Trust-based belief change Emiliano Lorini Guifei Jiang , Laurent - - PowerPoint PPT Presentation

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Trust-based belief change

Emiliano Lorini✶ Guifei Jiang✶,✷ Laurent Perrussel✶

✶IRIT – Université de Toulouse – France ✷AIRG – University of Western Sydney – Penrith – Australia

ECAI-2014 talk

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Motivation (1/3)

Communication among agents Agents are autonomous

Two agents may react differently while facing new information

Deciding to adapt Adapting belief

Input s❡♥❞❡r, ❝♦♥t❡♥t

Receiver adapts its belief with respect to its trust in the sender about the content.

1

Should it trust the sender (about the content)?

2

Up to which ”trust degree” it should consider new information?

Our goal: Exhibiting the interplay between trust and belief.

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Motivation (2/3)

Typical Interplay between trust and belief change Impact of the trust degree If NY Times informs Bill that Luigi’s Burger is the best burger restaurant (❧❜) in NYC and Bill strongly trusts NYT about ❧❜, then he should strongly believe ❧❜ Cumulative impact of the trust degree Jane trusts Trip Advisor, Hotels.com and Ebookers about Pine Hotel quality in a reasonable way. Trip Advisor, Hotels.com and Ebookers informs Jane that Pine Hotel-NYC is a bad hotel. Jane may strongly believe that Pine Hotel is a bad hotel. ⇒ Needs for a modular definition of the interplay

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Motivation (3/3)

How do we do that? How to represent ❜❡❧✐❡❢(✐, ❝♦♥t❡♥t, ❞❡❣r❡❡) and tr✉st(✐, , ❥, ❝♦♥t❡♥t, ❞❡❣r❡❡) (Static aspect)

Epistemic Logic with a trust operator Extension with degree

How to represent ✐♥❢♦r♠(✐, ❥, ❝♦♥t❡♥t) and r❡✈✐s❡(❦, ✐♥❢♦r♠(✐, ❥, ❝♦♥t❡♥t)) (Dynamic aspect)

Dynamic Epistemic Logic Iterated Belief Change a la Spohn.

⇒ DL-BT: Dynamic Logic of Graded Belief and Trust

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Plan

1

Motivation

2

DL-BT: Syntax

3

L-BT: Semantics Structure Truth conditions

4

DL-BT Semantics Change Policies Additive Policy Compensatory Policy

5

Axiomatics L-BT: Proof theory DL-BT: Reduction axioms Change Policies: Reduction axioms

6

Conclusion

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT: Syntax - Knowledge and Belief operators (1/2)

❑✐ ϕ: agent ✐ knows ϕ What is possible for agent ✐? ❇≥α

✐

ϕ: agent ✐ believes that ϕ is true with strength at least α "What is possible" is structured as an epistemic state. Scale for beliefs: numerical scale ◆✉♠ = {✵, . . . , ♠❛①} Scale is finite and can be viewed as an encoding of a qualitative scale Example:

◆✉♠ = {✵, ✶, ✷, ✸, ✹, ✺} s.t. ✵ stands for ’null’ and ✺ for ’very high’. Bill believes (at least weakly) that Luigi’s Burger is the best one in NYC: ¬❑❜✐❧❧❧✉✐❣✐_❜❡st_❜✉r❣❡r ∧ ❇≥✶

❜✐❧❧❧✉✐❣✐_❜❡st_❜✉r❣❡r

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT: Syntax - Knowledge and Belief operators (2/2)

Knowledge Shortcut

• ❑✐ ϕ =❞❡❢ ¬❑✐¬ϕ

Belief shortcut ❇✐ ϕ =❞❡❢ ❇≥✶

✐

ϕ

• ❇✐ ϕ =❞❡❢ ¬❇✐¬ϕ

❯✐ ϕ =❞❡❢ ¬❇✐ ϕ ∧ ¬❇✐¬ϕ ❇α

✐ ϕ =❞❡❢ ❇≥α ✐

ϕ ∧ ¬❇≥(α+✶)

✐

ϕ ❇♠❛①

✐

ϕ =❞❡❢ ❇≥♠❛①

✐

ϕ ❇✵

✐ ϕ =❞❡❢ ¬❇✐ ϕ

Example:

Bill believes that Luigi’s Burger is the best one in NYC: ❇❜✐❧❧❧✉✐❣✐_❜❡st_❜✉r❣❡r Bill only weakly believes that Luigi is the best one: ❇✶

❜✐❧❧❧✉✐❣✐_❜❡st_❜✉r❣❡r

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT: Syntax - Trust Operator

❚α

✐,❥ ϕ: agent ✐ trusts agent ❥’s judgement on formula ϕ with

strength α.

1

α ✵

2

Trust degree is exactly α (and not a lower bound)

3

Scale is shared with the belief scale

Shortcut ❚✐,❥ ϕ =❞❡❢

• α∈◆✉♠\{✵}

❚α

✐,❥ ϕ

Example:

Bill strongly trusts NYT about Luigi: ❚✹

❜✐❧❧,♥②t❧✉✐❣✐_❜❡st_❜✉r❣❡r

L-BT: Logic of Graded Belief and Trust

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT: Syntax - Revision Operator

Interplay between trust and change may be specific to each agent: ❢ is a policy change function. [∗❢

✐ ψ]ϕ: after agent ✐ has publicly announced that ψ is true

and each agent ❥ has revised her beliefs according to the trust-based belief change policy ❢ (❥), ϕ is true.

Example: Bill believes that Luigi is the best after NYT announces it (w.r.t. some ❢ (❜✐❧❧)): [∗❢

♥②t❧✉✐❣✐_❜❡st_❜✉r❣❡r]❇❜✐❧❧❧✉✐❣✐_❜❡st_❜✉r❣❡r

DL-BT: L-BT logic + revision operator

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

L-BT: Semantics - Structure (1/3)

Kripke semantics For each agent ✐:

possible states: equivalence relation E✐ Example: E✐(✇✵) = {✇✵, ✇✶, ✇✷, ✇✸, ✇✹, ✇✺} ranking of the states: κ function

0 is the best value κ(✇, ✐): how exceptional is ✇ for agent ✐

Example: 2 ✇✸, ✇✹, ✇✺ 1 ✇✷ ✇✵, ✇✶

for any state and agent, there is always a possible state with a 0 value (consistency).

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

L-BT: Semantics - Structure (2/3)

Neighbourhood semantics for trust Non normal operator for handling trust about contradicting statements. For each agent ✐ and state ✇:

❥-trustable states: function N✐,❥(✇, α) Example: N✐,❥(✇✵, ✶) = {{✇✶}} N✐,❥(✇✵, ✷) = {{✇✸, ✇✹}} N✐,❥(✇✵, ✸) = {{✇✵, ✇✷}{✇✵, ✇✺}}

constraint on N✐,❥

no two states with different degrees two equivalent possible states must lead to the trustable states (and values) trustable states must be possible states

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

L-BT: Semantics - Structure (3/3)

Valuations V on each state To sum up, a model ▼ = (❲ , {E✐}✐∈❆❣t, κ, {N✐,❥}✐,❥∈❆❣t, V) Exceptionality degree of a formula ϕ (w.r.t. some ✇ and ✐):

κ✇,✐(ϕ) = ♠✐♥✈∈ϕ✇,✐ κ(✈, ✐) if ϕ✇,✐ = ∅ κ✇,✐(ϕ) = ♠❛① if ϕ✇,✐ = ∅

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

L-BT: Semantics - Truth conditions

Truth conditions are defined with respect to some model ▼ and a state ✇. ▼, ✇ | = ❑✐ ϕ iff ∀✈ ∈ E✐(✇) : ▼, ✈ | = ϕ ▼, ✇ | = ❇≥α

✐

ϕ iff κ✇,✐(¬ϕ) ≥ α ▼, ✇ | = ❚α

✐,❥ ϕ iff ϕ▼ ∈ N✐,❥(✇, α)

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT: - Embedding policies

Policy: how to change degrees (κ values). Input: an L-BT model ▼ = (❲ , {E✐}✐∈❆❣t, κ, {N✐,❥}✐,❥∈❆❣t, V) Function ❢ maps a policy to each agent. Output: new degrees κ∗❢

✐ ϕ(✇, ❥): κ values for agent ❥ revised w.r.t. ▼, ❢ and

initial announcement ϕ by ✐. ... and a new model ▼∗❢

✐ ϕ = (❲ , {E✐}✐∈❆❣t, κ∗❢ ✐ ϕ, {N✐,❥}✐,❥∈❆❣t, V)

Semantics of the revision ▼, ✇ | = [∗❢

✐ ϕ]ψ iff ▼∗❢

✐ ϕ, ✇ |

= ψ

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT Semantics: - Additive policy (1/4)

Agent ✐ announces ϕ Agent ❥ changes its belief only if it trusts ✐ about ϕ If so (❚❥,✐ ϕ holds at ✇)

ϕ also holds on ✇: agent ❥ decreases the degree of exceptionality of ✇: κ∗❛❞❞

✐

ϕ(✇, ❥) = κ(✇, ❥) − κ✇,❥(ϕ)

ϕ does not hold on ✇: agent ❥ increases the degree of exceptionality of ✇ w.r.t. to its degree of trust (❚α

❥,✐ ϕ holds

at ✇): κ∗❛❞❞

✐

ϕ(✇, ❥) = κ(✇, ❥) + α bounded to ▼❛①

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT Semantics: - Additive policy (2/4)

Example

Suppose three states ✇✵, ✇✶ and ✇✷, 2 agents ❜✐❧❧ and ♥②t and statement ▲✉✐❣✐_❇❡st_❘❡st❛✉r❛♥t (lbr). ❧❜r holds in ✇✵ but not in ✇✶ and ✇✷. κ(✇✵, ❜✐❧❧) = ✵; κ(✇✶, ❜✐❧❧) = ✶; κ(✇✷, ❜✐❧❧) = ✷ κ✇✵,❜✐❧❧(❧❜r) = ✵; κ✇✶,❜✐❧❧(❧❜r) = ✵; κ✇✷,❜✐❧❧(❧❜r) = ✵ κ✇✵,❜✐❧❧(¬❧❜r) = ✶; κ✇✶,❜✐❧❧(¬❧❜r) = ✶; κ✇✷,❜✐❧❧(¬❧❜r) = ✶ N❜✐❧❧,♥②t(✷, ✇✵) = N❜✐❧❧,♥②t(✷, ✇✶) = N❜✐❧❧,♥②t(✷, ✇✷) = {{✇✵}} revising κ for an ❧❜r state κ∗❛❞❞

♥②t ❧❜r(✇✵, ❜✐❧❧) = κ(✇✵, ❜✐❧❧) − κ✇✵,❜✐❧❧(❧❜r) = ✵ − ✵ = ✵

revising κ for an ¬❧❜r state κ∗❛❞❞

♥②t ❧❜r(✇✶, ❜✐❧❧) = κ(✇✶, ❜✐❧❧) + α = ✶ + ✷ = ✸ ⇒ ✷

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT Semantics: - Additive policy (3/4)

Key Properties Syntax independence: if | = ϕ✶ ↔ ϕ✷ then | = [∗❢

✐ ϕ✶]ψ ↔ [∗❢ ✐ ϕ✷]ψ

Keeping beliefs (objective formulas only) | = (❇α

❥ ψ ∧ ¬❚❥,✐ ϕ) → [∗❢ ✐ ϕ]❇α ❥ ψ

| = ❚❥,✐ψ → [∗❢

✐ ψ]❇❥ψ

Increasing degrees (objective formulas only) | = (❚α

❥,✐ ϕ ∧ ❇β ❥ ϕ) → [∗❢ ✐ ϕ]❇α+β ❥

ϕ Priority to last input (objective formulas only) | = ❚❥,✐✷ψ → [∗❢

✐✶¬ψ][∗❢ ✐✷ψ]❇❥ψ

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT Semantics: - Additive policy (4/4)

Example

Bill has to decide whether he buys a certain stock and he is currently uncertain: ❍②♣✶ =❞❡❢ ❯❇✐❧❧st♦❝❦❯♣ Bill trusts fairly Mary’s judgement on st♦❝❦❯♣, and Bill trusts very weakly Jack’s judgement on st♦❝❦❯♣: ❍②♣✷ =❞❡❢ ❚✸

❇✐❧❧,▼❛r②st♦❝❦❯♣ ∧ ❚✶ ❇✐❧❧,❏❛❝❦st♦❝❦❯♣

Mary and Jack announces that stock will go up: ❍②♣✶ ∧ ❍②♣✷ → [∗❢

▼❛r②st♦❝❦❯♣][∗❢ ′ ❏❛❝❦st♦❝❦❯♣]❇✹ ❇✐❧❧st♦❝❦❯♣.

But, it also holds ❍②♣✶ ∧ ❍②♣✷ → [∗❢

▼❛r②¬st♦❝❦❯♣][∗❢ ′ ❏❛❝❦st♦❝❦❯♣]❇❇✐❧❧st♦❝❦❯♣.

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT Semantics: - Compensatory policy (1/3)

Avoiding the priority to last announcement Agent ❥ changes its belief only if it trusts ✐ about ϕ If so (❚❥,✐ ϕ holds at ✇)

ϕ also holds on ✇: agent ❥ decreases the degree of exceptionality of ✇ w.r.t. to the degree α of trust κ∗❛❞❞

✐

ϕ(✇, ❥) = κ(✇, ❥) − α bounded to ✵.

ϕ does not hold on ✇: agent ❥ increases the degree of exceptionality of ✇ w.r.t. to its degree of trust (❚α

❥,✐ ϕ holds at

✇): κ∗❛❞❞

✐

ϕ(✇, ❥) = κ(✇, ❥) + α bounded to ▼❛①

Case is limited to the situation where ❇✐ ϕ also holds

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT Semantics: - Compensatory policy (2/3)

Key Properties Cumulative effect | = (❚α

❥,✐ ϕ ∧ ❇β ❥ ϕ) → [∗❢ ✐ ϕ]❇α+β ❥

ϕ Compensatory effect (one shot) | = (❚α

❥,✐¬ϕ ∧ ❇β ❥ ϕ) → [∗❢ ✐ ¬ϕ]❇β−α ❥

ϕ multiple change and compensatory effect | = (❚α✶

❥,✐✶ ϕ ∧ ❚α✷ ❥,✐✷¬ϕ ∧ ❯❥ ϕ) → [∗❢ ✐✶ ϕ][∗❢ ′ ✐✷ ¬ϕ]❇α✶−α✷ ❥

ϕ

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

DL-BT Semantics: - Compensatory policy (3/3)

Bill has to decide whether he buys a certain stock and he is currently uncertain: ❍②♣✶ =❞❡❢ ❯❇✐❧❧st♦❝❦❯♣ Suppose that Mary and Jack provide contradictory information about proposition st♦❝❦❯♣. Bill trusts fairly Mary’s judgement on st♦❝❦❯♣ while Bill trusts very weakly Jack’s judgement on ¬st♦❝❦❯♣: ❍②♣✷ =❞❡❢ ❚✸

❇✐❧❧,▼❛r②st♦❝❦❯♣ ∧ ❚✶ ❇✐❧❧,❏❛❝❦¬st♦❝❦❯♣

Now, assume Mary announces that st♦❝❦❯♣ is true and Jack announces that st♦❝❦❯♣ is false (❢ (❇✐❧❧), ❢ ′(❇✐❧❧) = {❝♦♠♣}). | = (❍②♣✶ ∧ ❍②♣✷ ′) → [∗❢

▼❛r②st♦❝❦❯♣][∗❢ ′ ❏❛❝❦¬st♦❝❦❯♣]❇✷ ❇✐❧❧st♦❝❦❯♣.

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Axiomatics

2 steps definition

1

Axiomatics for L-BT logics

2

Reduction axiom for DL-BT logics

Additional axioms for the specifications of the policies

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Proof Theory for L-BT logic (1/2)

K,T, 4 and 5 for ❑✐ K, D for ❇≥α

✐

Interplay axioms between ❑✐ and ❇≥α

✐

❇≥α

✐

ϕ → ❑✐❇≥α

✐

ϕ ¬❇≥α

✐

ϕ → ❑✐¬❇≥α

✐

ϕ ❑✐ ϕ → ❇≥α

✐

ϕ ❇≥α+✶

✐

ϕ → ❇≥α

✐

ϕ

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Proof Theory for L-BT logic (2/2)

Graded Trust axioms

❚α

✐,❥ ϕ → ¬❚β ✐,❥ ϕ if α = β

❚α

✐,❥ ϕ → ❑✐❚α ✐,❥ ϕ

❚α

✐,❥ ϕ →

❑✐ ϕ

Inference rules for Graded Trust

From ϕ ↔ ψ infer ❚α

✐,❥ ϕ ↔ ❚α ✐,❥ψ

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Reduction axioms for DL-BT logic

Reduction axioms are similar to DEL: [∗❢

❥ ϕ]♣

↔ ♣ [∗❢

❥ ϕ]¬ψ

↔ ¬[∗❢

❥ ϕ]ψ

[∗❢

❥ ϕ](ψ✶ ∧ ψ✷)

↔ ([∗❢

❥ ϕ]ψ✶ ∧ [∗❢ ❥ ϕ]ψ✷)

[∗❢

❥ ϕ]❑✐ψ

↔ ❑✐[∗❢

❥ ϕ]ψ

[∗❢

❥ ϕ]❚α ✐,❦ψ

↔ ❚α

✐,❦[∗❢ ❥ ϕ]ψ

Theorem: Soundness and completeness for DL-BT

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Reduction axioms for Change Policy (Illustration)

Additive policy [∗❢

❥ ϕ]❇≥α ✐

ψ ↔

• (¬❚✐,❥ ϕ → ❇≥α

✐

[∗❢

❥ ϕ]ψ)∧

• β∈◆✉♠\{✵},γ✶∈◆✉♠

(❚β

✐,❥ ϕ ∧ ❇γ✶ ✐ ¬ϕ) →

(❇≥α+γ✶

✐

(ϕ → [∗❢

❥ ϕ]ψ)∧

❇≥α−β

✐

(¬ϕ → [∗❢

❥ ϕ]ψ))

• Theorem: Soundness and completeness for DL-BT❛❞❞ (idem for

DL-BT❝♦♠♣ and thus DL-BT❛❞❞,❝♦♠♣

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change

Motivation DL-BT: Syntax L-BT: Semantics DL-BT Semantics Axiomatics Conclusion

Conclusion

Key results

Modular definition of interplay between trust and belief change. Family of logics DL-BT Sound and complete definition for two policies (additive and compensatory)

Future work

In depth definition of the interplay (loop of announcements) New policies Representation theorems

• E. Lorini, G. Jiang, L. Perrussel

Trust-based belief change