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

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

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

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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

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Motivation Modeling Variant with cameras Discussion and conclusion

Outline

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Motivation

2

Modeling

3

Variant with cameras

4

Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion

Scenario: agents equipped with vision devices, positioned in the plane / space.

c a b d

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Motivation Modeling Variant with cameras Discussion and conclusion

Scenario: agents equipped with vision devices, positioned in the plane / space.

c a b d (E.g., robots that cooperate)

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Motivation Modeling Variant with cameras Discussion and conclusion

Scenario: agents equipped with vision devices, positioned in the plane / space.

c a b d (E.g., robots that cooperate) Aim: To represent and compute visual-epistemic reasoning of the agents.

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Outline

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Motivation

2

Modeling Axiomatization Model checking

3

Variant with cameras

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Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Modeling

Each agent has a sector (cone) of vision. a b c d e Assumptions (common knowledge): Agents are transparent points in the plane All objects of interest are agents Agents see infinite sectors Angles of vision are the same α No obstacles (yet)

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Possible worlds

Let U be the set of unit vectors of R2. Definition A geometrical possible world is a tuple w = (pos, dir) where: pos : Agt → R2 dir : Agt → U dir(a) is the bisector of the sector of vision with angle α: pos(a) dir(a) α

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Possible worlds

Let U be the set of unit vectors of R2. Definition A geometrical possible world is a tuple w = (pos, dir) where: pos : Agt → R2 dir : Agt → U dir(a) is the bisector of the sector of vision with angle α: pos(a) dir(a) α Cp,u,α: the closed sector with vertex at the point p, angle α and bisector in direction u. The region seen by a is Cpos(a),dir(a),α.

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

An agent sees another one

Definition a sees b in w = (pos, dir) if pos(b) ∈ Cpos(a),dir(a),α. pos(a) pos(b) pos(c) dir(a) Example a sees a, a sees b. a does not see c.

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Epistemic model Mflatland

Definition Mflatland = (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}. d b a c e ∼a b d e a c w u

In Hintikka’s World: Flatland

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Outline

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Motivation

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Modeling Axiomatization Model checking

3

Variant with cameras

4

Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Disjunctive surprises!

| = (Kaa sees b) ∨ (Kaa✘✘ sees b); | = Ka(b sees c ∨ d sees e) ↔ Ka(b sees c) ∨ Ka(d sees e);

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Some formulas are... Boolean

KaKbCKc,d,e(f sees g) a b c d e f g

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

In 1D, only qualitative positions matter

a d c b a d c b 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);

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Abstraction of the Kripke model in 1D

Definition abs(w) = {b sees c | Mrobots,1D, w | = b sees c} w u abs(w) abs(u) ∼a ∼abs

a

abstraction abstraction

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Axiomatization in 1D

Propositional tautologies; (sameDir(a, b) ↔ sameDir(b, c)) → sameDir(a, c); ¬(a isBetween b, c) ∨ ¬(b isBetween a, c); (Kaa sees b) ∨ (Kaa✘✘ sees b); a sees b → ((Kab sees c) ∨ (Kab✘✘ sees c)); χ → ˆ Kaψ where χ and ψ are completely descriptions with χ ∼abs

a

ψ; Kaϕ → 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.

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

In 2D, the qualitative representation is a open issue

Example Kb(a sees b ∧ a sees d → a sees c) a b c d a b c d true false

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Abstraction of the Kripke model in 2D

Definition abs(w) = {b sees c | Mrobots,2D, w | = b sees c} w u abs(w) abs(u) ∼a ∼abs

a

abstraction abstraction

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Outline

1

Motivation

2

Modeling Axiomatization Model checking

3

Variant with cameras

4

Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Model checking

Input: a description of a world w (and not a WHOLE Kripke model!); a formula ϕ. Output: yes if w | = ϕ.

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Complexity

lineland flatland PSPACE-complete PSPACE-hard, in EXPSPACE translation to R-FO-theory

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Motivation Modeling Variant with cameras Discussion and conclusion Axiomatization Model checking

Reduction to R-FO-theory

Standard translation from modal logic to first-order logic Kap rewrites in ∀u, (wRu) → p(u) [Blackburn et al., modal logic, 2001] Adapted translation from modal logic with seeing to the R-FO-theory Ka(b sees c) rewrites in ∀pos′

a∀pos′ b...∀dir ′ a∀dir ′ b...

{

b∈AGT[(posb ∈ Cpos(a),dir(a),α) → (pos′ b = posb ∧ dir ′ b = dirb)]∧

[(posb ∈ Cpos(a),dir(a),α) → (pos′

b ∈ Cpos(a),dir(a),α)}

→ (pos′

c ∈ Cpos(b),dir(b),α)

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Outline

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Motivation

2

Modeling

3

Variant with cameras Semantics Abstraction works! A PDL variant for cameras Model checking

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Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Outline

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Motivation

2

Modeling

3

Variant with cameras Semantics Abstraction works! A PDL variant for cameras Model checking

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Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Agents are cameras

Cameras Can turn; Can not move. Common knowledge

  • f the positions of agents;
  • f the abilities of perception;

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Semantics: restricted set of worlds

Set of worlds Given a fixed pos′ : AGENTS → R2, worlds are w = (pos, dir) s. th. pos = pos′ a b c d e

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Semantics: restricted set of worlds

Set of worlds Given a fixed pos′ : AGENTS → R2, worlds are w = (pos, dir) s. th. pos = pos′ a b c d e

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Semantics: restricted set of worlds

Set of worlds Given a fixed pos′ : AGENTS → R2, worlds are w = (pos, dir) s. th. pos = pos′ a b c d e

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Semantics: restricted set of worlds

Set of worlds Given a fixed pos′ : AGENTS → R2, worlds are w = (pos, dir) s. th. pos = pos′ a b c d e

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Semantics: Mcameras

Definition Mcameras is Mflatland where we publicly announced the current positions

  • f the agents.

d b a c e ∼a d b e a c w u In Hintikka’s World: Flatland with common knowledge of the positions

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Outline

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Motivation

2

Modeling

3

Variant with cameras Semantics Abstraction works! A PDL variant for cameras Model checking

4

Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Abstraction of the Kripke model Mcameras

Definition abs(w) = {b sees c | Mcameras, w | = b sees c} w u abs(w) abs(u) ∼a ∼abs

a

abstraction abstraction

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Spectrum of vision

Family of vision sets of agent a Sa = {{b}, ∅, {c}, {d}, {d, f }, {d, f , e}, {f , e}, {e}}. a b c d e f NB: each Sa is computed in O(k log k) steps, where k = #(Agt).

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Outline

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Motivation

2

Modeling

3

Variant with cameras Semantics Abstraction works! A PDL variant for cameras Model checking

4

Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

PDL Language

Grammar for formulas ϕ, ψ, . . . ::= a sees b | ¬ϕ | ϕ ∨ ψ | [π]ϕ [π]ϕ: after all executions of program π, ϕ holds.

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Programs

Grammar for programs π . . . ::=

  • a

| ϕ? | π; π′ | π ∪ π′ | π∗

  • a : a turns;

ϕ?: the program succeeds when ϕ is true;

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Translating epistemic operators in programs

Ka is simulated by:    

  • a sees b1? ∪ (a✘✘

sees b1?; b1 )

  • ; . . . ;
  • a sees bn? ∪ (a✘✘

sees bn?; bn )

  • πa

   

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Outline

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Motivation

2

Modeling

3

Variant with cameras Semantics Abstraction works! A PDL variant for cameras Model checking

4

Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion Semantics Abstraction works! A PDL variant for cameras Model checking

Model checking

Theorem Model checking of PDL for cameras is PSPACE-complete. [Gasquet, Goranko, et al. Big Brother Logic: Logical modeling and reasoning about

agents equipped with surveillance cameras in the plane, AAMAS 2014]

[JAAMAS2015]

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Motivation Modeling Variant with cameras Discussion and conclusion

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Modeling

3

Variant with cameras

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Discussion and conclusion

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Motivation Modeling Variant with cameras Discussion and conclusion

Summary: Visual-epistemic reasoning of agents

Epistemic language involving atomic propositions ‘a sees b’. Semantics in geometric and Kripke models. 1D case and 2D case with cameras (spectrum of vision):

Finite abstraction in the 1D case and in the 2D case with cameras (spectrum of vision). Optimal PSPACE model checking.

Open problem for the full 2D case: finite abstraction?

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Future work

Obstacles; Moving agents/cameras in the plane: mathematically more complex, finite abstractions may not work; Agents/cameras in the 3D space.

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