An Introduction to Abstract Argumentation

Dr. Pierpaolo Dondio,DIT – School of Computing

2012/2013 1

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Agenda

» Introduction» What is argumentation theory?

» Abstract Argumentation Frameworks (Dung 1995)» Stable, Grounded and Preferred Semantics» Instantiating Abstract Argumentation» Applications

» Probabilistic and Uncertain Argumentation

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

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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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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

B

EDC

A Paul is a good at Maths

Paul got 90% in his final Maths test

If you get high marks in a Maths test you are good at Maths

Paul is not good at Maths

Paul was never able to help me with my Maths homework

Exam result is not a valid evidence

Mary said Paul copied the exam

Exam result is not a valid evidence

Exam was very easy this year

What Mary said is not trustworthy

Mary is a well-known layer

Argumentations are nonmonotonic

A B

C D E

Abstract Argumentation Frameworks

(Dung 1995) An argumentation framework is a pair

where is a set of arguments, and is a binary relation on , i.e. .» For two arguments A,B, the meaning of 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

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Starting Point

» Given we define a labelling function as a total function over :

» We also define the

» Starting basic idea. We label all arguments according to these simple rules:1. An argument in each labelling is either IN or OUT2. 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’.2012/2013 - DT228/4 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!

» Our rule works fine. We expect» A in» B out» C in» This is called reinstatement; A is reinstated

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A B C

Example 2» D: Sarah says that John is honest!

» 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

A B C D

Complete Semantics

» Grounded (Pollock, Dung)» Preferred (Dung)» Stable (Dung, Caminada)

» Many more.. Semi-stable, CF2..

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Complete Semantics: Conflict-free

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S A

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

Arg

(A S & attack(A,B)) = > B S

Complete Semantics - Admissibility

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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

S

A

C

(A Arg & attacks(C,A))

B

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 in2012/2013 - DT228/4 16

Complete Semantics

» Many sub-semantics have been defined over a complete labeling. Let

Grounded» L is a complete labellings such as undec(L)

is maximal w.r.t. to set inclusionPreferred» L is a complete labellings such as in(L) is

maximalw.r.t. to set inclusionStable» L is a complete labelling such that

undec(L) = ∅

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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

» Preferred:» IN={b} ; OUT={c,a}» IN={c} ; OUT={b,a}

» Stable: same as preferred

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A

B C

The Nixon Diamonds & 3-Cycle» Grounded: undec» Preferred:

» IN={a}, OUT={b}» IN={b}, OUT={a}

» Stable: same as preferred

» Grounded: undec» Preferred: undec» Stable: none

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B C

A

B C

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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Instanciating an AA

» Nothing is said about argument internal arguments structure

» Arguments as modus-ponens rules

» How can these rule be attacked?» Rebuttals: attack » Undercutting: attack » Undermining : attack

» 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 whose atoms is a finite

set and its connectives are . » The symbol means logical consequence.» for and a formula in , is an argument iff and it

does not exits so that » Given argument , we call the support of and

the claim of . » Given two arguments and , we define rebuttal

and undercut attacks in the following way:» rebuts if » undercuts if there is such that

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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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Trust as a form of Argumentation

» if agent has high level of past performance, then trust it» if agent is similar to Carol, then trust him» if agent has low reputation, then distrust him» if task context is new to the agent, then invalidate argument

PP» if past performance are high an reputation is low, then prefer

past performance» if past-performance are out of date, then invalidate

argument PP2012/2013 - DT228/4 26

+ - +

== = A2

PP+

R-

Sim+

A1 A3

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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Towards Argumentation with Strengths» First attempt: Pollock

» Arguments are modus ponens rules» Two premises: fact and assumption » Strength are numbers in [0,1] » Strength of a conclusion – Weakest Link: » If C attacks B, B strength is » Multiple attacks. No accrual, chose the max

only» What is this strength? Ad-hoc?» Rejection of probability, is this justified?

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A

B DC

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 flu30

Probabilistic Argumentation

» A PAF is a tuple (AF,P) where » AF = (Ar,R) is an abstract argumentation

framework and » is a joint probability over arguments. If

statistical independence holds, is a scalar function:

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0.5 0.7 0.4

How to compute a PAF?

» We need to find the probability that an argument is labelled IN, OUT, UNDEC

» Probabilistic arguments implies multiple scenarios () 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

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Notation for group of sub-graphs

» = 2 sub-graphs, all the sub-graphs containing and not

» = 5 sub-graphs not containing or all the sub-graphs containing togheter

» = 1 sub-graph,

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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 is labelled

» We call these sets

» We compute the probabilities of these sets summing up the probabilities of all the sub-graphs in each set.

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Brute Force

Brute force approach for

Given Chose argument in for each sub-graph of

assign a label to in using the chosen semantics

if add to if add to if add to

are computed using and they are the probabilities we were looking for

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2-Layer Approach

» is not » is the probability that argument a claim

holds in isolation» is the probability that the argument claims

holds after the argumentation process

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Example / Grounded Case

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Scenario (Sub-graph)

A B C P(s)

U U UU U OI O II O O O O IO I O O O I O O O

» Not a very clever approach (w.r.t. computation)» We can skip many scenarios and reduce the

problem space

Notation for Grounded labelling

» , , , is the set of all the sub-graphs of » Under grounded semantic there is one unique

labelling for each sub-graph called . Sets of interests:

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(2 sub-graphs) (5 sub-graphs) (1 subgraph)

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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Preferred Sets

» = set of labelings for a scenario s» Preferred credulous sets:

» While the skeptical sets are:

»

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Preferred Case

» (note that also in )» (6 scenarios)

» Necessity is = 0.25» Possibility is = 0.375» Indifferent probability = 0.3125

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Brute force is not efficient

» w.r.t. computational time» w.r.t. the length of the expression of

» 56 sub-graphs in » It can be reduced to 3 clauses only

» 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)

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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 and imposes the following two rules:1.

» Terminal conditions» if is required to be in then » if node is required to be out then

» Cycles» If a cycle is detected, end the recursive step and

return » Some optimizations

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Recursive Algorithm / Example

  Node, label

Constraint Parent List

Comment

1↓ is present and =OUT2↓ is out when b is not present

or exists and = in or = in 3= =IN when is present and

=OUT. Cycle with , 4= is initial5↑ 6↑

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Some Optimizations

1. Generate non-overlapping solutions can be rewritten as disjoint sets in the

form , condition 2 is rewritten as:

2. Optimizing condition 1: returning empty set

3. Exploiting Rebuttalsit is instead of when rebutts

4. Re-using computations if

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1 Condtion 1 1.1 Condition 2b (with

reordering)2b after rebuttals detection. Since c rebuts b, c cannot

label b.1.1.1 Terminal node1.1.2 Terminal node

1.1 Solution of the recursive step 1.1

1.2 Condition 2bRebuttals

optimization applied, cannot

defeat 1 Final Solution

Recursive Algorithm Example /2

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 w.r.t. , called , is defined as follows:» If is initial, is the number of arguments that are defeated

by plus the arguments that result disconnected from once the arguments defeated by are removed from .

» Note that, if directly attacks , then .» If x is not initial, is the number of arguments that are

disconnected from after is removed.

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ADT /2 - Example

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𝐴𝐼𝑁=𝐹 +𝐹 𝐵𝐶 𝐷𝐸

Applications / Legal Reasoning

Paul and John are under trial for the assassination of Sam. Evidence collected:

» 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

» 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

» The medical test is void [0.1] → nothing can be said on Sam’s time of death

» We also , since Sam either died btw. 1 and 3 or btw. 3 and 5. » The fingerprints are Paul’s [0.7]→ Paul shot Sam btw 3 and 5 » The weapon was tampered and the test is void [0.5] → fingerprints

are not a valid evidence » 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)

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Argumentation Graph for the Legal Case

Argument Probability  0.60.40.1 All arguments

independent0.80.50.7

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» John is guilty when argument is in or is in. Therefore:

How is changing if changes?

Good points of PAF» Solid axioms» Much richer

computation» Maybe useful for

prob. reasoning?

Merging Probabilistic KB

» Probabilistic KB contains facts with their own probability, such as

» And conditional rules of the form

» 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)

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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

» (In)famous Example» Doc. 1: 1% tumor, 99% nothing» Doc. 2: 1% tumor, 99% meningitis» DS Merging

» 100% tumor» P. Smets adjustment

» 0.01% Tumor, 99.9% unknown52

t1n1

m2t2

Mass 1

Mass 2

t1n1

m2t2

Mass 1

Mass 2

Dempster-Schafter Rule and Semantics

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P

Which semantic is this?Not grounded, not preferredAn 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?

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 , with a degree of truth the negation holds

» Example:» A: witness 1 said the murder was tall» B: witness 2 said witness 1 is very

unreliable

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A B𝝁𝒂 𝝁𝒃

Problems

» Computational problems» Middle excluded not guaranteed» No similar properties as total probability:

» Semantics Problem with rebuttal attacks» B: It is Blue! A: It is Red!» Do they contradict? It depends…» Fine if » But if » Weak Conjunciton » Strong conj.

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B R

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

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Thanks for your attentionQ&A?

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