On The Complexity of Bounded Time Reachability for Piecewise Affine - - PowerPoint PPT Presentation

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on the complexity of bounded time reachability

On The Complexity of Bounded Time Reachability for Piecewise Affine - - PowerPoint PPT Presentation

Apr 16, 2023 191 likes •1.22k views

On The Complexity of Bounded Time Reachability for Piecewise Affine Systems H. Bazille 3 O. Bournez 1 W. Gomaa 2 , 4 A. Pouly 1 1 cole Polytechnique, LIX, 91128 Palaiseau Cedex, France 2 Egypt Japan University of Science and Technology, CSE,

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On The Complexity of Bounded Time Reachability for Piecewise Affine Systems∗

H. Bazille3
O. Bournez1
W. Gomaa2,4
A. Pouly1

1École Polytechnique, LIX, 91128 Palaiseau Cedex, France 2Egypt Japan University of Science and Technology, CSE, Alexandria, Egypt 3ENS Cachan/Bretagne et Université Rennes 1, France 4Faculty of Engineering, Alexandria University, Alexandria, Egypt

September 23, 2014

∗This work was partially supported by DGA Project CALCULS. Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 −∞ / 17

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Outline

1

Introduction Piecewise Affine Systems Problems

2

Proof Complexity Hardness

3

Conclusion

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 −∞ / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd piecewise affine function f : H → H

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd piecewise affine function f : H → H f(x) = Aix + bi for x ∈ Hi Ai ∈ Qd×d, bi ∈ Qd

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd piecewise affine function f : H → H f(x) = Aix + bi for x ∈ Hi Ai ∈ Qd×d, bi ∈ Qd trajectory: x, f(x), f [2](x), . . . , f [i](x), . . .

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd piecewise affine function f : H → H f(x) = Aix + bi for x ∈ Hi Ai ∈ Qd×d, bi ∈ Qd trajectory: x, f(x), f [2](x), . . . , f [i](x), . . . ⇒ Discrete time dynamical system

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd piecewise affine function f : H → H f(x) = Aix + bi for x ∈ Hi Ai ∈ Qd×d, bi ∈ Qd trajectory: x, f(x), f [2](x), . . . , f [i](x), . . . ⇒ Discrete time dynamical system Three cases: K = N: integer case K = [0, 1]: continuous bounded case K = R: continuous unbounded case

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd piecewise affine function f : H → H f(x) = Aix + bi for x ∈ Hi Ai ∈ Qd×d, bi ∈ Qd trajectory: x, f(x), f [2](x), . . . , f [i](x), . . . ⇒ Discrete time dynamical system Three cases: K = N: integer case → Very different from [0, 1] and R K = [0, 1]: continuous bounded case K = R: continuous unbounded case

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd piecewise affine function f : H → H f(x) = Aix + bi for x ∈ Hi Ai ∈ Qd×d, bi ∈ Qd trajectory: x, f(x), f [2](x), . . . , f [i](x), . . . ⇒ Discrete time dynamical system Three cases: K = N: integer case → Very different from [0, 1] and R K = [0, 1]: continuous bounded case → Our case K = R: continuous unbounded case

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (1)

General Model vector space: H = Kd partition of the space: H = ∪m

i=1Hi

Hi = convex polyhedron = {x | Mix vi} Mi ∈ Qd×d, vi ∈ Qd piecewise affine function f : H → H f(x) = Aix + bi for x ∈ Hi Ai ∈ Qd×d, bi ∈ Qd trajectory: x, f(x), f [2](x), . . . , f [i](x), . . . ⇒ Discrete time dynamical system Three cases: K = N: integer case → Very different from [0, 1] and R K = [0, 1]: continuous bounded case → Our case K = R: continuous unbounded case → Similarish to [0, 1] ?

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 1 / 17

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Piecewise Affine System (2)

f continuous f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 2 / 17

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Piecewise Affine System (2)

f continuous f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

f discontinuous f(x) =

2x

if x ∈ [0, 1

2[

2x − 1 if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 2 / 17

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Piecewise Affine System (2)

f continuous f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

→ Our case f discontinuous f(x) =

2x

if x ∈ [0, 1

2[

2x − 1 if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 2 / 17

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Piecewise Affine System (2)

f continuous f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

→ Our case f discontinuous f(x) =

2x

if x ∈ [0, 1

2[

2x − 1 if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

→ Quite different

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 2 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Trajectory

1 2

1

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Trajectory

1 2

1 x x = 0.5625

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Trajectory

1 2

1 x f(x) x = 0.5625 f(x) = 0.875

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Trajectory

1 2

1 x f(x) f [2](x) x = 0.5625 f(x) = 0.875 f [2](x) = 0.25

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Trajectory

1 2

1 x f(x) f [2](x) f [3](x) x = 0.5625 f(x) = 0.875 f [2](x) = 0.25 f [3](x) = 0.5

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Trajectory

1 2

1 x f(x) f [2](x) f [3](x) f [4](x) x = 0.5625 f(x) = 0.875 f [2](x) = 0.25 f [3](x) = 0.5 f [4](x) = 1

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Trajectory

1 2

1 x f(x) f [2](x) f [3](x) f [4](x) f [5](x) x = 0.5625 f(x) = 0.875 f [2](x) = 0.25 f [3](x) = 0.5 f [4](x) = 1 f [5](x) = 0

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Trajectory

1 2

1 x f(x) f [2](x) f [3](x) f [4](x) f [5](x) x = 0.5625 f(x) = 0.875 f [2](x) = 0.25 f [3](x) = 0.5 f [4](x) = 1 f [n](x) = 0 n 5

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Example

Function f(x) =

2x

if x ∈ [0, 1

2]

2 − 2x if x ∈ [ 1

2, 1]

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Remark Trajectory depends on the bi- nary expansion of x Trajectory

1 2

1 x f(x) f [2](x) f [3](x) f [4](x) f [5](x) x = 0.5625 f(x) = 0.875 f [2](x) = 0.25 f [3](x) = 0.5 f [4](x) = 1 f [n](x) = 0 n 5

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

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Existings Results

Problem: REACH-REGION Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

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Existings Results

Problem: REACH-REGION Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

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Existings Results

Problem: REACH-REGION Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d Question: ∃x ∈ R0, ∃t ∈ N, f [t](x) ∈ R ?

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

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Existings Results

Problem: REACH-REGION Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d Question: ∃x ∈ R0, ∃t ∈ N, f [t](x) ∈ R ? Example R0 R x f(x) f [2](x) f [3](x)

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

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Existings Results

Problem: REACH-REGION Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d Question: ∃x ∈ R0, ∃t ∈ N, f [t](x) ∈ R ? Example R0 R x f(x) f [2](x) f [3](x) Theorem (Koiran, Cosnard, Garzon) REACH-REGION is undecidable for d 2

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

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Existings Results

Problem: REACH-REGION Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d Question: ∃x ∈ R0, ∃t ∈ N, f [t](x) ∈ R ? Example R0 R x f(x) f [2](x) f [3](x) Theorem (Koiran, Cosnard, Garzon) REACH-REGION is undecidable for d 2 Proof (Idea) Simulate a Turing Machine and re- duce from halting problem.

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

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Existings Results

Problem: REACH-REGION Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d Question: ∃x ∈ R0, ∃t ∈ N, f [t](x) ∈ R ? Example R0 R x f(x) f [2](x) f [3](x) Theorem (Koiran, Cosnard, Garzon) REACH-REGION is undecidable for d 2 Proof (Idea) Simulate a Turing Machine and re- duce from halting problem. Open Problem Decidability for d = 1.

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

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Existings Results

Problem: CONTROL-REGION Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d Question: ∀x ∈ R0, ∃t ∈ N, f [t](x) ∈ R ? Example R0 R x f(x) f [2](x) f [3](x) Theorem (Blondel, Bournez, Koiran, Tsitsiklis) CONTROL-REGION is undecidable for d 2 Proof (Idea) Harder simulation of a Turing Ma- chine Open Problem Decidability for d = 1.

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

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Our Results

Problem: REACH-REGION-TIME Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 5 / 17

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Our Results

Problem: REACH-REGION-TIME Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d, T ∈ N in unary

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 5 / 17

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Our Results

Problem: REACH-REGION-TIME Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d, T ∈ N in unary Question: ∃x ∈ R0, ∃t T, f [t](x) ∈ R ?

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 5 / 17

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Our Results

Problem: REACH-REGION-TIME Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d, T ∈ N in unary Question: ∃x ∈ R0, ∃t T, f [t](x) ∈ R ? Theorem REACH-REGION-TIME is NP-complete for d 2

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 5 / 17

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Our Results

Problem: REACH-REGION-TIME Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d, T ∈ N in unary Question: ∃x ∈ R0, ∃t T, f [t](x) ∈ R ? Theorem REACH-REGION-TIME is NP-complete for d 2 Open Problem Complexity for d = 1.

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 5 / 17

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Our Results

Problem: CONTROL-REGION-TIME Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d, T ∈ N in unary Question: ∀x ∈ R0, ∃t T, f [t](x) ∈ R ? Theorem CONTROL-REGION-TIME is coNP-complete for d 2 Open Problem Complexity for d = 1.

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 5 / 17

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Statement

Problem: REACH-REGION-TIME Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d, T ∈ N in unary Question: ∃x ∈ R0, ∃t T, f [t](x) ∈ R ? Theorem REACH-REGION-TIME is in NP .

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 6 / 17

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Signature

Example R0 R1 R2 R3 R4 R5 Definition The signature σ(x) ∈ {0, . . . , n}N of x is defined by: σi(x) = j ⇔ f [i](x) ∈ Rj

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

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Signature

Example R0 R1 R2 R3 R4 R5 x σ(x) = (0, . . .) Definition The signature σ(x) ∈ {0, . . . , n}N of x is defined by: σi(x) = j ⇔ f [i](x) ∈ Rj

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

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Signature

Example R0 R1 R2 R3 R4 R5 x f(x) σ(x) = (0, 1, . . .) Definition The signature σ(x) ∈ {0, . . . , n}N of x is defined by: σi(x) = j ⇔ f [i](x) ∈ Rj

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

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Signature

Example R0 R1 R2 R3 R4 R5 x f(x) f [2](x) σ(x) = (0, 1, 4, . . .) Definition The signature σ(x) ∈ {0, . . . , n}N of x is defined by: σi(x) = j ⇔ f [i](x) ∈ Rj

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

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Signature

Example R0 R1 R2 R3 R4 R5 x f(x) f [2](x) f [3](x) σ(x) = (0, 1, 4, 3, . . .) Definition The signature σ(x) ∈ {0, . . . , n}N of x is defined by: σi(x) = j ⇔ f [i](x) ∈ Rj

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

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Signature

Example R0 R1 R2 R3 R4 R5 x f(x) f [2](x) f [3](x) σ(x) = (0, 1, 4, 3, . . .) Definition The signature σ(x) ∈ {0, . . . , n}N of x is defined by: σi(x) = j ⇔ f [i](x) ∈ Rj Lemma If σ(x) = (r1, r2, . . . , rt, . . .) then f [t](x) = Art(· · · (Ar1x + br1) · · · ) + brt = Cσ + dσ

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

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Signature

Example R0 R1 R2 R3 R4 R5 x f(x) f [2](x) f [3](x) σ(x) = (0, 1, 4, 3, . . .) Definition The signature σ(x) ∈ {0, . . . , n}N of x is defined by: σi(x) = j ⇔ f [i](x) ∈ Rj Lemma If σ(x) = (r1, r2, . . . , rt, . . .) then f [t](x) = Art(· · · (Ar1x + br1) · · · ) + brt = Cσ + dσ Furthermore (s(X) =coeff size): s(Cσ, dσ) = poly(s(A), s(b), t)

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

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Algorithm

Given f, R0, R = Rn and T:

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

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Algorithm

Given f, R0, R = Rn and T: Guess t T ← Nondeterministic polynomial

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

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SLIDE 51

Algorithm

Given f, R0, R = Rn and T: Guess t T ← Nondeterministic polynomial Guess signature r1, . . . , rt−1 ← Nondeterministic polynomial

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

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SLIDE 52

Algorithm

Given f, R0, R = Rn and T: Guess t T ← Nondeterministic polynomial Guess signature r1, . . . , rt−1 ← Nondeterministic polynomial Guess x ∈ Qd of polynomial size ← Nondeterministic polynomial

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

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SLIDE 53

Algorithm

Given f, R0, R = Rn and T: Guess t T ← Nondeterministic polynomial Guess signature r1, . . . , rt−1 ← Nondeterministic polynomial Guess x ∈ Qd of polynomial size ← Nondeterministic polynomial Check that f [i](x) ∈ Rri for all i ∈ {0, . . . , t}:

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

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SLIDE 54

Algorithm

Given f, R0, R = Rn and T: Guess t T ← Nondeterministic polynomial Guess signature r1, . . . , rt−1 ← Nondeterministic polynomial Guess x ∈ Qd of polynomial size ← Nondeterministic polynomial Check that f [i](x) ∈ Rri for all i ∈ {0, . . . , t}: f [i](x) ∈ Rri ⇔ Mri(Cix + di) vi ← Polynomial size

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

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SLIDE 55

Algorithm

Given f, R0, R = Rn and T: Guess t T ← Nondeterministic polynomial Guess signature r1, . . . , rt−1 ← Nondeterministic polynomial Guess x ∈ Qd of polynomial size ← Nondeterministic polynomial Check that f [i](x) ∈ Rri for all i ∈ {0, . . . , t}: f [i](x) ∈ Rri ⇔ Mri(Cix + di) vi ← Polynomial size Accept if all systems are satisfied

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

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Algorithm

Given f, R0, R = Rn and T: Guess t T ← Nondeterministic polynomial Guess signature r1, . . . , rt−1 ← Nondeterministic polynomial Guess x ∈ Qd of polynomial size ← Nondeterministic polynomial Check that f [i](x) ∈ Rri for all i ∈ {0, . . . , t}: f [i](x) ∈ Rri ⇔ Mri(Cix + di) vi ← Polynomial size Accept if all systems are satisfied Theorem (Koiran) Every satisfiable rational linear system Ax b has a rational solution

f polynomial size.

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

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SLIDE 57

Statement

Problem: REACH-REGION-TIME Input: f : [0, 1]d → [0, 1]d continuous, piecewise affine Input: R0, R: convex regions of [0, 1]d, T ∈ N in unary Question: ∃x ∈ R0, ∃t T, f [t](x) ∈ R ? Theorem REACH-REGION-TIME is NP-hard for d 2.

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 9 / 17

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SLIDE 58

General idea

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 59

General idea

Consider L a NP-hard problem

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 60

General idea

Consider L a NP-hard problem Consider L′ in P such that: L =

x | ∃y, |y| poly(|x|) and (x, y) ∈ L′

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 61

General idea

Consider L a NP-hard problem Consider L′ in P such that: L =

x | ∃y, |y| poly(|x|) and (x, y) ∈ L′

Define f a piecewise affine function which simulates L′:

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 62

General idea

Consider L a NP-hard problem ψ = encoding function Consider L′ in P such that: L =

x | ∃y, |y| poly(|x|) and (x, y) ∈ L′

Define f a piecewise affine function which simulates L′: (x, y) ∈ L′ ⇔ ∃t poly(|x|, |y|), f [t](ψ(x, y)) ∈ R

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 63

General idea

Consider L a NP-hard problem ψ = encoding function Consider L′ in P such that: L =

x | ∃y, |y| poly(|x|) and (x, y) ∈ L′

Define f a piecewise affine function which simulates L′: (x, y) ∈ L′ ⇔ ∃t poly(|x|, |y|), f [t](ψ(x, y)) ∈ R Define region Rx =

ψ(x, y) | |y| poly(|x|)
Amaury Pouly et al.

Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 64

General idea

Consider L a NP-hard problem ψ = encoding function Consider L′ in P such that: L =

x | ∃y, |y| poly(|x|) and (x, y) ∈ L′

Define f a piecewise affine function which simulates L′: (x, y) ∈ L′ ⇔ ∃t poly(|x|, |y|), f [t](ψ(x, y)) ∈ R Define region Rx =

ψ(x, y) | |y| poly(|x|)
Reduce L to REACH-REGION-TIME:

x ∈ L ⇔ ∃t poly(|x|), ∃u ∈ Rx, f [t](u) ∈ R

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 65

General idea

Consider L a NP-hard problem ψ = encoding function Consider L′ in P such that: L =

x | ∃y, |y| poly(|x|) and (x, y) ∈ L′

Define f a piecewise affine function which simulates L′: (x, y) ∈ L′ ⇔ ∃t poly(|x|, |y|), f [t](ψ(x, y)) ∈ R Define region Rx =

ψ(x, y) | |y| poly(|x|)
Reduce L to REACH-REGION-TIME:

x ∈ L ⇔ ∃t poly(|x|), ∃u ∈ ˜ Rx, f [t](u) ∈ R Tricky points Rx is not a convex polyhedron: replace it with its convex hull ˜ Rx

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 66

General idea

Consider L a NP-hard problem ψ = encoding function Consider L′ in P such that: L =

x | ∃y, |y| poly(|x|) and (x, y) ∈ L′

Define f a piecewise affine function which simulates L′: (x, y) ∈ L′ ⇔ ∃t poly(|x|, |y|), f [t](ψ(x, y)) ∈ R Define region Rx =

ψ(x, y) | |y| poly(|x|)
Reduce L to REACH-REGION-TIME:

x ∈ L ⇔ ∃t poly(|x|), ∃u ∈ ˜ Rx, f [t](u) ∈ R Tricky points Rx is not a convex polyhedron: replace it with its convex hull ˜ Rx Choice of L ?

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

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SLIDE 67

More on tricky points

Rx = {initial configuration} ˜ Rx = convex hull of Rx

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

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SLIDE 68

More on tricky points

Rx = {initial configuration} ˜ Rx = convex hull of Rx Problem ˜ Rx \ Rx contains bizarre points

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

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SLIDE 69

More on tricky points

Rx = {initial configuration} ˜ Rx = convex hull of Rx Problem ˜ Rx \ Rx contains bizarre points Example Take u ∈ ˜ Rx \ Rx, assume x / ∈ L

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

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SLIDE 70

More on tricky points

Rx = {initial configuration} ˜ Rx = convex hull of Rx Problem ˜ Rx \ Rx contains bizarre points Example Take u ∈ ˜ Rx \ Rx, assume x / ∈ L u = ψ(x, y) for all x, y → point normally inacessible

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

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SLIDE 71

More on tricky points

Rx = {initial configuration} ˜ Rx = convex hull of Rx Problem ˜ Rx \ Rx contains bizarre points Example Take u ∈ ˜ Rx \ Rx, assume x / ∈ L u = ψ(x, y) for all x, y → point normally inacessible f(u) may be uncontrolled

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

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SLIDE 72

More on tricky points

Rx = {initial configuration} ˜ Rx = convex hull of Rx Problem ˜ Rx \ Rx contains bizarre points Example Take u ∈ ˜ Rx \ Rx, assume x / ∈ L u = ψ(x, y) for all x, y → point normally inacessible f(u) may be uncontrolled if ∃t, f [t](u) ∈ R, system wrongly accepts x

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

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SLIDE 73

More on tricky points

Rx = {initial configuration} ˜ Rx = convex hull of Rx Problem ˜ Rx \ Rx contains bizarre points Example Take u ∈ ˜ Rx \ Rx, assume x / ∈ L u = ψ(x, y) for all x, y → point normally inacessible f(u) may be uncontrolled if ∃t, f [t](u) ∈ R, system wrongly accepts x So what ? The simulation of L′ has to be studied for bizarre points too

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

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SLIDE 74

More on tricky points

Rx = {initial configuration} ˜ Rx = convex hull of Rx Problem ˜ Rx \ Rx contains bizarre points Example Take u ∈ ˜ Rx \ Rx, assume x / ∈ L u = ψ(x, y) for all x, y → point normally inacessible f(u) may be uncontrolled if ∃t, f [t](u) ∈ R, system wrongly accepts x So what ? The simulation of L′ has to be studied for bizarre points too This is difficult for most languages

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

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SLIDE 75

And the winner is...

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

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SLIDE 76

And the winner is...

Problem SUBSEM-SUM Input: a goal B ∈ N and integers A1, . . . , An ∈ N

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

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SLIDE 77

And the winner is...

Problem SUBSEM-SUM Input: a goal B ∈ N and integers A1, . . . , An ∈ N Question: ∃I ⊆ {1, . . . , n},

i∈I Ai = B ?

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

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SLIDE 78

And the winner is...

Problem SUBSEM-SUM Input: a goal B ∈ N and integers A1, . . . , An ∈ N Question: ∃I ⊆ {1, . . . , n},

i∈I Ai = B ?

Simulation (1) Configuration: (i, σ, εi, . . . , εn) i ∈ {1, . . . , n + 1}, εi ∈ {0, 1}

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

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SLIDE 79

And the winner is...

Problem SUBSEM-SUM Input: a goal B ∈ N and integers A1, . . . , An ∈ N Question: ∃I ⊆ {1, . . . , n},

i∈I Ai = B ?

Simulation (1) Configuration: (i, σ, εi, . . . , εn) i ∈ {1, . . . , n + 1}, εi ∈ {0, 1} i = current number σ = current sum εi = pick Ai ?

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

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SLIDE 80

And the winner is...

Problem SUBSEM-SUM Input: a goal B ∈ N and integers A1, . . . , An ∈ N Question: ∃I ⊆ {1, . . . , n},

i∈I Ai = B ?

Simulation (1) Configuration: (i, σ, εi, . . . , εn) i ∈ {1, . . . , n + 1}, εi ∈ {0, 1} i = current number σ = current sum εi = pick Ai ? Transition: (i, σ, ε1, . . . , εn) (i + 1, σ + εiAi, εi+1, . . . , εn)

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

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SLIDE 81

And the winner is...

Problem SUBSEM-SUM Input: a goal B ∈ N and integers A1, . . . , An ∈ N Question: ∃I ⊆ {1, . . . , n},

i∈I Ai = B ?

Simulation (1) Configuration: (i, σ, εi, . . . , εn) i ∈ {1, . . . , n + 1}, εi ∈ {0, 1} i = current number σ = current sum εi = pick Ai ? Transition: (i, σ, ε1, . . . , εn) (i + 1, σ + εiAi, εi+1, . . . , εn) Simulation lemma (1) Instance is satisfiable ⇔ ∃ε1, . . . εn ∈ {0, 1} such that (1, 0, ε1, . . . , εn) n (n + 1, B)

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

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SLIDE 82

Tell me more...

Why SUBSET-SUM ?

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

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SLIDE 83

Tell me more...

Why SUBSET-SUM ? Configuration encoding: c = (i, σ, εi, . . . , εn)

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

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SLIDE 84

Tell me more...

Why SUBSET-SUM ? Configuration encoding: c = (i, σ, εi, . . . , εn) ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn  

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

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SLIDE 85

Tell me more...

Why SUBSET-SUM ? Configuration encoding: c = (i, σ, εi, . . . , εn) ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   =

i2−p + σ2−q

εi2−1 + εi+12−2 + · · ·

Amaury Pouly et al.

Complexity of bounded PAF reachability September 23, 2014 13 / 17

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SLIDE 86

Tell me more...

Why SUBSET-SUM ? Configuration encoding: c = (i, σ, εi, . . . , εn) ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   =

i2−p + σ2−q

εi2−1 + εi+12−2 + · · ·

Transitions: ψ(c) ψ(c′)

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

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SLIDE 87

Tell me more...

Why SUBSET-SUM ? Configuration encoding: c = (i, σ, εi, . . . , εn) ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   =

i2−p + σ2−q

εi2−1 + εi+12−2 + · · ·

Transitions: ψ(c) ψ(c′)
εi = 0 :

 0. i σ

0. 0 . . . 0 εi+1. . . εn

 

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

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SLIDE 88

Tell me more...

Why SUBSET-SUM ? Configuration encoding: c = (i, σ, εi, . . . , εn) ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   =

i2−p + σ2−q

εi2−1 + εi+12−2 + · · ·

Transitions: ψ(c) ψ(c′)
εi = 0 :

 0. i σ

0. 0 . . . 0 εi+1. . . εn

   0. i + 1 σ

0. 0 . . . 0 εi+1. . . εn

 

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

slide-89

SLIDE 89

Tell me more...

Why SUBSET-SUM ? Configuration encoding: c = (i, σ, εi, . . . , εn) ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   =

i2−p + σ2−q

εi2−1 + εi+12−2 + · · ·

Transitions: ψ(c) ψ(c′)
εi = 0 :

 0. i σ

0. 0 . . . 0 εi+1. . . εn

   0. i + 1 σ

0. 0 . . . 0 εi+1. . . εn

 

εi = 1 :

 0. i σ

0. 0 . . . 1 εi+1. . . εn

 

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

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SLIDE 90

Tell me more...

Why SUBSET-SUM ? Configuration encoding: c = (i, σ, εi, . . . , εn) ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   =

i2−p + σ2−q

εi2−1 + εi+12−2 + · · ·

Transitions: ψ(c) ψ(c′)
εi = 0 :

 0. i σ

0. 0 . . . 0 εi+1. . . εn

   0. i + 1 σ

0. 0 . . . 0 εi+1. . . εn

 

εi = 1 :

 0. i σ

0. 0 . . . 1 εi+1. . . εn

   0. i + 1 σ + Ai

0. 0 . . . 0 εi+1. . . εn

 

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

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SLIDE 91

And then were the regions...

ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn  

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 14 / 17

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SLIDE 92

And then were the regions...

ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   1 2−i+1 Ri,0 Ri,1 i εi = 0 εi = 1

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 14 / 17

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SLIDE 93

And then were the regions...

ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   1 2−i+1 Ri,0 Ri,1 i i + 1 εi = 0 εi = 1 Transition on Ri,0 f x y

=

x + 2−p y

Amaury Pouly et al.

Complexity of bounded PAF reachability September 23, 2014 14 / 17

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SLIDE 94

And then were the regions...

ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   1 2−i+1 Ri,0 Ri,1 i i + 1 εi = 0 εi = 1 Transition on Ri,0 f x y

=

x + 2−p y

Transition on Ri,1

f x y

=

x + 2−p + Ai2−q y − 2−i

Amaury Pouly et al.

Complexity of bounded PAF reachability September 23, 2014 14 / 17

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SLIDE 95

And then were the regions...

ψ(c) =  0. i σ

0. 0 . . . εi

. . . εn   1 2−i+1 Ri,0 Ri,1 i i + 1 εi = 0 εi = 1 Transition on Ri,0 f x y

=

x + 2−p y

Transition on Ri,1

f x y

=

x + 2−p + Ai2−q y − 2−i

But this doesn’t work, right ?

f is not continuous

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 14 / 17

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SLIDE 96

Ok, the actual proof is slightly more complicated...

1 1 R0 Rn+1 R1 R2

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 15 / 17

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SLIDE 97

...horribly more complicated

i2−p i2−p + 2−p−1 β−i+1 i2−p + (B + 1 − Ai)2−q Ri,0 : (a + 2−p, 0) Ri,0⋆ : (a + 2−p, b − 0⋆β−i) Ri,2 : (a + 2−p, 3β−i − b) Rlin

i,3 : (a + 2−p + Ai2−q(bβi − 3), 0)

Rsat

i,3 : (⋆) Rlin

i,1⋆:(a+2−p+Ai2−q,b−1⋆β−i) Rsat i,1⋆ : ((i+1)2−p+(B+1)2−q, b−1⋆β−i )

β−i 2β−i 3β−i 4β−i

(⋆):((i+1)2−p+2−p−1−(bβi−3)(2−p−1−(B+1)2−q),0)

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 16 / 17

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SLIDE 98

Conclusion

Reachability in piecewise affine systems:

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 17 / 17

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SLIDE 99

Conclusion

Reachability in piecewise affine systems: undecidable for d 2

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 17 / 17

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SLIDE 100

Conclusion

Reachability in piecewise affine systems: undecidable for d 2 NP-complete for d 2 (bounded time variant)

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 17 / 17

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SLIDE 101

Conclusion

Reachability in piecewise affine systems: undecidable for d 2 NP-complete for d 2 (bounded time variant)

pen problem for d = 1

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 17 / 17

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SLIDE 102

Questions ?

Do you have any questions ?

Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 ∞ / 17