mod09]).
• a1: “Today will be dry in London since the BBC forecast sunshine”;
• a2: “Today will be wet in London since CNN forecast rain”;
• a3: “But the BBC are more trustworthy than CNN”;
• a4: “However, statistically CNN are more accurate forecasters than15
the BBC”;
• a5: “Basing a comparison on statistics is more rigorous and ratio-
nal than basing a comparison on your instincts about their relative
trustworthiness”.
a1 and a2 are mutually conﬂicting; a3 is a preference in favour of a1,20
a4 is a preference in favour of a2. a3 and a4 are mutually conﬂicting. a5
is a preference in favour of a4. ♥

AFRA: Argumentation Framework with Recursive Attacks
[Bar+11; Bar+09]
Example 3 ([Bar+11; Bar+09]). Suppose Bob is deciding about his Christ-
mas holidays.
• a1: There is a last minute offer for Gstaad: therefore I should go to5
Gstaad;
• a2: There is a last minute offer for Cuba: therefore I should go to
Cuba;
• a3: I do like to ski;
• a4: The weather report informs that in Gstaad there were no snow-10
falls since one month: therefore it is not possible to ski in Gstaad;
• a5: It is anyway possible to ski in Gstaad, thanks to a good amount
of artificial snow. ♥
Definition 38 (AFRA). An Argumentation Framework with Recursive
Attacks (AFRA) is a pair 〈A ,R〉 where:15
• A is a set of arguments;
• R is a set of attacks, namely pairs (a1,X ) s.t. a1 ∈ A and (X ∈ R
or X ∈ A ).
Given an attack α = (a1,X ) ∈ R, we say that a1 is the source of α, denoted
as src(α) = a1 and X is the target of α, denoted as trg(α) = X .20
When useful, we will denote an attack to attack explicitly showing
all the recursive steps implied by its definition; for instance (a1,(a2,a3))
means (a1,α) where α = (a2,a3). ♠

Definition 39 (Semantics). Let Γ = 〈A ,R〉 be an AFRA. A set S ⊆ A ∪R
is:
• a complete extension if and only if S is admissible and every el-
ement of A ∪ R which is acceptable w.r.t. S belongs to S , i.e.
FΓ(S ) ⊆ S ;5
• the grounded extension of Γ iff is the least fixed point of FΓ;
• a preferred extension of Γ iff it is a maximal (w.r.t. set inclusion)
admissible set;
• a stable extension of Γ if and only if S is conflict-free and ∀V ∈
A ∪R,V ∉ S , ∃α ∈ S s.t. α →R V . ♠10
Theorem 3.

In the case where an AFRA is also an AF, a bijective corre-
spondence between the semantics notions according to the two formalisms
hold. ♣
Theorem 4.

Moreover, in the case where an AFRA is not an AF, it is
possible to rewrite it as an AF with extra arguments. ♣15
Bipolar Argumentation Framework [CL05]
Example 4 ([CL05, Example 1]). A murder has been performed and the
suspects are Liz, Mary and Peter. The following pieces of information
have been gathered:
• The type of murder suggests us that the killer is a female (f );20
• The killer is certainly small (s);
• Liz is tall and Mary and Peter are small;
• The killer has long hair and uses a lipstick (l);
• A witness claims that he saw the killer who was tall;
• The witness is reliable (w);25
• Moreover we are told that the witness is short-sighted, so he is no
more reliable (b).
The following arguments can be formed:
• a1 in favour of m, with premises {s, f ,(s∧ f ) → m};
• a2 in favour of ¬s, with premises {w,w → ¬s};30
• a3 in favour of ¬w, with premises {b,b → ¬w};
• a4 in favour of f , with premises {l,l → f }

Frameworks • Deterministic Structured Argumentation
a3 a2 a1
a4
Figure 4.4: Graphical representation of Ex. 4: rounded arrows represent
the support relationship.
a3 defeats a2; a2 defeats a1. But, the argument a4 confirms one of the
premises of a1, thus strengthening it. ♥
4.2 Deterministic Structured Argumentation
Defeasible Logic Programming (DeLP) [Sim89; SL92; GS04;
GS14]5
A defeasible logic program (DeLP) is a set of:
• facts, i.e. ground literals representing atomic information or the
negation of atomic information using strong negation ¬;
• strict rules, Lo ←− L1,...,Ln, represent non-defeasible information.
Lo is the head, the body {Li}i>0 is a non-empty set of ground literals;10
• defeasible rules, Lo −< L1,...,Ln, represent tentative information.
Lo is the head, the body {Li}i>0 is a non-empty set of ground literals.
A DeLP program is denoted by 〈Π,∆〉, where Π is the subset of non-
defeasible knowledge (strict rules and facts); and ∆ is the subset of defea-
sible knowledge.15
A defeasible derivation of a literal Q from a DeLP program 〈Π,∆〉 |∼ Q,
is a finite sequence of ground literals L1,L2,...,Ln = Q where either:
1. Li is a fact;
2. there exists a rule Ri in 〈Π,∆〉 (either strict or defeasible) with head
Li and body B1,...,Bk, and every literal of the body is an element20
L j of the sequence appearing before Li (j < i).
A derivation from 〈Π, 〉 is called a strict derivation.
Definition 40. Let H be a ground literal, 〈Π,∆〉 a DeLP program, and
A ⊆ ∆. The pair 〈A ,H〉 is an argument structure if:
• there exists a defeasible derivation for H from 〈Π,A 〉;25
• there are no defeasible derivations from 〈Π,A 〉 of contradictory lit-
erals;

• and there is no proper subset A ⊂ A such that A satisfies (1) and
(2). ♠
Definition 41. An argument 〈B,S〉 is a counter-argument for 〈A ,H〉 at
literal P, if there exists a sub-argument 〈C ,P〉 of 〈A ,H〉 such that P
and S disagree, that is, there exist two contradictory literals that have a5
strict derivation from Π∪{S,P}. The literal P is referred as the counter-
argument point and 〈C ,P〉 as the disagreement sub-argument. ♠
Let assume an argument comparison criterion .
Definition 42. Let 〈B,S〉 be a counter-argument for 〈A ,H〉 at point P,
and 〈C ,P〉 the disagreement sub-argument.10
If 〈B,S〉 〈C ,P〉, then 〈B,S〉 is a proper defeater for 〈A ,H〉.
If 〈B,S〉 〈C ,P〉 and 〈C ,P〉 〈B,S〉, then 〈B,S〉 is a blocking de-
feater for 〈A ,H〉.
〈B,S〉 is a defeater for 〈A ,H〉 if 〈B,S〉 is either a proper or blocking
defeater for 〈A ,H〉. ♠15
Example 5. Let 〈Π1,∆1〉 be a DeLP-program such that:
Π1 =



monday
cloudy
dry_season
waves
grass_grown
hire_gardener
vacation
¬working ←− vacation
few_surfers ←− ¬many_surfers
¬surf ←− ill



∆1 =



surf −< nice,spare_time
nice −< waves
spare_time −< ¬busy
¬busy −< ¬working
¬nice −< rain
rain −< cloudy
¬rain −< dry_season
...



From 〈Π1,∆1〉, these are some arguments that can be derived:
〈A0,surf〉 =



surf −< nice,spare_time
nice −< waves
spare_time −< ¬busy
¬busy −< ¬working



,surf
〈A1,¬nice〉 = 〈{¬nice −< rain; rain −< cloudy},¬nice〉20
〈A2,nice〉 = 〈{nice −< waves},nice〉
〈A3,rain〉 = 〈{rain −< cloudy},rain〉
〈A4,¬rain〉 = 〈{¬rain −< dry_season},¬rain〉

Figure 4.5: Arguments and their interactions from Example 5
〈A9,¬busy〉 = 〈{¬busy −< ¬working},¬busy〉
♥
Assumption Based Argumentation (ABA) [BTK93; Bon+97;
Ton12; Ton14; DT10]
Definition 43. An ABA is a tuple 〈L ,R,A , 〉 where:5
• 〈L ,R〉 is a deductive system, with L the language and R a set of
rules, that we assume of the form σ0 ←− σ1,...,σm (m ≥ 0), with
σi ∈ L ; σ0 is referred to as the head and σ1,...,σm as the body of
the rule σ0 ←− σ1,...,σm;
• A ⊆ L is a (non-empty) set, referred to as assumptions;10
• is a total mapping from A into L ; a is referred to as the contrary
of a.
♠
Definition 44. A deduction for σ ∈ L supported by S ⊆ L and R ⊆ R,
denoted as S
R
σ, is a (finite) tree with nodes labelled by sentences in15
L or by τ ∉ L , the root labelled by σ, leaves either τ or sentences in S,
non-leaves σ with, as children, the elements of the body of some rule in
R with head σ , and R the set of all such rules. ♠
Definition 45. An argument for the claim σ ∈ L supported by A ⊆ A
(A σ) is a deduction for σ supported by A (and some R ⊆ R). ♠20
Definition 46. An argument A1 σ1 attacks an argument A2 σ2 iff σ1
is the contrary of one of the assumptions in A2. ♠

Example 6.
R = { innocent(X) ←− notGuilty(X);
killer(oj) ←− DNAshows(oj),DNAshows(X) ⊃ killer(X);
DNAshows(X) ⊃ killer(X) ←− DNAfromReliableEvidence(X);
evidenceUnreliable(X) ←− collected(X,Y ),racist(Y );
DNAshows(oj) ←−;
collected(oj,mary) ←−;
racist(mary) ←− }
A = { notGuilty(oj);
DNAfromReliableEvidence(oj) }
Moreover, notGuilty(oj) = killer(oj), and
DNAfromReliableEvidence(oj) = evidenceUnreliale(oj). ♥5
ASPIC+ [Pra10; MP13; MP14]
Given a logical language L , and a set of strict or defeasible inference
rules — resp. ϕ1,...,ϕn −→ ϕ and ϕ1,...,ϕn =⇒ ϕ. A strict rule inference
always holds — i.e. if the antecedents ϕ1,...,ϕn hold, the consequent ϕ
holds as well — while a defeasible inference “usually” holds. Arguments10
are constructed w.r.t. a knowledge base with two types of formulae.
Deﬁnition 47. An argumentation system is as tuple AS = 〈L ,R,ν〉 where:
• : L → 2L
is a contrariness function s.t. if ϕ ∈ ψ and:
– ψ ∉ ϕ, then ϕ is a contrary of ψ;
– ψ ∈ ϕ, then ϕ is a contradictory of ψ (ϕ = –ψ);15
• R = Rd ∪ Rs is a set of strict (Rs) and defeasible (Rd) inference
rules such that Rd ∩Rs = ;
• ν : Rd → L , is a partial function.1
1Informally, ν(r) is a wff in L which says that the defeasible rule r is applicable.

For any P ⊆ L , Cl(P ) denotes the closure of P under strict rules, viz. the
smallest set containing P and any consequent of any consequent of any
strict rule in Rs whose antecedents are in Cl(P ).
P ⊆ L is consistent iff ϕ,ψ ∈ P s.t. ϕ ∈ ψ, otherwise is inconsistent.
A knowledge base in an AS is a set Kn ∪ Kp = K ⊆ L ; {Kn,Kp} is a5
partition of K ; Kn contains axioms that cannot be attacked; Kp contains
ordinary premises that can be attacked.
An argumentation theory is a pair AT = 〈AS,K 〉. ♠
Deﬁnition 48.

An argument a on the basis of a AT = 〈AS,K 〉, AS =
〈L ,R,ν〉 is:10
1. ϕ if ϕ ∈ K with: Prem(a) = {ϕ}; Conc(a) = ϕ; Sub(a) = {ϕ}; Rules(a) =
DefRules(a) = ; TopRule(a) = undefined.
2. a1,...,an −→ / =⇒ ψ if a1,...,an, with n ≥ 0, are arguments such
that there exists a strict/defeasible rule r = Conc(a1),...,Conc(an) −→
/ =⇒ ψ ∈ Rs/Rd.15
Prem(a) = n
i=1
Prem(ai); Conc(a) = ψ;
Sub(a) = n
i=1
Sub(ai)∪{a};
Rules(a) = n
i=1
Rules(ai)∪{r};
DefRules(a) = {d | d ∈ Rules(a)∩Rd};
TopRule(a) = r20
a is strict if DefRules(a) = , otherwise defeasible; firm if Prem(a) ⊆ Kn,
otherwise plausible.
P A ϕ iff ∃a strict argument s.t. Conc(a) = ϕ and P ⊇ Prem(a). ♠
An argument can be attacked in its premises (undermining), conclu-
sion (rebuttal), or inference step (undercut). The definition of defeats25
takes into account an argument ordering : a b iff a is “less preferred”
than b (a b iff a b and b a).
Definition 49.

Given a and b arguments, a defeats b iff a undercuts,
successfully rebuts or successfully undermines b, where:
• a undercuts b (on b ) iff Conc(a) ∉ ν(r) for some b ∈ Sub(b) s.t. r =30
TopRule(b ) ∈ Rd;
• a successfully rebuts b (on b ) iff Conc(a) ∉ ϕ for some b ∈ Sub(b) of
the form b1,...,bn =⇒ –ϕ, and a b ;
• a successfully undermines b (on ϕ) iff Conc(a) ∉ ϕ, and ϕ ∈ Prem(b)∩
Kp, and a ϕ. ♠35

Definition 50. AF is the abstract argumentation framework defined by
AT = 〈AS,K 〉, AS = 〈L ,R,ν〉 if A is the smallest set of all finite argu-
ments constructed from K satisfying Def. 48; and → is the defeat relation
on A as defined in Def. 49. ♠
Definition 51 (Rationality postulates [CA07; MP14]). Given ∆, an AF5
defined by an AT, and a semantic σ. ∀S ∈ E∆(σ), ∆ satisfies :
P1: direct consistency iff {Conc(a) | a ∈ S} is consistent;
P2: indirect consistency iff Cl({Conc(a) | a ∈ S}) is consistent;
P3: closure iff {Conc(a) | a ∈ S} = Cl({Conc(a) | a ∈ S});
P4 : sub-argument closure iff ∀a ∈ S, Sub(a) ⊆ S. ♠10
Note that P2 follows from P1 and P3.
An AT satisfies the postulates (i.e. it is Well-Formed) iff (let us con-
sider classical negation here instead of contrariness function) [MP13; MP14]:
• it is close under transposition2
or under contraposition;3
• Cl(Kn) is consistent;15
• the argument ordering is reasonable, namely:
– ∀a,b, if a is strict and firm, and b is plausible or defeasible,
then a b;
– ∀a,b, if b is strict and firm, then b a;
– ∀a,a ,b such that a is a strict continuation of {a}, if a b20
then a b, and if b a, then b a ;
– given a finite set of arguments {a1,...,an}, let a+i
be some
strict continuation of {a1,...,ai−1,ai+1,...,an}. Then it is not
the case that ∀i,a+i
ai.
An argument a is a strict continuation of a set of arguments {a1,...,an}25
iff (Prem(a)∩Kp) = n
i=1
(Prem(ai)∩Kp); DefRules(a) = n
i=1
DefRules(ai);
Rules(a) ⊇ n
i=1
Rules(ai) and (Prem(a)∩Kn) ⊆ n
i=1
(Prem(ai∩Kn)).
Example 7. It is well known that (1) birds normally fly; while (2) pen-
guins are known not to fly, although (3) all penguins are birds. In these
terms, one can say that (4) penguins are abnormal birds with respect to30
flying. (5) Tweety is observed to be a penguin, and (6) animals that are
observed to be penguins normally are penguins.
d1 : bird =⇒ canfly; d2 : penguin =⇒ ¬canfly; d3 : observed_penguin =⇒
penguin; f1 : penguin ⊃ bird; f2 : penguin ⊃ ¬d1; f3 : observed_penguin. The
2If ϕ1,...,ϕn −→ ψ ∈ Rs, then ∀i = 1...n, ϕ1,...,ϕi−1,¬ψ,ϕi+1,...,ϕn =⇒ ¬ϕi ∈ Rs.
3∀P ⊆ L , l ∈ P , if P A ϕ, then P {l}∪{¬ϕ} A ¬l

derived arguments are: a1 : observed_penguin; a2 : a1 =⇒ penguin; a3 :
penguin ⊃ bird; a4 : a2,a3 =⇒ canfly; b1 : a2 =⇒ ¬canfly; c1 : a2 =⇒ ¬ν(d1).
♥
Deductive Argumentation [BH01; BH08; GH11; BH14]
Focus on simple logic and classical logic, but other options include5
non-monotonic logics, conditional logics, temporal logics, description log-
ics, and paraconsistent logics.
Definition 52 (Base Logic). Let L be a language for a logic, and let i
be the consequence relation for that logic. If α is an atom in L , then α is
a positive literal in L and ¬α is a negative literal in L .10
For a literal β, the complement of β is defined as follows:
• If β is a positive literal, i.e. it is of the form α, then the complement
of β is the negative literal ¬α,
• if β is a negative literal, i.e. it is of the form ¬α, then the comple-
ment of β is the positive literal α. ♠15
Definition 53 (Deductive Argument). A deductive argument is an or-
dered pair 〈Φ,α〉 where Φ i α. Φ is the support, or premises, or assump-
tions of the argument, and α is the claim, or conclusion, of the argument.
For an argument a = 〈Φ,α〉, the function Support(a) returns Φ and the
function Claim(a) returns α. ♠20
Definition 54 (Constraints). An argument 〈Φ,α〉 satisfies the:
• consistency constraint when Φ is consistent (not essential, cf.
paraconsistent logic).
• minimality constraint when there is no Ψ ⊂ Φ such that Ψ α.
♠25
Definition 55 (Classical Logic Argument). A classical logic argument
from a set of formulae ∆ is a pair 〈Φ,α〉 such that
1. Φ ⊆ ∆
2. Φ ⊥
3. Φ α30
4. there is no Φ ⊂ Φ such that Φ α. ♠
Definition 56 (Counterargument). If 〈Φ,α〉 and 〈Ψ,β〉 are arguments,
then
• 〈Φ,α〉 rebuts 〈Ψ,β〉 iff α ¬β

• 〈Φ,α〉 undercuts 〈Ψ,β〉 iff α ¬∧Ψ ♠
Deﬁnition 57 (Direct undercut). Let a and b be two classical arguments.
We deﬁne the following types of classical attack.
• a is a direct undercut of b if ¬Claim(a) ∈ Support(b)
• a is a classical defeater of b if Claim(a) ¬ Support(b).5
• a is a classical direct defeater of b if ∃φ ∈ Support(b) s.t. Claim(a)
¬φ
• a is a classical undercut of b if ∃Ψ ⊆ Support(b) s.t. Claim(a) ≡
¬ Ψ
• a is a classical direct undercut of b if ∃φ ∈ Support(b) s.t. Claim(a) ≡10
¬φ
• a is a classical canonical undercut of b if Claim(a) ≡ ¬ Support(b).
• a is a classical rebuttal of b if Claim(a) ≡ ¬Claim(b).
• a is a classical defeating rebuttal of b if Claim(a) ¬Claim(b).
♠15
An arrow from D1 to D2 indicates that D1 ⊆ D2.
Defeater
Direct defeat Undercut Direct rebut
Direct undercut
Canonical
undercut
Rebut

bp(high)
ok(diuretic)
bp(high) ∧ok(diuretic)
→ give(diuretic)
¬ok(diuretic) ∨¬ok(betablocker)
give(diuretic) ∧¬ok(betablocker)
bp(high)
ok(betablocker)
bp(high) ∧ok(betablocker)
→ give(betablocker)
¬ok(diuretic) ∨¬ok(betablocker)
give(betablocker) ∧¬ok(diuretic)
symptom(emphysema),
symptom(emphysema) → ¬ok(betablocker)
¬ok(betablocker)
Figure 4.7: Example of argumentation with classical logic.
A Logic for Clinical Knowledge [GHW09; HW12; Wil+15]
Evidence on
treatments
T1 and T2
Inference rules for
inductive arguments
and meta-arguments
Arguments
Preferences on
outcomes and
their magnitude
Argument
graph
(T1 > T2) or (T1 = T2) or (T1 < T2)
Let us assume a set of evidence EVIDENCE = {e1,..., en}.
Deﬁnition 58 (Inductive Arguments). Given treatments τ1 and τ2, X ⊆
EVIDENCE, there are three kinds of inductive argument that can be formed.5
1. 〈X,τ1 > τ2〉, meaning the evidence in X supports the claim that
treatment τ1 is superior to τ2.
2. 〈X,τ1 ∼ τ2〉, meaning the evidence in X supports the claim that
treatment τ1 is equivalent to τ2
3. 〈X,τ1 < τ2〉, meaning the evidence in X supports the claim that10
treatment τ1 is inferior to τ2.

Frameworks • Probabilistic Argumentation
♠
Given an inductive argument a = 〈X, 〉, support(a) = X.
ARG(EVIDENCE) denotes the set of inductive arguments that can be
generated from the evidence in EVIDENCE.
Definition 59 (Conflicts). If the claim of argument ai is i and the claim5
of argument aj is j then we say that ai conflicts with aj whenever:
1. i = τ1 > τ2, and ( j = τ1 ∼ τ2 or j = τ1 < τ2 ).
2. i = τ1 ∼ τ2, and ( j = τ1 > τ2 or j = τ1 < τ2 ).
3. i = τ1 < τ2, and ( j = τ1 > τ2 or j = τ1 ∼ τ2 ). ♠
Definition 60 (Attack). For any pair of arguments ai and aj, and a pref-10
erence relation R, ai attacks aj with respect to R iff ai conflicts with aj
and it is not the case that aj is strictly preferred to ai according to R. ♠
A domain-specific benefit preference relation is defined in [HW12].
Definition 61 (Meta-Arguments). For a ∈ ARG(EVIDENCE), if there is an
e ∈ SUPPORT(a) such that:15
• e is not statistically significant, and the outcome indicator of e is not
a side-effect, then the following is a meta-argument that attacks a:
〈Not statistically significant〉;
• e is a non-randomised and non-blind trial, then the following is
a meta-argument that attacks a: 〈Non-randomized & non-blind20
trials〉;
• e is a meta-analysis that concerns a narrow patient group then the
following is a meta-argument that attacks a: 〈Meta-analysis for
a narrow patient group〉. ♠
Example 8. Example where CP is contraceptive pill and NT is no treat-25
ment. Fictional data.
ID Left Right Indicator Risk ratio Outcome p
e1 CP NT Pregnancy 0.05 superior 0.01
e2 CP NT Ovarian cancer 0.99 superior 0.07
e3 CP NT Breast cancer 1.04 inferior 0.01
e4 CP NT DVT 1.02 inferior 0.05
♥

〈{e1},CP > NT〉
〈{e2},CP > NT〉
〈{e1, e2},CP > NT〉
〈{e3},CP < NT〉
〈{e4},CP < NT〉
〈{e3, e4},CP < NT〉
〈Notstatistically
significant〉
Figure 4.8: Arguments derived from Ex. 8, with preferences and meta
arguments.
4.3 Probabilistic Argumentation
Epistemic Approach [Thi12; Hun13; HT14; BGV14]
Deﬁnition 62. Probability distribution over models of the language M
A function P : M → [0,1] such that
m∈M
P(m) = 15
♠
Deﬁnition 63. Probability of a formula φ, cf. [Par94]
P(φ) =
m∈Models(φ)
P(m)
♠
Example 9.10
Model a b P
m1 true true 0.8
m2 true false 0.2
m3 false true 0.0
m4 false false 0.0
• P(a) = 1
• P(a∧ b) = 0.8
• P(b ∨¬b) = 1
• P(¬a∨¬b) = 0.215
♥

Definition 64. Probability of an argument The probability of an argu-
ment 〈Φ,α〉, denoted P(〈Φ,α〉), is P(φ1∧...∧φn), where Φ = {φ, ...,φn}. ♠
Example 10. Consider the following probability distributions over mod-
els
Model a b Agent 1 Agent 2
m1 true true 0.5 0.0
m2 true false 0.5 0.0
m3 false true 0.0 0.6
m4 false false 0.0 0.4
5
Below is the probability of each argument according to each participant.
Argument Agent 1 Agent 2
a1 = 〈{a},a〉 1.0 0.0
a2 = 〈{b,b → ¬a},¬a〉 0.0 0.6
a3 = 〈{¬b},¬b〉 0.5 0.4
♥
Definition 65. For an argumentation framework AF = 〈A ,→〉 and a
probability assignment P, the epistemic extension is10
{a ∈ A | P(a) > 0.5}
♠
Definition 66 (From [Thi12; Hun13; BGV14]). Given an argumentation
framework 〈A ,→〉, a probability function:
COH P is coherent if for every a,b ∈ A , if a attacks b then P(a) ≤ 1−P(b).15
SFOU P is semi-founded if P(a) ≥ 0.5 for every unattacked a ∈ A .
FOU P is foundedif P(a) = 1 for every unattacked a ∈ A .
SOPT P is semi-optimistic if P(a) ≥ 1− b∈a− P(b) for every a ∈ A with at
least one attacker.
OPT P is optimistic if P(a) ≥ 1− b∈a− P(b) for every a ∈ A .20
JUS P is justifiableif P is coherent and optimistic.
TER P is ternary if P(a) ∈ {0,0.5,1} for every a ∈ A .
RAT P is rational if for every a,b ∈ A , if a attacks b then P(a) > 0.5
implies P(b) ≤ 0.5.
NEU P is neutral if P(a) = 0.5 for every a ∈ A .25

Frameworks • Structural Approach [Hun14]
INV P is involutary if for every a,b ∈ A , if a attacks b, then P(a) =
1− P(b).
Let the event “a is accepted” be denoted as a, and let be Eac(S) =
{a|a ∈ S}. Then P is weakly p-justifiable iff ∀a ∈ A , ∀b ∈ a−
, P(a) ≤ 1 −
P(b). ♠5
Proposition 3 ([BGV14]). For every argumentation framework, there is
at least one P that it is de Finetti coherent [Fin74] and weakly p-justifiable.
♣
Definition 67. Correspondences between probabilistic and classical se-
mantics10
Restriction on complete probability function P Classical semantics
No restriction complete extensions
No arguments a such that P(a) = 0.5 stable
Maximal no. of a such that P(a) = 1 preferred
Maximal no. of a such that P(a) = 0 preferred
Maximal no. of a such that P(a) = 0.5 grounded
Minimal no. of a such that P(a) = 1 grounded
Minimal no. of a such that P(a) = 0 grounded
Minimal no. of a such that P(a) = 0.5 semi-stable
♠
4.4 Structural Approach [Hun14]
Definition 68. Subframework For G = 〈A ,→〉 and G = 〈A ,→ 〉,
G G iff A ⊆ A and → = {〈a,b〉 ∈→| a,b ∈ A }15
♠
Definition 69. Graphs giving an extension For an argument framework
G = 〈A ,→〉, a set of arguments Γ ⊆ A , and a semantics σ,
QX (Γ) = {G G | G σ Γ}
where G σ Γ denotes that Γ is an σ extension of G . ♠20
Definition 70. Probability of a set being an extension The probability
that a set of arguments Γ is an σ extension, denoted Pσ(Γ), is
PX (Γ) =
G ∈Qσ(Γ)
P(G )
where P is a probability distribution over subframeworks of G. ♠

Frameworks • A Computational Framework
Example 11.
Subframework Probability
G1 a ↔ b 0.09
G2 a 0.81
G3 b 0.01
G4 0.09
PGR({a,b}) = = 0.00
PGR({a}) = P(G2) = 0.81
PGR({b}) = P(G3) = 0.01
PGR({}) = P(G1)+ P(G4) = 0.18
♥
4.5 A Computational Framework5
Deﬁnition 71 ([LON12; Li15]). A Li-PAF is a tuple 〈A ,PA ,→,P→〉, where
〈A ,→〉 is an argumentation framework, PA : A → (0..1] and P→ :→→
(0..1]. ♠
Deﬁnition 72 ([LON12; Li15]). Given a Li-PAF 〈A ,PA ,→,P→〉, AFI
=
〈A I
,→I
〉 is said to be induced iff A I
⊆ A ; and →I
⊆→ ∩(A T
× A T
); and10
∀a ∈ A s.t. PA (a) = 1,a ∈ A I
; and ∀〈a,b〉 ∈→ where P(a) = P(b) = 1 if
P→(〈a,b〉) = 1, then 〈a,b〉 ∈→I
. ♠
Under an assumption of independence, the probability of an inducible
∆I
= 〈A I
,→I
〉, denoted PI
PrAF
(∆I
), by the following equation:
PI
PrAF
(∆I
) = a∈A I PA (a) a∈A A I (1− PA (a)) 〈a,b〉∈→I P→(〈a,b〉)
〈a,b〉∈(→∪(A I ×A I ))→I (1− P→(〈a,b〉))
Assumption relaxed in [LON13; Li15] by relying on a bipolar argu-15
mentation framework, i.e. the evidential argumentation framework [ON08].
A correspondence with ASPIC+
is also drawn in [Li15], see Figure 4.9.

Frameworks • A Computational Framework
Convert to
ASPIC+ Argumentation
System
• Logical Language
• Inference Rules
• Contrariness Function
• ......
Structured
Argumentation
Framework
(SAF)
DAF
DAFEAF
Extended
Evidential
Framework
(EEAF)
Probabilistic
Extended
Evidential
Framework
Convert to
Convert to
Extended
Evidential
Framework
(EEAF)
Model
Probabilistic
Extended
Evidential
Framework
Associate
Probabilities
Convert toPrEAF
Associate
Probabilities
Semantics
Preserved
PrAF
Associate
Probabilities
Figure 4.9: [Li15]’s probabilistic argumentation architecture.

5 A novel synthesis: Collaborative
Intelligence Spaces (CISpaces)
Acknowledgement
Alice Toniolo and Timothy J. Norman. Main reference: [Ton+15].5
5.1 Introduction
Problem
• Intelligence analysis is critical for making well-informed decisions
• Complexities in current military operations increase the amount of
information available to intelligence analysts10
CISpaces (Collaborative Intelligence Spaces)
• A toolkit developed to support collaborative intelligence analysis
• CISpaces aims to improve situational understanding of evolving sit-
uations
5.2 Intelligence Analysis15
Deﬁnition 73 ([DCD11]). The directed and coordinated acquisition and
analysis of information to assess capabilities, intent and opportunities for
exploitation by leaders at all levels. ♠
Fig. 5.1 summarises the Pirolli and Card Model [PC05].
Table 5.1 illustrates the problems of individual analysis and how col-20
laborative analysis can improve it.

CISpaces • Intelligence Analysis
External
Data
Sources
Presentation
Search
and Filter
Schematize
Build Case
Tell Story
Reevaluate
Search
for support
Search
for evidence
Search for
information
FORAGING LOOP
SENSE-MAKING LOOP
Structure
Effort
inf
Shoebox
Ev
Ev
EvEv Ev
Ev
Ev
Ev
Ev
Ev
Ev
Evidence File
Hyp1 Hyp2
Hypotheses
Pirolli & Card Model
Figure 5.1: The Pirolli & Card Model [PC05]
Individual analysis Collaborative analysis
• Scattered Information &
Noise
• Hard to make connections
• Missing Information
• Cognitive biases
• Missing Expertise
• More effective and reliable
• Brings together different
expertise, resources
• Prevent biases
Table 5.1: Individual vs. Collaborative Analysis

CISpaces • Intelligence Analysis
Harbour
Kish Farm
KISH
River
Water pipe
Aqueduct
KISHSHIRE
Kish Hall
Hotel
Illness among young and
elderly people in
Kishshire caused by
bacteria
Unidentified illness is
affecting the local
livestock in Kishshire,
the rural area of Kish
Figure 5.2: Initial information assigned to Joe
PEOPLE and
LIVESTOCK
illness
Water TEST
shows a
BACTERIA in the
water supply
Answer to POI:
"GER-MAN" seen
in Kish
Explosion in KISH
Hall Hotel
TIME
Tests on people/livestock POI for suspicious people
Figure 5.3: Further events happening in Kish
Example of Intelligence Analysis Process
Goal: discover potential threats in Kish
Analysts: Joe, Miles and Ella
What Joe knows is summarised by Figs. 5.2 and 5.3
Main critical points and possible conclusions during the analysis:5
• Causes of water contamination → waterborne/non-waterborne
bacteria;
• POI responsible for water contamination;
• Causes of hotel explosion.

CISpaces • Reasoning with Evidence
5.3 Reasoning with Evidence
• Identify what to believe happened from the claims constructed upon
information (the sensemaking process);
• Derive conclusions from data aggregated from explicitly requested
information (the crowdsourcing process);5
• Assess what is credible according to the history of data manipula-
tion (the provenance reasoning process).
5.4 Arguments for Sensemaking
Formal Linkage for Semantics Computation
A CISpace graph, WAT, can be transformed into a corresponding ASPIC-10
based argumentation theory. An edge in CISpaces is represented textu-
ally as →, an info/claim node is written pi and a link node is referred to
as type where type = {Pro,Con}. Then, [p1,...,pn → Pro → pφ] indicates
that the Pro-link has p1,..., pn as incoming nodes and an outgoing node
pφ.15
Definition 74. A WAT is a tuple 〈K, AS〉 such that AS= 〈L ,¯,R〉 is con-
structed as follows:
• L is a propositional logic language, and a node corresponds to a
proposition p ∈ L . The WAT set of propositions is Lw.
• The set R is formed by rules ri ∈ R corresponding to Pro links20
between nodes such that: [p1,..., pn → Pro → pφ] is converted to
ri : p1,..., pn ⇒ pφ
• The contrariness function between elements is defined as: i) if [p1 →
Con → p2] and [p2 → Con → p1], p1 and p2 are contradictory; ii)
[p1 → Con → p2] and p1 is the only premise of the Con link, then p125
is a contrary of p2; iii) if [p1, p3 → Con → p2] then a rule is added
such that p1 and p3 form an argument with conclusion ph against
p2, ri : p1, p3 ⇒ ph and ph is a contrary of p2. ♠
Definition 75. K is composed of propositions pi,
K = {pj, pi,...}, such that: i) let a set of rules r1,...,rn ∈ R indicate a cycle30
such that for all pi that are consequents of a rule r exists r containing pi
as antecedent, then pi ∈ K if pi is an info-node; ii) otherwise, pi ∈ K if pi
is not consequent of any rule r ∈ R. ♠

CISpaces • Arguments for Provenance
An Example of Argumentation Schemes for Intelligence
Analysis
Intelligence analysis broadly consists of three components: Activities
(Act) including actions performed by actors, and events happening in the
world; Entities (Et) including actors as individuals or groups, and objects5
such as resources; and Facts (Ft) including statements about the state of
the world regarding entities and activities.
A hypothesis in intelligence analysis is composed of activities and events
that show how the situation has evolved. The argument from cause to ef-
fect (ArgCE) forms the basis of these hypotheses. The scheme, adapted10
from [WRM08], is:
Argument from cause to effect
Premises:
• Typically, if C (either a fact Fti or an ac-
tivity Acti) occurs, then E (either a fact
Fti or an activity Acti) will occur
• In this case, C occurs
Conclusions:
In this case E will occur
Critical questions:
CQCE1 Is there evidence for C to occur?
CQCE1 Is there a general rule for C causing E ?
CQCE3 Is the relationship between C and E
causal?
CQCE4 Are there any exceptions to the causal
rule that prevent the effect E from occur-
ring?
CQCE5 Has C happened before E ?
CQCE6 Is there any other C that caused E ?
Formally:
rCE : rule(R,C ,E ),occur(C ),before(C ,E ),15
ruletype(R,causal),noexceptions(R) ⇒ occur(E )
5.5 Arguments for Provenance
Provenance can be used to annotate how, where, when and by whom some
information was produced [MM13]. Figure 5.4 depicts the core model for

WasInformedBy
Used
WasGeneratedBy
WasAssociatedWith
ActedOnBehalfOf
WasAttributedTo
WasDerivedFrom
Entity
Actor
Activity
Figure 5.4: PROV Data Model [MM13]
Lab Water
Testing
wasGeneratedBy
Used
wasAssociatedWith
pjID:Bacteria
contaminates
local water
Water
Sample
Generate
Requirement
Water
monitoring
Requirement
wasDerivedFrom
Used
wasGeneratedBy
wasInformedBy
Monitoring of
water supply
used
water
contamination
report
Report
generation
Used wasGeneratedBy
wasAssociatedWith
wasDerivedFrom
?a1Pattern Pg
Goal
NGO
lab
assistant
NGO
Chemical
Lab
PrimarySource
Time2014-11-13T08-16-45Z
Time2014-11-12T10-14-40Z
Time2014-11-14T05-14-10Z
?a2
?p
?ag
LEGEND
p-Agent
p-Entity
p-Activity
Node
Older p-elements Newer
Figure 5.5: Provenance of Joe’s information
representing provenance, and Figure 5.5 shows an example of provenance
for the pieces of information for analyst Joe w.r.t. the water contamination
problem in Kish.
Patterns representing relevant provenance information that may war-
rant the credibility of a datum can be integrated into the analysis by ap-5
plying the argument scheme for provenance (ArgPV) [Ton+14]:

Argument Scheme for Provenance
Premises:
• Given pj about activity Acti, entity Eti, or
fact Fti (ppv1)
• GP(pj) includes pattern Pm of p-entities
Apv, p-activities Ppv, p-agents Agpv in-
volved in producing pj (ppv2)
• GP(pj) infers that information pj is true
(ppv3)
Conclusions:
Acti/Eti/Fti in pj may plausibly be true
(ppvcn)
Critical questions:
CQPV1 Is pj consistent with other information?
CQPV2 Is pj supported by evidence?
CQPV3 Does GP(pj) contain p-elements that lead
us not to believe pj?
CQPV4 Is there any other p-element that should
have been included in GP(pj) to infer that
pj is credible?

6 Implementations
Acknowledgement
Massimiliano Giacomin, Mauro Vallati, and Stefan Woltran.
Comprehensive survey recently published in [Cha+15].5
6.1 Ad Hoc Procedures
NAD-Alg [NDA12; NAD14]
6.2 Constraint Satisfaction Programming
A Constraint Satisfaction Problem (CSP) P [BS12; RBW08] is a triple
P = 〈X,D,C〉 such that:10
• X = 〈x1,...,xn〉 is a tuple of variables;
• D = 〈D1,...,Dn〉 a tuple of domains such that ∀i,xi ∈ Di;
• C = 〈C1,...,Ct〉 is a tuple of constraints, where ∀j,Cj = 〈RSj
,Sj〉,
Sj ⊆ {xi|xi is a variable}, RSj
⊆ SD
j
× SD
j
where SD
j
= {Di|Di is a
domain, and xi ∈ Sj}.15
A solution to the CSP P is A = 〈a1,...,an〉 where ∀i,ai ∈ Di and ∀j,RSj
holds on the projection of A onto the scope Sj. If the set of solutions is
empty, the CSP is unsatisﬁable.

Implementations • Answer Set Programming
CONArg2 [BS12]
In [BS12], the authors propose a mapping from AFs to CSPs.
Given an AF Γ, they first create a variable for each argument whose
domain is always {0,1} — ∀ai ∈ A ,∃xi ∈ X such that Di = {0,1}.
Subsequently, they describe constraints associated to different defi-5
nitions of Dung’s argumentation framework: for instance {a1,a2} ⊆ A is
D-conflict-free iff ¬(x1 = 1∧ x2 = 1).
6.3 Answer Set Programming
Answer Set Programming (ASP) [Fab13] is a declarative problem solving
paradigm. In ASP, representation is done using a rule-based language,10
while reasoning is performed using implementations of general-purpose
algorithms, referred to as ASP solvers.
AspartixM [EGW10; Dvo+11]
AspartixM [Dvo+11] expresses argumentation semantics in Answer Set
Programming (ASP): a single program is used to encode a particular ar-15
gumentation semantics, and the instance of an argumentation framework
is given as an input database. Tests for subset-maximality exploit the
metasp optimisation frontend for the ASP-package gringo/claspD.
Given an AF Γ, Aspartix encodes the requirements for a “semantics”
(e.g. the D-conflict-free requirements) in an ASP program whose database20
considers:
{arg(a) | a ∈ A }∪{defeat(a1,a2) | 〈a1,a2〉 ∈→}
The following program fragment is thus used to check the D-conflict-
freeness [Dvo+11]:
πcf = { in(X) ← not out(X),arg(X);
out(X) ← not in(X),arg(X);
← in(X),in(Y ),defeat(X,Y )}.
25
πS T = { in(X) ← not out(X),arg(X);
out(X) ← not in(X),arg(X);
← in(X),in(Y ),defeat(X,Y );
defeated(X) ← in(Y ),defeat(Y , X);
← out(X),not defeated(X)}.
6.4 Propositional Satisfiability Problems
In the propositional satisfiability problem (SAT) the goal is to determine
whether a given Boolean formula is satisfiable. A variable assignment
that satisfies a formula is a solution.30

Implementations • Propositional Satisfiability Problems
In SAT, formulae are commonly expressed in Conjunctive Normal Form
(CNF). A formula in CNF is a conjunction of clauses, where clauses are
disjunctions of literals, and a literal is either positive (a variable) or neg-
ative (the negation of a variable). If at least one of the literals in a clause
is true, then the clause is satisfied, and if all clauses in the formula are5
satisfied then the formula is satisfied and a solution has been found.
PrefSAT [Cer+14b]
Requirements for complete labelling as a CNF [Cer+14b]: for each argu-
ment ai ∈ A , three propositional variables are considered: Ii (which is
true iff L ab(ai) = in), Oi (which is true iff L ab(ai) = out), Ui (which is10
true iff L ab(ai) = undec). Given |A | = k and φ : {1,...,k} → A .
i∈{1,...,k}
(Ii ∨Oi ∨Ui)∧(¬Ii ∨¬Oi)∧(¬Ii ∨¬Ui)∧(¬Oi ∨¬Ui) (6.1)
{i|φ(i)−= }
Ii (6.2)
{i|φ(i)−= }
Ii ∨
{j|φ(j)→φ(i)}
(¬Oj) (6.3)
{i|φ(i)−= } {j|φ(j)→φ(i)}
¬Ii ∨Oj (6.4)15
{i|φ(i)−= } {j|φ(j)→φ(i)}
¬I j ∨Oi (6.5)
{i|φ(i)−= }
¬Oi ∨
{j|φ(j)→φ(i)}
I j (6.6)
{i|φ(i)−= } {k|φ(k)→φ(i)}
Ui ∨¬Uk ∨
{j|φ(j)→φ(i)}
I j (6.7)
{i|φ(i)−= } {j|φ(j)→φ(i)}
(¬Ui ∨¬I j) ∧ ¬Ui ∨
{j|φ(j)→φ(i)}
Uj (6.8)
i∈{1,...k}
Ii (6.9)20

As noticed in [Cer+14b], the conjunction of the above formulae is re-
dundant. However, the non-redundant CNFs are not equivalent from an
empirical evaluation [Cer+14b]: the overall performance is signiﬁcantly
affected by the chosen conﬁguration pair CNF encoding–SAT solver.

Algorithm 1 Enumerating the D-preferred extensions of an AF
PrefSAT(∆)
1: Input: ∆ = Γ
2: Output: Ep ⊆ 2A
3: Ep :=
4: cnf := Π∆
5: repeat
6: cnf df := cnf
7: pref cand :=
8: repeat
9: lastcompf ound := SatS(cnf df )
10: if lastcompf ound ! = ε then
11: pref cand := lastcompf ound
12: for a1 ∈ I-ARGS(lastcompf ound) do
13: cnf df := cnf df ∧ Iφ−1(a1)
14: end for
15: remaining := F ALSE
16: for a1 ∈ A I-ARGS(lastcompf ound) do
17: remaining := remaining ∨ Iφ−1(a1)
18: end for
19: cnf df := cnf df ∧ remaining
20: end if
21: until (lastcompf ound ! = ε∧I-ARGS(lastcompf ound) ! = A )
22: if pref cand ! = then
23: Ep := Ep ∪{I-ARGS(pref cand)}
24: oppsolution := F ALSE
25: for a1 ∈ A I-ARGS(pref cand) do
26: oppsolution := oppsolution∨ Iφ−1(a1)
27: end for
28: cnf := cnf ∧ oppsolution
29: end if
30: until (pref cand ! = )
31: if Ep = then
32: Ep = { }
33: end if
34: return Ep

Parallel-SCCp [Cer+14a; Cer+15]
Based on the SCC-Recursiveness Schema [BGG05].
ab
ef
cdgh

Algorithm 1 Computing D-preferred labellings of an AF
P-PREF(∆)
1: Input: ∆ = Γ
2: Output: Ep ∈ 2L(∆)
3: return P-SCC-REC(∆,A )
Algorithm 2 Greedy computation of base cases
GREEDY(L,C)
1: Input: L = (L1
,...,Ln
:= {Sn
1 ,...,Sn
h
}),C ⊆ A
2: Output: M = {...,(Si,Bi),...}
3: M :=
4: for S ∈ n
i=1
Li
do in parallel
5: B := B-PR(∆↓S,S ∩C)
6: M = M ∪{(S,B)}
7: end for
8: return M
BOUNDCOND(∆,Si,L ab) returns (O, I) where O = {a1 ∈ Si | ∃a2 ∈
S ∩ a−
1 : L ab(a2) = in} and I = {a1 ∈ Si | ∀ a2 ∈ S ∩ a−
1 ,L ab(a2) = out},
with S ≡ S1 ∪...∪ Si−1.

Algorithm 3 Determining the D-grounded labelling of an AF in a set C
GROUNDED(∆,C)
1: Input: ∆ = Γ, C ⊆ A
2: Output: (L ab,U) : U ⊆ A ,L ab ∈ LA U
3: L ab :=
4: U := A
5: repeat
6: initial f ound := ⊥
7: for a1 ∈ C do
8: if {a2 ∈ U | a2 → a1} = then
9: initial f ound :=
10: L ab := L ab ∪{(a1,in)}
11: U := U a1
12: C := C a1
13: for a2 ∈ (U ∩a+
1 ) do
14: L ab := L ab ∪{(a2,out)}
15: U := U a2
16: C := C a2
17: end for
18: end if
19: end for
20: until (initial f ound)
21: return(L ab,U)

Algorithm 4 Computing D-preferred labellings of an AF in C
P-SCC-REC(∆,C)
1: Input: ∆ = Γ, C ⊆ A
2: Output: Ep ∈ 2L(∆)
3: (L ab,U) = GROUNDED(∆,C)
4: Ep := {L ab}
5: ∆ = ∆↓U
6: L:= (L1
:= {S1
1,...,S1
k
},...,Ln
:= {Sn
1 ,...,Sn
h
})
= SCCS-LIST(∆)
7: M := {...,(Si,Bi),...} = GREEDY(L,C)
8: for l ∈ {1,...,n} do
9: El := {E
S1
l
:= (),...,E
Sk
l
:= ()}
10: for S ∈ Ll
do in parallel
11: for L ab ∈ Ep do in parallel
12: (O, I) := L-COND(∆,S,Ll
,L ab)
13: if I = then
14: ES
l
[L ab] ={{(a1,out) | a1 ∈ O} ∪{(a1,undec) | a1 ∈ S O}}
15: else
16: if I = S then
17: ES
l
[L ab] = B where (S,B) ∈ M
18: else
19: if O = then
20: ES
l
[L ab] = B-PR(∆↓S, I ∩C)
21: else
22: ES
l
[L ab]={{(a1,out) | a1 ∈ O}}
23: ES
l
[L ab] = ES
l
[L ab]⊗P-SCC-REC(∆↓SO, I ∩C)
24: end if
25: end if
26: end if
27: end for
28: end for
29: for S ∈ Ll
do
30: Ep :=
31: for L ab ∈ Ep do in parallel
32: Ep = Ep ∪({L ab}⊗ ES
l
[L ab])
33: end for
34: Ep := Ep
35: end for
36: end for
37: return Ep

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