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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion On the Complexity of Dynamic Epistemic Logic Guillaume Aucher 1 Francois Schwarzentruber 2 Workshop Believing, planning, acting, revising July 5, 2013 1 Universit

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion

On the Complexity of Dynamic Epistemic Logic

Guillaume Aucher1 Francois Schwarzentruber2

Workshop ”Believing, planning, acting, revising”

July 5, 2013

1Universit´

e de Rennes 1 - INRIA, France

2ENS Cachan, Brittany extension, France 1 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion

Our environment: now and in the future

socialnetwork AMAISON.fr

  • Distributed systems

Internet, robots Objects Video games Cooperation during a rescue in nuclear plant e-commerce e-voting

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion

A dream

Issues Synthesis of (a squeleton of) a program Planning Verification Logics Time and knowledge: ETL strategies and knowledge: ATEL description of an action and knowledge: DEL

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion

Dynamic epistemic logic

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Outline

1

Dynamic Epistemic Logic (Static) Epistemic logic Event models Product update DEL language

2

Model checking

3

Satisfiability problem

4

Conclusion

5 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Outline

1

Dynamic Epistemic Logic (Static) Epistemic logic Event models Product update DEL language

2

Model checking

3

Satisfiability problem

4

Conclusion

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Epistemic static Kripke models

M = (W , R1, . . . , Rn, V ) with W : possible worlds Ri ⊆ W × W : accessibility relation for agent i V : ATM → 2W : valuation Example b ¬b 1, 2 1, 2 1, 2

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

(Static) Epistemic language

L : ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | Baϕ Semantics Baϕ: agent a believes ϕ; M, w | = Baϕ iff for all u ∈ Ra(w), we have M, u | = ϕ. Example b ¬b 1, 2 1, 2 1, 2

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Outline

1

Dynamic Epistemic Logic (Static) Epistemic logic Event models Product update DEL language

2

Model checking

3

Satisfiability problem

4

Conclusion

9 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Event Kripke models

M′ = (W ′, R′

1, . . . , R′ n, Pre) with

W ′ : possible events R′

a ⊆ W ′ × W ′ : accessibility relation for agent a

Pre : W ′ → L : preconditions

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Example 1 of an event model

b 1, 2

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Example 2 of an event model

b ⊤ 2 1, 2 1

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Outline

1

Dynamic Epistemic Logic (Static) Epistemic logic Event models Product update DEL language

2

Model checking

3

Satisfiability problem

4

Conclusion

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Our expectation

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Updated models

Given M M′ We define the updated model M ⊗ M′ = (W ⊗, R⊗, V ⊗) by: (v, v′) ∈ W ⊗ if v ∈ W , v′ ∈ W ′ and M, v | = Pre(v′) (v, v′)R⊗

i (u, u′)

if vRiu and v′R′

i u′,

(v, v′) ∈ V ⊗(p) if M, v | = p.

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Pointed updated models

Given M, w M′, w′ the pointed updated model M ⊗ M′, (w, w′) is defined iff M, w | = Pre(w′)

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Example

b ¬b 1, 2 1, 2 1, 2 ⊗ b ⊤ 2 1, 2 1 = b b ¬b 2 1, 2 1 2 1, 2 1, 2

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

Outline

1

Dynamic Epistemic Logic (Static) Epistemic logic Event models Product update DEL language

2

Model checking

3

Satisfiability problem

4

Conclusion

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion (Static) Epistemic logic Event models Product update DEL language

DEL language

[van Ditmarsch et al. 2007, van der Hoek, Kooi, Dynamic Epistemic Logic] ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | Baϕ | [π]ϕ π ::= M′, w′ | π ∪ π Semantics [π]ϕ: after the event π, ϕ is true M, w | = [M′, w′]ϕ iff if M ⊗ M′, (w, w′) is defined then M ⊗ M′, (w, w′) | = ϕ; M, w | = [π1 ∪ π2]ϕ iff M, w | = [π1]ϕ and M, w | = [π2]ϕ.

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Outline

1

Dynamic Epistemic Logic

2

Model checking Definition A PSPACE procedure PSPACE-hardness

3

Satisfiability problem

4

Conclusion

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Outline

1

Dynamic Epistemic Logic

2

Model checking Definition A PSPACE procedure PSPACE-hardness

3

Satisfiability problem

4

Conclusion

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Model checking

model checking M, w ϕ yes, if M, w | = ϕ (no otherwise)

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Outline

1

Dynamic Epistemic Logic

2

Model checking Definition A PSPACE procedure PSPACE-hardness

3

Satisfiability problem

4

Conclusion

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

A PSPACE procedure for model checking

Specification M-Check M, w, M′

1, w′ 1,

. . . , M′

i, w′ i

ϕ

such that M, w ⊗ M′

1, w′ 1, . . . , ⊗M′ i , w′ i is defined

yes, if M, w ⊗ M′

1, w′ 1, . . . , ⊗M′ i, w′ i |

= ϕ (no otherwise) Procedure function M-Check(M, w M′

1, w′ 1; . . . ; M′ i, w′ i

ϕ) . . . endFunction

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

A PSPACE procedure for model checking

function M-Check(M, w M′

1, w′ 1; . . . ; M′ i, w′ i

ϕ) match (ϕ) case p: return w ∈ V (p); case ¬ψ: return not M-Check(M, w M′

1, w′ 1; . . . ; M′ i, w′ i

ψ); case ψ1 ∧ ψ2: . . . case Baψ: . . . case [M′, w′]ψ: . . . case [π ∪ γ]ψ: . . . endMatch endFunction

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

A PSPACE procedure for model checking

function M-Check(M, w M′

1, w′ 1; . . . ; M′ i, w′ i

ϕ) match (ϕ) . . . case [M′, w′]ψ: if M-Check(M, w M′

1, w′ 1; . . . ; M′ i, w′ i

Pre(w′)) return M-Check(M, w M′

1, w′ 1; . . . ; M′ i, w′ i ; M′, w′ ψ);

endIf return true ; . . . endMatch endFunction

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

A PSPACE procedure for model checking

function M-Check(M, w M′

1, w′ 1; . . . ; M′ i, w′ i

ϕ) match (ϕ) . . . case Baψ: for u ∈ Ra(w), u′

1 ∈ R′ a(w′ 1), . . . , u′ i ∈ R′ a(w′ i )

if M-Check(w, Pre(u′

1)) and . . .

M-Check(M, u M′

1, u′ 1; . . . ; M′ i−1, u′ i−1 Pre(u′ i))

if not M-Check(M, u M′

1, u′ 1; . . . ; M′ i, u′ i ψ);

return false ; endIf endIf endFor return true ; . . . endMatch endFunction

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Model checking in PSPACE

Theorem The model checking problem is in PSPACE. Proof. Number of nested recursive calls is bounded by |M| + i

1 |M′ i| + |ϕ|;

Memory for local variables of one call is polynomial.

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Outline

1

Dynamic Epistemic Logic

2

Model checking Definition A PSPACE procedure PSPACE-hardness

3

Satisfiability problem

4

Conclusion

29 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

PSPACE-hardness

Theorem The model checking problem is PSPACE-hard. Proof. model checking M, w, ϕ yes/no QBF-SAT reduction ∀p∃q . . . ψ

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Reduction from QBF-SAT to DEL model checking

∀p1∃p2∀p3ψ ⇓ | =

  • ∪

∪

    ∪      ψ   p1 := BaBa⊥, p2 := Ba2Ba⊥, p3 := Ba3Ba⊥  

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

A valuation interpreted as a model

Interpretation pk: there is a path from root of length k. p1, ¬p2, p3, p4

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Product updates = valuation revisions

⊗ ⊗ ⊗ ⊗ ⊗

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A PSPACE procedure PSPACE-hardness

Reduction from QBF-SAT to DEL model checking

∀p1∃p2∀p3ψ ⇓ | =

  • ∪

∪

    ∪      ψ   p1 := BaBa⊥, p2 := Ba2Ba⊥, p3 := Ba3Ba⊥  

34 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

Outline

1

Dynamic Epistemic Logic

2

Model checking

3

Satisfiability problem Definition A NEXPTIME procedure NEXPTIME-hard lower bound

4

Conclusion

35 / 60

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SLIDE 36

Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

Outline

1

Dynamic Epistemic Logic

2

Model checking

3

Satisfiability problem Definition A NEXPTIME procedure NEXPTIME-hard lower bound

4

Conclusion

36 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

Satisfiability problem

SAT ϕ yes, if there exists M, w such that M, w | = ϕ (no otherwise)

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

Outline

1

Dynamic Epistemic Logic

2

Model checking

3

Satisfiability problem Definition A NEXPTIME procedure NEXPTIME-hard lower bound

4

Conclusion

38 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

M′

1

w′

1 : p

u′

1 : ⊤

1 2 1,2

M′

2

w′

2 : B2p

u′

2 : ⊤

2 1 1,2

¬[M′

1, w′ 1][M′ 2, w′ 2]B2B1B2p

39 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

M′

1

w′

1 : p

u′

1 : ⊤

1 2 1,2

M′

2

w′

2 : B2p

u′

2 : ⊤

2 1 1,2

¬[M′

1, w′ 1][M′ 2, w′ 2]B2B1B2p

M′

1, w′ 1 :

M′

1, w′ 1 : ¬[M′ 2, w′ 2]B2B1B2p

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

M′

1

w′

1 : p

u′

1 : ⊤

1 2 1,2

M′

2

w′

2 : B2p

u′

2 : ⊤

2 1 1,2

¬[M′

1, w′ 1][M′ 2, w′ 2]B2B1B2p

M′

1, w′ 1 :

M′

1, w′ 1 : ¬[M′ 2, w′ 2]B2B1B2p

p

41 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

M′

1

w′

1 : p

u′

1 : ⊤

1 2 1,2

M′

2

w′

2 : B2p

u′

2 : ⊤

2 1 1,2

¬[M′

1, w′ 1][M′ 2, w′ 2]B2B1B2p

M′

1, w′ 1 :

M′

1, w′ 1 : ¬[M′ 2, w′ 2]B2B1B2p

p M′

1, w′ 1, M′ 2, w′ 2 :

M′

1, w′ 1, M′ 2, w′ 2 : ¬B2B1B2p

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SLIDE 43

Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

M′

1

w′

1 : p

u′

1 : ⊤

1 2 1,2

M′

2

w′

2 : B2p

u′

2 : ⊤

2 1 1,2

¬[M′

1, w′ 1][M′ 2, w′ 2]B2B1B2p

M′

1, w′ 1 :

M′

1, w′ 1 : ¬[M′ 2, w′ 2]B2B1B2p

p M′

1, w′ 1, M′ 2, w′ 2 :

M′

1, w′ 1, M′ 2, w′ 2 : ¬B2B1B2p

M′

1, w′ 1 : B2p

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

M′

1

w′

1 : p

u′

1 : ⊤

1 2 1,2

M′

2

w′

2 : B2p

u′

2 : ⊤

2 1 1,2

¬[M′

1, w′ 1][M′ 2, w′ 2]B2B1B2p

M′

1, w′ 1 :

M′

1, w′ 1 : ¬[M′ 2, w′ 2]B2B1B2p

p M′

1, w′ 1, M′ 2, w′ 2 :

M′

1, w′ 1, M′ 2, w′ 2 : ¬B2B1B2p

M′

1, w′ 1 : B2p

M′

1, w′ 1, M′ 2, u′ 2 : ¬B1B2p

M′

1, w′ 1, M′ 2, u′ 2 :

2

44 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

M′

1

w′

1 : p

u′

1 : ⊤

1 2 1,2

M′

2

w′

2 : B2p

u′

2 : ⊤

2 1 1,2

¬[M′

1, w′ 1][M′ 2, w′ 2]B2B1B2p

M′

1, w′ 1 :

M′

1, w′ 1 : ¬[M′ 2, w′ 2]B2B1B2p

p M′

1, w′ 1, M′ 2, w′ 2 :

M′

1, w′ 1, M′ 2, w′ 2 : ¬B2B1B2p

M′

1, w′ 1 : B2p

M′

1, w′ 1, M′ 2, u′ 2 : ¬B1B2p

M′

1, w′ 1, M′ 2, u′ 2 :

M′

1, w′ 1 :

M′

1, w′ 1 : ⊤

2

45 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

M′

1

w′

1 : p

u′

1 : ⊤

1 2 1,2

M′

2

w′

2 : B2p

u′

2 : ⊤

2 1 1,2

¬[M′

1, w′ 1][M′ 2, w′ 2]B2B1B2p

M′

1, w′ 1 :

M′

1, w′ 1 : ¬[M′ 2, w′ 2]B2B1B2p

p M′

1, w′ 1, M′ 2, w′ 2 :

M′

1, w′ 1, M′ 2, w′ 2 : ¬B2B1B2p

M′

1, w′ 1 : B2p

M′

1, w′ 1, M′ 2, u′ 2 : ¬B1B2p

M′

1, w′ 1, M′ 2, u′ 2 :

M′

1, w′ 1 :

M′

1, w′ 1 : ⊤

p 2

46 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

A NEXPTIME procedure: a tableau method

@M′

1, u′ 1, . . . , ψ

@M′

3, b′ 3, . . . , ψ′

. . .

≤ |ϕ| ≤ |ϕ|

content ≤ exp(|ϕ|)

Tree in construction exponential size ⇒ NEXPTIME

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

Outline

1

Dynamic Epistemic Logic

2

Model checking

3

Satisfiability problem Definition A NEXPTIME procedure NEXPTIME-hard lower bound

4

Conclusion

48 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

Theorem Our satisfiability problem is NEXPTIME-hard. Proof. SAT ϕ yes/no TILING reduction

k

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

The NEXPTIME-complete tiling problem

k

Input: a finite set T of tile types; k = 2n ∈ N. Output: yes iff we can fill a 2n × 2n grid with tiles from T such that : If two tiles are in contact, the colors match

50 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

Trick of the reduction

1 We build a tree; 2 We encode two non-constrained tilings in a tree;

ϕ (1, 2), (3, 1)

3 We enforce the equality of the two tilings and the tiling

constraints. =

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

1) We build a tree

Let ϕ be a K-formula that enforces the existence of a tree a leave per a valuation over p0, . . . , p4n−1 leaves.

  • m<4n

Bj m

  • (Bjpm ∧ Bj¬pm) ∧
  • i<m

((pi → Bjpi) ∧ (¬pi → Bj¬pi))

  • .

[Blackburn et al., Modal logic.]

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

1) We build a tree

ϕ (3, 4), (12, 2) ¬p0, p1, ¬p2, ¬p3

  • 4

¬p4, ¬p5, p6, p7

  • 3

p8, p9, ¬p10, ¬p11

  • 12

¬p12, ¬p13, p14, ¬p15

  • 2

The tile for the tiling nr. 1 are represented by extra propositions 1t, 1t′, . . . . The tile for the tiling nr. 2 are represented by extra propositions 2t, 2t′, . . . .

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

2) We encode two non-constrained tilings in a tree

ϕ (3, 4), (3, 4), ϕ (3, 4), (3, 4),

What we want to express For all valuations ν over p0, . . . , p2k−1, all ν-leaves satisfy the same proposition 1t.

54 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

2) We encode two non-constrained tilings in a tree

ϕ (3, 4), (3, 4), ϕ (3, 4), (3, 4),

          . . . p0 ∪ . . . ¬p0           . . .            . . . p2k−1 ∪ . . . ¬p2k−1           

  • t∈T B4n

a 1t

55 / 60

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion Definition A NEXPTIME procedure NEXPTIME-hard lower bound

3) We enforce the equality of the two tilings and the constraints of a tiling:

ϕ (3, 4), (3, 4)

B4n

a

  • (x1 = x2) ∧ (y1 = y2) →

t∈T(1t ↔ 2t)

  • ;

ϕ (3, 5), (3, 4)

B4n

a

  • (x1 = x2) ∧ (y1 = y2 + 1)

→

t∈T

{1t → {2t′ | t′ ∈ T, down(t′) = up(t)}}

  • ϕ

(4, 4), (3, 4)

B4n

a

  • (x1 = x2 + 1) ∧ (y1 = y2)

→

t∈T {1t → {2t′ | t′ ∈ T, left(t′) = right(t)}}

  • .

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Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion

Outline

1

Dynamic Epistemic Logic

2

Model checking

3

Satisfiability problem

4

Conclusion

57 / 60

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SLIDE 58

Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion

Conclusion

undecidable decidable NEXPTIME PSPACE NP P

PAL-mc PAL-sat PAL, *, ∪-sat DEL-CK-sat ∪-mc ∪-sat ∪-mc ∪-mc ∪, ∗-mc ∗-mc ∪-sat 58 / 60

slide-59
SLIDE 59

Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion

A plethora of open problems

1 Language to specify event models [M4M2011, JELIA2012] A

whole Dynamic logic? complexity? ApolyEXPTIME as RML [JELIA2012]?

2 Expressive logic but less that entire DEL [LORI2013]. More

expressive?

3 Common knowledge... complexity? automata theory? 4 Constraints over frames (S4, S5...) decidability? 5 Planning instead of verification. undecidable in general.

Fragments?

6 Parametric complexity: modal depths, etc. 7 post-conditions, etc. 59 / 60

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SLIDE 60

Dynamic Epistemic Logic Model checking Satisfiability problem Conclusion

Thank you for your attention!

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