Knowledge and seeing Franois Schwarzentruber (joint work with - PowerPoint PPT Presentation

Motivation Modeling Variant with cameras Discussion and conclusion Knowledge and seeing François Schwarzentruber (joint work with Philippe Balbiani, Olivier Gasquet and Valentin Goranko) École Normale Supérieure Rennes May 13, 2019 1 / 62

Motivation Modeling Variant with cameras Discussion and conclusion Outline 1 Motivation 2 Modeling Variant with cameras 3 Discussion and conclusion 4 2 / 62

Motivation Modeling Variant with cameras Discussion and conclusion Scenario: agents equipped with vision devices, positioned in the plane / space. a b c d 3 / 62

Motivation Modeling Variant with cameras Discussion and conclusion Scenario: agents equipped with vision devices, positioned in the plane / space. a b c d (E.g., robots that cooperate) 4 / 62

Motivation Modeling Variant with cameras Discussion and conclusion Scenario: agents equipped with vision devices, positioned in the plane / space. a b c d (E.g., robots that cooperate) Aim: To represent and compute visual-epistemic reasoning of the agents. 5 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Outline 1 Motivation 2 Modeling Axiomatization Model checking 3 Variant with cameras Discussion and conclusion 4 6 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Modeling Each agent has a sector (cone) of vision. Assumptions (common knowledge): Agents are transparent points in the e plane All objects of interest are agents c d b Agents see infinite sectors Angles of vision are the same α a No obstacles (yet) 7 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Possible worlds Let U be the set of unit vectors of R 2 . Definition A geometrical possible world is a tuple w = (pos , dir) where: pos : Agt → R 2 dir : Agt → U dir( a ) is the bisector of the sector of vision with angle α : α dir( a ) pos(a) 8 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Possible worlds Let U be the set of unit vectors of R 2 . Definition A geometrical possible world is a tuple w = (pos , dir) where: pos : Agt → R 2 dir : Agt → U dir( a ) is the bisector of the sector of vision with angle α : α dir( a ) pos(a) C p , u ,α : the closed sector with vertex at the point p , angle α and bisector in direction u . The region seen by a is C pos( a ) , dir( a ) ,α . 9 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion An agent sees another one Definition a sees b in w = (pos , dir) if pos( b ) ∈ C pos( a ) , dir( a ) ,α . pos(c) pos(b) dir( a ) pos(a) Example a sees a , a sees b . a does not see c . 10 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Epistemic model M flatland Definition M flatland = ( W , ( ∼ a ) a ∈ AGT , V ) with: W is the set of all geometrical possible worlds; w ∼ a u if agents a see the same agents in both w and u and these agents have the same position and direction in both w and u ; V ( w ) = { a sees b | agent a sees b in w } . b d e e a a c c b d ∼ a w u In Hintikka’s World: Flatland 11 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Outline 1 Motivation 2 Modeling Axiomatization Model checking 3 Variant with cameras Discussion and conclusion 4 12 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Disjunctive surprises! | = ( K a a sees b ) ∨ ( K a a ✘✘ sees b ); | = K a ( b sees c ∨ d sees e ) ↔ K a ( b sees c ) ∨ K a ( d sees e ); 13 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Some formulas are... Boolean K a K b CK c , d , e ( f sees g ) d g a e b f c 14 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion In 1D, only qualitative positions matter a c b d a c b d Expressivity Qualitative positions are expressible in the language. sameDir( a , b ) := ( a sees b ↔ b ✘✘ sees a ) a isBetween b , c := ( a sees b ↔ a ✘✘ sees c ); 15 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Abstraction of the Kripke model in 1D Definition abs ( w ) = { b sees c | M robots , 1 D , w | = b sees c } abstraction w abs ( w ) ∼ a ∼ abs a abs ( u ) u abstraction 16 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Axiomatization in 1D Propositional tautologies; (sameDir( a , b ) ↔ sameDir( b , c )) → sameDir( a , c ); ¬ ( a isBetween b , c ) ∨ ¬ ( b isBetween a , c ); ( K a a sees b ) ∨ ( K a a ✘✘ sees b ); a sees b → (( K a b sees c ) ∨ ( K a b ✘✘ sees c )); χ → ˆ K a ψ where χ and ψ are completely descriptions with χ ∼ abs ψ ; a K a ϕ → ϕ . [ Balbiani et al. Agents that look at one another. Logic Journal of IGPL. 2012 ] Definition A complete decription is a conjunction that: contains a sees b or a ✘✘ sees b for all agents a , b ; is satisfiable. 17 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion In 2D, the qualitative representation is a open issue Example K b ( a sees b ∧ a sees d → a sees c ) a a c c b d b d true false 18 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Abstraction of the Kripke model in 2D Definition abs ( w ) = { b sees c | M robots , 2 D , w | = b sees c } abstraction w abs ( w ) ∼ a ∼ abs a u abs ( u ) abstraction 19 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Outline 1 Motivation 2 Modeling Axiomatization Model checking 3 Variant with cameras Discussion and conclusion 4 20 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Model checking Input: a description of a world w (and not a WHOLE Kripke model!); a formula ϕ . Output: yes if w | = ϕ . 21 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Complexity lineland flatland PSPACE-complete PSPACE-hard, in EXPSPACE translation to R -FO-theory 22 / 62

Motivation Modeling Axiomatization Variant with cameras Model checking Discussion and conclusion Reduction to R -FO-theory Standard translation from modal logic to first-order logic K a p rewrites in ∀ u , ( wRu ) → p ( u ) [ Blackburn et al., modal logic, 2001 ] Adapted translation from modal logic with seeing to the R -FO-theory K a ( b sees c ) rewrites in ∀ pos ′ a ∀ pos ′ b ... ∀ dir ′ a ∀ dir ′ b ... { � b ∈ AGT [(pos b ∈ C pos( a ) , dir( a ) ,α ) → ( pos ′ b = pos b ∧ dir ′ b = dir b )] ∧ [( pos b �∈ C pos( a ) , dir( a ) ,α ) → ( pos ′ b �∈ C pos( a ) , dir( a ) ,α ) } → ( pos ′ c �∈ C pos( b ) , dir( b ) ,α ) 23 / 62

Motivation Semantics Modeling Abstraction works! Variant with cameras A PDL variant for cameras Discussion and conclusion Model checking Outline Motivation 1 Modeling 2 3 Variant with cameras Semantics Abstraction works! A PDL variant for cameras Model checking 4 Discussion and conclusion 24 / 62

Motivation Semantics Modeling Abstraction works! Variant with cameras A PDL variant for cameras Discussion and conclusion Model checking Outline Motivation 1 Modeling 2 3 Variant with cameras Semantics Abstraction works! A PDL variant for cameras Model checking 4 Discussion and conclusion 25 / 62

Motivation Semantics Modeling Abstraction works! Variant with cameras A PDL variant for cameras Discussion and conclusion Model checking Agents are cameras Cameras Can turn; Can not move. Common knowledge of the positions of agents; of the abilities of perception; 26 / 62

Motivation Semantics Modeling Abstraction works! Variant with cameras A PDL variant for cameras Discussion and conclusion Model checking Semantics: restricted set of worlds Set of worlds Given a fixed pos ′ : AGENTS → R 2 , worlds are w = ( pos , dir ) s. th. pos = pos ′ c d a b e 27 / 62

Motivation Semantics Modeling Abstraction works! Variant with cameras A PDL variant for cameras Discussion and conclusion Model checking Semantics: restricted set of worlds Set of worlds Given a fixed pos ′ : AGENTS → R 2 , worlds are w = ( pos , dir ) s. th. pos = pos ′ c d a b e 28 / 62

Motivation Semantics Modeling Abstraction works! Variant with cameras A PDL variant for cameras Discussion and conclusion Model checking Semantics: restricted set of worlds Set of worlds Given a fixed pos ′ : AGENTS → R 2 , worlds are w = ( pos , dir ) s. th. pos = pos ′ c d a b e 29 / 62

Motivation Semantics Modeling Abstraction works! Variant with cameras A PDL variant for cameras Discussion and conclusion Model checking Semantics: restricted set of worlds Set of worlds Given a fixed pos ′ : AGENTS → R 2 , worlds are w = ( pos , dir ) s. th. pos = pos ′ c d a b e 30 / 62