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?