T14Argumentation for agent societies

1

Argumentation for Agent Societies Part I

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2

Introduction to the tutorial

Argumentation for Agent Societies

(Some) answers to the following two questions:

1.  What’s argumentation?

! mainly today

2.  What is argumentation good for (in the MAS context)?

! mainly tomorrow

Let’s start with the first question…

3

An informal example (1)

The reason

The conclusion

We are justified in believing that we should run LHC !

We should run Large Hadron Collider

LHC allows us to understand the Laws

of the Universe

Understanding the Laws of the Universe is good

4

An informal example (2)

The reason

The conclusion

We are justified in believing that we should run LHC !

We should run Large Hadron Collider

LHC allows us to understand the Laws

of the Universe

Understanding the Laws of the Universe is good

In Argumentation (and in real life as well): - reasons are not necessary “conclusive” (they don’t logically entail conclusions) - arguments and conclusions can be “retracted” in front of new information, i.e. counterarguments

BUT

5

An informal example (3)

We should run Large Hadron Collider

LHC allows us to understand the Laws

of the Universe

Understanding the Laws of the Universe is good

We should not run LHC

LHC will generate black holes

destroying Earth

Destroying Earth is bad

Now we are justified in believing that we should not run LHC "

6

An informal example (4)

We should run Large Hadron Collider

LHC allows us to understand the Laws

of the Universe

Understanding the Laws of the Universe is good

We should not run LHC

LHC will generate black holes

destroying Earth

Destroying Earth is bad

Black holes will not destroy Earth

Black holes will evaporate because

of Hawking radiation

Now we are again justified in believing that we should run LHC !

7

An informal example (5)

We should run Large Hadron Collider

LHC allows us to understand the Laws

of the Universe

Understanding the Laws of the Universe is good

We should not run LHC

LHC will generate black holes

destroying Earth

Destroying Earth is bad

Black holes will not destroy Earth

Black holes will evaporate because

of Hawking radiation

Hawking radiation does not exist

Dr Azzeccagarbugli says so

Now we are again justified in believing that we should not run LHC "

8

An informal example (6)

We should run Large Hadron Collider

LHC allows us to understand the Laws

of the Universe

Understanding the Laws of the Universe is good

We should not run LHC

LHC will generate black holes

destroying Earth

Destroying Earth is bad

Black holes will not destroy Earth

Black holes will evaporate because

of Hawking radiation

Hawking radiation does not exist

Dr Azzeccagarbugli says so

Dr Azzeccagarbugli is not expert in physics

He is a lawyer

Now we are again justified in believing that we should

run LHC !

9

What’s argumentation? (1)

[Prakken 2011] Argumentation is the process of supporting claims with grounds and defending them against attack. [van Eemeren et al, 1996] Argumentation is a verbal and social activity of reason aimed at increasing (or decreasing) the acceptability of a controversial standpoint for the listener or reader, by putting forward a constellation of propositions intended to justify (or refute) the standpoint before a rational judge.

•  A framework for practical and uncertain reasoning able to cope with partial and inconsistent knowledge - philosophical roots: Aristotle, Toulmin (1958) - in AI: R.P. Loui (1987), J. Pollock (1987), G. Simari & Loui (1992)

10

What’s argumentation? (2)

The elements of an argumentation system

•  The definition of argument (possibly including an underlying logical language + a notion of logical consequence) •  The notion of attack and defeat (successful attack) between arguments

•  An argumentation semantics selecting acceptable (justified) arguments

11

Definition of argument: several possibilities (1)

•  ASSUMPTION-BASED ARGUMENTATION

Given a knowledge base (K, Ass)

Consistent theory Set of assumptions

ARGUMENT for p:

(A, p) such that

- A " Ass - A # K is consistent and entails p - There is no A’$A such that A’ # K entails p

ATTACKS to an argument: on its assumptions

[see Besnard&Hunter, Dung-Kowalski-Toni]

12

Definition of argument: several possibilities (2)

•  ARGUMENT SCHEMES

- correspond to recurring patterns of reasoning - have associated “critical questions”

Example: Expert Testimony

E is expert on D E says P P is in D Therefore, P is the case

Critical questions: Is E biased? Is P consistent with what other experts say? Is P consistent with known evidence?

[WALTON 1996]

13

Definition of argument: several possibilities (3)

•  ARGUMENT SCHEMES IN A MEDICAL APPLICATION

[Tolchinsky et al, 2006]

Organ O of donor D is available No contraindications are known for donating O to recipient R Therefore, organ O is viable

CRITICAL QUESTIONS:

Does donor D have a contraindication for donating organ O?

Viability Scheme

Nonviability Scheme

Donor D of organ O has condition C C is a contraindication are for donating O Therefore, organ O is nonviable

14

•  STABLE MARRIAGE PROBLEM - Arguments of the kind <Alice, John> - <Barbara, John> attacks <Alice, John> if John prefers Barbara to Alice

Definition of argument: several possibilities (4)

… … …

In general

Arguments take different forms (domain-independent vs. domain dependent)

Today examples will refer to rule-based approaches…

•  PLANNING - Plans as arguments (that a goal will be achieved) - Defeat between plans as attacks

15

•  ARGUMENT

a tree made up of rules of inference constructed from a set of premises to reach a conclusion

•  Two kinds of rules:

  A % B: deductive - “indefeasible”

  A ! B: non-deductive - “defeasible”

% ¬C (0.7) A (0.7) B (0.9)

D (0.9) ! C (0.8)

Rule-based approaches

•  A strength value may be associated to premises and rules, giving rise to argument strength

See [J.Pollock, 1992], [G. Vreeswijk, 1997], …

16

Notion of conflict

A % ¬C

D ! C B

E!(D&C) [Pollock 92]

–  Rebutting: an argument attacks another one by denying its [possibly intermediate] conclusion

–  Undercutting: an argument attacks the applicability of a defeasible rule of inference

Notion of defeat

An argument ' defeats ( iff: - ' undercuts (, or - ' rebuts ( and ' is not weaker than (

Rule-based approaches (2)

17

Smith says it’s raining

It’s raining

Bob says it’s not raining

It’s not raining

REBUTTING DEFEAT

Bob is drunk

Bob is unreliable

UNDERCUTTING DEFEAT

EXAMPLE

Rule-based approaches (3)

18

The ASPIC framework

•  One result of the European ASPIC Project (2004-2006)

•  Generalizes Pollock’s rule-based approach in several respects:

- any logical language (and an associated ‘contrariness function’

generalizing classical negation) can be adopted

- can be instantiated by a partial preorder on defeasible rules

- premises are distinguished into necessary, ordinary and assumption

premises (ordinary and assumption premises partially preordered)

- a partial preorder is assumed between arguments

•  Besnard & Hunter’s approach, Pollock’s system… can be obtained as

instances of ASPIC framework

•  See [H. Prakken, “An abstract framework for argumentation with

structured arguments”, Argument and Computation, 2010] for details.

19

Advantageous features

•  Several kinds of arguments can be represented

- epistemic reasoning

- practical reasoning

•  Able to handle uncertain and partial knowledge

- nonmonotonic notion of warrant:

1) wrt further information

2) wrt further reasoning steps (anytime reasoning framework)

•  A “natural” representation + justification of choices

(in terms of ‘argument’, ‘rebuttal’, ‘counterargument’…)

•  Argumentation has a dialogical side

(in terms of ‘argument’, ‘attack’, ‘defence’…)

Argumentation in the context of MAS (1)

20

The uses of argumentation (examples)

Argumentation in the context of MAS (2)

AUTONOMOUS REASONING

MULTI-AGENT INTERACTION

EPISTEMIC REASONING

- Belief Revision (arguing over beliefs) - Trust management (arguing over other agents reputation)

- Dialectics in Multiagent Interaction PRATICAL

REASONING - Decision making (arguing about the expected value of possible actions)

21

What’s argumentation? (3)

The elements of an argumentation system

•  The definition of argument (possibly including an underlying logical language + a notion of logical consequence) •  The notion of attack and defeat (successful attack) between arguments

•  An argumentation semantics selecting acceptable (justified) arguments

22

What’s abstract argumentation?

The elements of an argumentation system

Usually “abstract” stands for a difficult thing… Here it means “simple”!

Abstract argumentation focuses on this aspect

•  The definition of argument (possibly including an underlying logical language + a notion of logical consequence) •  The notion of attack and defeat (successful attack) between arguments

•  An argumentation semantics selecting acceptable (justified) arguments

23

Dung’s argumentation framework

AF = <A, %>

Arguments [origin and structure not specified]

attack (or defeat) relation [unspecified definition]

•  Graphical representation as a directed graph [defeat graph], e.g.

[Dung ’95]

Representation of LHC example

Representation of weather example

24

Dung’s argumentation framework (2)

ARGUMENT EVALUATION:

GIVEN AN ARGUMENTATION FRAMEWORK, DETERMINE THE JUSTIFICATION STATE (ALSO CALLED DEFEAT STATUS) OF ARGUMENTS, IN PARTICULAR: WHAT ARGUMENTS EMERGE UNDEFEATED FROM THE CONFLICT, I.E. ARE ACCEPTABLE?

So, what remains to be done?

25

•  Specification of a method for argument evaluation, or of criteria to determine, given a set of arguments, their “defeat status”

Argumentation Framework

Semantics

Defeat status

Defeat status

Undefeated

Defeated

Provisionally Defeated

Argumentation semantics

26

Labelling vs. extension-based semantics

LABELLING-BASED SEMANTICS

- Based on the notion of labelling [assignment to each argument of a label from a predefined set] - Specifies how to derive from an argumentation framework a set of labellings - Justification of arguments derived from the set of labellings

EXTENSION-BASED SEMANTICS

- Less general (at least in theory), but more common kind of semantics -  Based on the notion of extension [set of arguments “collectively acceptable”]

27

Extension-based semantics

Set of extensions S(AF) Argumentation framework AF

Semantics S

28

From extensions to defeat status

Set of extensions S(AF) Defeat/Justification Status

A common definition

•  Skeptically justified argument: belongs to all of the extensions •  Credulously justified argument: belongs to at least one •  Indefensible argument: does not belong to any extension

29

Unique-status vs. multiple-status semantics

()') ()')

()')

Unique-Status Semantics

()')Unique extension: empty set ' and ( directly unjustified (provisionally defeated)

Multiple-Status Semantics

! ' and ( unjustified (provisionally defeated)

30

Relationship between labelling and extension-based approaches

•  Almost all approaches adopt the set {IN, OUT, UNDEC} - IN = belonging to the extension - OUT = attacked by the extension - UNDEC= not belonging to nor attacked by the extension

()') ()')

()')

Unique-Status Semantics

()')

Multiple-Status Semantics

UNDEC UNDEC

IN OUT OUT IN

31

The core of Dung’s theory: complete “semantics”

Acceptability

' acceptable w.r.t. (“defended by”) S

•  all attackers of ' are attacked by S

Admissible set S

•  conflict-free •  every element acceptable w.r.t. S (defends all of its elements)

'

S

IF also includes all acceptable elements w.r.t. itself

Complete extension

Complete semantics

All traditional semantics select complete extensions

32

Complete “semantics”: examples

') () *)

Chain Admissible sets:

ø, {'}, {', *} Only one complete extension:

CO(AF) = {{', *}}

Nixon Diamond

()')

()')

()')

()')

All admissible sets are complete

CO(AF) =

{ ø, {'}, {(} }

33

Complete “semantics”: examples (2)

Nixon Diamond + node

()')

()')

()')

Admissible sets:

ø, {'}, {(}, {', *}

CO(AF) = {

ø

{', *},

{(} }

()') *)

CO(AF)

*)

*)

*)

34

The Grounded Semantics: a unique status approach

Undefeated

Defeated

Provisionally Defeated

Grounded extension GE(AF):

Least complete extension

Defeat status

included in all extensions of any traditional semantics

Grounded semantics is the “most skeptical” one

35

Grounded semantics: examples

') () *)

Chain

GE(AF) = {', *}

Nixon Diamond

()') GE(AF) = ø

Nixon Diamond + node

()') *) GE(AF) = ø

36

Floating arguments: a problem for grounded semantics

()

')

*) +)

()

')

*) +) VS

What we (may) want Grounded Semantics

•  Actually, grounded semantics is polynomially computable •  But sometimes a more discriminative behavior is desirable

THE CASE OF FLOATING ARGUMENTS

•  A problem for all possible unique status approaches

Let us consider multiple status approaches!

37

Stable Semantics

Stable extension = conflict-free set attacking all outside arguments

()

')

*) +)

()

')

*) +)

THE CASE OF FLOATING ARGUMENTS

ODD-LENGTH CYCLES: A PROBLEM FOR STABLE SEMANTICS

')

*)

() No stable extension exists! (and also imposing ø is not satisfactory)

ST(AF) = { {', +}, {(, +} } ! + is justified

38

Stable Semantics: an unsatisfactory patch

Stable extensions = - conflict-free sets attacking all outside arguments, if there is one

- {ø}, otherwise

'1)

'2)

'3)

() *)

ST(AF) = {ø } ! ( NOT justified!!!

39

Preferred semantics

Preferred semantics

Preferred extension

Maximal complete extension = max Set: •  is conflict-free •  defends all of its elements

[P.M. Dung, ’95]

Stable extensions are maximal complete extensions

•  conflict-free: by definition •  admissible: every argument attacking an extension is outside

! attacked by the extension itself •  maximal: no argument can be included!

40

Preferred semantics and floating arguments

()

')

*) +)

()

')

*) +)

()

')

*) +)

()

')

*) +)

()

')

*) +)

Grounded semantics:

PR(AF) = ST(AF) = { {', +}, {(, +} } ! + is justified

41

Preferred semantics and odd-length cycles

')

*)

()

No argument justified w.r.t. grounded and preferred semantics

PR(AF) = {ø}

ST(AF) = ø A big difference, isn’t it?

GE(AF) = {ø}

•  As stable semantics, preferred semantics handles the case of floating arguments (differently wrt grounded semantics) •  W.r.t. stable semantics it behaves “better” in the case of odd-length cycles (as the grounded semantics)

So, what remains to be done?

42

•  Stable semantics - clashes in some cases (odd-length cycles), however: - a widely applied approach (default logic, stable models of logic programming, answer set programming, etc.) - a very credulous approach: stable extensions are preferred but not viceversa ! justified arguments w.r.t. stable semantics are a (sometimes strict) superset of arguments justified w.r.t. preferred semantics, e.g.

()')

,)

+)

*)

PR(AF)={{', !}, {"}} ST(AF) = {{', !}}

()')

,)

+)

*)

Semi-stable semantics (1)

43

Semi-stable semantics (2)

•  Aims at guaranteeing existence of extensions (differently from stable semantics) + coinciding with stable semantics when stable extensions exist (differently from preferred semantics)

•  Definition:

E - SST(AF) iff

E is a complete extension such that (E U {'| E% '}) is maximal

•  Main properties:

[Verheij’96, Caminada’06]

-  A semistable extension always exists (in the finite case!) since a maximization requirement replaces “aggressive attack” -  If a stable extension E exists, then (E U {'| E% '}) includes all arguments, therefore semistable extensions # stable extensions -  In any case, semistable extensions are preferred extensions, but the opposite is not always true

44

Semi-stable semantics: examples

')

*)

()

()')

,)

+)

*)

Example for backward compatibility (and difference w.r.t. preferred semantics)

PR(AF)={{', !}, {"}}

Example for existence

The unique admissible set is empty ! trivially maximizes (E U {'| E% '} )

SST(AF)={{', !}} )= ST(AF)

45

CF2 semantics: motivation

Preferred/stable/semistable semantics and cycles

()')

()')

()')

')

*)

()

')

*)

()

A different treatment for even and odd-length cycles. Is it just a matter of symmetry and elegance?

46

Preferred/Semistable Semantics and cycles

()') +1) +2)

VS

PR(AF) =

{{', +1}, {', +2},

{(, +2} }

47

Preferred/Semistable Semantics and cycles

()

')

*) +1) +2)

VS

()') +1) +2)

VS

PR(AF) =

{{', +1}, {', +2},

{(, +2} }

PR(AF) = {{+2}}

48

Preferred/Semistable Semantics and cycles

()

')

*) +1) +2)

+1) +2)*)')

()

+)

VS

()') +1) +2)

VS

PR(AF) =

{{', +1}, {', +2},

{(, +2} }

PR(AF) = {{+2}}

PR(AF) =

{{', *, +2},

{(, +, +1}, {(, +, +2} }

NOTE: grounded semantics yields the empty set in all cases

49

Pollock example revisited (1)

Rob says Jones unrel.

Jones unreliable

Smith says Rob unrel.

Rob unreliable

Jones says Smith unrel.

Smith unreliable

Smith says it’s raining

It’s raining

Bob says it’s not raining

It’s not raining

50

Pollock example revisited (2)

Rob says Fred unrel.

Fred unreliable

Smith says Rob unrel.

Rob unreliable

Jones says Smith unrel.

Smith unreliable

Smith says it’s raining

It’s raining

Bob says it’s not raining

Fred says Jones unrel.

Jones unreliable

It’s not raining

51

Preferred Semantics and Floating Arguments again…

()

')

*) +)

VS

()

')

*) +) .)

[ two preferred extensions]

[empty set is the unique preferred extension]

NB: same behavior for semistable semantics, stable semantics clashes, grounded semantics yields the empty set in both cases

52

Strongly connected components (SCCs)

Equivalence classes under the relation of path-equivalence (mutual reachability)

()

')*) .1) .2)

()

')*) .1) .2)

()

')*) .1) .2)

53

Strongly connected components (SCCs)

SCCs form an acyclic graph

S1 and S2 are initial SCCs S1 is sccparent of S3, S4 and S5 all other SCCs precede S7

S1

S3

S5

S6

S7

S4

S2

54

CF2 semantics: the definition

E- CF2(AF) iff:

- E - MCF(AF) if |SCCSAF| = 1

- / S - SCCSAF (E0S) - CF2(AF UP_AF(S,E)) otherwise

S

UP_AF(S,E)

55

()

')

*)

()

')

*)

()

')

*)

()

')

*)

Maximal conflict-free sets

CF2 semantics and odd-length cycles (1)

56

()

')*) .1) .2)

Yields several extensions ! all arguments not justified in both cases

.1) .2)*)')

()

+)

{*,.2}, {',.1}, {',.2}, {(,.1}, {(,.2}

{',*,.2}, {(,+,.1}, {(,+,.2}

CF2 semantics and odd-length cycles (2)

57

Floating arguments with a three-length cycle

()

')*) +) .)

Extensions: {*,.}, {',.}, {(,.}

()

')*) +) .)

()

')*) +) .)

()

')*) +) .)

Defeat status

58

A problem of CF2 semantics (1)

•  Considering some examples with structured arguments, it turns out that

conflict-freeness does not entail consistency, e.g.

5.2 Semantics for reasoning with rules of thumb

When applying reasoning for constraint satisfaction, one starts with a problem that is well

understood, and then aims to write a perfect representation in a particular constraint satisfaction

formalism (like Answer Set Programming), so that the original problem can be solved in an

automated way. However, in many cases, one would like to reason about issues that are perhaps

not perfectly understood (like for instance which treatment to give to a patient) and where one has

to deal with rules of thumb, which can give reasons in favor of or against drawing a particular

conclusion. These rules of thumb are not necessarily perfect, nor do they have to be complete. The

challenge, then, is to come up with a reasonable position one can adopt based on imperfect

information. This makes it desirable to apply a semantics that satisfies crash resistance and non-

interference, since we do not want problems in one part of the knowledge base to affect other,

possibly totally unrelated parts of the knowledge base. Stable semantics is therefore not an option.

Would the semantics have to be admissibility based? That is, is it desirable that each extension

(labelling) is an admissible (or even complete) one? Again, it is difficult to provide an ultimate

answer in general: one has to refer to specific contexts. In particular, in the context of instantiated

arguments generated from an underlying logical knowledge base, admissibility can be regarded as

advantageous in relation to consistency requirements, as explained in the following.

Suppose one generates an argumentation framework based on a set of propositional formulas

P and a set of defeasible rules D. The idea is that the propositional formulas express information

that is beyond doubt and the defeasible rules express rules of thumb that can be subject to

exceptions. Now consider the following knowledge base:

P ! fjw; mw; sw; :"jt ^mt ^ st#gD ! fjw ) jt; mw ) mt; sw ) stg

This example can be interpreted as follows: John, Mary, and Suzy want to go cycling on a

tandem. The fact that John wants to get on the tandem (jw) is a reason to believe that John will be

on the tandem (jt). The same holds for Mary and Suzy. However, since the tandem only has two

seats, they cannot be on it with the three of them: :(jt4mt4 st). From this knowledge base, we

can then construct the 10 following arguments, based on an argument construction scheme as

presented in Caminada and Amgoud (2007) and Prakken (2010):

A15:(jt4mt4 st)

A25 jw

A35mw

A45 sw

A55A2 ) jt

A65A3)mt

A75A4) st

A85A6, A7, A1-:jtA95A5, A7, A1-:mt

A105A5, A6, A1-:st

Assuming the principle of restricted rebutting23 it would then follow that A8 attacks A5, A9, and

A10, that A9 attacks A6, A8, and A10, and that A10 attacks A7, A8, and A9. This yields the

argumentation framework of Figure 20.

In the argumentation framework of Figure 20, there are four complete extensions: {A1, A2, A3,

A4}, {A1, A2, A3, A4, A6, A7, A8}, {A1, A2, A3, A4, A5, A7, A9}, and {A1, A2, A3, A4, A5, A6, A10}.

The first of these is the grounded extension; the other three are stable extensions (and therefore

23 Restricted rebutting basically means that conclusion-based attacks can only be done against a conclusionthat is the consequent of a defeasible reasoning step. Thus, in our example, A8 attacks A5 but A5 does notattack A8. The reader may refer to Caminada and Amgoud (2007) for more details.

406 P . BARON I ET AL .

5.2 Semantics for reasoning with rules of thumb

When applying reasoning for constraint satisfaction, one starts with a problem that is well

understood, and then aims to write a perfect representation in a particular constraint satisfaction

formalism (like Answer Set Programming), so that the original problem can be solved in an

automated way. However, in many cases, one would like to reason about issues that are perhaps

not perfectly understood (like for instance which treatment to give to a patient) and where one has

to deal with rules of thumb, which can give reasons in favor of or against drawing a particular

conclusion. These rules of thumb are not necessarily perfect, nor do they have to be complete. The

challenge, then, is to come up with a reasonable position one can adopt based on imperfect

information. This makes it desirable to apply a semantics that satisfies crash resistance and non-

interference, since we do not want problems in one part of the knowledge base to affect other,

possibly totally unrelated parts of the knowledge base. Stable semantics is therefore not an option.

Would the semantics have to be admissibility based? That is, is it desirable that each extension

(labelling) is an admissible (or even complete) one? Again, it is difficult to provide an ultimate

answer in general: one has to refer to specific contexts. In particular, in the context of instantiated

arguments generated from an underlying logical knowledge base, admissibility can be regarded as

advantageous in relation to consistency requirements, as explained in the following.

Suppose one generates an argumentation framework based on a set of propositional formulas

P and a set of defeasible rules D. The idea is that the propositional formulas express information

that is beyond doubt and the defeasible rules express rules of thumb that can be subject to

exceptions. Now consider the following knowledge base:

P ! fjw; mw; sw; :"jt ^mt ^ st#gD ! fjw ) jt; mw ) mt; sw ) stg

This example can be interpreted as follows: John, Mary, and Suzy want to go cycling on a

tandem. The fact that John wants to get on the tandem (jw) is a reason to believe that John will be

on the tandem (jt). The same holds for Mary and Suzy. However, since the tandem only has two

seats, they cannot be on it with the three of them: :(jt4mt4 st). From this knowledge base, we

can then construct the 10 following arguments, based on an argument construction scheme as

presented in Caminada and Amgoud (2007) and Prakken (2010):

A15:(jt4mt4 st)

A25 jw

A35mw

A45 sw

A55A2 ) jt

A65A3)mt

A75A4) st

A85A6, A7, A1-:jtA95A5, A7, A1-:mt

A105A5, A6, A1-:st

Assuming the principle of restricted rebutting23 it would then follow that A8 attacks A5, A9, and

A10, that A9 attacks A6, A8, and A10, and that A10 attacks A7, A8, and A9. This yields the

argumentation framework of Figure 20.

In the argumentation framework of Figure 20, there are four complete extensions: {A1, A2, A3,

A4}, {A1, A2, A3, A4, A6, A7, A8}, {A1, A2, A3, A4, A5, A7, A9}, and {A1, A2, A3, A4, A5, A6, A10}.

The first of these is the grounded extension; the other three are stable extensions (and therefore

23 Restricted rebutting basically means that conclusion-based attacks can only be done against a conclusionthat is the consequent of a defeasible reasoning step. Thus, in our example, A8 attacks A5 but A5 does notattack A8. The reader may refer to Caminada and Amgoud (2007) for more details.

406 P . BARON I ET AL .

also semi-stable and preferred extensions). It should be mentioned that the sets of conclusions

associated with these extensions are consistent: {jw; mw; sw; :(jt4mt4 st)}, {jw; mw; sw;

:(jt4mt4 st); mt; st; :jt}, {jw; mw; sw; :(jt4mt4 st); jt; st; :mt}, and {jw; mw; sw; :(jt4mt4 st); jt; mt; :st}. Now let us examine what happens if one lowers the requirement of

admissibility to the mere property of conflict-freeness. In that case, we lose consistency, since the

set of arguments {A5, A6, A7, A1} is conflict-free and yet its associated set of conclusions {jt; mt; st;

:(jt4mt4 st)} is inconsistent. It is therefore important to notice that conflict-freeness by itself

does not imply consistency; in order to yield consistent conclusions, something stronger is needed.

In Caminada and Amgoud (2007), it is proved that, under the right procedure of argument

construction, admissibility of a set of arguments is a condition that is strong enough to yield

consistent conclusions of this set.

What does this mean for non-admissibility-based semantics, such as stage or CF2? First of all, it

should be mentioned that the above-described example is not a counter example against stage

semantics or CF2 semantics. This is because the stage extensions (and CF2 extensions) are {A1, A2,

A3, A4, A6, A7, A8}, {A1, A2, A3, A4, A5, A7, A9}, and {A1, A2, A3, A4, A5, A6, A10}, which yield

consistent conclusions.

For stage semantics, it is possible to come up with a slightly more complex example where a stage

extension does yield inconsistent conclusions. Such an example24 could be constructed by adopting

the argumentation framework of Figure 20 and adding three self-attacking arguments A8, A9, and

A10, where A8 is also attacked by A5, A9 by A6, and A10 by A7. Such arguments could be constructed

by using the notion of self-undercut, like is done in Caminada (2005).

As for CF2 semantics, a counterexample against its consistency has been discovered by

Wolfgang Dvorak (private communication). Basically, the idea is to add three defeasible rules

) :jt, ) :mt and ) :st to D. This yields three additional arguments in the argumentation

framework of Figure 20: A11 :) :jt, A12 :) :mt and A13 :) :st. It then follows that A5 attacks

A11, and A11 attacks A5, A9 and A10, and A6 attacks A12, and A12 attacks A6, A8 and A10, and A7

attacks A13, and A13 attacks A7, A8 and A9. In the resulting argumentation framework,

fA5;A6;A7;A1;A2;A3;A4g will be one of the CF2 extensions, yielding the inconsistent set of

conclusions fjt;mt; st;:!jt ^mt ^ st"; jw;mw; swg. For those applications where consistency is

A1

A3A2

A4

A5

A6A7

A8

A9A10

Figure 20 Conflict-freeness is not enough to obtain consistent conclusions

24 This counter example was presented at COMMA 2010 and is available at: http://www.ing.unibs.it/comma2010/presentations/P15-Caminada.pdf

An introduction to argumentation semantics 407

59

A problem of CF2 semantics (2)

•  By slightly complicating example, one can find a CF2 extension

which is not consistent

•  On the other hand, admissibility entails consistency

(proved by Caminada & Amgoud 2007, AIJ)

ADMISSIBILITY CAN BE A DESIRED REQUIREMENT

60

TO CONCLUDE…

GROUNDED

PREFERRED STABLE

SEMISTABLE

IDEAL

STAGE PRUDENT

ROBUST

CF2 TOLERANT

SUSTAINABLE

EACH SEMANTICS HAS ITS OWN ROLE… … WHICH ONE IS A GOOD RESEARCH QUESTION…!

61

What about general principles?

Grounded Preferred CF2 Semistable

CF-principle Yes Yes Yes Yes Admissibility Yes Yes No Yes Reinstatement Yes Yes No Yes Weak reinstatement Yes Yes Yes Yes CF-reinstatement Yes Yes Yes Yes I-maximality Yes Yes Yes Yes Directionality Yes Yes Yes No Weak Skepticism Adequacy

Yes [all forms]

No Yes No

Weak Resolution Adequacy

No Yes [all forms]

No Yes

Here we consider only some semantics - see [Baroni & Giacomin ’06]

62

Applications and principles

PRINCIPLES

APPLICATION DOMAINS

TO BE STUDIED

SEMANTICS

TO BE DEEPENED

63

Semantics and attitude

/ E2 -E2, 1 E1 -E1 : E1" E2

: E W E2 E1

Fig. 3. !S!+ , !S

!! and !S! relations for any argumentation framework.

! "

#

$ %

Fig. 4. A case showing that GRP !S" COP, GR !S

" GRP , GRP !S" PRP , and

GR !S! GRP .

!

%

" $

Fig. 5. A case showing that ID !S" GR, ID !S

" CF2, ID !S! GR, and CF2 !S

! PR.

$

!

"

%

Fig. 6. A case showing that PR !S" ID.

this is not the case, starting from the relations !S"+ , !S

"! and !S": due to the

order of implication between them, it is su!cient to prove that no additionalrelationships hold for !S

". First, considering Figure 4, we have ECOP(AF) ={{!, "}, {#, "}} (note in particular that # indirectly conflicts with !), whileGPE(AF) = {{!, "}}, which entails GRP !S

" COP . Since it also holds thatGE(AF) = {!, ", #}, it is easy to see that GE(AF) " GPE(AF), i.e. GR !S

"GRP . Going up in the diagram, the example of Figure 5 shows that ID !S

"GR, since ID(AF) = {$, !} while GE(AF) = ". The well known example ofFigure 6, concerning so-called floating arguments, shows that PR !S

" ID,since EPR(AF) = {{#, !}, {$, !}} while ID(AF) = ". The fact that SST !S

"

16

SKEPTICAL REASONING CREDULOUS REASONING

/ E1 -E1, 1 E2 -E2 : E1" E2

: E C E2 E1

64

MANY THANKS

FOR YOUR KIND ATTENTION

65

Selected references (1)

S. Toulmin, “The Uses of Argument” Cambridge University Press, 1958.

Landmark argumentation papers and books

R. P. Loui, “Defeat Among Arguments: a System of Defeasible Inference”, Computational Intelligence, vol. 3(3), 1987.

J. Pollock, “Defeasible Reasoning”, Cognitive Science, vol. 11(4), 1987.

G. Simari & R. P. Loui, “A mathematical treatment of defeasible reasoning and its implementation”, Artificial Intelligence, vol. 53(2-3), 1992.

Argumentation surveys

H. Prakken & G.A.W. Vreeswijk, “Logics for Defeasible Argumentation”, in Handbook of Philosophical Logic, 2nd Edition, Kluwer Academic Publishers, 2001.

C.I. Chesnevar, A.G. Maguitman, R.P. Loui, “Logical models of argument”, ACM Computing Surveys, vol. 32(4), 2000.

66

Selected references (2)

Books

P. Besnard & A. Hunter, “Elements of Argumentation”, MIT Press, 2008.

“Argumentation in Artificial Intelligence”, edited by I. Rahwan and G. R. Simari, Springer, 2009.

D. Walton, “Fundamentals of critical argumentation”, Cambridge University Press, 2006.

Dung’s influential paper on abstract argumentation

P.M. Dung, “On the Acceptability of Arguments and Its Fundamental Role in Nonmonotonic Reasoning, Logic Programming, and n-Person Games”, Artificial Intelligence, vol. 77(2), 1995.

Argumentation semantics survey

P. Baroni, M. Caminada, M. Giacomin “An introduction to argumentation semantics”, The Knowledge Engineering Review, vol. 26(4),2011.

67

P. Baroni, M. Giacomin, G. Guida, “SCC-recursiveness: a general schema for argumentation semantics” Artificial Intelligence, vol. 168(1-2), 2005.

Semantics

Selected references (3)

B. Verheij, “Two approaches to dialectical argumentation:admissible sets and argumentation stages”, Proc. of the 8th Dutch Conference on Artificial Intelligence, 1996

M. Caminada, “Semi-Stable Semantics”, Proc. of 1st International Conference on Computational Models of Arguments (COMMA 2006), 2006

P.M. Dung, P. Mancarella, F. Toni, “A dialectic procedure for sceptical, assumption-based argumentation”, Proc. of 1st International Conference on Computational Models of Arguments (COMMA 2006), 2006

S. Coste-Marquis, C. Devred, P. Marquis, "Prudent Semantics for Argumentation Frameworks", Proc. of 17th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2005), 2005

68

Semantics

P. Baroni, M. Giacomin, G. Guida, “SCC-recursiveness: a general schema for argumentation semantics” Artificial Intelligence, vol. 168(1-2), 2005.

P. Baroni, M. Giacomin, “Resolution-based argumentation semantics”, Proc. of 2nd International Conference on Computational Models of Arguments (COMMA 2008), 2008

G.A. Bodanza, F.A. Tohmé, “Two approaches to the problems of self-attacking arguments and general odd-length cycles of attack” Journal of Applied Logic, to appear.

H. Jakobovits & D. Vermeir, "Robust Semantics for Argumentation Frameworks", Journal of Logic and Computation 9(2), 1999

Selected references (4)

P. Baroni, P. Dunne, M. Giacomin, “Computational Properties of Resolution-based Grounded Semantics”, IJCAI 2009, to appear.

69

General criteria for semantics evaluation and comparison

M. Caminada & L. Amgoud, “On the evaluation of argumentation formalisms”, Artificial Intelligence, vol. 171(5-6), 2007.

P. Baroni, M. Giacomin, G. Guida, “On principle-based evaluation of extension-based argumentation semantics”, Artificial Intelligence, vol. 171(10-15), 2007.

P. Baroni, M. Giacomin, “Skepticism relations for comparing argumentation semantics”, International Journal of Approximate Reasoning, vol. 50(6), 2009.

Selected references (5)

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TO)%H5&,

If John did not kill Mary, then John is innocent.

Judge

Wit2

Corner

Wit1

I saw John killing Mary, thus John killed Mary.

John was at the theater with me when Mary was killed, thus John did not kill Mary.

Mary was killed before 6 p.m., thus when Mary was killed the show was still to begin

A

B

C

D

TO)%H5&,

If John did not kill Mary, then John is innocent.

Judge

Wit2

Corner

Wit1

I saw John killing Mary, thus John killed Mary.

John was at the theater with me when Mary was killed, thus John did not kill Mary.

Mary was killed before 6 p.m., thus when Mary was killed the show was still to begin

A

B

C

DA(D) = 0.3

A(C) = 1.0

A(B) = 0.2

A(A) = 1.0

TO)%H5&,

If John did not kill Mary, then John is innocent.

Judge

Wit2

Corner

Wit1

I saw John killing Mary, thus John killed Mary.

John was at the theater with me when Mary was killed, thus John did not kill Mary.

Mary was killed before 6 p.m., thus when Mary was killed the show was still to begin

A

B

C

Dα(D) = 0.3

α(C) = 0.7

α(B) = 0.2

α(A) = 0.8

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