Argumentation Semantics for Contextual Defeasible Logic
Antonis BikakisUniversity College London
Based on the joint work withGrigoris Antoniou
The London Argumentation Forum 2012,King’s College London
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Overview Background Contextual Defeasible Logic
Representation Model Argumentation Semantics Properties
More about CDL Operational Semantics, Applications, Future Work…
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Background Context in AI
A partial and approximate theory of the world from some individual’s perspective (McCarthy, 1987)
A logical theory – a set of axioms and inference rules
Multi-Context Systems (Giunchiglia &Trento group) Distributed context theories connected through mappings that enable
information flow between different contexts Mappings modeled as inference rules with premises and consequences
in different contexts
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Background Nonmonotonic MCS
MCS enriched with nonmonotonic features to handle imperfections, e.g. incomplete knowledge, inconsistencies
Context C
¬kkContext A
Context B
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Background Nonmonotonic MCS (Vienna Group)
Bridge rules modeled as default rules Diagnoses / Explanations to resolve inconsistency Centralized inconsistency resolution (global monitoring)
Contextual Defeasible Logic Bridge rules modeled as defeasible rules Preference information on contexts to resolve inconsistency Distributed inconsistency resolution (local view)
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Overview Background Contextual Defeasible Logic
Representation Model Argumentation Semantics Properties
Future Steps
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Representation Model
A Defeasible MCS C is a collection of distributed defeasible theories Ci
Each context Ci is a tuple (Vi , Ri , Ti ) Vi : vocabulary used by Ci Ri : set of rules Ti : preference ordering on C
Vi : a set of positive literals and their negations
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Representation Model
Three types of rules in Ri
Strict local rules
ril : ai
1 , ai2 ,…, ai
n-1→ ain
Defeasible local rules
rid : ai
1 , ai2 ,…, ai
n-1 ain
Mapping rules
rim : ai
1 , aj2 ,…, ak
n-1 aln
Ti is a partial preference ordering on C modeled as a Directed Acyclic Graph
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Argumentation Semantics Extends the argumentation semantics of Defeasible Logic
Distribution of available knowledge Preference information
Main Features Arguments with local range Arguments made by different contexts associated through mapping rules Partial preference preorder on the set of arguments
Variants Blocking / Propagating Ambiguity With / Without Team Defeat
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Support Relation (SRC)
Nodes of PTpi labeled by literals: Root labeled by pi
For every node with label q If q in Vi and a1, a2,…, an label the children of q then there is a rule
ri in Ci with body a1, a2,…, an and head q If q does not belong to Vi then this is a leaf node, and there is a
triple of the form (Cj , PTq , q) in SRC
Arcs of PTpi labeled by the rules used to obtain them
Set of triples of the form (Ci , PTpi , pi) Ci : context in C, pi : literal in Vi ,,
PTpi : proof tree for pi
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Argument
pi is the conclusion of A Any literal labeling a node in A is a conclusion of A A’ is a (proper) subargument of A if its proof tree is a (proper) subtree
of the proof tree of A A is a local argument of Ci if it contains only literals from Vi –
otherwise it is a mapping argument of Ci
Strict local arguments contain only strict local rules Defeasible local arguments contain at least one defeasible local
rule ArgsCi is the set of all arguments in Ci
ArgsC is the set of all arguments in C
An argument A for pi is a triple (Ci , PTpi , pi) in SRC
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Example 1Consider the following context theory C1
r11l : a1 → x1 r15
d : b1
r12m
: a2 a1 r16l : d1 → ¬b1
r13m
: a3 , a4 ¬a1 r17l : → d1
r14d
: b1 x1
Arguments in ArgsC1
x1
A1
a1
a2
¬a1
B1
a3 a4
x1
A3
b1
¬b1
A4
d1
r11
r12
r13 r13 r14
r15
r16
r17
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PreferenceAn argument A is preferred to argument B in context Ci iff one of the following conditions hold A is a strict local argument of Ci and B is not A is a local argument of Ci and B is not Both arguments are mapping arguments of Ci and
for all nodes labeled by a foreign literal ak in A (ak in Vk ≠ Vi) there is a node labeled by a foreign literal bl in B (bl in Vl ≠ Vi) such that ak is preferred to bl in Ci
- ak is preferred to bl in Ci iff there is a path from Cl to Ck in Ti
Partial Order on Contexts => Partial Preorder on Arguments
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Attack
An argument A attacks an argument B at p if p is a conclusion of B, ¬p is a conclusion of A, and B’ is not preferred to A’
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Example 1 (cont’d)
x1
A1
a1
a2
¬a1
B1
a3 a4
x1
A3
b1
¬b1
A4
d1
r11
r12
r13 r13 r14
r15
r16
r14
Assuming T1 = {[C2 , C4]} A1 attacks B1 at ¬a1 B1 does not attack A1 at a1
A4 attacks A3 at b1
A3 does not attack A4 (strict local argument)
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Argumentation Line
Head of argumentation line AL is the argument added in step 1 p is called the conclusion of AL
AL is a finite argumentation line if the number of steps required to build it is finite
An argumentation line AL for a literal p is a sequence of arguments constructed in steps as follows
In the first step add in AL one argument for p In each next step, for each distinct literal qj labeling a leaf node of the proof trees of the arguments added in the previous step, add one argument with conclusion qj
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Support - Undercut An argument A is supported by a set of arguments S if
Every proper subargument of A is in S and There is a finite argumentation line AL with head A such that every argument in AL – {A} is in S
An argument A is undercut by a set of arguments S if for every argumentation line AL with head A there is an argument B s.t.
B is supported by S and B attacks a proper subargument of A or an argument in AL – {A}
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Example 2
C1
a2
x1
A1
a1
¬a1
B1
a3 a4
C2
A1
a2
a5
A2
¬a2
a6
B2
C3a3
B3 C4a4
B4 C5a5
A5 C6a6
B6
¬a6
A6
Argumentation lines: AL1={A1, A2, A5}, BL1={B1, B3, B4} , BL2={B2, B6} Assuming that S={A5, A6}, A2 supported by S, B2 undercut by S Assuming that S={A5, A6 , B3 , B4 , A2}, B1, A1’ supported by S, A1’not undercut
by S
T2 = [C6 , C5]
A1’
T1 = [C3 , C2 , C4]
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Acceptability - Justifiability
An argument A is acceptable w.r.t. a set of arguments S if A is a strict local argument or A is supported by S and every argument attacking A is undercut by S
The set of justified arguments is defined as JArgsC = UJi
C where J0
C = {} Ji+1
C = {A | A is acceptable w.r.t. JiC}
A literal pi is justified if it is a conclusion of an argument in JArgsC
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Refutability
An argument A is rejected by a set of arguments S when A is undercut by S or A is attacked by an argument that is supported by S
rejected arguments (RArgsC): set of arguments rejected by JArgsC
A literal p is rejected if there is no argument for p in ArgsC-RArgsC
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Example 2 (cont’d)
C1
a2
x1
A1
a1
¬a1
B1
a3 a4
C2
A1
a2
a5
A2
¬a2
a6
B2
C3a3
B3 C4a4
B4 C5a5
A5 C6a6
B6
¬a6
A6
J0C={}
J1C={B3, B4, A5, A6}
J2C={B3, B4, A5, A6 , A2}
J3C={B3, B4, A5, A6 , A2 , A1’}
J4C={B3, B4, A5, A6, A2 , A1’, A1} = JArgsC
RArgsC={B6, B2, B1}
T2 = [C6 , C5]
A1’
T1 = [C2 , C4]
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Properties of Argumentation System The sequence Ji
C is monotonically increasing No argument is both justified and rejected. No literal is both justified and rejected If the set of justified arguments JArgsC contains two arguments with
contradictory conclusions, then both are strict local arguments⇒ Assuming consistency in the strict local rules of each context, the entire
framework is consistent
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More about CDL Operational Semantics
Algorithms for distributed query evaluation Alternative strategies for conflict resolution Implemented in Logic Programming
Applications Mobile Social Networks Ambient Intelligence (Internet of Things)
Future Work Relation with Abstract Argumentation Frameworks
Preference-based Afs, Context Argumentation Systems Access Control Layer Large-scale applications
Argumentation Semantics for Contextual Defeasible Logic
Thank you for your attention!Questions?
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