Epistemic reasoning in Flatland and Linelandpeople.irisa.fr/Francois.Schwarzentruber/publications/... · 2011. 9. 1. · Flatland, Edwin Abbott Abbott, 1884: in the story The square

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MotivationsFlatland

LinelandDifference between Lineland and Flatland

Conclusion

Epistemic reasoning in Flatland and Lineland

P. Balbiani, O. Gasquet and F. Schwarzentruber

AiML 2010

August 25, 2010

1/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland

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MotivationsFlatland


Conclusion

Our aim

Provide a modal logic combining:epistemic reasoning;perception of agents.


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Conclusion

Intuitions

A sees B;

B sees C;

A does not see C;

B knows that C sees B;

B does not know that Asees B.

“model”


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Conclusion

Intuitions

A sees B;

B sees C;

A does not see C;


B does not know that Asees B.

“model”


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Conclusion

Intuitions

A sees B;

B sees C;

A does not see C;


B does not know that Asees B. “model”


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Conclusion

Plan

1 Motivations

2 Flatland

3 Lineland

4 Difference between Lineland and Flatland

5 Conclusion


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Conclusion

Outline

1 Motivations

2 Flatland

3 Lineland


5 Conclusion


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Conclusion

Video games. Example: Super Mario World

IF the ghost knows that Mario does not see him THENthe ghost begins attacking Mario.


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Conclusion

Robotics: Mars exploration...

IF the robot A does not know that C sees B THENrobot A emits a signal.


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Conclusion

ReferenceSyntaxSemanticsResults

Outline

1 Motivations

2 FlatlandReferenceSyntaxSemanticsResults

3 Lineland


5 Conclusion8/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland

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Conclusion


Outline

1 Motivations


3 Lineland



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Conclusion


Flatland, Edwin Abbott Abbott, 1884


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Conclusion


Flatland, Edwin Abbott Abbott, 1884: the story

A square does not believe that the third dimension exists.


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MotivationsFlatland


Conclusion


Outline

1 Motivations


3 Lineland



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Conclusion


Syntax ∼ standard epistemic logic

ϕ ::= aBb︸︷︷︸a sees b

(litteral)

| ⊥ | ¬ϕ | (ϕ∨ϕ) | Kaϕ︸︷︷︸a knows ϕ


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Conclusion




(litteral)

| ⊥ | ¬ϕ | (ϕ∨ϕ)

| Kaϕ︸︷︷︸a knows ϕ


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Conclusion




(litteral)

| ⊥ | ¬ϕ | (ϕ∨ϕ) | Kaϕ︸︷︷︸a knows ϕ


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Conclusion


Outline

1 Motivations


3 Lineland



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Conclusion


Idea of the semantics

Give a semantics foraBb and Kaϕ...

... in terms of positions ofagents and vision cones.


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Conclusion


Idea of the semantics


... in terms of positions ofagents and vision cones.


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Conclusion


Restrictions

no obstacles;

agents are points;

agents are transparent;

agents are in a plane;

an agent sees an openhalf-plane;

the world is static.16/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland

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MotivationsFlatland


Conclusion


A flatworld

Definition (flatworld)

〈pos, ~dir〉 where:pos : AGT → R2;~dir : AGT → U.

where R2 is the plane;U is the unit circle.


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Conclusion


Epistemic relation

Two flatworlds areundistinguable for agent c~w�

agent c exactly sees the samething.


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MotivationsFlatland


Conclusion


A Kripke structure made up of flatworlds


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Conclusion


Outline

1 Motivations


3 Lineland



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Conclusion


Results

The model-checking and the satisfiability problem is:in EXPSPACE;

translation

epistemic logic −→ FO-real numbers theory

aBb ~dir(a)�pos(a)pos(b) > 0︸︷︷︸scalar product

PSPACE-hard;Reduction QBF-SAT

Axiomatization unknown.


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Conclusion


Results


translation






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Conclusion


Results


translation






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Conclusion

ReferenceSemanticsAxiomatization

Outline

1 Motivations

2 Flatland

3 LinelandReferenceSemanticsAxiomatization



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Conclusion


Outline

1 Motivations

2 Flatland




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Conclusion


Flatland, Edwin Abbott Abbott, 1884


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Conclusion


Flatland, Edwin Abbott Abbott, 1884: in the story

The square dreams about Lineland: he is surprised becauseinhabitants of Lineland do not believe that the seconddimension exists.


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Conclusion


Outline

1 Motivations

2 Flatland




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Conclusion


Idea


(((((Flatworld

Lineworld


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Conclusion


Idea


(((((Flatworld

Lineworld


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Conclusion


Idea


(((((Flatworld

Lineworld


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Conclusion


A lineworld

Definition

〈<, ~dir〉 where:< is a strict total order over AGT ;~dir : AGT → {Left,Right}.

B < A < C


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Conclusion


Kripke model made up of lineworlds


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Conclusion


Truth conditions

b. . .

c. . .

a

a. . .

c. . .

b

Definition

w |= aBb iff

{either ( ~dir(a) = Left and b < a)

or ( ~dir(a) = Right and a < b)

};

w |= Kaϕ iff for all u such that wRau we have u |= ϕ.


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Conclusion


Model-checking and Satisfiability problem

TheoremThe model-checking and the satisfiability problem isPSPACE-complete.


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Conclusion


Outline

1 Motivations

2 Flatland




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Conclusion


Axiomatization of perception

Boolean tautologies;modus ponens;

(Ax1) ¬(aBa);(Ax2) ¬(aBb) ∧ aBc → (b Ba↔ b Bc);

b. . .

a. . .

c(Ax4) aBb, c ∧ (c Ba↔ c Bb)→ (¬b Ba↔ b Bc);(Ax3) ¬aBb∧¬aBc∧(cBa↔ ¬cBb)→ (bBc ↔ bBa);(Ax5) ¬aBb ∧ ¬aBc ∧ ¬aBd ∧ (b Ba↔

b Bc) ∧ (c Ba↔ c Bd)→ (b Ba↔ b Bd);(Ax6) aBb ∧ aBc ∧ aBd ∧ (b Ba↔ ¬b Bc) ∧ (c Ba↔

¬c Bd)→ (b Ba↔ ¬b Bd)


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Conclusion




(Ax1) ¬(aBa);

(Ax2) ¬(aBb) ∧ aBc → (b Ba↔ b Bc);

b. . .

a. . .





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Conclusion





b. . .

a. . .

c

(Ax4) aBb, c ∧ (c Ba↔ c Bb)→ (¬b Ba↔ b Bc);(Ax3) ¬aBb∧¬aBc∧(cBa↔ ¬cBb)→ (bBc ↔ bBa);(Ax5) ¬aBb ∧ ¬aBc ∧ ¬aBd ∧ (b Ba↔




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Conclusion





b. . .

a. . .





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Conclusion


Axiomatization of epistemic operator

necessitation rules;(AxK ) axiom K: Ka(ϕ→ ψ)→ (Kaϕ→ Kaψ);

(Ax7) aBb → KaaBb ;(Ax8) ¬aBb → Ka¬aBb;(Ax9) aBb ∧ b Bc → Kab Bc;

c. . .

a. . .

b(Ax10) aBb ∧ ¬b Bc → Ka¬b Bc;


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Conclusion



necessitation rules;(AxK ) axiom K: Ka(ϕ→ ψ)→ (Kaϕ→ Kaψ);(Ax7) aBb → KaaBb ;

(Ax8) ¬aBb → Ka¬aBb;(Ax9) aBb ∧ b Bc → Kab Bc;

c. . .

a. . .



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Conclusion



necessitation rules;(AxK ) axiom K: Ka(ϕ→ ψ)→ (Kaϕ→ Kaψ);(Ax7) aBb → KaaBb ;(Ax8) ¬aBb → Ka¬aBb;

(Ax9) aBb ∧ b Bc → Kab Bc;

c. . .

a. . .



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Conclusion



necessitation rules;(AxK ) axiom K: Ka(ϕ→ ψ)→ (Kaϕ→ Kaψ);(Ax7) aBb → KaaBb ;(Ax8) ¬aBb → Ka¬aBb;(Ax9) aBb ∧ b Bc → Kab Bc;

c. . .

a. . .

b

(Ax10) aBb ∧ ¬b Bc → Ka¬b Bc;


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Conclusion



necessitation rules;(AxK ) axiom K: Ka(ϕ→ ψ)→ (Kaϕ→ Kaψ);(Ax7) aBb → KaaBb ;(Ax8) ¬aBb → Ka¬aBb;(Ax9) aBb ∧ b Bc → Kab Bc;

c. . .

a. . .



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Conclusion


Axiomatization: axiom of “imagination”

(Ax11) ϕ→ K̂aψ

s.th.∃G ⊆ AGT , G finite and∃V ⊆ G s.th.:

ϕ

is a conjunction of litterals s.th:for all b ∈ V , aBb ∈ ϕ;for all b ∈ G \ V , ¬aBb ∈ ϕ;for all b ∈ V , for all c ∈ G,either b Bc ∈ ϕ or ¬b Bc ∈ ϕ;ϕ is satisfiable.

ψ

is a maximal satisfiableconjunction of litterals b Bc or¬b Bc where b, c ∈ G,subsuming ϕ.


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Conclusion



(Ax11) ϕ→ K̂aψ s.th.∃G ⊆ AGT , G finite and∃V ⊆ G s.th.:

ϕ is a conjunction of litterals s.th:for all b ∈ V , aBb ∈ ϕ;for all b ∈ G \ V , ¬aBb ∈ ϕ;for all b ∈ V , for all c ∈ G,either b Bc ∈ ϕ or ¬b Bc ∈ ϕ;ϕ is satisfiable.

ψ

is a maximal satisfiableconjunction of litterals b Bc or¬b Bc where b, c ∈ G,subsuming ϕ.


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Conclusion



(Ax11) ϕ→ K̂aψ s.th.∃G ⊆ AGT , G finite and∃V ⊆ G s.th.:

ϕ is a conjunction of litterals s.th:for all b ∈ V , aBb ∈ ϕ;for all b ∈ G \ V , ¬aBb ∈ ϕ;for all b ∈ V , for all c ∈ G,either b Bc ∈ ϕ or ¬b Bc ∈ ϕ;ϕ is satisfiable.

ψ is a maximal satisfiableconjunction of litterals b Bc or¬b Bc where b, c ∈ G,subsuming ϕ.


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Conclusion

Outline

1 Motivations

2 Flatland

3 Lineland


5 Conclusion


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Conclusion

In Lineland: characterization

Theorem (Lineland)The true formulas only depend of the truths of the litterals aBb.

Example

B A Cand

C A Bsatisfy the same formulas.


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Conclusion

In Flatland: no characterization

Those flatworlds satisfy the same litterals:

¬Kb[(aBb ∧ aBd)→ aBc)] Kb[(aBb ∧ aBd)→ aBc)]


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Conclusion



¬Kb[(aBb ∧ aBd)→ aBc)]

Kb[(aBb ∧ aBd)→ aBc)]


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Conclusion



¬Kb[(aBb ∧ aBd)→ aBc)] Kb[(aBb ∧ aBd)→ aBc)]


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Conclusion

Outline

1 Motivations

2 Flatland

3 Lineland


5 Conclusion


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Conclusion

Perspectives

What about a finite axiomatization of Lineland?

What about an axiomatization of Flatland?Complexity of the model-checking and satisfiabilityproblem of Flatland?Capturing the difference between dimension 2 and 3?What about myopic agents etc.?What about mobility? time?


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MotivationsFlatland


Conclusion

Perspectives

What about a finite axiomatization of Lineland?What about an axiomatization of Flatland?

Complexity of the model-checking and satisfiabilityproblem of Flatland?Capturing the difference between dimension 2 and 3?What about myopic agents etc.?What about mobility? time?


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MotivationsFlatland


Conclusion

Perspectives

What about a finite axiomatization of Lineland?What about an axiomatization of Flatland?Complexity of the model-checking and satisfiabilityproblem of Flatland?

Capturing the difference between dimension 2 and 3?What about myopic agents etc.?What about mobility? time?


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MotivationsFlatland


Conclusion

Perspectives

What about a finite axiomatization of Lineland?What about an axiomatization of Flatland?Complexity of the model-checking and satisfiabilityproblem of Flatland?Capturing the difference between dimension 2 and 3?

What about myopic agents etc.?What about mobility? time?


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MotivationsFlatland


Conclusion

Perspectives

What about a finite axiomatization of Lineland?What about an axiomatization of Flatland?Complexity of the model-checking and satisfiabilityproblem of Flatland?Capturing the difference between dimension 2 and 3?What about myopic agents etc.?

What about mobility? time?


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MotivationsFlatland


Conclusion

Perspectives

What about a finite axiomatization of Lineland?What about an axiomatization of Flatland?Complexity of the model-checking and satisfiabilityproblem of Flatland?Capturing the difference between dimension 2 and 3?What about myopic agents etc.?What about mobility? time?


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Conclusion

Thank you!

Questions?


Documents

Epistemic reasoning in Flatland and Linelandpeople.irisa.fr/Francois.Schwarzentruber/publications/... · 2011. 9. 1. · Flatland, Edwin Abbott Abbott, 1884: in the story The square