1/41
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
2/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
Our aim
Provide a modal logic combining:epistemic reasoning;perception of agents.
2/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
3/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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”
3/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
3/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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”
3/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
3/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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”
3/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
4/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
Plan
1 Motivations
2 Flatland
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion
4/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
5/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
Outline
1 Motivations
2 Flatland
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion
5/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
6/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
Video games. Example: Super Mario World
IF the ghost knows that Mario does not see him THENthe ghost begins attacking Mario.
6/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
7/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
Robotics: Mars exploration...
IF the robot A does not know that C sees B THENrobot A emits a signal.
7/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
8/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Outline
1 Motivations
2 FlatlandReferenceSyntaxSemanticsResults
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion8/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
9/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Outline
1 Motivations
2 FlatlandReferenceSyntaxSemanticsResults
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion9/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
10/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Flatland, Edwin Abbott Abbott, 1884
10/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
11/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Flatland, Edwin Abbott Abbott, 1884: the story
A square does not believe that the third dimension exists.
11/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
12/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Outline
1 Motivations
2 FlatlandReferenceSyntaxSemanticsResults
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion12/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
13/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Syntax ∼ standard epistemic logic
ϕ ::= aBb︸ ︷︷ ︸a sees b
(litteral)
| ⊥ | ¬ϕ | (ϕ∨ϕ) | Kaϕ︸︷︷︸a knows ϕ
13/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
13/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Syntax ∼ standard epistemic logic
ϕ ::= aBb︸ ︷︷ ︸a sees b
(litteral)
| ⊥ | ¬ϕ | (ϕ∨ϕ)
| Kaϕ︸︷︷︸a knows ϕ
13/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
13/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Syntax ∼ standard epistemic logic
ϕ ::= aBb︸ ︷︷ ︸a sees b
(litteral)
| ⊥ | ¬ϕ | (ϕ∨ϕ) | Kaϕ︸︷︷︸a knows ϕ
13/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
14/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Outline
1 Motivations
2 FlatlandReferenceSyntaxSemanticsResults
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion14/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
15/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Idea of the semantics
Give a semantics foraBb and Kaϕ...
... in terms of positions ofagents and vision cones.
15/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
15/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Idea of the semantics
Give a semantics foraBb and Kaϕ...
... in terms of positions ofagents and vision cones.
15/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
16/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
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
17/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
A flatworld
Definition (flatworld)
〈pos, ~dir〉 where:pos : AGT → R2;~dir : AGT → U.
where R2 is the plane;U is the unit circle.
17/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
18/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Epistemic relation
Two flatworlds areundistinguable for agent c~w�
agent c exactly sees the samething.
18/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
19/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
A Kripke structure made up of flatworlds
19/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
20/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
Outline
1 Motivations
2 FlatlandReferenceSyntaxSemanticsResults
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion20/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
21/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
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.
21/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
21/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
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.
21/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
21/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSyntaxSemanticsResults
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.
21/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
22/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Outline
1 Motivations
2 Flatland
3 LinelandReferenceSemanticsAxiomatization
4 Difference between Lineland and Flatland
5 Conclusion22/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
23/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Outline
1 Motivations
2 Flatland
3 LinelandReferenceSemanticsAxiomatization
4 Difference between Lineland and Flatland
5 Conclusion23/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
24/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Flatland, Edwin Abbott Abbott, 1884
24/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
25/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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.
25/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
26/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Outline
1 Motivations
2 Flatland
3 LinelandReferenceSemanticsAxiomatization
4 Difference between Lineland and Flatland
5 Conclusion26/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
27/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Idea
Give a semantics foraBb and Kaϕ...
(((((Flatworld
Lineworld
27/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
27/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Idea
Give a semantics foraBb and Kaϕ...
(((((Flatworld
Lineworld
27/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
27/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Idea
Give a semantics foraBb and Kaϕ...
(((((Flatworld
Lineworld
27/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
28/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
A lineworld
Definition
〈<, ~dir〉 where:< is a strict total order over AGT ;~dir : AGT → {Left,Right}.
B < A < C
28/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
29/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Kripke model made up of lineworlds
29/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
30/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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.
30/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
31/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Model-checking and Satisfiability problem
TheoremThe model-checking and the satisfiability problem isPSPACE-complete.
31/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
32/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
Outline
1 Motivations
2 Flatland
3 LinelandReferenceSemanticsAxiomatization
4 Difference between Lineland and Flatland
5 Conclusion32/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
33/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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)
33/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
33/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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)
33/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
33/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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)
33/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
33/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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)
33/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
34/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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;
34/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
34/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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;
34/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
34/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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;
34/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
34/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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;
34/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
34/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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;
34/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
35/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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.
35/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
35/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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.
35/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
35/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
ReferenceSemanticsAxiomatization
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.
35/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
36/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
Outline
1 Motivations
2 Flatland
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion
36/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
37/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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.
37/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
38/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
In Flatland: no characterization
Those flatworlds satisfy the same litterals:
¬Kb[(aBb ∧ aBd)→ aBc)] Kb[(aBb ∧ aBd)→ aBc)]
38/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
38/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
In Flatland: no characterization
Those flatworlds satisfy the same litterals:
¬Kb[(aBb ∧ aBd)→ aBc)]
Kb[(aBb ∧ aBd)→ aBc)]
38/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
38/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
In Flatland: no characterization
Those flatworlds satisfy the same litterals:
¬Kb[(aBb ∧ aBd)→ aBc)] Kb[(aBb ∧ aBd)→ aBc)]
38/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
39/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
Outline
1 Motivations
2 Flatland
3 Lineland
4 Difference between Lineland and Flatland
5 Conclusion
39/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
40/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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?
40/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
40/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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?
40/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
40/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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?
40/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
40/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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?
40/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
40/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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?
40/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
40/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
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?
40/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland
41/41
MotivationsFlatland
LinelandDifference between Lineland and Flatland
Conclusion
Thank you!
Questions?
41/41 P. Balbiani, O. Gasquet and F. Schwarzentruber Epistemic reasoning in Flatland and Lineland