A DYNAMIC ARGUMENTATION

FRAMEWORK

ND Rotstein, MO Moguillansky,AJ García, GR Simari

The DAF, Introduction

Extension of Dung’s AFs

Representation of argumental structures

Modelling knowledge dynamics

AFs as an instance of a DAF

Dung’s AFs

The most abstract viewpoint on argumentation

Simple yet meaningful

Suitable for grounding argumentation-based theoretical approaches

Main focus: argumentation semantics

Why Another AF?

Composition of arguments (argumental substructures)

How to determine defeat (conflict + preference)

Consideration of evidence (dynamics)

Evidence; Arguments

In the DAF:

Arguments have an interface: premises and claims

Arguments may chain: argumental structures

Arguments might not hold now: evidence

Arguments

An argument A defined upon a domain language L:

pr(A) ! 2L, cl(A) ! L

cl(A) " pr(A)

cl(A) " pr(A)

Pieces of evidence also belong to L

Coherency

A set of evidence is a consistent set of facts representing the current environment

Given a set ArgsL of arguments, the conflict relation ⋈:ArgsL×ArgsL # ⋈ # {(A,B) | cl(A) = cl(B) or cl(A) ! pr(B)}

Coherent argument wrt. a set E of evidence:(consistent) cl(A) " E, (non-redundant) cl(A) " E

Active Arguments; Support

Aa b a b

c

A

B

xx

E = {a, b} E = {a, c}

Active Arguments; Support

Aa b a b

c

A

B

xx

E = {a, b} E = {a, c}

support

Active Arguments

a b cA B

x

E = {a, b, c}

b

Argumental Structures

Trees of arguments linked from claims to premises

Each premise is supported by at most one argument

Top argument: top(S) - Set of arguments: args(S)

Claim: cl(S) - Premises: pr(S)

Well-formed Arg. Structure

(Premise consistency) no a,b ! pr(S) s.t. a = b

(Consistency) no A,B s.t. A⋈B

(Non-circularity) if cl(B) ! pr(A) then A does not transitively support B

(Uniformity) if A supports B through b, then A supports every Bi with b as a premise

Non-WF Arg. Structures, e.g.

b

a

¬a

c

c

a

b

a

b c

a

x b

b c

a

x b

y

Non-WF Arg. Structures, e.g.

b

a

¬a

c

c

a

b

a

b c

a

x b

b c

a

x b

y

Non-WF Arg. Structures, e.g.

b

a

¬a

c

c

a

b

a

b c

a

x b

b c

a

x b

y

Non-WF Arg. Structures, e.g.

b

a

¬a

c

c

a

b

a

b c

a

x b

b c

a

x b

y

Non-WF Arg. Structures, e.g.

b

a

¬a

c

c

a

b

a

b c

a

x b

b c

a

x b

y

Conflict + Preference = Defeat

Si ⊑ S iff Si is a structure and args(Si) $ args(S)

S1 ≍ S2 iff top(S1) ⋈ top(S2)

Preference function pref(S1,S2) = [S1 | S2 | !]S1 % S2 iff Sk ⊑ S2, Sk ≍ S1 and pref(S1,Sk) = S1

The Framework

DAF F = ⟨E, W, ⋈, pref⟩

E: current evidence

W: working set of arguments

⋈: conflict relation upon pairs of arguments

pref: function defined over pairs of arg. structures

The DAF, e.g.

many_copsA2

good_security

thieves andpoor_security

A3

dangerous_route

underpaid_copsA1

volunteer_copsB1

good_security

unacquaintedB2

poor_security

B3foreign_cops

unacquaintedB2

poor_security

Active Arg. Structure

S is active wrt. E iff S is well-formed, pr(S) $ E and &A ! args(S) is coherent

If S is active then &A ! args(S) is active

A is active iff 'S s.t. top(S) = A and S is active

(Minimality) If S is active then ∄Si ⋤ S s.t. Si is active

Active Arg. Structure, e.g.

cA

d

a bB

cA

d

S1 S2

well-formed

arguments

are activeE = {a, b}

Active Instance

F = ⟨E,W,⋈,pref⟩, the active instance is (S,R)

S: set of active arg. structures

R: active attack relation over S

The active instance of a DAF is a Dung’s AF

Active Instance, e.g.

mcA2

gs

th psA3

dr

upcA1 vc

B1

gs

unB2

ps

B3fc

unB2

ps

Active Instance, e.g.

mcA2

gs

th psA3

dr

upcA1 vc

B1

gs

unB2

ps

B3fc

unB2

ps

E1 = {mc, upc, th}

Active Instance, e.g.

mcA2

gs

th psA3

dr

upcA1 vc

B1

gs

unB2

ps

B3fc

unB2

ps

E1 = {mc, upc, th}E2 = {mc, upc, th, vc}

Active Instance, e.g.

mcA2

gs

th psA3

dr

upcA1 vc

B1

gs

unB2

ps

B3fc

unB2

ps

E2 = {mc, upc, th, vc}E3 = {upc, th, vc, un, fc}

Active Instance, e.g.

mcA2

gs

th psA3

dr

upcA1 vc

B1

gs

unB2

ps

B3fc

unB2

ps

E3 = {upc, th, vc, un, fc} E4 = {upc, th, vc, fc}

Changing the DAF

Evidence changes

Old arguments no longer reasonable

New arguments can be taken into account

Non-syntactic conflicts can be added/deleted

...any combination of the above

Argument Contraction, e.g.

mcA2

gs

th psA3

dr

upcA1 vc

B1

gs

unB2

ps

B3fc

unB2

ps

E4 = {upc, th, vc, fc}

Argument Contraction, e.g.

mcA2

gs

th psA3

dr

upcA1 vc

B1

gsunB2

ps

E4 = {upc, th, vc, fc}

Semantics

Plenty of argumentation semantics have been defined over Dung’s AF

These results can be reutilised for the DAF through its active instance

Conclusions

Richer representation for arguments

Consideration of argumental structures

Evidence as a separate entity

Active/Inactive knowledge

Change operations and dynamics

Related/Future Work

Boella et al. approach on dynamics: framework changes that do not affect extensions

Our focus is KR; both approaches are combinable

Using the DAF for hypothetical/abductive reasoning

Thank you

Questions?