An Introduction to Abstract
Argumentation
Dr. Pierpaolo Dondio,
DIT – School of Computing
2012/2013
1
Agenda
» Introduction
» What is argumentation theory?
» Abstract Argumentation Frameworks (Dung 1995)
» Stable, Grounded and Preferred Semantics
» Instantiating Abstract Argumentation
» Applications
» Probabilistic and Uncertain Argumentation
2
What is argumentation Theory
» the interdisciplinary study of how conclusions can be
reached through logical reasoning
» Key Questions:
» How arguments are built?
» How humans negotiate, discuss, argue?
» Who wins? i.e. how can we identify acceptable valid
arguments and discard invalid?
» We focus on AI developments
» Computational Logic & Non-monotonic reasoning
» Abstract Argumentation
3
Nonmonotonic logic
» Standard logic is monotonic:
» If S |-  and S  S’ then S’ |- 
» But commonsense reasoning is often nonmonotonic:
» John is an adult, Adults are usually employed, so John
is presumably employed
» But suppose also that John is a student and students
are usually not employed …
» We often reason with rules that have exceptions
» We apply the general rule if we have no evidence of
exceptions
» But must retract our conclusion if we learn evidence of
an exception
Source of non-monotonicity
»
»
»
»
»
»
Exceptions
Moral Rules
Legal Rules
(False) Generalizations
Limited knowledge
…
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5
Some nonmonotonic logics
»
»
»
»
Default logic (Ray Reiter)
Logic programming (Robert Kowalski)
…
Argumentation logics
» Argumentation Logics, and in particular Abstract
Argumentation Frameworks studied here, have
the same expressive power as Default Logic
Argumentations are nonmonotonic
Paul is a good at Maths
A
Paul is not good at Maths
B
Paul got 90%
in his final
Maths test
Exam result is
not a valid
evidence
C
Exam was
very easy this
year
If you get high marks in
a Maths test you are
good at Maths
Exam result is
not a valid
evidence
Paul was never able to help
me with my Maths
homework
What Mary said
is not
trustworthy
D
E
Mary said
Paul copied
the exam
Mary is a
well-known
layer
Abstract Argumentation Frameworks
A
C
B
D
E
(Dung 1995) An argumentation framework is a pair
𝐴𝐹 = < 𝐴𝑟, 𝑅 >
where 𝐴𝑟 is a set of arguments, and 𝑅 is a binary
relation on 𝐴𝑟, i.e. 𝑅  𝐴𝑟 × 𝐴𝑟.
» For two arguments A,B, the meaning of 𝑅(𝐴, 𝐵) is that
A represents an attack against B.
The Key Problem
» I want to say something about arguments :
» Which arguments are acceptable? Which are not?
» When to abstain?
» A argumentation semantics sets the rules
(postulates) used to answer the above questions
» In the labelling approach, we label each
argument
» IN – Argument is accepted
» OUT – Argument is rejected
» UNDEC – Nothing can be said on argument status
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Starting Point
» Given 𝐴𝐹 = < 𝐴𝑟, 𝑅 > we define a labelling
function as a total function over 𝐴𝑟:
𝐿: 𝐴𝑟 → 𝑖𝑛, 𝑜𝑢𝑡, 𝑢𝑛𝑑𝑒𝑐
» We also define the 𝑖𝑛(𝐿) = 𝑎 ∈ 𝐴𝑟 𝐿 𝑎 = 𝑖𝑛 }
» Starting basic idea. We label all arguments
according to these simple rules:
1. An argument in each labelling is either IN or OUT
2. An argument is ‘in’ iff all arguments defeating it are ‘out’.
3. An argument is ‘out’ iff it is defeated by an argument that
is ‘in’.
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10
Example
» A: Mark is good at Math, he got 90%!
» B: John said Mark copied the test!
» C: John is a well-known layer!
A
B
C
» Our rule works fine. We expect
»
»
»
»
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A in
B out
C in
This is called reinstatement; A is reinstated by C.
11
Example 2
» D: Sarah says that John is honest!
A
B
C
D
» And now? The graph is cyclic! Our basic rule does not work
anymore!
» Multiple solutions:
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A
B
C
D
A
B
C
D
A
B
C
D
12
Complete Semantics
» Grounded (Pollock, Dung)
» Preferred (Dung)
» Stable (Dung, Caminada)
» Many more.. Semi-stable, CF2..
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Complete Semantics: Conflict-free
A set S of arguments is said to be conflict-free if there
are no arguments A,B in S such that A attacks B .
B
S
A
Arg
(A  S
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&
attack(A,B))
=>BS
14
Complete Semantics - Admissibility
An argument A is admissible with respect to a set S if S
can defend A with an argument B  S against all
attacks C on A.
We want to accept arguments for which there is an
admissibility set
A
S
B
C
(A  Arg & attacks(C,A))
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Complete Semantics
» It accepts all the conflict-tree and admissible arguments
» In general multiple labelings are valid
» It can be proven that the following labelling rules
exactly compute the complete semantics
» if A is labelled in then all attackers of A are labelled out
» if all attackers of A are labelled out then A is labelled in
» if A is labelled out then A has a attacker that is labelled
in, and
» if A has a attacker that is labelled in then A is labelled
out
» A is labelled undec iff at least one attacker is undec and
thre is no attacker labelled in
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Complete Semantics
» Many sub-semantics have been defined over a
complete labeling. Let 𝐴𝐹 = (𝐴𝑟 , 𝑅)
Grounded
» L is a complete labellings such as undec(L) is
maximal w.r.t. to set inclusion
Preferred
» L is a complete labellings such as in(L) is
maximalw.r.t. to set inclusion
Stable
» L is a complete labelling such that undec(L) = ∅
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17
Hierarchies of Semantics (Caminada)
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Example
» Grounded: all undecided
» Stable: IN={b,d} ; OUT={a,c,e}
» Preferred:
» IN={b,d} ; OUT={a,c,e}
» IN={a} ; OUT={b} ; UNDEC={c,d,e}
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The example of floating assignment
» Grounded: all undecided
A
» Preferred:
» IN={b} ; OUT={c,a}
» IN={c} ; OUT={b,a}
B
C
» Stable: same as preferred
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The Nixon Diamonds & 3-Cycle
» Grounded: undec
» Preferred:
» IN={a}, OUT={b}
» IN={b}, OUT={a}
C
B
» Stable: same as preferred
» Grounded: undec
» Preferred: undec
» Stable: none
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A
B
C
21
Credulous vs Sceptical Acceptability
» After each labelling, we are left with three set of
arguments In general, there are multiple labelings (one
for grounded, maybe many for preferred or stable)
How can I accept arguments?
» Credulous acceptance.
» If there is at least one labelling where argument A is labeled IN,
accept it
» Sceptical acceptance
» An argument must have the same labels in all the labelings
» Grounded acceptance implies sceptical preferred or
stable acceptance
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22
Instanciating an AA
» Nothing is said about argument internal
arguments structure
» Arguments as modus-ponens rules
𝐴, 𝐴 → 𝐵 ⊢ 𝐵
» How can these rule be attacked?
» Rebuttals: attack 𝐵
» Undercutting: attack 𝐴 → 𝐵
» Undermining : attack 𝐴
» The red light example
» If an object looks red, it is red
» What if the object is illuminated by red light?
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Arguments as logical consequences
» propositional language 𝐿 whose atoms is a finite set
{a, b, c … } and its connectives are {∧,∨, →, ¬}.
» The symbol ⊢ means logical consequence.
» for 𝜙 ⊆ 𝐿 and a formula 𝛼 in 𝐿, 𝜙, 𝛼 is an argument
iff 𝜙 ⊢ 𝛼 and it does not exits 𝜙 ′ ⊆ 𝜙 so that 𝜙 ′ ⊢ 𝛼
» Given argument 𝑎 = 𝜙, 𝛼 , we call 𝜙 the support of 𝑎
and 𝛼 the claim of 𝑎.
» Given two arguments 𝑎 and 𝑏, we define rebuttal and
undercut attacks in the following way:
» 𝑎 rebuts 𝑏 if 𝑐𝑙𝑎𝑖𝑚(𝑎) ⊢ ¬𝑐𝑙𝑎𝑖𝑚(𝑏)
» 𝑎 undercuts 𝑏 if there is 𝜑 ⊆ 𝑠𝑢𝑝𝑝𝑜𝑟𝑡(𝑏) such that
𝑐𝑙𝑎𝑖𝑚(𝑎) ≡ ¬ ∧ 𝜑
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Some Applications
» Legal Reasoning (Prakken)
» Computational Trust
» Dondio (2007 – 2013, Phd)
» Multi-Agents Conflict Resolution
» Decision Support Systems
» Healtchare (Longo 2012)
» … many more
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25
Trust as a form of Argumentation
PP+
+
-
+
Sim+
R-
=
A1 =
=
A3
A2
»
»
»
»
»
𝑷𝑷+ : if agent has high level of past performance, then trust it
𝑺𝒊𝒎+ : if agent is similar to Carol, then trust him
𝑹− : if agent has low reputation, then distrust him
𝑨𝟏 : if task context is new to the agent, then invalidate argument PP
𝑨𝟐 : if past performance are high an reputation is low, then prefer past
performance
» 𝑨𝟑 ∶ if past-performance are out of date, then invalidate argument PP
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Problems with Abstract Argumentation
» Nothing is said about arguments. There is the
danger to model impossible situations or derive
useless conclusions
» Too coarse!
» Many times, you are left with no arguments or multiple
labelings and nothing to choose about
» Arguments are perceived with difference strength,
importance, maybe based on their likelihood or
certainty level or subjective preferences..
» How can we build a numerical argumentation?
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27
Towards Argumentation with Strengths
» First attempt: Pollock
»
»
»
»
Arguments are modus ponens rules
Two premises: fact and assumption 𝐴, 𝐴 → 𝐵 ⊢ 𝐵
Strength 𝑆𝐴 , 𝑆𝐴→𝐵 , 𝑆𝐵 are numbers in [0,1]
Strength of a conclusion – Weakest Link: 𝑆𝐵 =
𝑚𝑖𝑛(𝑆𝐴 , 𝑆𝐴→𝐵 )
» If C attacks B, B strength is 𝑆𝐵 = 𝑚𝑎𝑥(𝑆𝐵 − 𝑆𝐶 , 0)
» Multiple attacks. No accrual, chose the max only
» What is this strength? Ad-hoc?
» Rejection of probability, is this justified?
A
B
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C
D
28
Weighted Argumentation
» Abstract Argumentation with weights
» Inconsistency Tolerance (Dunne 2009)
» Baroni / Toni 2013 proposal (ordinal functions
for attack and support), Pollock-like
» Same criticism: what’s the meaning of the numbers
used to quantify argument importance?
» Social Argumentation
» Weights (importance) attached to each arguments
come form a voting systems (online forums etc..)
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Probabilistic Argumentation
Li (2011), Hunter (2012), Dondio (2012)
» Real world arguments are clearly affected by
uncertainty.
» Probabilistic uncertainty is well studied (probability
calculus) and clear understood (maybe….)
» Allow arguments to be probabilistic in nature.
Source of probability:
» Randomness, stochastic processes
» Statistical information
» Subjective Beliefs
» Example: If you have fever, 80% is flu
30
Probabilistic Argumentation
» A PAF is a tuple (AF,P) where
» AF = (Ar,R) is an abstract argumentation
framework and
» 𝑃: 2𝐴𝑟 → [0,1] is a joint probability over
arguments. If statistical independence holds, 𝑃
is a scalar function: 𝑃: 𝐴𝑟 → [0,1]
0.5
0.7
0.4
𝒑𝑨
𝒑𝑩
𝒑𝑪
31
How to compute a PAF?
» We need to find the probability that an argument is
labelled IN, OUT, UNDEC
» Probabilistic arguments implies multiple scenarios
(2𝑁 ) each obtained by assuming that each argument
claim hold or not.
» Each scenario has its own probability (computed
using P)
» Each scenario corresponds to a sub-graph of the
argumentation framework
32
Notation for group of sub-graphs
» 𝐴𝐵 = 2 sub-graphs, all the sub-graphs
containing 𝑎 and not 𝑏
» 𝐴 + 𝐴𝐵𝐶 = 5 sub-graphs not containing 𝑎 or all
the sub-graphs containing 𝑎, 𝑏, 𝑐 togheter
» 𝐴𝐵𝐶 = 1 sub-graph, 𝑃 𝐴𝐵𝐶 = 𝑃 𝑎 𝑃 𝑏 𝑃(𝑐)
33
Computing PAF
» We can label arguments in each sub-graph,
assigning the OUT label if the argument is not
present in the sub-graph (defeated by its own)!
» We then group all the sub-graphs where a generic
argument 𝑎 is labelled 𝑖𝑛 , 𝑜𝑢𝑡, 𝑢𝑛𝑑𝑒𝑐
» We call these sets 𝐴𝐼𝑁 , 𝐴𝑂𝑈𝑇 , 𝐴𝑈
» We compute the probabilities of these sets summing
up the probabilities of all the sub-graphs in each set.
34
Brute Force
Brute force approach for 𝑨𝑰𝑵 , 𝑨𝑶𝑼𝑻 , 𝑨𝑼
Given 𝑃𝐴𝐹 = ( 𝐴𝑟, 𝑅 , 𝑃). Chose argument 𝑎 in 𝐴𝑟
for each sub-graph 𝑔 of 𝐺 = (𝐴𝑟, 𝑅)
assign a label 𝑙(𝑎) to 𝑎 in 𝑔 using the
chosen semantics
if 𝑙 𝑎 = 𝑖𝑛 add 𝑔 to 𝐴𝐼𝑁
if 𝑙 𝑎 = 𝑜𝑢𝑡 add 𝑔 to 𝐴𝑂𝑈𝑇
if 𝑙 𝑎 = 𝑢𝑛𝑑𝑒𝑐 add 𝑔 to 𝐴𝑈
𝑃 𝐴𝐼𝑁 , 𝑃 𝐴𝑂𝑈𝑇 , 𝑃 𝐴𝑈 are computed using 𝑃 and they
are the probabilities we were looking for
35
2-Layer Approach
» 𝑃 𝑎 is not 𝑃 𝑎𝐼𝑁
» 𝑃 𝑎 is the probability that argument a claim holds
in isolation
» 𝑃 𝑎𝐼𝑁 is the probability that the argument claims
holds after the argumentation process
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36
Example / Grounded Case
Scenario (Sub-graph)
A
U
B
U
C
U
U
U
O
I
O
I
I
O
O
O
O
I
O
I
O
O
O
I
O
O
O
P(s)
» 𝑃 𝐴𝐼𝑁 = 𝑃 𝑠2 + 𝑃 𝑠3 = 0.25
» Not a very clever approach (w.r.t. computation)
» We can skip many scenarios and reduce the problem
space
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37
Notation for Grounded labelling
» 𝐴𝐹 = 𝐴𝑟, 𝑅 , 𝐴𝑟 = 𝑛, 𝐺 =< 𝐴𝑟, 𝑅 >, 𝐻 is the set of all
the sub-graphs of 𝐺
» Under grounded semantic there is one unique labelling for
each sub-graph 𝑔 called ℒ 𝑔 (𝑔). Sets of interests:
𝐴𝐼𝑁 = 𝑔 ∈ 𝐻: 𝑎 ∈ 𝑖𝑛(ℒ 𝑔 (𝑔)) ;
𝐴𝑂𝑈𝑇 = 𝑔 ∈ 𝐻: 𝑎 ∈ 𝑜𝑢𝑡(ℒ 𝑔 (𝑔))
𝐴𝑈 = 𝑔 ∈ 𝐻: 𝑎 ∈ 𝑢𝑛𝑑𝑒𝑐(ℒ 𝑔 (𝑔))
𝐴𝐼𝑁 = 𝐴𝐵 (2 sub-graphs)
𝐴𝑂𝑈𝑇 = 𝐴 + 𝐴𝐵𝐶 (5 sub-graphs)
𝐴𝑈 = 𝐴𝐵𝐶 (1 subgraph)
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Preferred Case
» There are multiple labelings for each sub-graph
» An argument can be IN and OUT in the same
scenario (but in different labelings)
» Solution 1: give a probability for credulous
acceptance (possibility) and 1 for sceptical
acceptance (necessity)
» Solution 2: use principle of indifference, splitting
the probability of a single sub-graph among all the
valid preferred labeling
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39
Preferred Sets
» 𝓵𝑝𝑟 𝑠 = set of labelings for a scenario s
» Preferred credulous sets:
𝑝𝑟+
» 𝐴𝐼𝑁 = 𝑠 ∈ 𝑆: ∃ ℒ 𝑝𝑟 𝑠 ∈ 𝓵𝑝𝑟 𝑠 ∶ 𝑎 ∈ 𝑖𝑛 ℒ 𝑝𝑟 , 𝑠
𝑝𝑟+
» 𝐴𝑂𝑈𝑇 = 𝑠 ∈ 𝑆: ∃ ℒ 𝑝𝑟 𝑠 ∈ 𝓵𝑝𝑟 𝑠 ∶ 𝑎 ∈ 𝑜𝑢𝑡 ℒ 𝑝𝑟 , 𝑠
𝑝𝑟+
» 𝐴𝑈
= 𝑠 ∈ 𝑆: ∃ ℒ 𝑝𝑟 𝑠 ∈ 𝓵𝑝𝑟 𝑠 ∶ 𝑎 ∈ 𝑢𝑛𝑑𝑒𝑐 ℒ 𝑝𝑟 , 𝑠
» While the skeptical sets are:
𝑝𝑟−
𝑝𝑟
𝑝𝑟−
𝑝𝑟
𝑝𝑟
𝑝𝑟
» 𝐴𝐼𝑁 = 𝐴𝐼𝑁 ∖ 𝐴𝑂𝑈𝑇 ⋃𝐴𝑈
𝑝𝑟
𝑝𝑟
» 𝐴𝑂𝑈𝑇 = 𝐴𝑂𝑈𝑇 ∖ (𝐴𝐼𝑁 ⋃𝐴𝑈 )
𝑝𝑟−
» 𝐴𝑈
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𝑝𝑟
𝑝𝑟
𝑝𝑟
= 𝐴𝑈 ∖ (𝐴𝑂𝑈𝑇 ⋃𝐴𝐼𝑁 )
40
Preferred Case
𝑝𝑟+
𝑝𝑟+
» 𝐴𝐼𝑁 = 𝐴𝐵 + 𝐴𝐵𝐶 (note that 𝐴𝐵𝐶 also in 𝐴𝑂𝑈𝑇 )
𝑝𝑟+
» 𝐴𝑂𝑈𝑇 = 𝐴 + 𝐴𝐵𝐶 + 𝐴𝐵𝐶 (6 scenarios)
𝑝𝑟−
» 𝐴𝐼𝑁 = 𝐴𝐵
𝑝𝑟−
» 𝐴𝑂𝑈𝑇 = 𝐴 + 𝐴𝐵𝐶
𝑝𝑟−
» Necessity is 𝑃(𝐴𝐼𝑁 ) = 0.25
𝑝𝑟+
» Possibility is 𝑃(𝐴𝐼𝑁 ) = 0.375
» Indifferent probability 𝑃
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𝐴𝑖𝐼𝑁
= 𝑃 𝐴𝐵
1
+ P(ABC)
2
= 0.3125
41
Brute force is not efficient
» w.r.t. computational time
» w.r.t. the length of the expression of 𝐴𝐼𝑁
» 56 sub-graphs in 𝐴𝐼𝑁
» It can be reduced to 3 clauses only
𝐴𝐼𝑁 = 𝐴𝐵𝐷 + 𝐴𝐵𝐸 𝐷 + 𝐴𝐵𝐸𝐺 𝐷
» Wasted of computational time. Some cases do not need to
be comptued / recomputed
» Idea: assign sets of sub-graphs at each computational step
» Recursive and decision-tree-like argument (Dondio 2013)
42
A Recursive Algorithm (grounded)
» It decomposes the grounded semantics computation. It
applies the rules of a complete labelling + maximise undec.
» The algorithm. Visit the transpose graph from root 𝑎 and
imposes the following two rules:
1. 𝐴𝐼𝑁 = 𝐴 ∩ 𝑋1 𝑂𝑈𝑇 ∩ 𝑋2 𝑂𝑈𝑇 ∩ ⋯ ∩ 𝑋𝑛 𝑂𝑈𝑇 𝑋𝑖 = 𝑎𝑡𝑡𝑎𝑐𝑘𝑒𝑟𝑠 𝑜𝑓 𝑎
2. A𝑂𝑈𝑇 = 𝐴 ∪ 𝐴 ∩ X1 𝐼𝑁 ∪ X2 𝐼𝑁 ∪ ⋯ ∪ X𝑛 𝐼𝑁
» Terminal conditions
» if 𝑥𝑇 is required to be in then 𝑋𝑇 𝐼𝑁 = 𝑋𝑇
» if node 𝑥𝑇 is required to be out then 𝑋𝑇 𝑂𝑈𝑇 = 𝑋𝑇
» Cycles
» If a cycle is detected, end the recursive step and return ∅
» Some optimizations
43
Recursive Algorithm / Example
Node, Constraint
label
1↓
2↓
3=
4=
5↑
6↑
Parent Comment
List
is present and =OUT
is out when b is not present or
exists and = in or = in
=IN when
is present and
=OUT. Cycle with ,
is initial
44
Some Optimizations
1. Generate non-overlapping solutions
𝐴 + 𝐵 + 𝐶 + ⋯ can be rewritten as disjoint sets in the form
𝐴 + 𝐴𝐵 + 𝐴 𝐵𝐶 + ⋯, condition 2 is rewritten as:
𝐴𝑂𝑈𝑇 = 𝐴 ∪ 𝐴 ∩ 𝑋1 𝐼𝑁 ∪ 𝑋1 𝐼𝑁 𝑋2
𝐼𝑁
∪ 𝑋1 𝐼𝑁 𝑋2 𝐼𝑁 𝑋3 𝐼𝑁 ∪ ⋯ ∪ 𝑋1 𝐼𝑁 … 𝑋𝑛−1 𝐼𝑁 𝑋𝑛 𝐼𝑁
2. Optimizing condition 1: returning empty set
𝐴𝐼𝑁 = 𝐴 ∩ 𝑋1 𝑂𝑈𝑇 ∩ 𝑋2 𝑂𝑈𝑇 ∩ ⋯ ∩ 𝑋𝑛 𝑂𝑈𝑇
3. Exploiting Rebuttals
it is 𝐴𝑂𝑈𝑇 = 𝐴 instead of 𝐴𝑂𝑈𝑇 = 𝐴 + 𝐴𝐵𝐼𝑁
when 𝑏 rebutts 𝑎
4. Re-using computations if 𝐶1 (𝑥) = 𝐶2 (𝑥)
2012/2013 - DT228/4
45
Recursive Algorithm Example /2
1
1.1.1
Condtion 1
Condition 2b (with
reordering)
2b after rebuttals
detection. Since c
rebuts b, c cannot
label b.
Terminal node
1.1.2
Terminal node
1.1
Solution of the
recursive step 1.1
1.2
Condition 2b
Rebuttals optimization
applied, cannot
defeat
Final Solution
1.1
1
ADT – Arguments Decision Tree
» Decision Tree-like Algorithm. Select an argument, split,
analyse the two spit sub-graphs
» Which is the criteria for selecting
the splitting argument?
» Dialectical Strength. the dialectical strength of an
argument 𝑥 w.r.t. 𝑎, called 𝐷𝑆𝑎 (𝑥), is defined as follows:
» If 𝑥 is initial, 𝐷𝑆𝑎 (𝑥) is the number of arguments that are
defeated by 𝑥 plus the arguments that result disconnected from 𝑎
once the arguments defeated by 𝑥 are removed from 𝐺.
» Note that, if 𝑥 directly attacks 𝑎, then 𝐷𝑆𝑎 𝑥 = |𝐴𝑟|.
» If x is not initial, 𝐷𝑆𝑎 (𝑥) is the number of arguments that are
disconnected from 𝑎 after 𝑥 is removed.
47
ADT /2 - Example
𝐴𝐼𝑁 = 𝐹 + 𝐹 𝐵𝐶 𝐷𝐸
48
Applications / Legal Reasoning
Paul and John are under trial for the assassination of Sam. Evidence collected:
» 𝑹𝑱 : John was alone in the room between 1 to 3; the medical test says that Sam
died between 1 and 3 [0.6] → John shoot Sam
» 𝑹𝑷 : Paul was alone in the room between. 3 and 5; the medical test says that
Sam died between 3 and 5 [0.4] → Paul shoot Sam
» 𝑴𝒕 : The medical test is void [0.1] → nothing can be said on Sam’s time of
death
» We also 𝑃 𝑅𝐽 ∧ 𝑅𝑃 = 0, since Sam either died btw. 1 and 3 or btw. 3 and 5.
» 𝑭𝑷 : The fingerprints are Paul’s [0.7]→ Paul shot Sam btw 3 and 5
» 𝑻𝑭 : The weapon was tampered and the test is void [0.5] → fingerprints are
not a valid evidence
» 𝑾: A witness heard a shot at 2pm [0.8], John was in the room at 2 → John
shot Sam and Sam died between 1 and 3
» The number in square brackets is the probability of each premise (assume 1 if
no number is specified)
2012/2013 - DT228/4
49
Argumentation Graph for the Legal Case
Argument
Probability
0.6
0.4
0.1
0.8
0.5
0.7
All arguments
independent
» John is guilty when argument R J is
in or W is in. Therefore:
» P(Gj ) = P(R J
IN
∨ WIN ) = 0.6278
» GP = R P IN ∨ FP IN = 0.284
How is 𝑃(𝐺𝑗 ) changing if 𝑃(𝑊) changes?
»
𝜕PIN (GP )
𝜕P(W)
= −0.519
Good points of PAF
» Solid axioms
» Much richer
computation
» Maybe useful for
prob. reasoning?
50
Merging Probabilistic KB
» Probabilistic KB contains facts with their own
probability, such as 𝑃 𝑎 = 0.6
» And conditional rules of the form 𝑃 𝑎 𝑏 = 0.8
» Due to uncertainty, PKB could be inconsistent, i.e. there
is no probability distribution able to satisfies all facts
and rules. This happens especially when 2 PKB merge
» Probabilistic argumentation can be used to convert
rules and fact into arguments and then resolve conflicts
and produce necessity/possibility measures for all the
KB statements (see Thrimm 2012)
51
Evidential Reasoning, DS Rule
» How can I combine two independent belief assignments?
» DS rules
» It ignores contradictions (no belief assigned). All the mass
(belief) assigned to the agreement
Mass 1
»
»
»
»
(In)famous Example
Doc. 1: 1% tumor, 99% nothing
Doc. 2: 1% tumor, 99% meningitis
DS Merging
t1
n1
m2
t2
» 100% tumor
» P. Smets adjustment
» 0.01% Tumor, 99.9% unknown
Mass 2
52
Dempster-Schafter Rule and Semantics
Mass 1
t1
n1
m2
t2
P
Which semantic is this?
Mass 2
Not grounded, not preferred
An implicit undercutting argument in favour of the statements
that agrees. Smets correction behaves like grounded sem.
Can we use probabilistic argumentation semantics to propose
an alternative?
53
Fuzzy Argumentation
» My ongoing proposal
» Basic idea: use the same scenarios (sub-graph)
approach used for probabilistic argumentation
» If each argument has a degree of truth 𝝁𝒂 , with
a degree of truth 𝟏 − 𝝁𝒂 the negation holds
» Example:
» A: witness 1 said the murder was tall
» B: witness 2 said witness 1 is very unreliable
𝝁𝒂
A
B
𝝁𝒃
54
Problems
» Computational problems
» Middle excluded not guaranteed: 𝝁𝒂 ∧ 𝝁𝒂 ≠ 𝟎
» No similar properties as total probability: 𝝁𝒂 ∨ 𝝁𝒂 ≠ 𝟏
» Semantics Problem with rebuttal attacks
»
»
»
»
»
»
B: It is Blue! A: It is Red!
Do they contradict? It depends…
Fine if 𝜇𝐵𝐿𝑈𝐸 + 𝜇𝑅𝐸𝐷 ≤ 1
R
B
But if 𝜇𝐵𝐿𝑈𝐸 + 𝜇𝑅𝐸𝐷 > 1
Weak Conjunciton 𝜇𝐵𝐿𝑈𝐸 ∧ 𝜇𝑅𝐸𝐷 = max(𝜇𝐵𝐿𝑈𝐸 , 𝜇𝑅𝐸𝐷 )
Strong conj. 𝜇𝐵𝐿𝑈𝐸 ∧ 𝜇𝑅𝐸𝐷 = m𝑖𝑛(𝜇𝐵𝐿𝑈𝐸 + 𝜇𝑅𝐸𝐷 − 1,0)
55
What’s next
» Applications: any use in data analytics?
» Any use in Bayesian Network?
» I am focusing on Uncertain Argumentation, try
to publish some proposals
» Revisiting Dempster-Schafter Rule
» First proposal of fuzzy abstract argumentation
» A unified approach for uncertainty management in
argumentation
56
Thanks for your attention
Q&A?
57