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

Computing Science

Argumentation in Artificial Intelligence:20 Years after Dung’s Work

Federico Cerutti

Department of Computing Science July 2015University of AberdeenKing’s CollegeAberdeen AB24 3UE

Copyright © 2015, The University of Aberdeen

Argumentation in Artificial Intelligence: 20Years 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 [email protected] for reporting any.

Keywords: argumentation; tutorial; IJCAI2015

University of Aberdeen, 2015 Page 1

[email protected]

1 Dung’s ArgumentationFramework

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 drawn 10

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 set 20

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 ∈ A 25

b → a ⇒ S → b.

The function FAF : 2A 7→ 2A which, given a set S ⊆ A , returns the

set of the D-acceptable arguments w.r.t. S, is called the D-characteristicfunction of AF. ♠

Definition 4. Given an argumentation framework AF = ⟨A ,→⟩, a set 30

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) 6= ;}

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 as 5

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-maximalityprinciple 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 6 ∃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 framework 20

⟨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 anargumentation framework AF = ⟨A ,→⟩, let S ⊆ A be a D-admissible setof arguments, and a,b be arguments which are acceptable with respect toS. 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 frameworkAF = ⟨A ,→⟩, the set of all D-admissible sets of ⟨A ,→⟩ form a completepartial 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).+ Given an argumentation frame-

work AF = ⟨A ,→⟩. The grounded extension of AF is the least complete 20

extension of AF. GR denotes the grounded semantics. ♠

Definition 17 (Preferred Extension).+ Given an argumentation frame-

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 YesD-admissibility Yes Yes Yes YesD-strongly admissibility No Yes No NoD-reinstatement Yes Yes Yes YesD-I-maximality No Yes Yes YesD-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 labelling 10

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 preferredlabelling 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, let 15

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 ∈ S 20

• L ab(a1)= out⇔∃ a2 ∈ S s.t. a2 → a1



σ=C O σ=GR σ=P R σ=S T

EXISTSσ trivial trivial trivial NP-cCAσ NP-c polynomial NP-c NP-cSAσ polynomial polynomial Π

p2 -c coNP-c

VERσ polynomial polynomial coNP-c polynomialNEσ 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


Dung’s AF • Skepticism Relationships [BG09b]

GR

C O

P R

GR

C O

P RS T

Figure 1.2: ¹S⊕ relation for any argumentation framework (left) and forargumentation framework where stable extensions exist (right).

1.5 Skepticism Relationships [BG09b]

E1 ¹E E2 denotes that E1 is at least as skeptical as E2.

Definition 23. Let ¹E be a skepticism relation between sets of exten-

sions. The skepticism relation between argumentation semantics ¹S is

such that for any argumentation semantics σ1 and σ2, σ1 ¹S σ2 iff ∀AF ∈ 5

Dσ1 ∩Dσ2 , EAF (σ1)¹E EAF (σ2). ♠

Definition 24. Given two sets of extensions E1 and E2 of an argumenta-

tion framework AF:

• E1 ¹E∩+ E2 iff ∀E2 ∈ E2, ∃E1 ∈ E1: E1 ⊆ E2;

• E1 ¹E∪+ E2 iff ∀E1 ∈ E1, ∃E2 ∈ E2: E1 ⊆ E2. ♠ 10

Lemma 2. Given two argumentation semantics σ1 and σ2, if for anyargumentation framework AF EAF (σ1) ⊆ EAF (σ2), then σ1 ¹E

∩+ σ2 andσ1 ¹E

∪+ σ2 (σ1 ¹E⊕ σ2). ♣

1.6 Signatures [Dun+14]

Let A be a countably infinite domain of arguments, and 15

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} 6∈ S then there exists an b ∈ S such that ⟨a,b⟩ 6∈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 6= ; | 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 ,→), a 10

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 input 15

(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 σ 20

(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)= out 30

• 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 6=i LP j )↓Piinp ,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 NoTop-down decomposability Yes Yes Yes YesBottom-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 contexts 5

(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 rational 10

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 lack 15

of knowledge, using the conclusions tentatively accepted.

2.1 An example: Walton et al. ’s ArgumentationSchemes 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 can 30

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 governing 5

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 comments 15



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 that 15

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, while 25

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, whereas 30

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 practical 10

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; and 15

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. The 5

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 explain 10

how and why the maker of a proposal needs to be open to questioning and

to requests for justification.


3 A Semantic-Web View ofArgumentation

Acknowledgement


Chris Reed. An overview can also be find at [Bex+13]. 5

3.1 The Argument Interchange Format [Rah+11]

Node Graph(argumentnetwork)

has-a

InformationNode

(I-Node)

is-a

Scheme NodeS-Node

has-a

Edge

is-a

Rule of inferenceapplication node

(RA-Node)

Conflict applicationnode (CA-Node)

Preferenceapplication node

(PA-Node)

Derived conceptapplication node (e.g.

defeat)

is-a

...

ContextScheme

Conflictscheme

contained-in

Rule of inferencescheme

Logical inference scheme

Presumptiveinference scheme ...

is-a

Logical conflictscheme

is-a

...

Preferencescheme

Logical preferencescheme

is-a

...Presumptivepreference 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.owl 10

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)


http://protege.stanford.edu/

http://www.arg.dundee.ac.uk/wp-content/uploads/AIF.owl



Semantic Web Argumentation • AIF-OWL

PracticalInference

Bringing about is my goal

Sn

Si

In order to bring about I need to bring about

Sn

Therefore I needto bring about Si

hasConcDeschasPremiseDesc

hasPremiseDesc

Bringing about being richis my goal

In order to bring about being richI need to bring about having a job

fulfilsPremiseDesc

fulfilsPremiseDesc

fulfilsScheme

supports

supports

Therefore I needto bring abouthaving a job

hasConclusion

fulfils

Figure 3.2: An argument network linking instances of argument andscheme components

Symmetric attack

r → p

r pMP2A1

A2p → q

p

qMP1

neg1

Undercut attack

r MP2A3

A2 s → v

s

vMP1

cut1p

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 v Statement

GoalPlan_Statement

GoalPlan_Statement v Statement 5

Goal_Statement

Goal_Statement v Statement

I-node

I-node ≡ Statement

I-node v Node 10

I-node v ¬ S-node

Inference

Inference ≡ RA-node

Inference v ∃ fulfils Inference_Scheme

Inference v ≥ 1 hasPremise Statement 15

Inference v Scheme_Application

Inference v = hasConclusion (Scheme_Application t Statement)

Inference_Scheme

Inference_Scheme v Scheme u ≥1 hasPremise_Desc Statement_Description u = hasConclusion_Desc 20

(Scheme t Statement_Description)

PracticalReasoning_Inference

PracticalReasoning_Inference ≡ Presumptive_Inference u ∃ hasCon-

clusion EncouragedAction_Statement u ∃ hasGoalPlan_Premise Goal-

Plan_Statement u ∃ hasGoal_Premise Goal_Statement 25

RA-node

RA-node ≡ Inference

RA-node v S-node

S-node

S-node ≡ Scheme_Application 30

S-node v Node

S-node v ¬ I-node



Scheme

Scheme v Form

Scheme v ¬ Statement_Description

Scheme_Application

Scheme_Application ≡ S-node 5

Scheme_Application v ∃ fulfils Scheme

Scheme_Application v Thing

Scheme_Application v ¬ Statement

Statement

Statement ≡ NegStatement 10

Statement ≡ I-node

Statement v Thing

Statement v ∃ fulfils Statement_Description

Statement v ¬ Scheme_Application

Statement_Description 15

Statement_Description v Form

Statement_Description v ¬ Scheme

fulfils

∃ fulfils Thing v Node

hasConclusion_Desc 20

∃ hasConclusion_Desc Thing v Inference_Scheme

hasGoalPlan_Premise

v hasPremise

hasGoal_Premise

v hasPremise 25

claimText

∃ claimText DatatypeLiteral v Statement

> v ∀ claimText DatatypeString

Individuals of EncouragedAction_Desc

EncouragedAction_Desc : Statement_Description 30

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_Desc 5

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 Graphs 15

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 in 20

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, he 25

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

a2LC , FC

a3LC , FH

a1LC

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 than 15

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 to 5

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 formalismshold. ♣

Theorem 4.+ Moreover, in the case where an AFRA is not an AF, it ispossible 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 representthe 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 {L i}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 {L i}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. L i is a fact;

2. there exists a rule Ri in ⟨Π,∆⟩ (either strict or defeasible) with head

L i and body B1, . . . ,Bk, and every literal of the body is an element 20

L j of the sequence appearing before L i ( 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 Pand S disagree, that is, there exist two contradictory literals that have a 5

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 Â.

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⟩ Â ⟨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=

mondaycloudydry_seasonwavesgrass_grownhire_gardenervacation¬working←− vacationfew_surfers←−¬many_surfers¬surf ←− ill

∆1=

surf −<nice,spare_timenice−<wavesspare_time−<¬busy¬busy−<¬working¬nice−< rainrain−< cloudy¬rain−<dry_season. . .

From ⟨Π1,∆1⟩, these are some arguments that can be derived:

⟨A0,surf ⟩ =⟨

surf −<nice,spare_timenice−<wavesspare_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 contraryof a.

♠

Definition 44. A deduction for σ ∈ L supported by S ⊆ L and R ⊆ R,

denoted as SR` σ, is a (finite) tree with nodes labelled by sentences in 15

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 ϕholds as well — while a defeasible inference “usually” holds. Arguments 10

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 7→ 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 7→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 6∈ψ, otherwise is inconsistent.A knowledge base in an AS is a set Kn ∪Kp = K ⊆ L ; {Kn,Kp} is a 5

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)=⋃ni=1Prem(ai); Conc(a)=ψ;

Sub(a)=⋃ni=1Sub(ai)∪ {a};

Rules(a)=⋃ni=1Rules(ai)∪ {r};

DefRules(a)= {d | d ∈ Rules(a)∩Rd};

TopRule(a)= r 20

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 defeats 25

takes into account an argument ordering ¹: a ¹b iff a is “less preferred”

than b (a ≺b iff a ¹b 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 b′′1, . . . ,b′′

n =⇒ –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 byAT = ⟨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 AF 5

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 ¹ 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 ⊀ b 20

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)=⋃ni=1(Prem(ai)∩Kp); DefRules(a)=⋃n

i=1DefRules(ai);

Rules(a)⊇⋃ni=1Rules(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 to 30

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 include 5

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 argumentfrom a set of formulae ∆ is a pair ⟨Φ,α⟩ such that

1. Φ⊆∆

2. Φ 6` ⊥

3. Φ`α 30

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 undercutCanonical

undercutRebut



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 that 10

treatment τ1 is inferior to τ2.


Frameworks • Probabilistic Argumentation

♠

Given an inductive argument a = ⟨X ,ε⟩, 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 εi and the claim 5

of argument a j is ε j then we say that ai conflicts with a j whenever:

1. εi = τ1 > τ2, and ( ε j = τ1 ∼ τ2 or ε j = τ1 < τ2 ).

2. εi = τ1 ∼ τ2, and ( ε j = τ1 > τ2 or ε j = τ1 < τ2 ).

3. εi = τ1 < τ2, and ( ε j = τ1 > τ2 or ε j = τ1 ∼ τ2 ). ♠

Definition 60 (Attack). For any pair of arguments ai and a j, and a pref- 10

erence relation R, ai attacks a j with respect to R iff ai conflicts with a j

and it is not the case that a j 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-blind 20

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⟩

⟨Not

stat

isti

call

ysi

gnif

ican

t⟩Figure 4.8: Arguments derived from Ex. 8, with preferences and metaarguments.

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)= 1 5

♠

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)= 0.2 15

♥



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 is 10

{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 isat least one P that it is de Finetti coherent [Fin74] and weakly p-justifiable.

♣

Definition 67. Correspondences between probabilistic and classical se-

mantics 10

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′ vG 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′ vG |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 Framework 5

Definition 71 ([LON12; Li15]). A Li-PAF is a tuple ⟨A ,PA ,→,P→⟩, where

⟨A ,→⟩ is an argumentation framework, PA : A 7→ (0..1] and P→ :→7→(0..1]. ♠

Definition 72 ([LON12; Li15]). Given a Li-PAF ⟨A ,PA ,→,P→⟩, AF I =⟨A I ,→I⟩ is said to be induced iff A I ⊆ A ; and →I⊆→∩(A T ×A T ); and 10

∀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 P IPrAF(∆I ), by the following equation:

P IPrAF(∆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+ ArgumentationSystem

• Logical Language• Inference Rules• Contrariness Function• ......

StructuredArgumentation

Framework(SAF)

DAF

DAFEAFExtendedEvidential

Framework(EEAF)

ProbabilisticExtendedEvidential

Framework

Convert to

Convert to

ExtendedEvidential

Framework(EEAF)

Model

ProbabilisticExtendedEvidential

FrameworkAssociate

ProbabilitiesConvert toPrEAF

AssociateProbabilities

Semantics

Preserved

PrAF

AssociateProbabilities

Figure 4.9: [Li15]’s probabilistic argumentation architecture.


5 A novel synthesis: CollaborativeIntelligence 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 analysts 10

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 Analysis 15

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

Searchand Filter

Schematize

Build Case

Tell Story

Reevaluate

Search for support

Search for evidence

Search for information

FORAGING LOOP

SENSE-MAKING LOOP

Stru

ctur

e

Effort

inf

Shoebox

Ev

Ev

EvEv Ev

EvEv

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 differentexpertise, resources

• Prevent biases

Table 5.1: Individual vs. Collaborative Analysis


CISpaces • Intelligence Analysis

HarbourKish 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 7→, an info/claim node is written pi and a link node is referred to

as `type where type= {Pro,Con}. Then, [p1, . . . ,pn 7→ `Pro 7→ 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 r i ∈ R corresponding to Pro links 20

between nodes such that: [p1, . . . , pn 7→ `Pro 7→ pφ] is converted to

r i : p1, . . . , pn ⇒ pφ

• The contrariness function between elements is defined as: i) if [p1 7→`Con 7→ p2] and [p2 7→ `Con 7→ p1], p1 and p2 are contradictory; ii)[p1 7→ `Con 7→ p2] and p1 is the only premise of the Con link, then p1 25

is a contrary of p2; iii) if [p1, p3 7→ `Con 7→ p2] then a rule is added

such that p1 and p3 form an argument with conclusion ph against

p2, r i : p1, p3 ⇒ ph and ph is a contrary of p2. ♠

Definition 75. K is composed of propositions pi,

K = {p j, pi, . . . }, such that: i) let a set of rules r1, . . . , rn ∈R indicate a cycle 30

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 IntelligenceAnalysis

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 objects 5

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, adapted 10

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 WaterTesting

wasGeneratedByUsed

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

NGOlab

assistant

NGOChemical

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 p j about activity Acti, entity Eti, or

fact Fti (ppv1)• GP (p j) includes pattern P ′

m of p-entities

Apv, p-activities Ppv, p-agents Agpv in-

volved in producing p j (ppv2)• GP (p j) infers that information p j is true

(ppv3)

Conclusions:Acti/Eti/Fti in p j may plausibly be true

(ppvcn)

Critical questions:CQPV1 Is p j consistent with other information?

CQPV2 Is p j supported by evidence?

CQPV3 Does GP (p j) contain p-elements that lead

us not to believe p j?CQPV4 Is there any other p-element that should

have been included in GP (p j) to infer that

p j 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 ∈ D i;

• C = ⟨C1, . . . ,Ct⟩ is a tuple of constraints, where ∀ j,C j = ⟨RS j ,S j⟩,S j ⊆ {xi|xi is a variable}, RS j ⊆ SD

j ×SDj where SD

j = {D i|D i is a

domain, and xi ∈ S j}. 15

A solution to the CSP P is A = ⟨a1, . . . ,an⟩ where ∀i,ai ∈ D i and ∀ j,RS j

holds on the projection of A onto the scope S j. 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 D i = {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 database 20

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]:

πc f = { 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 are 5

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: I i (which is

true iff L ab(ai) = in), Oi (which is true iff L ab(ai) = out), Ui (which is 10

true iff L ab(ai)= undec). Given |A | = k and φ : {1, . . . ,k} 7→A .

∧i∈{1,...,k}

((I i ∨Oi ∨Ui)∧ (¬I i ∨¬Oi)∧(¬I i ∨¬Ui)∧ (¬Oi ∨¬Ui)

)(6.1)

∧{i|φ(i)−=;}

I i (6.2)

∧{i|φ(i)− 6=;}

(I i ∨

( ∨{ j|φ( j)→φ(i)}

(¬O j)

))(6.3)

∧{i|φ(i)− 6=;}

( ∧{ j|φ( j)→φ(i)}

¬I i ∨O j

)(6.4) 15

∧{i|φ(i)− 6=;}

( ∧{ j|φ( j)→φ(i)}

¬I j ∨Oi

)(6.5)

∧{i|φ(i)− 6=;}

(¬Oi ∨

( ∨{ j|φ( j)→φ(i)}

I j

))(6.6)

∧{i|φ(i)− 6=;}

( ∧{k|φ(k)→φ(i)}

(Ui ∨¬Uk ∨

( ∨{ j|φ( j)→φ(i)}

I j

)))(6.7)

∧{i|φ(i)− 6=;}

(( ∧{ j|φ( j)→φ(i)}

(¬Ui ∨¬I j)

)∧

(¬Ui ∨

( ∨{ j|φ( j)→φ(i)}

U j

)))(6.8)

∨i∈{1,...k}

I i (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 AFPrefSAT(∆)

1: Input: ∆= Γ2: Output: Ep ⊆ 2A

3: Ep := ;4: cnf := Π∆

5: repeat6: cnf d f := cnf

7: pre f cand := ;8: repeat

9: lastcompf ound := SatS(cnf d f )

10: if lastcompf ound != ε then

11: pre f cand := lastcompf ound

12: for a1 ∈ I-ARGS(lastcompf ound) do

13: cnf d f := cnf d f ∧ 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 d f := cnf d f ∧ remaining

20: end if

21: until (lastcompf ound != ε∧ I-ARGS(lastcompf ound) != A )

22: if pre f cand != ; then

23: Ep := Ep ∪ {I-ARGS(pre f cand)}

24: oppsolution := F ALSE

25: for a1 ∈A \ I-ARGS(pre f cand) do

26: oppsolution := oppsolution∨ Iφ−1(a1)

27: end for

28: cnf := cnf ∧ oppsolution

29: end if30: until (pre f cand != ;)

31: if Ep =; then32: Ep = {;}

33: end if34: return Ep



Parallel-SCCp [Cer+14a; Cer+15]

Based on the SCC-Recursiveness Schema [BGG05].

ab

ef

cd

gh



Algorithm 1 Computing D-preferred labellings of an AFP-PREF(∆)

1: Input: ∆= Γ2: Output: Ep ∈ 2L(∆)

3: return P-SCC-REC(∆,A )

Algorithm 2 Greedy computation of base casesGREEDY(L,C)

1: Input: L = (L1, . . . ,Ln := {Sn1 , . . . ,Sn

h}),C ⊆A

2: Output: M = {. . . , (Si,Bi), . . .}3: M :=;4: for S ∈⋃n

i=1 Li do in parallel5: B := B-PR(∆↓S ,S∩C)6: M = M∪ {(S,B)}7: end for8: 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 CGROUNDED(∆,C)

1: Input: ∆= Γ, C ⊆A

2: Output: (L ab,U) : U ⊆A ,L ab ∈LA \U3: L ab := ;4: U := A

5: repeat6: initial f ound := ⊥7: for a1 ∈ C do8: if {a2 ∈U | a2 → a1}=; then9: 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 for18: end if19: end for20: until (initial f ound)21: return(L ab,U)



Algorithm 4 Computing D-preferred labellings of an AF in CP-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 := {S11, . . . ,S1

k}, . . . ,Ln := {Sn1 , . . . ,Sn

h})

=SCCS-LIST(∆)7: M := {. . . , (Si,Bi), . . .}=GREEDY(L,C)

8: for l ∈ {1, . . . ,n} do9: E l := {ES1

l := (), . . . ,ESkl := ()}

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: ESl [L ab]={{(a1,out) | a1 ∈O} ∪ {(a1,undec) | a1 ∈ S \O}}

15: else

16: if I = S then

17: ESl [L ab]= B where (S,B) ∈ M

18: else

19: if O =; then

20: ESl [L ab]=B-PR(∆↓S , I ∩C)

21: else

22: ESl [L ab]={{(a1,out) | a1 ∈O}}

23: ESl [L ab]= ES

l [L ab]⊗P-SCC-REC(∆↓S\O, I ∩C)

24: end if

25: end if

26: end if

27: end for

28: end for

29: for S ∈ Ll do

30: E′p := ;

31: for L ab ∈ Ep do in parallel

32: E′p = E′

p ∪ ({L ab}⊗ESl [L ab])

33: end for

34: Ep := E′p

35: end for36: end for37: return Ep


Implementations • Which One?

6.5 Which One?

We need to be smartHolger H. Hoos, Invited Keynote Talk at ECAI2014

Features for AFs [VCG14; CGV14]

Directed Graph (26 features) 5

Structure:

# vertices ( |A | )

# edges ( |→ | )

# vertices / #edges ( |A |/|→ | )

# edges / #vertices ( |→ |/|A | )

density

average

Degree: stdev

attackers max

min

#

average

stdev

max

SCCs:

min

Structure:

# self-def

# unattacked

flow hierarchy

Eulerian

aperiodic

CPU-time: . . .



Undirected Graph (24 features)

Structure:

# edges

# vertices / #edges

# edges / #vertices

density

Degree:

average

stdev

max

min

SCCs:

#

average

stdev

max

min

Structure: Transitivity

3-cycles:

#

average

stdev

max

min

CPU-time: . . .

Average CPU-time, stdev, needed for extracting the featuresDirect Graph Features (DG)

Class CPU-Time # featMean stdDev

Graph Size 0.001 0.009 5

Degree 0.003 0.009 4

SCC 0.046 0.036 5

Structure 2.304 2.868 5

Undirect Graph Features (UG)Class CPU-Time # feat

Mean stDev

Graph Size 0.001 0.003 4

Degree 0.002 0.004 4

SCC 0.011 0.009 5

Structure 0.799 0.684 1

Triangles 0.787 0.671 5

5

Best Features for Runtime Prediction [CGV14]

Determined by a greedy forward search based on the Correlation-based

Feature Selection (CFS) attribute evaluator.



Solver B1 B2 B3

AspartixM num. arguments density (DG) size max. SCC

PrefSAT density (DG) num. SCCs aperiodicity

NAD-Alg density (DG) CPU-time density CPU-time Eulerian

SSCp density (DG) num. SCCs size max SCC

Predicting the (log)Runtime [CGV14]

RSME of Regression (Lower is better)

B1 B2 B3 DG UG SCC All

AspartixM 0.66 0.49 0.49 0.48 0.49 0.52 0.48PrefSAT 1.39 0.93 0.93 0.89 0.92 0.94 0.89NAD-Alg 1.48 1.47 1.47 0.77 0.57 1.61 0.55SSCp 1.36 0.80 0.78 0.75 0.75 0.79 0.74

Log runtime is defined as

√∑ni=1

(log10( ti )− log10( yi )

)2

n5

Best Features for Classification [CGV14]

Determined by a greedy forward search based on the Correlation-based

Feature Selection (CFS) attribute evaluator.

C-B1 C-B2 C-B3

num. arguments density (DG) min attackers

Classification [CGV14] 10

Classification (Higher is better)

C−B1 C-B2 C-B3 DG UG SCC All

Accuracy 48.5% 70.1% 69.9% 78.9% 79.0% 55.3% 79.5%Prec. AspartixM 35.0% 64.6% 63.7% 74.5% 74.9% 42.2% 76.1%Prec. PrefSAT 53.7% 67.8% 68.1% 79.6% 80.5% 60.4% 80.1%

Prec. NAD-Alg 26.5% 69.2% 69.0% 81.7% 85.1% 35.3% 86.0%Prec. SSCp 54.3% 73.0% 72.7% 76.6% 76.8% 57.8% 77.2%

Selecting the Best Algorithm [CGV14]

Metric: Fastest

(max. 1007)

AspartixM 106

NAD-Alg 170

PrefSAT 278

SSCp 453

EPMs Regression 755

EPMs Classification 788



Metric: IPC

(max. 1007)

NAD-Alg 210.1

AspartixM 288.3

PrefSAT 546.7

SSCp 662.4

EPMs Regression 887.7

EPMs Classification 928.1

IPC score1: for each AF, each system gets a score of T∗/T, where Tis its execution time and T∗ the best execution time among the compared

systems, or a score of 0 if it fails in that case. Runtimes below 0.01 seconds

get by default the maximal score of 1. The IPC score considers, at the 5

same time, the runtimes and the solved instances

1 http://ipc.informatik.uni-freiburg.de/ .


http://ipc.informatik.uni-freiburg.de/

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