argumentation semantics for contextual defeasible logic

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

The London Argumentation Forum, April 2012

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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?

The London Argumentation Forum 2012,King’s College London

argumentation semantics for contextual defeasible logic

Documents

argumentation semanticsextends

ain defeasible local

rules ti

types of rules

form ci

label q

different contexts mappings

set of axioms