Abstract Machines for Argumentation Logic and Interactions 2012, - - PowerPoint PPT Presentation

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Abstract Machines for Argumentation Logic and Interactions 2012, - - PowerPoint PPT Presentation

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Abstract Machines for Argumentation Logic and Interactions 2012, Week 2 Kurt Ranalter SIAG Bolzano/Bozen CIRM, 10/02/2012 Overview Introduction 1 Abstract machines 2 Argumentation 3 Conclusion 4 Introduction Summary of content

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SLIDE 1

Abstract Machines for Argumentation

Logic and Interactions 2012, Week 2 Kurt Ranalter

SIAG Bolzano/Bozen

CIRM, 10/02/2012

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SLIDE 2

Overview

1

Introduction

2

Abstract machines

3

Argumentation

4

Conclusion

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SLIDE 3

Introduction

Summary of content

Related work and motivations Aim of talk and contributions

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SLIDE 4

Introduction

Summary of content

Related work and motivations Aim of talk and contributions

Lecomte and Quatrini

Ludics and its applications to natural language semantics (in LNAI 5514, 2009) A theory of meaning that is based on ludics

convergence via daimon meaning via orthogonality

Match between rules of ludics and moves in dialogue

rules of ludics: positive vs negative roles in dialogue: speaker vs hearer actions in dialogue: sender vs receiver

Put these aspects together by means of normalisation

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SLIDE 5

Introduction

Curien and Herbelin

Abstract machines for dialogue games (in Panoramas et Synthèses 27, 2009) Proofs in ludics regarded as abstract Böhm trees Various abstract machines for computing with ABTs

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SLIDE 6

Introduction

Curien and Herbelin

Abstract machines for dialogue games (in Panoramas et Synthèses 27, 2009) Proofs in ludics regarded as abstract Böhm trees Various abstract machines for computing with ABTs

Combining these strands

Want to extend duality to abstract Böhm trees

rules of ludics: positive vs negative roles in dialogue: speaker vs hearer actions in dialogue: sender vs receiver abstract Böhm trees: replies vs queries

Towards computational account for modelling dialogue

normalisation by means of geometric abstract machine

ABTs more expressive than MLL-based variant of ludics

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SLIDE 7

Introduction

Basaldella and Faggian

Ludics with repetitions: exponentials, interactive types and completeness (in LMCS 7, 2011) An extension of ludics that deals with exponentials Add pointers to trace occurrences of subformulae

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SLIDE 8

Introduction

Basaldella and Faggian

Ludics with repetitions: exponentials, interactive types and completeness (in LMCS 7, 2011) An extension of ludics that deals with exponentials Add pointers to trace occurrences of subformulae

Relation to our framework

Relevant differences mostly of technical nature

normalisation via view abstract machine pointer interaction not a primary concern main focus on repetition of actions

Should be possible to translate all of our examples Pointer interaction one of the central topics of this talk

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SLIDE 9

Abstract machines

Summary of content

Sketch of formal definitions How does GAM actually work?

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SLIDE 10

Abstract machines

Summary of content

Sketch of formal definitions How does GAM actually work?

General considerations

Operational account of concepts from game semantics Crisp graphical representation for abstract Böhm trees

interaction may be seen as interleaved tree traversal graphical representation vs concrete implementation

Small number of rules leads to compact implementation Rapid prototyping as main benefit of implementation

a potential framework for developing applications why not abstract Böhm trees as data structures?

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SLIDE 11

Abstract machines

Abstract Böhm trees

  • a7
  • .

. . . . . a6

  • a3
  • a5
  • a2
  • a4
  • a1
  • a0
  • (∗)

Two types of moves

queries: a0, a2, a4, a6 replies: a1, a3, a5, a7

Pointer conditions

from reply to query

  • nly within branch

Branching condition

  • nly after replies

(Counter-)strategies

(∗) is counterstrategy strategy when 1) a0 = ⋆ and 2) no pointers to ⋆

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SLIDE 12

Abstract machines

Geometric abstract machine

(1) → {1 ← ⋆} hd(Γ)={2n ← q[a, −]} (2n)f Γ → Γ{2n ← a} hd(Γ)={2n−1 ← q}, φ(q)=[a, κ] (2n) Γ → Γ{2n ← q[a, κ]} hd(Γ)={2n ← q[a, ι]}, π(pop ι(q))=2k −1, Γ•2k −1=r (2n)b Γ → Γ{2n ← ra} hd(Γ)={2n ← q}, ψ(q)=[a, κ] (2n+1) Γ → Γ{2n+1 ← q[a, κ]} hd(Γ)={2n+1 ← q[a, ι]}, π(pop ι(q))=2k, Γ•2k =r (2n+1) Γ → Γ{2n ← ra}

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SLIDE 13

Abstract machines

GAM at work: outline

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 14

Abstract machines

GAM at work: step 1

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 15

Abstract machines

GAM at work: step 2

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 16

Abstract machines

GAM at work: step 2

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 17

Abstract machines

GAM at work: step 3

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 18

Abstract machines

GAM at work: step 3

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 19

Abstract machines

GAM at work: step 4

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 20

Abstract machines

GAM at work: step 4

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 21

Abstract machines

GAM at work: step 5

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 22

Abstract machines

GAM at work: step 5

. . . . . .

  • a2
  • 1
  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 23

Abstract machines

GAM at work: step 6

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 24

Abstract machines

GAM at work: step 6

. . . . . .

  • a2

1

  • a3
  • a3
  • a2
  • a2
  • a1
  • a1
  • ⋆
  • 1 *

2 *[a1,-] 2 a1 3 a1[a2,0] 3 *[a1,-]a2 4 *[a1,-]a2[a3,0] 4 a1[a2,0]a3 5 a1[a2,0]a3[a2,1] 5 *[a1,-]a2 6 *[a1,-]a2[a3,0] 6 a1[a2,0]a3[a2,1]· · ·

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SLIDE 25

Argumentation

Summary of content

Example dialogue about burden of proof Synthesised dialogue and formalisation

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SLIDE 26

Argumentation

Summary of content

Example dialogue about burden of proof Synthesised dialogue and formalisation

Prakken, Reed and Walton

Dialogues about the burden of proof (in ICAIL, 2005) Combine persuasion dialogue with burden of proof

argumentation schemes, critical questions technical solution based on dialogue levels

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SLIDE 27

Argumentation

Summary of content

Example dialogue about burden of proof Synthesised dialogue and formalisation

Prakken, Reed and Walton

Dialogues about the burden of proof (in ICAIL, 2005) Combine persuasion dialogue with burden of proof

argumentation schemes, critical questions technical solution based on dialogue levels

Use of pointer interaction

Embedded dialogues and concept of backtracking

backtracking: returning to earlier point in dialogue

Dialogue as product of normalisation via GAM

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SLIDE 28

Argumentation

Example of legal dispute

u1:CLAIM C v1:WHY C u2:C SINCE says(e, C) ∧ expert(e, C) v2:WHY ¬biased(e) u3:WHY biased(e) v3:BOP (¬biased(e), u) SINCE ¬biased(e) → trusted(e) u4:WHY ¬biased(e) → trusted(e) v4:WHY ¬(¬biased(e) → trusted(e)) u5:¬(¬biased(e) → trusted(e)) SINCE presumed(¬biased(e)) v5:RETRACT ¬biased(e) → trusted(e) v6:biased(e) SINCE paid(e, c) ∧ testifies(e, c) u6:CONCEDE biased(e) u7:RETRACT C

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SLIDE 29

Argumentation

u’s & v’s point of view

v3

· · ·

  • v5
  • u2
  • v2
  • u3
  • ⋆
  • u1

v1

  • v6
  • u7
  • u6
  • v3
  • · · ·
  • v5
  • u2
  • v2
  • u3
  • u1
  • v1
  • v6
  • u7

u6

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SLIDE 30

Conclusion

General considerations

Dialogue regarded as product of interaction Pointer interaction crucial for backtracking Lots of other applications indeed possible

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SLIDE 31

Conclusion

General considerations

Dialogue regarded as product of interaction Pointer interaction crucial for backtracking Lots of other applications indeed possible

Ongoing and future work

Syntax versus semantics

grammars as abstract Böhm trees compositional theory of meaning?

Analysis versus synthesis

modular approach to abstract Böhm trees abstract Böhm trees as data structures?

Rationality, decision making

implementation of selection functions?