Michael SchroederRalf Schweimeier Department of ComputingCity University, London, UKmsch@soi.city.ac.ukhttp://www.soi.city.ac.uk/~msch

Arguments and Misunderstandings:A Fuzzy Approach to Conflict Resolution in Open Systems

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Overview

Motivation: Expressive Knowledge Representation Part I: Argumentation as LP semantics

Notions of attack and justified arguments Hierarchy of semantics Proof procedure

Part II: Fuzzy unification and argumentation Fuzzy negation Fuzzy argumentation Fuzzy unification (work done together with David Gilbert)

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Knowledge representation

Pete earns 500.000$ p.a. earns(pete,500000).

Cross the street if there are no cars cross not car cross car

The fridge is quite cheap cheap(fridge):70%

Does Mike live in Londn? address(mike,london) = address(mike,londn): 95%

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Knowledge System Cube

rFB

fDB

fdFB

rDB

dDB

fdDB

dFB

fFB

r: relational f: fuzzy d: deductive

DB: database FB: factbase

ded

uct

ive

negation

fuzz

y

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Part I:Argumentation as semantics for Extended Logic Programs

rFB

fDB

fdFB

rDB

dDB

fdDB

dFB

fFB

r: relational f: fuzzy d: deductive

DB: database FB: factbase

ded

uct

ive

negation

fuzz

y

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Extended Logic Programming

Logic Programming with 2 negations Default negation:

not p : true if all attempts to prove p fail. Explicit negation:

p : falsehood of a literal may be stated explicitly. Coherence principle:

p not p

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Argumentation Interaction between agents in order to

gain knowledge revise existing knowledge convince the opponent solve conflicts

Elegant way to define semantics for (extended) logic programming Dung Kowalski, Toni, Sadri Prakken & Sartor Etc.

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Arguments

An argument is a partial proof, with implicitly negated literals as assumptions.

Formally: Argument for objective literal L: sequence of rules [ r1, …, rn ] such that

L is the head of r1 ;

no two rules have the same head ; for each objective literal L’ in the body of a rule ri, there is

a rule rj (j > i) such that L’ is the head of rj.

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Attacking arguments

Two fundamental kinds of attack: A undercuts B = A invalidates premise of B A rebuts B = A contradicts B

Derived notions of attack used in Literature:

A attacks B = A u B or A r B

A defeats B = A u B or (A r B and not B u A)

A strongly attacks B = A a B and not B u A

A strongly undercuts B = A u B and not B u A

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Proposition: Hierarchy of attacks

Undercuts = u

Strongly undercuts = su = u - u -1

Strongly attacks = sa = (u r ) - u -1

Defeats = d = u ( r - u -1)

Attacks = a = u r

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Fixpoint Semantics Argumentation:

game between proponent and opponent argument A is acceptable if opponent’s x-attack is countered by

proponent’s y-attack, which proponent already accepted earlier. Acceptable

Let x,y be notions of attack. An argument A is x,y-acceptable w.r.t. a set of arguments S iff

for every argument B, such that (B,A) x, there is a C S such that (C,B) y

Fixpoint semantics Fx/y (S) = { A | A is x,y-acceptable w.r.t. S }

x/y-justified arguments = Least Fixpoint of Fx/y.

x/y-overruled arguments = x-attacked by a justified argument. x/y-defensible iff neither justified nor overruled

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Theorem: Relationship of semantics Weakening opponent or strengthening proponent increases justified

arguments Different notions of acceptability give rise to different argumentation

semantics

sa/u=sa/d=sa/a

sa/su=sa/sa

d/su=d/u=d/a=d/d=d/sa

u/su=u/u

su/su

su/u

su/a=su/d

su/sa

u/a=u/d=u/sa

a/su=a/u=a/a=a/d=a/sa

Dung’s groundedargumentation semantics

Prakken and Sartor’ssemantics w/o priorities

WFSX

If opponent is allowed to attack,type of defense does not matter

If opponent is allowed defeat,type of defense does not matter

If opponent is allowed undercut,defense with (a,u,sa) or without(su,u) rebut makes a difference

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Proof procedure Dialogues:

x/y-dialogue is sequence of moves such that Proponent and Opponent alternate Players cannot repeat arguments Opponent x-attacks Proponent’s last argument Proponent y-attacks Opponent’s last argument

Player wins dialogue if other player cannot move Argument A is provably justified if proponent wins all branches of

dialogue tree with root A Concrete implementation SLXA:

Since u/a=u/d=u/sa=WFSX

compute justified arguments with top-down proof procedure SLXA for WFSX [Alferes, Damasio, Pereira]

SLXA can be adapted for other notions

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Part II:Fuzzy unification and argumentation

rFB

fDB

fdFB

rDB

dDB

fdDB

dFB

fFB

r: relational f: fuzzy d: deductive

DB: database FB: factbase

ded

uct

ive

negation

fuzz

y

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Classical Fuzzy Logic

Solution: Truth values in [0,1] instead of {0,1}. Assertions:

p:V (p a formula, V a truth value). Conjunction:

p:V, q:W p q : min(V,W) Disjunction:

p:V, q:W p q : max(V,W) Inference:

p q1, …, qn ; q1:V1, …, qn:Vn p : min(V1, …, Vn)

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Fuzzy Negation

Classical fuzzy negation: L:V L: 1-V (Zadeh)

Our setting (fuzzy adaptation of WFSX): L:V and L:V’ with V’ 1-V possible L and L not directly related.

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Fuzzy Coherence Principle

If L:V and V > 0, and not L:V’,

then V’ > V. “If there is some explicit evidence that L is false, then

there is at least the same evidence that L is false by default.”

If L:V and V > 0,

then not L: 1.

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Law of excluded... ...contradiction ...middle

p p :V V > 0 possible Contradictory programs!

not p p : V V > 0 possible By coherence principle!

Contradiction removal

not p p : V V > 0

p p : V V = 0 possible p is unknown

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Strength of an argument

Strength st of an argument: st ( L:V ) = V st ( L L1,…,Ln ) = min { st (L1 ),…,st (Ln ) }

st ( [r1,…,rn] ) = st ( r1 )

Least fuzzy value of the facts contributing to the argument.

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Theorems

Theorem (Soundness and Completeness)There is a justified argument of strength V for L

iffThere is a successful T-tree of truth value V for L

Theorem (Conservative Extension)

Argumentation semantics is a conservative extension of WFSX.

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Application: Fuzzy unification

Open systems: knowledge and ontologies may not match interaction with humans “Does Mike live in Londn?”

Approach: address(mike,london) = address(mike,londn): 95% adapt unification algorithm

(normalised edit distance over trees net) embed into argumentation framework

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Finding Mismatches: Edit distance

Edit distance between strings A and B: minimal number of delete, add, replace operations to

convert A into B. efficient implementation with dynamic programming

Example: e(address,adresse)=2, e(007,aa7)=2

Normalise: ne(A,B) = e(A,B) / max{ |A|, |B| }

Trees: net = sum of all mismatches divided by sum of all

max lengths

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Fuzzy unification and arguments

net is conservative extension of MGU (most general unifier)

net(t,t’) ne(t,t’)

V-argument: for all L in a body, there is L’ in head such that net(L,L’) 1-V

A V-undercuts B if A contains not L and B’s head is L’ and net(L,L’) 1-V

A V-rebuts B if A’s head is L and B’s head is L’ and net(L,L’) 1-V

Adapt previous definitions accordingly

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Comparison: Argumentation

Our framework allows us to relate existing and new argumentation semantics: Dung= a/su=a/u=a/a=a/d=a/sa Prakken&Sartor = d/su=d/u=d/a=d/d=d/sa WFSX = u/a = u/d = u/sa Dung Prakken&Sartor WFSX

Proof Theory and Top-down Proof Procedure adapted from Alferes, Damasio, Pereira’s SLXA

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Comparison: Fuzzy Argumentation

Wagner: Scale: -1 to +1 Unlike WFSX, he relates F and F:

F: -V iff F:V We adopted his interpretation for not:

not F:1 if F:V, V>0 Relates his work to stable models, but there is no

top-down proof procedure for stable models [Alferes&Pereira]

Our approach conservatively extends WFSX, hence we can adapt proof procedure SLXA

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Comparison: Fuzzy unification

Arcelli, Formato, Gerla define abstract fuzzy unification/resolution framework cannot deal with missing parameters (common

problem [Fung et al.]) no conservative extension of classical unification we use concrete distance: edit distance

Evaluated idea on bioinfo DB

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Conclusion “A database needs two kinds of negation” (Wagner) Argumentation is an elegant way of defining semantics Our framework allows classification of various new and

existing semantics Efficient top-down proof procedure for justified arguments Argumentation as basis for belief revision (REVISE) We cover the whole knowledge system cube including

fuzzy argumentation Defined fuzzy unification, which is useful in open systems

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Other relevant work

ACA: Arguing and Cooperating Agents multi-agent argumentation engine

Demo online at www.soi.city.ac.uk/~msch

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Other relevant work(together with Gerd Wagner)

Vivid Agents action and reaction rules executable specifications Implemented in simulation and with PVM-Prolog

Demo online at www.soi.city.ac.uk/~msch

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Other relevant work(together with Carlos Damasio, Luis Pereira)

REVISE Contradiction removal Applied to many domains like circuit diagnosis,

information integration in bioinformatics, alarm correlation in cellular phone network

Demo online at www.soi.city.ac.uk/~msch