Assumption-based Argumentation Dialogues
Xiuyi Fan
Department of Computing, Imperial College London
1 / 23
Outline
Introduction
Background (ABA)
Dialogue
Results
Conclusion
2 / 23
Key Aspects
Formal modelling of (two-agent) argumentation dialogues
Generic framework
Dened with legal-move functions and outcome functions
Connect dialogues with argumentation semantics
3 / 23
J
A
Motivation, Jenny ( ) and Amy ( ) - lm dialogue
Let's see if Terminator is a good movie to watch.
OK.
J: I would like a movie that is fun and has a good screening time.
A: OK.
J: To me, a movie is fun if it is an action movie.
A: OK.
J: And, Terminator is an action movie.
A: OK.
J: I also believe Terminator starts at the right time.
A: Are you sure it is not going to be too late?
J: Why?
A: I don't know. I am just afraid so.
J: It won't be too late if it nishes by 10 o'clock.
A: I see. Indeed Terminator nishes by 10 o'clock.
J: OK.
A: OK.
J:
A:
4 / 23
Background ABA
Formally, an ABA framework is a tuple hL, R, A, Ci where
hL, Ri is a deductive system, with a language L and a set of
inference rules R of the form s0 ← s1 , ..., sm (m ≥ 0),
A ⊆ L is a (non-empty) set, whose elements are referred to as
assumptions,
C is a total mapping from A into L, C(α) is a contrary of α.
Arguments are deductions of claims supported by sets of
assumptions.
Attacks are directed at the assumptions in the support of
arguments.
A set of arguments, Args , is conict-free if and only if the
union of all sets of assumptions that support arguments in
Args does not attack itself.
A set of arguments, Args , is admissible if and only if it is
conict-free and it attacks all sets of arguments which attack
Args .
5 / 23
ABA - Example
Jenny's ABA framework:
Rules:
watchMovie(X) ← fun(X), goodScreenTime(X)
fun(X) ← actionMovie(X)
actionMovie(Terminator)
Assumptions:
goodScreenTime(X)
late(X)
Contraries:
C (goodScreenTime(X)) = late(X)
C (late(X)) = nishByTen(X)
6 / 23
ABA - Example (2)
watchMovie(O Terminator
kWWWWW )
W
WWWWW
WWWWW
WWWWW
goodScreenTime(Terminator )
fun(Terminator
)
O
actionMovie(O Terminator )
τ
Figure: Jenny's argument for watching Terminator
.
7 / 23
Agent
Agent1 (Jenny) with ABA framework
AF1 = hL, R1 , A1 , C1 i
and Agent (Amy) with ABA framework
2
AF2 = hL, R2 , A2 , C2 i
The agents share the same language L while having potentially
dierent rules, assumptions and contraries of assumptions.
8 / 23
Utterance & Dialogue
Utterance:
hai , aj , Target , Content , ID i
Content:
clm(claim), rl (rule ), asm(assumption), ctr (contrary ), π
Related Utterance:
rl (p ← q ), rl (q ← r )
rl (p ← a), asm(a)
asm(a), ctr (a, c )
...
Dialogue:
start with a claim;
all utterances are related;
ends with two passes.
9 / 23
Dialogue - Example
J:
A:
J:
A:
J:
A:
J:
A:
J:
A:
J:
A:
J:
A:
J:
A:
hJ , A, 0, clm(watchMovie (t )), 1i
hA, J , 0, π, 2i
hJ , A, 1, rl (watchMovie (t ) ← fun(t ), goodScreenTime (t )), 3i
hA, J , 0, π, 4i
hJ , A, 3, rl (fun(t ) ← actionMovie (t )), 5i
hA, J , 0, π, 6i
hJ , A, 5, rl (actionMovie (t )), 7i
hA, J , 0, π, 8i
hJ , A, 3, asm(goodScreenTime (t )), 9i
hA, J , 9, ctr (goodScreenTime (t ), late (t )), 10i
hJ , A, 0, π, 11i
hA, J , 10, asm(late (t )), 12i
hJ , A, 12, ctr (late (t ), nishByTen(t )), 13i
hA, J , 13, rl (nishByTen(t )), 14i
hJ , A, 0, π, 15i
hA, J , 0, π, 16i
10 / 23
ABA Framework Drawn from a Dialogue
Given a dialogue, we drawn a framework by extracting its rules,
assumptions, and contraries.
The framework drawn from a dialogue δ = hu , . . . , un i is
hL, Rδ , Aδ , Cδ i where
1
Rδ = {r |rl (ρ) is the content of some ui in δ};
Aδ = {a|asm(a) is the content of some ui in δ};
Cδ is a mapping such that, for any a ∈ Aδ , Cδ (a) = c such
that ctr (a, c ) is the content of some ui in δ , if one exists, and
is undened otherwise.
11 / 23
ABA Framework Drawn from a Dialogue - Example
Rules:
watchMovie(X) ← fun(X), goodScreenTime(X)
fun(X) ← actionMovie(X)
actionMovie(Terminator)
nishByTen(Terminator)
Assumptions:
goodScreenTime(X)
late(X)
Contraries:
C (goodScreenTime(X)) = late(X)
C (late(X)) = nishByTen(X)
12 / 23
Dialectical Trees
We use the notions of legal-move and outcome functions to
guarantee that dialogues compute admissible arguments. These
renements are given using dialectical trees.
Nodes are either proponent or opponent.
Nodes are labelled with pairs of multi-sets of sentences
(marked, unmarked).
Nodes are associated with an utterance in the dialogue from
which the tree is extracted.
Ignore all passes.
13 / 23
Dialectical Trees - Example
wM (t )O : P [1]
fun(t ), goodScreenTime
(t ) : P [3]
O
actionMovie (t ), goodScreenTime
(t ) : P [5]
O
goodScreenTime
(t ) : P [7]
O
M
goodScreenTime
O (t ) : P [9]
late (t ) O : O [10]
late (t )MO : O [12]
nishByTenO (t ) : P [13]
{} : P [14]
14 / 23
Legal-move Function
Works like a protocol - denes what can be uttered.
A legal-move function is a mapping λ : D 7→ U .
The base legal-move function denes follows:
the rst utterance is the claim;
no repeated utternaces to the same target;
one contrary per assumption.
15 / 23
Legal-move Function (2)
Built on top of the base legal-move function, we further dene the
following legal-moves.
1 Flat legal-move functions:
if a sentence p is declared as an assumption, then there is no
rule such that p ← q.
Vice Versa.
2
Patient legal-move functions:
an argument is attacked only after it has been fully
constructed.
3
Focused legal-move functions:
only one way of supporting a proponent node is allowed in a
dialogue.
16 / 23
Outcome Function
Base: an outcome function is a mapping ω : D 7→ {true , false }.
1 ABA outcome function:
the framework drawn from the dialogue is an ABA framework.
2
Exhaustive outcome function:
if a sentence has been uttered, then it must be uttered at all
occasions that are suitable.
3
Last-word outcome function:
the proponent is able to counter-attack all attacks from the
opponent.
4
admissible outcome function:
no overlap between proponent assumptions and opponent
assumptions.
17 / 23
Results
Admissible outcome function: ωADM (δ)
Dialectical tree drawn from a dialogue δ : T (δ)
The union of all assumptions in proponent nodes: Def (T (δ))
Theorem:
a
Given a successful dialogue Da (s ) = δ , if ωADM (δ) = true , then
there exists an argument S ` s that belongs to an admissible
extension supported by Def (T (δ)) w.r.t. the ABA framework
drawn from δ .
j
i
Proof:
If ωADM (δ) = true , then T (δ) can be mapped to a equivalent
admissible concrete dispute tree, as dened in [Dung2006].
18 / 23
Key Aspects
Formal modelling of (two-agent) argumentation dialogues
Generic framework
Dened with legal-move functions and outcome functions
Connect dialogues with argumentation semantics
Questions?
19 / 23
Utterance
An utterance from agent ai to agent aj (i , j = 1, 2, i 6= j ) is a tuple
hai , aj , InReply , C , ID i, where:
C (the content) is of one of the following forms:
clm(s ) for some s ∈ L (a claim),
rl (s0←s1 ,. . . ,sm ) for some s0 ,. . . ,sm ∈ L (a rule),
asm(a) for some a ∈ L (an assumption),
ctr (a, s ) for some a, s ∈ L (a contrary),
a pass sentence π , such that π ∈
/ L.
ID ∈ N (the identier).
InReply ∈ N ∪ {0} (the target); InReply < ID .
20 / 23
Related Utterances
For any two utterances ui 6= uj , ui = hFi , Ti , _, Ci , ID i and
uj = hFj , Tj , ID , Cj , _i, we say that uj is related to ui i one of the
following cases holds:
1 Ci = rl (ρi ), Cj = rl (ρj ), and Head (ρj ) ∈ Body (ρi );
2 Ci = rl (ρ), Cj = asm (a), and a ∈ Body (ρ);
3 Ci = asm (a) and Cj = ctr (a, s );
4 Ci = ctr (a, s ) and Cj = asm (s );
5 Ci = ctr (a, s ), Cj = rl (ρ) and Head (ρ) = s ;
6 Ci = clm (s ), Cj = rl (ρ), and Head (ρ) = s ;
7 Ci = clm (s ) and Cj = asm (s ).
21 / 23
Dialogue
A dialogue Daa (s ) (between agents ai and aj , for claim s ∈ L),
i , j = 1, 2, i 6= j , is a nite sequence hu , . . . , un i, n ≥ 0, where
each ul , l = 1, . . . , n, is an utterance from ai when l is odd, and aj
when l is even, and:
i
j
1
1
2
3
4
the content of the rst utterance, u1 , is clm(s );
no other utterance has clm(_) as its content;
each non-pass-utterance other than the claim utterance is
related to its target utterance;
the target of pass- and claim utterances is 0;
no two consecutive utterances are pass-utterances, other than
possibly the last two utterances, un− and un .
1
22 / 23
Legal-move Function
Dialogues: D
Utterances: U
Turn-taking function: θsi
A legal-move function is a mapping λ : D 7→ U such that, given
δ = Daa (s ) = hu , . . . , un i:
(
hai , aj , 0, clm(s ), 1i
if n = 0
λ(δ) =
where
hax , ay , t , content (Cx ), n + 1i if n > 0
ax = θsi (δ), ay 6= ax , and
1 No repeated utterance to the same target.
2 One contrary per assumption.
i
j
1
23 / 23