Computationally Viable Handling of Beliefs inArguments for Persuasion

Emmanuel Hadoux and Anthony HunterNovember 6, 2016

University College LondonEPSRC grant Framework for Computational Persuasion

Introduction

Persuasion problems

• One agent (the proponent) tries to persuade the other(the opponent)

• e.g., doctor persuading a patient to quit smoking, asalesman, a politician, ...

• the agents exchange arguments during a persuasiondialogue

• These arguments are connected by an attack relation

1

Persuasion problems

• One agent (the proponent) tries to persuade the other(the opponent)

• e.g., doctor persuading a patient to quit smoking, asalesman, a politician, ...

• the agents exchange arguments during a persuasiondialogue

• These arguments are connected by an attack relation

1

Persuasion problems

• One agent (the proponent) tries to persuade the other(the opponent)

• e.g., doctor persuading a patient to quit smoking, asalesman, a politician, ...

• the agents exchange arguments during a persuasiondialogue

• These arguments are connected by an attack relation

1

Abstract argumentation framework

A1 A2 A3

Figure 1: Argument graph with 3 arguments

Based on Dung’s abstract argumentation framework [1]

Example (Figure 1)A1 = “It will rain, take an umbrella”A2 = “The sun will shine, no need for an umbrella”A3 = “Weather forecasts say it will rain”

2

Purpose of the work

The objective for the proponent:

1. Have an argument or a set of arguments holding at theend of the dialogue

2. Have these arguments believed by the opponent

Need to maintain and update a belief distribution → to positthe right argument

3

Purpose of the work

The objective for the proponent:

1. Have an argument or a set of arguments holding at theend of the dialogue

2. Have these arguments believed by the opponent

Need to maintain and update a belief distribution → to positthe right argument

3

Purpose of the work

The objective for the proponent:

1. Have an argument or a set of arguments holding at theend of the dialogue

2. Have these arguments believed by the opponent

Need to maintain and update a belief distribution

→ to positthe right argument

3

Purpose of the work

The objective for the proponent:

1. Have an argument or a set of arguments holding at theend of the dialogue

2. Have these arguments believed by the opponent

Need to maintain and update a belief distribution → to positthe right argument

3

Belief distribution

Epistemic approach to probabilistic argumentation (e.g., [2])DefinitionLet G = ⟨A, R⟩ be an argument graph.Each X ⊆ A is called a model.A belief distribution P over 2A is such that

∑X⊆A P(X) = 1 and

P(X) ∈ [0, 1], ∀X ⊆ A.The belief in an argument A is P(A) =

∑X⊆A s.t. A∈X P(X).

If P(A) > 0.5, argument A is accepted.

Example (of a belief distribution)Let A = {A, B} where P({A, B}) = 1/6, P({A}) = 2/3, andP({B}) = 1/6 is a belief distribution.Then, P(A) = 5/6 > 0.5 and P(B) = 2/6 < 0.5.

4

Belief distribution

Epistemic approach to probabilistic argumentation (e.g., [2])DefinitionLet G = ⟨A, R⟩ be an argument graph.Each X ⊆ A is called a model.A belief distribution P over 2A is such that

∑X⊆A P(X) = 1 and

P(X) ∈ [0, 1], ∀X ⊆ A.The belief in an argument A is P(A) =

∑X⊆A s.t. A∈X P(X).

If P(A) > 0.5, argument A is accepted.

Example (of a belief distribution)Let A = {A, B} where P({A, B}) = 1/6, P({A}) = 2/3, andP({B}) = 1/6 is a belief distribution.Then, P(A) = 5/6 > 0.5 and P(B) = 2/6 < 0.5.

4

Belief distribution

Epistemic approach to probabilistic argumentation (e.g., [2])DefinitionLet G = ⟨A, R⟩ be an argument graph.Each X ⊆ A is called a model.A belief distribution P over 2A is such that

∑X⊆A P(X) = 1 and

P(X) ∈ [0, 1], ∀X ⊆ A.The belief in an argument A is P(A) =

∑X⊆A s.t. A∈X P(X).

If P(A) > 0.5, argument A is accepted.

Example (of a belief distribution)Let A = {A, B} where P({A, B}) = 1/6, P({A}) = 2/3, andP({B}) = 1/6 is a belief distribution.Then, P(A) = 5/6 > 0.5 and P(B) = 2/6 < 0.5.

4

Refinement of a belief distribution

Each time a new argument is added to the dialogue, thedistribution needs to be updated.

A

B

Figure 2

AB P H1A(P) H0.75A (P)

11 0.6 0.7 0.67510 0.2 0.3 0.27501 0.1 0.0 0.02500 0.1 0.0 0.025

Table 1: Examples of Belief Redistribution

We can modulate the update to take into account differenttypes users (skeptical, credulous, etc.)

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Splitting the distribution

Splitting the distribution

• 30 arguments → 230 = 1, 073, 741, 824 models → 8.6 gB iftreated as a double type

• Fortunately, they are not all directly linked to each other• We can group related arguments into flocks which arethemselves linked to each other

• We create a split distribution from the metagraph, asopposed to the joint distribution

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Splitting the distribution

• 30 arguments → 230 = 1, 073, 741, 824 models → 8.6 gB iftreated as a double type

• Fortunately, they are not all directly linked to each other

• We can group related arguments into flocks which arethemselves linked to each other

• We create a split distribution from the metagraph, asopposed to the joint distribution

6

Splitting the distribution

• 30 arguments → 230 = 1, 073, 741, 824 models → 8.6 gB iftreated as a double type

• Fortunately, they are not all directly linked to each other• We can group related arguments into flocks which arethemselves linked to each other

• We create a split distribution from the metagraph, asopposed to the joint distribution

6

Splitting the distribution

• 30 arguments → 230 = 1, 073, 741, 824 models → 8.6 gB iftreated as a double type

• Fortunately, they are not all directly linked to each other• We can group related arguments into flocks which arethemselves linked to each other

• We create a split distribution from the metagraph, asopposed to the joint distribution

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Metagraphs

A1A2

A3

A4

A5 A6

A7

A8 A9 A10(a)

A1 A2, A3

A4

A5 A6

A7, A8, A9, A10

(b)

Figure 3: Argument graph and possible metagraph

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Creating a split distribution

A1 A2, A3

A4

A5 A6

A7, A8, A9, A10

Figure 4: Metagraph

We define three assumptions forthe split to be clean:1. Arguments from non directlyconnected flocks areconditionaly independent

2. Arguments in a flock areconsidered connected

3. Arguments in a flock areconditionally dependent

No bayesian networks because: not probabilities, users are notrational, etc.

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Creating a split distribution

• We can define an optimal, irreducible, split w.r.t. the graph

• However, an irreducible split may not be computable• Only the irreducible split is unique, we therefore need torank the others.

9

Creating a split distribution

• We can define an optimal, irreducible, split w.r.t. the graph• However, an irreducible split may not be computable

• Only the irreducible split is unique, we therefore need torank the others.

9

Creating a split distribution

• We can define an optimal, irreducible, split w.r.t. the graph• However, an irreducible split may not be computable• Only the irreducible split is unique, we therefore need torank the others.

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Ranking the splits

Definition (valuation of a split)

x =∑A∈A

∑Pi∈S s.t. A∈E(P)

|Pi|

and P ≻ P′ iff x < x′.

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Ranking the splits

Definition (valuation of a split)

x =∑A∈A

∑Pi∈S s.t. A∈E(P)

|Pi|

and P ≻ P′ iff x < x′.

Example (of valuation and ranking)Let P be the joint distribution for Figure 3a. Value ofP : 10× 210 = 10, 240

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Ranking the splits

Definition (valuation of a split)

x =∑A∈A

∑Pi∈S s.t. A∈E(P)

|Pi|

and P ≻ P′ iff x < x′.

A1 A2, A3

A4

A5 A6

A7, A8, A9, A10

Figure 5: Metagraph

Example (of valuation andranking)P1 = (P(A5), P(A6 | A5),P(A4 | A5,A6), P(A2,A3 | A4,A7),P(A1 | A2,A3), P(A7,A8,A9,A10)):21+22+23+2×24+23+4×24 = 118

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Ranking the splits

Definition (valuation of a split)

x =∑A∈A

∑Pi∈S s.t. A∈E(P)

|Pi|

and P ≻ P′ iff x < x′.

Example (of valuation and ranking)P2 = (P(A1,A2,A3,A4,A5,A6 | A7), P(A7,A8,A9,A10)):6× 27 + 4× 24 = 832.

We then see that P1 ≻ P2 ≻ P.

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Experiments

Splitting the distribution

A1 A2, A3

A4

A5 A6

A7, A8, A9, A10

Figure 5: Metagraph

• Original graph: 10arguments → 1,024 values→ 8kB

• Metagraph: 10 argumentsin 6 flocks → 54 values →432B

• And the time taken toupdate.

• An argument can beupdated by updating onlyits flock.

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Splitting the distribution

A1 A2, A3

A4

A5 A6

A7, A8, A9, A10

Figure 5: Metagraph

• Original graph: 10arguments → 1,024 values→ 8kB

• Metagraph: 10 argumentsin 6 flocks → 54 values →432B

• And the time taken toupdate.

• An argument can beupdated by updating onlyits flock.

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Splitting the distribution

A1 A2, A3

A4

A5 A6

A7, A8, A9, A10

Figure 5: Metagraph

• Original graph: 10arguments → 1,024 values→ 8kB

• Metagraph: 10 argumentsin 6 flocks → 54 values →432B

• And the time taken toupdate.

• An argument can beupdated by updating onlyits flock.

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Experiments with flocks of different sizes

# flocks # links 1 update 50 updates

2 flocks10 links 2ms 107ms30 links 6ms 236ms

4 flocks10 links 1ms 45ms30 links 3ms 114ms

10 flocks10 links 0.03ms 1.6ms30 links 0.06ms 2.5ms

Table 2: Computation Time for Updates in Different Graphs of 50Arguments (in ms)

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Experiments with different numbers of arguments

# args Time for 20 updates Comparative %

25 497ns +0%50 517ns +4%75 519ns +4%100 533ns +7%

Table 3: Computation Time for 20 Updates (in ns)

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Experiments

A new version of the library is currently begin developped inC++ and is available at: https://github.com/ComputationalPersuasion/splittercell.

As a rule of thumb, we should keep flocks to less than 25arguments each.

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Conclusion

We have presented:

1. A framework to represent the belief of the opponent inthe arguments

2. How to create a split distribution using a metagraph3. How to rank the splits in order to choose the mostappropriate one w.r.t. the problem

4. Experiments showing the viability of the approach

Next step: adapt this work to the whole project to scale.

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Conclusion

We have presented:

1. A framework to represent the belief of the opponent inthe arguments

2. How to create a split distribution using a metagraph3. How to rank the splits in order to choose the mostappropriate one w.r.t. the problem

4. Experiments showing the viability of the approach

Next step: adapt this work to the whole project to scale.

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Thank you!

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Phan Minh Dung.On the acceptability of arguments and its fundamentalrole in nonmonotonic reasoning, logic programming, andn-person games.Artificial Intelligence, 77:321–357, 1995.

Anthony Hunter.A probabilistic approach to modelling uncertain logicalarguments.International Journal of Approximate Reasoning,54(1):47–81, 2013.

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