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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,

  1. Example Trajectory Function f [ 5 ] ( x ) f [ 3 ] ( x ) f [ 4 ] ( x ) � if x ∈ [ 0 , 1 2 x 2 ] x 0 f [ 2 ] ( x ) 1 1 f ( x ) f ( x ) = 2 if x ∈ [ 1 2 − 2 x 2 , 1 ] x = 0 . 5625 1 f ( x ) = 0 . 875 0 . 8 f [ 2 ] ( x ) = 0 . 25 0 . 6 0 . 4 f [ 3 ] ( x ) = 0 . 5 0 . 2 f [ 4 ] ( x ) = 1 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 f [ n ] ( x ) = 0 n � 5 Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

  2. Example Function Trajectory � if x ∈ [ 0 , 1 2 x f [ 5 ] ( x ) f [ 3 ] ( x ) f [ 4 ] ( x ) 2 ] f ( x ) = if x ∈ [ 1 2 − 2 x 2 , 1 ] x 0 f [ 2 ] ( x ) 1 1 f ( x ) 2 1 0 . 8 x = 0 . 5625 0 . 6 f ( x ) = 0 . 875 0 . 4 f [ 2 ] ( x ) = 0 . 25 0 . 2 f [ 3 ] ( x ) = 0 . 5 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 f [ 4 ] ( x ) = 1 Remark f [ n ] ( x ) = 0 n � 5 Trajectory depends on the bi- nary expansion of x Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 3 / 17

  3. 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

  4. Existings Results Problem: REACH-REGION Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

  5. Existings Results Problem: REACH-REGION Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d Question: ∃ x ∈ R 0 , ∃ t ∈ N , f [ t ] ( x ) ∈ R ? Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

  6. Existings Results Problem: REACH-REGION Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d Question: ∃ x ∈ R 0 , ∃ t ∈ N , f [ t ] ( x ) ∈ R ? Example f ( x ) f [ 3 ] ( x ) R x R 0 f [ 2 ] ( x ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

  7. Existings Results Problem: REACH-REGION Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d Question: ∃ x ∈ R 0 , ∃ t ∈ N , f [ t ] ( x ) ∈ R ? Example Theorem (Koiran, Cosnard, Garzon) f ( x ) f [ 3 ] ( x ) REACH-REGION is undecidable for d � 2 R x R 0 f [ 2 ] ( x ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

  8. Existings Results Problem: REACH-REGION Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d Question: ∃ x ∈ R 0 , ∃ t ∈ N , f [ t ] ( x ) ∈ R ? Example Theorem (Koiran, Cosnard, Garzon) f ( x ) f [ 3 ] ( x ) REACH-REGION is undecidable for d � 2 R Proof (Idea) Simulate a Turing Machine and re- x duce from halting problem. R 0 f [ 2 ] ( x ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

  9. Existings Results Problem: REACH-REGION Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d Question: ∃ x ∈ R 0 , ∃ t ∈ N , f [ t ] ( x ) ∈ R ? Example Theorem (Koiran, Cosnard, Garzon) f ( x ) f [ 3 ] ( x ) REACH-REGION is undecidable for d � 2 R Proof (Idea) Simulate a Turing Machine and re- x duce from halting problem. R 0 Open Problem f [ 2 ] ( x ) Decidability for d = 1. Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

  10. Existings Results Problem: CONTROL-REGION Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d Question: ∀ x ∈ R 0 , ∃ t ∈ N , f [ t ] ( x ) ∈ R ? Example Theorem (Blondel, Bournez, Koiran, Tsitsiklis) f ( x ) f [ 3 ] ( x ) is undecidable CONTROL-REGION R for d � 2 Proof (Idea) Harder simulation of a Turing Ma- x chine R 0 Open Problem f [ 2 ] ( x ) Decidability for d = 1. Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 4 / 17

  11. 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

  12. Our Results Problem: REACH-REGION-TIME Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , 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

  13. Our Results Problem: REACH-REGION-TIME Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d , T ∈ N in unary Question: ∃ x ∈ R 0 , ∃ t � T , f [ t ] ( x ) ∈ R ? Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 5 / 17

  14. Our Results Problem: REACH-REGION-TIME Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d , T ∈ N in unary Question: ∃ x ∈ R 0 , ∃ 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

  15. Our Results Problem: REACH-REGION-TIME Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d , T ∈ N in unary Question: ∃ x ∈ R 0 , ∃ 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

  16. Our Results Problem: CONTROL-REGION-TIME Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d , T ∈ N in unary Question: ∀ x ∈ R 0 , ∃ 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

  17. Statement Problem: REACH-REGION-TIME Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d , T ∈ N in unary Question: ∃ x ∈ R 0 , ∃ 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

  18. Signature Example R 1 Definition R 3 R 2 The signature σ ( x ) ∈ { 0 , . . . , n } N of x is defined by: f [ i ] ( x ) ∈ R j σ i ( x ) = j ⇔ R 0 R 4 R 5 Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

  19. Signature Example R 1 R 3 R 2 Definition The signature σ ( x ) ∈ { 0 , . . . , n } N of x is defined by: x f [ i ] ( x ) ∈ R j R 0 R 4 σ i ( x ) = j ⇔ R 5 σ ( x ) = ( 0 , . . . ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

  20. Signature Example f ( x ) R 1 R 3 R 2 Definition The signature σ ( x ) ∈ { 0 , . . . , n } N of x is defined by: x f [ i ] ( x ) ∈ R j R 0 R 4 σ i ( x ) = j ⇔ R 5 σ ( x ) = ( 0 , 1 , . . . ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

  21. Signature Example f ( x ) R 1 R 3 R 2 Definition The signature σ ( x ) ∈ { 0 , . . . , n } N of x is defined by: x f [ i ] ( x ) ∈ R j R 0 R 4 σ i ( x ) = j ⇔ f [ 2 ] ( x ) R 5 σ ( x ) = ( 0 , 1 , 4 , . . . ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

  22. Signature Example f ( x ) f [ 3 ] ( x ) R 1 R 3 R 2 Definition The signature σ ( x ) ∈ { 0 , . . . , n } N of x is defined by: x f [ i ] ( x ) ∈ R j R 0 R 4 σ i ( x ) = j ⇔ f [ 2 ] ( x ) R 5 σ ( x ) = ( 0 , 1 , 4 , 3 , . . . ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

  23. Signature Example Definition The signature σ ( x ) ∈ { 0 , . . . , n } N of f ( x ) f [ 3 ] ( x ) x is defined by: R 1 R 3 R 2 f [ i ] ( x ) ∈ R j σ i ( x ) = j ⇔ Lemma x If σ ( x ) = ( r 1 , r 2 , . . . , r t , . . . ) then R 0 R 4 f [ t ] ( x ) = A r t ( · · · ( A r 1 x + b r 1 ) · · · ) + b r t f [ 2 ] ( x ) = C σ + d σ R 5 σ ( x ) = ( 0 , 1 , 4 , 3 , . . . ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

  24. Signature Definition Example The signature σ ( x ) ∈ { 0 , . . . , n } N of f ( x ) f [ 3 ] ( x ) x is defined by: R 1 f [ i ] ( x ) ∈ R j R 3 σ i ( x ) = j ⇔ R 2 Lemma If σ ( x ) = ( r 1 , r 2 , . . . , r t , . . . ) then x f [ t ] ( x ) = A r t ( · · · ( A r 1 x + b r 1 ) · · · ) + b r t R 0 R 4 = C σ + d σ f [ 2 ] ( x ) R 5 Furthermore ( s ( X ) = coeff size): s ( C σ , d σ ) = poly ( s ( A ) , s ( b ) , t ) σ ( x ) = ( 0 , 1 , 4 , 3 , . . . ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 7 / 17

  25. Algorithm Given f , R 0 , R = R n and T : Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

  26. Algorithm Given f , R 0 , R = R n and T : Guess t � T ← Nondeterministic polynomial Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

  27. Algorithm Given f , R 0 , R = R n and T : Guess t � T ← Nondeterministic polynomial Guess signature r 1 , . . . , r t − 1 ← Nondeterministic polynomial Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

  28. Algorithm Given f , R 0 , R = R n and T : Guess t � T ← Nondeterministic polynomial Guess signature r 1 , . . . , r t − 1 ← Nondeterministic polynomial Guess x ∈ Q d of polynomial size ← Nondeterministic polynomial Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

  29. Algorithm Given f , R 0 , R = R n and T : Guess t � T ← Nondeterministic polynomial Guess signature r 1 , . . . , r t − 1 ← Nondeterministic polynomial Guess x ∈ Q d of polynomial size ← Nondeterministic polynomial Check that f [ i ] ( x ) ∈ R r i for all i ∈ { 0 , . . . , t } : Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

  30. Algorithm Given f , R 0 , R = R n and T : Guess t � T ← Nondeterministic polynomial Guess signature r 1 , . . . , r t − 1 ← Nondeterministic polynomial Guess x ∈ Q d of polynomial size ← Nondeterministic polynomial Check that f [ i ] ( x ) ∈ R r i for all i ∈ { 0 , . . . , t } : f [ i ] ( x ) ∈ R r i ⇔ M r i ( C i x + d i ) � v i ← Polynomial size Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

  31. Algorithm Given f , R 0 , R = R n and T : Guess t � T ← Nondeterministic polynomial Guess signature r 1 , . . . , r t − 1 ← Nondeterministic polynomial Guess x ∈ Q d of polynomial size ← Nondeterministic polynomial Check that f [ i ] ( x ) ∈ R r i for all i ∈ { 0 , . . . , t } : f [ i ] ( x ) ∈ R r i ⇔ M r i ( C i x + d i ) � v i ← Polynomial size Accept if all systems are satisfied Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

  32. Algorithm Given f , R 0 , R = R n and T : Guess t � T ← Nondeterministic polynomial Guess signature r 1 , . . . , r t − 1 ← Nondeterministic polynomial Guess x ∈ Q d of polynomial size ← Nondeterministic polynomial Check that f [ i ] ( x ) ∈ R r i for all i ∈ { 0 , . . . , t } : f [ i ] ( x ) ∈ R r i ⇔ M r i ( C i x + d i ) � v i ← Polynomial size Accept if all systems are satisfied Theorem (Koiran) Every satisfiable rational linear system Ax � b has a rational solution of polynomial size. Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 8 / 17

  33. Statement Problem: REACH-REGION-TIME Input: f : [ 0 , 1 ] d → [ 0 , 1 ] d continuous, piecewise affine Input: R 0 , R : convex regions of [ 0 , 1 ] d , T ∈ N in unary Question: ∃ x ∈ R 0 , ∃ 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

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

  35. General idea Consider L a NP-hard problem Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

  36. General idea Consider L a NP-hard problem Consider L ′ in P such that: x | ∃ y , | y | � poly ( | x | ) and ( x , y ) ∈ L ′ � � L = Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

  37. General idea Consider L a NP-hard problem Consider L ′ in P such that: x | ∃ y , | y | � poly ( | x | ) and ( x , y ) ∈ L ′ � � L = Define f a piecewise affine function which simulates L ′ : Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

  38. General idea Consider L a NP-hard problem ψ = encoding function Consider L ′ in P such that: x | ∃ y , | y | � poly ( | x | ) and ( x , y ) ∈ L ′ � � 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

  39. General idea Consider L a NP-hard problem ψ = encoding function Consider L ′ in P such that: x | ∃ y , | y | � poly ( | x | ) and ( x , y ) ∈ L ′ � � 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 R x = � ψ ( x , y ) | | y | � poly ( | x | ) � Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

  40. General idea Consider L a NP-hard problem ψ = encoding function Consider L ′ in P such that: x | ∃ y , | y | � poly ( | x | ) and ( x , y ) ∈ L ′ � � 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 R x = � ψ ( x , y ) | | y | � poly ( | x | ) � Reduce L to REACH-REGION-TIME : x ∈ L ⇔ ∃ t � poly ( | x | ) , ∃ u ∈ R x , f [ t ] ( u ) ∈ R Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

  41. General idea Consider L a NP-hard problem ψ = encoding function Consider L ′ in P such that: x | ∃ y , | y | � poly ( | x | ) and ( x , y ) ∈ L ′ � � 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 R x = � ψ ( x , y ) | | y | � poly ( | x | ) � Reduce L to REACH-REGION-TIME : x ∈ L ⇔ ∃ t � poly ( | x | ) , ∃ u ∈ ˜ R x , f [ t ] ( u ) ∈ R Tricky points R x is not a convex polyhedron: replace it with its convex hull ˜ R x Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

  42. General idea Consider L a NP-hard problem ψ = encoding function Consider L ′ in P such that: x | ∃ y , | y | � poly ( | x | ) and ( x , y ) ∈ L ′ � � 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 R x = � ψ ( x , y ) | | y | � poly ( | x | ) � Reduce L to REACH-REGION-TIME : x ∈ L ⇔ ∃ t � poly ( | x | ) , ∃ u ∈ ˜ R x , f [ t ] ( u ) ∈ R Tricky points R x is not a convex polyhedron: replace it with its convex hull ˜ R x Choice of L ? Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 10 / 17

  43. More on tricky points ˜ R x = { initial configuration } R x = convex hull of R x Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

  44. More on tricky points ˜ R x = { initial configuration } R x = convex hull of R x Problem ˜ R x \ R x contains bizarre points Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

  45. More on tricky points ˜ R x = { initial configuration } R x = convex hull of R x Problem ˜ R x \ R x contains bizarre points Example Take u ∈ ˜ R x \ R x , assume x / ∈ L Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 11 / 17

  46. More on tricky points ˜ R x = { initial configuration } R x = convex hull of R x Problem ˜ R x \ R x contains bizarre points Example Take u ∈ ˜ R x \ R x , 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

  47. More on tricky points ˜ R x = { initial configuration } R x = convex hull of R x Problem ˜ R x \ R x contains bizarre points Example Take u ∈ ˜ R x \ R x , 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

  48. More on tricky points ˜ R x = { initial configuration } R x = convex hull of R x Problem ˜ R x \ R x contains bizarre points Example Take u ∈ ˜ R x \ R x , 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

  49. More on tricky points ˜ R x = { initial configuration } R x = convex hull of R x Problem ˜ R x \ R x contains bizarre points Example Take u ∈ ˜ R x \ R x , 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

  50. More on tricky points ˜ R x = { initial configuration } R x = convex hull of R x Problem ˜ R x \ R x contains bizarre points Example Take u ∈ ˜ R x \ R x , 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

  51. And the winner is... Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

  52. And the winner is... Problem SUBSEM-SUM Input: a goal B ∈ N and integers A 1 , . . . , A n ∈ N Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

  53. And the winner is... Problem SUBSEM-SUM Input: a goal B ∈ N and integers A 1 , . . . , A n ∈ N Question: ∃ I ⊆ { 1 , . . . , n } , � i ∈ I A i = B ? Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

  54. And the winner is... Problem SUBSEM-SUM Input: a goal B ∈ N and integers A 1 , . . . , A n ∈ N Question: ∃ I ⊆ { 1 , . . . , n } , � i ∈ I A i = B ? Simulation (1) i ∈ { 1 , . . . , n + 1 } , ε i ∈ { 0 , 1 } Configuration: ( i , σ, ε i , . . . , ε n ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

  55. And the winner is... Problem SUBSEM-SUM Input: a goal B ∈ N and integers A 1 , . . . , A n ∈ N Question: ∃ I ⊆ { 1 , . . . , n } , � i ∈ I A i = B ? Simulation (1) i ∈ { 1 , . . . , n + 1 } , ε i ∈ { 0 , 1 } Configuration: ( i , σ, ε i , . . . , ε n ) i = current number σ = current sum ε i = pick A i ? Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

  56. And the winner is... Problem SUBSEM-SUM Input: a goal B ∈ N and integers A 1 , . . . , A n ∈ N Question: ∃ I ⊆ { 1 , . . . , n } , � i ∈ I A i = B ? Simulation (1) i ∈ { 1 , . . . , n + 1 } , ε i ∈ { 0 , 1 } Configuration: ( i , σ, ε i , . . . , ε n ) i = current number σ = current sum ε i = pick A i ? Transition: ( i , σ, ε 1 , . . . , ε n ) � ( i + 1 , σ + ε i A i , ε i + 1 , . . . , ε n ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

  57. And the winner is... Problem SUBSEM-SUM Input: a goal B ∈ N and integers A 1 , . . . , A n ∈ N Question: ∃ I ⊆ { 1 , . . . , n } , � i ∈ I A i = B ? Simulation (1) i ∈ { 1 , . . . , n + 1 } , ε i ∈ { 0 , 1 } Configuration: ( i , σ, ε i , . . . , ε n ) i = current number σ = current sum ε i = pick A i ? Transition: ( i , σ, ε 1 , . . . , ε n ) � ( i + 1 , σ + ε i A i , ε i + 1 , . . . , ε n ) Simulation lemma (1) Instance is satisfiable ⇔ ∃ ε 1 , . . . ε n ∈ { 0 , 1 } such that ( 1 , 0 , ε 1 , . . . , ε n ) � n ( n + 1 , B ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 12 / 17

  58. Tell me more... Why SUBSET-SUM ? Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  59. Tell me more... Why SUBSET-SUM ? Configuration encoding: c = ( i , σ, ε i , . . . , ε n ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  60. Tell me more... Why SUBSET-SUM ? Configuration encoding: c = ( i , σ, ε i , . . . , ε n )   σ  0 . i ψ ( c ) =  0 . 0 . . . ε i . . . ε n Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  61. Tell me more... Why SUBSET-SUM ? Configuration encoding: c = ( i , σ, ε i , . . . , ε n )   σ i 2 − p + σ 2 − q  0 . i � �  = ψ ( c ) = ε i 2 − 1 + ε i + 1 2 − 2 + · · · 0 . 0 . . . ε i . . . ε n Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  62. Tell me more... Why SUBSET-SUM ? Configuration encoding: c = ( i , σ, ε i , . . . , ε n )   σ i 2 − p + σ 2 − q  0 . i � �  = ψ ( c ) = ε i 2 − 1 + ε i + 1 2 − 2 + · · · 0 . 0 . . . ε i . . . ε n Transitions: ψ ( c ) � ψ ( c ′ ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  63. Tell me more... Why SUBSET-SUM ? Configuration encoding: c = ( i , σ, ε i , . . . , ε n )   σ i 2 − p + σ 2 − q  0 . i � �  = ψ ( c ) = ε i 2 − 1 + ε i + 1 2 − 2 + · · · 0 . 0 . . . ε i . . . ε n Transitions: ψ ( c ) � ψ ( c ′ )   σ  0 . i • ε i = 0 :  � 0 . 0 . . . 0 ε i + 1 . . . ε n Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  64. Tell me more... Why SUBSET-SUM ? Configuration encoding: c = ( i , σ, ε i , . . . , ε n )   σ i 2 − p + σ 2 − q  0 . i � �  = ψ ( c ) = ε i 2 − 1 + ε i + 1 2 − 2 + · · · 0 . 0 . . . ε i . . . ε n Transitions: ψ ( c ) � ψ ( c ′ )     σ i + 1 σ  0 . i  0 . • ε i = 0 :  � 0 . 0 . . . 0 ε i + 1 . . . ε n 0 . 0 . . . 0 ε i + 1 . . . ε n  Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  65. Tell me more... Why SUBSET-SUM ? Configuration encoding: c = ( i , σ, ε i , . . . , ε n )   σ i 2 − p + σ 2 − q  0 . i � �  = ψ ( c ) = ε i 2 − 1 + ε i + 1 2 − 2 + · · · 0 . 0 . . . ε i . . . ε n Transitions: ψ ( c ) � ψ ( c ′ )     σ i + 1 σ  0 . i  0 . • ε i = 0 :  � 0 . 0 . . . 0 ε i + 1 . . . ε n 0 . 0 . . . 0 ε i + 1 . . . ε n    σ  0 . i • ε i = 1 :  � 0 . 0 . . . 1 ε i + 1 . . . ε n Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  66. Tell me more... Why SUBSET-SUM ? Configuration encoding: c = ( i , σ, ε i , . . . , ε n )   σ i 2 − p + σ 2 − q  0 . i � �  = ψ ( c ) = ε i 2 − 1 + ε i + 1 2 − 2 + · · · 0 . 0 . . . ε i . . . ε n Transitions: ψ ( c ) � ψ ( c ′ )     σ i + 1 σ  0 . i  0 . • ε i = 0 :  � 0 . 0 . . . 0 ε i + 1 . . . ε n 0 . 0 . . . 0 ε i + 1 . . . ε n      i + 1 σ σ + A i  0 . i  0 . • ε i = 1 :  � 0 . 0 . . . 1 ε i + 1 . . . ε n 0 . 0 . . . 0 ε i + 1 . . . ε n  Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 13 / 17

  67. And then were the regions...   σ  0 . i ψ ( c ) =  0 . 0 . . . ε i . . . ε n Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 14 / 17

  68. And then were the regions...   σ  0 . i ψ ( c ) =  0 . 0 . . . ε i . . . ε n 2 − i + 1 ε i = 1 R i , 1 R i , 0 ε i = 0 0 0 1 i Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 14 / 17

  69. And then were the regions...   σ  0 . i ψ ( c ) =  0 . 0 . . . ε i . . . ε n 2 − i + 1 Transition on R i , 0 R i , 1 ε i = 1 � x + 2 − p � x � � f = y y R i , 0 ε i = 0 0 0 1 i + 1 i Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 14 / 17

  70. And then were the regions...   σ  0 . i ψ ( c ) =  0 . 0 . . . ε i . . . ε n Transition on R i , 0 2 − i + 1 � x + 2 − p � x � � f = y y R i , 1 ε i = 1 Transition on R i , 1 � x + 2 − p + A i 2 − q � x � � f = y − 2 − i y R i , 0 ε i = 0 0 0 1 i + 1 i Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 14 / 17

  71. And then were the regions...   σ  0 . i ψ ( c ) = Transition on R i , 0  0 . 0 . . . ε i . . . ε n � x + 2 − p � x � � 2 − i + 1 f = y y R i , 1 Transition on R i , 1 ε i = 1 � x + 2 − p + A i 2 − q � x � � f = y − 2 − i y But this doesn’t work, right ? R i , 0 ε i = 0 f is not continuous 0 0 1 i + 1 i Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 14 / 17

  72. Ok, the actual proof is slightly more complicated... 1 R 0 R 1 R 2 0 0 1 R n + 1 Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 15 / 17

  73. ...horribly more complicated β − i + 1 i , 1 ⋆ : (( i + 1 ) 2 − p +( B + 1 ) 2 − q , R lin i , 1 ⋆ :( a + 2 − p + A i 2 − q , b − 1 ⋆ β − i ) Rsat b − 1 ⋆β − i ) 4 β − i R sat i , 3 : ( ⋆ ) i , 3 : ( a + 2 − p + A i 2 − q ( b β i − 3 ) , 0 ) R lin 3 β − i R i , 2 : ( a + 2 − p , 3 β − i − b ) 2 β − i R i , 0 ⋆ : ( a + 2 − p , b − 0 ⋆ β − i ) β − i R i , 0 : ( a + 2 − p , 0 ) 0 i 2 − p + ( B + 1 − A i ) 2 − q i 2 − p + 2 − p − 1 i 2 − p ( ⋆ ):(( i + 1 ) 2 − p + 2 − p − 1 − ( b β i − 3 )( 2 − p − 1 − ( B + 1 ) 2 − q ) , 0 ) Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 16 / 17

  74. Conclusion Reachability in piecewise affine systems: Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 17 / 17

  75. Conclusion Reachability in piecewise affine systems: undecidable for d � 2 Amaury Pouly et al. Complexity of bounded PAF reachability September 23, 2014 17 / 17

  76. 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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