Algorithmic Analysis of Polygonal Hybrid Systems G ERARDO S CHNEIDER - PowerPoint PPT Presentation

Solving the Reachability Prob- lem 1. From trajectories to simplified trajectories 2. From simplified trajectories to signatures 3. From signatures to factorized signatures Algorithmic Analysis of Polygonal Hybrid Systems – p.16/66

3. Factorization of Signatures For σ = e 1 e 2 e 3 . . . e 5 e 6 e 7 . . . e 13 e 6 e 7 e 8 e 15 e 9 e 8 R 8 R 9 R 7 e 10 x ′ e 7 e 15 R 10 e 14 R 6 e 11 R 12 e 6 R 11 R 5 e 12 e 13 R 1 e 5 e 1 e 4 x R 4 R 2 e 2 e 3 R 3 We obtain the representation: σ = e 1 e 2 e 3 ( e 4 e 1 e 2 e 3 ) 2 e 5 e 6 e 7 e 8 ( e 9 · · · e 13 e 6 e 7 e 8 ) 2 e 15 Algorithmic Analysis of Polygonal Hybrid Systems – p.17/66

3. Canonical Factorization of Signatures Representation Theorem: Any edge signature σ = e 1 , e 2 , . . . , e n can be represented as σ = r 1 ( s 1 ) k 1 r 2 ( s 2 ) k 2 . . . r n ( s n ) k n r n +1 Algorithmic Analysis of Polygonal Hybrid Systems – p.18/66

3. Canonical Factorization of Signatures Representation Theorem: Any edge signature σ = e 1 , e 2 , . . . , e n can be represented as σ = r 1 ( s 1 ) k 1 r 2 ( s 2 ) k 2 . . . r n ( s n ) k n r n +1 • Properties: • r i is a seq. of pairwise different edges; • s i is a simple cycle; • r i and r j are disjoint • s i and s j are different Proof based on topological properties of the plane Algorithmic Analysis of Polygonal Hybrid Systems – p.18/66

Solving the Reachability Prob- lem 1. From trajectories to simplified trajectories 2. From simplified trajectories to signatures 3. From signatures to factorized signatures 4. From factorized signatures to types of signatures Algorithmic Analysis of Polygonal Hybrid Systems – p.19/66

4. Types of Signatures Abstraction: Any edge signature σ = r 1 ( s 1 ) k 1 r 2 ( s 2 ) k 2 . . . r n ( s n ) k n r n +1 belongs to a type type ( σ ) = r 1 , s 1 , r 2 , s 2 , . . . r n , s n , r n +1 Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

4. Types of Signatures Abstraction: Any edge signature σ = r 1 ( s 1 ) k 1 r 2 ( s 2 ) k 2 . . . r n ( s n ) k n r n +1 belongs to a type type ( σ ) = r 1 , s 1 , r 2 , s 2 , . . . r n , s n , r n +1 s 1 s 2 s n r 1 r 2 r 3 r n r n +1 Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

4. Types of Signatures Abstraction: Any edge signature σ = r 1 ( s 1 ) k 1 r 2 ( s 2 ) k 2 . . . r n ( s n ) k n r n +1 belongs to a type type ( σ ) = r 1 , s 1 , r 2 , s 2 , . . . r n , s n , r n +1 In the previous example: type ( σ ) = e 1 e 2 e 3 , e 4 e 1 e 2 e 3 , e 5 e 6 e 7 e 8 , e 9 · · · e 13 e 6 e 7 e 8 , e 15 Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

4. Types of Signatures Abstraction: Any edge signature σ = r 1 ( s 1 ) k 1 r 2 ( s 2 ) k 2 . . . r n ( s n ) k n r n +1 belongs to a type type ( σ ) = r 1 , s 1 , r 2 , s 2 , . . . r n , s n , r n +1 Prop. The set of types of signatures is finite Algorithmic Analysis of Polygonal Hybrid Systems – p.20/66

Solving the Reachability Prob- lem 1. From trajectories to simplified trajectories 2. From simplified trajectories to signatures 3. From signatures to factorized signatures 4. From factorized signatures to types of signatures 5. Analysis of each type of signature (computing successors) Algorithmic Analysis of Polygonal Hybrid Systems – p.21/66

Computing Successors (for σ ) One step ( σ = e 1 e 2 ) e 3 e 2 [ a 1 x + b 1 , a 1 x + b 1 ] I 2 e 4 x e 1 e 9 e 13 e 12 e 10 e 11 e 5 e 8 e 7 e 6 Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e 13 Computing Successors (for σ ) e 8 e 1 2 ] ∩ e 3 x 2 x + b ′ e 7 e 2 I 3 = Succ σ ( x ) = [ a 2 x + b 2 , a ′ Several steps ( σ = e 1 e 2 e 3 ) e 12 e 11 e 9 e 10 e 6 e 3 I 3 e 4 e 5

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e 13 e 8 Computing Successors (for σ ) e 1 4 ] ∩ e 5 x 4 x + b ′ e 2 e 7 Several steps ( σ = e 1 e 2 e 3 e 4 e 5 ) I 5 = Succ σ ( x ) = [ a 4 x + b 4 , a ′ e 12 e 11 e 9 e 10 e 3 e 6 e 4 e 5

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e 13 e 8 Computing Successors (for σ ) e 1 8 ] ∩ e 1 x 8 x + b ′ One cycle ( σ = s = e 1 e 2 · · · e 8 e 1 ) I 9 e 2 I 9 = Succ σ ( x ) = [ a 8 x + b 8 , a ′ e 7 e 12 e 11 e 9 e 10 e 3 e 6 e 4 e 5

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66 ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ✁ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � e 13 u ∗ e 8 Computing Successors (for σ ) e 1 σ ( x ) = [ l ∗ , u ∗ ] ∩ e 1 u ∗ = a 2 u ∗ + b 2 I ∗ x One cycle ( σ = s = e 1 e 2 · · · e 8 e 1 ) e 2 e 7 l ∗ e 12 e 11 I ∗ = Succ ∗ e 9 e 10 e 3 e 6 l ∗ = a 1 l ∗ + b 1 e 5 e 4

Algorithmic Analysis of Polygonal Hybrid Systems – p.22/66 ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✂ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✁ ✂ ✂ ✁ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✁ ✁ ✂ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✂ ✂ ✁ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✁ ✁ ✂ � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � � � � � � � � � � � � � � � � � � � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ � ✁ ✁ ✁ � � � � ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ s ( x ) e 13 (acceleration) : I ′ = Succ e 8 e 13 ◦ Succ e 1 ··· e 8 ◦ Succ ∗ I ′ One cycle iterated: solution of fixpoint equation e 8 Computing Successors (for σ ) e 1 x σ = ( s ) ∗ e 13 ( s = e 1 e 2 · · · e 8 e 1 ) e 2 e 7 e 12 e 11 e 9 e 10 e 6 e 3 e 5 e 4

Computing Successors Lemma: Successors have the form Succ σ ( l , u ) = [ a 1 l + b 1 , a 2 u + b 2 ] ∩ J if [ l, u ] ⊆ S Lemma: Fixpoint equations [ a 1 l ∗ + b 1 , a 2 u ∗ + b 2 ] = [ l ∗ , u ∗ ] can be explicitely solved (without iterating). We have that ( I = [ l, u ] ): Succ ∗ σ ( I ) = [ l ∗ , u ∗ ] ∩ J Algorithmic Analysis of Polygonal Hybrid Systems – p.23/66

Reachability Algorithm for each type of signature τ do check whether Reach τ ( x 0 , x f ) To test whether Reach τ ( x 0 , x f ) for τ = r 1 ( s 1 ) ∗ · · · ( s n ) ∗ r n +1 Compute Succ r Accelerate ( Succ s ) ∗ Algorithmic Analysis of Polygonal Hybrid Systems – p.24/66

Reachability: Main Result • The capability of computing fixpoints for simple cycles (acceleration) • The set of types of signatures is finite Reachability is decidable for SPDI Algorithmic Analysis of Polygonal Hybrid Systems – p.25/66

SPeeDI: a Tool for SPDIs Algorithmic Analysis of Polygonal Hybrid Systems – p.26/66

Implementation: SPeeDI • We have implemented the reachability algorithm for SPDIs: SPeeDI (joint work with Gordon Pace) • Language: Haskell Algorithmic Analysis of Polygonal Hybrid Systems – p.27/66

Implementation: SPeeDI • We have implemented the reachability algorithm for SPDIs: SPeeDI (joint work with Gordon Pace) • Language: Haskell <file.spdi> <type_of_signature> <trace> simsig2fig simsig reachable <file.fig> <input interval> YES <exit interval> NO Algorithmic Analysis of Polygonal Hybrid Systems – p.27/66

Implementation: SPeeDI Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

Implementation: SPeeDI Animate Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

Implementation: SPeeDI Animate Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

Implementation: SPeeDI Animate Algorithmic Analysis of Polygonal Hybrid Systems – p.28/66

Phase Portrait of SPDIs Algorithmic Analysis of Polygonal Hybrid Systems – p.29/66

Phase Portrait Phase Portrait: a picture of important objects of a dynamical system Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

Phase Portrait Phase Portrait: a picture of important objects of a dynamical system e 2 e 3 e 4 e 1 e 9 e 12 e 11 e 10 e 5 e 8 e 7 e 6 Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

Phase Portrait Phase Portrait: a picture of important objects of a dynamical system e 2 e 3 e 4 e 1 e 9 e 12 e 11 e 10 e 5 e 8 e 7 e 6 Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

Phase Portrait Phase Portrait: a picture of important objects of a dynamical system e 2 e 3 e 4 e 1 e 9 e 12 e 11 e 10 e 5 e 8 e 7 e 6 Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

Phase Portrait Phase Portrait: a picture of important objects of a dynamical system e 2 e 3 e 4 e 1 e 9 e 12 e 11 e 10 e 5 e 8 e 7 e 6 Algorithmic Analysis of Polygonal Hybrid Systems – p.30/66

Viability Kernel Viab ( σ ) : Is the greatest set of initial points of trajectories which can cycle forever in σ Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

Viability Kernel Viab ( σ ) : Is the greatest set of initial points of trajectories which can cycle forever in σ Example: σ = e 1 e 2 . . . e 8 e 1 R 3 R 4 R 2 e 2 e 3 e 1 e 4 R 5 R 1 e 5 e 8 e 7 e 6 R 6 R 8 R 7 Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

Viability Kernel Viab ( σ ) : Is the greatest set of initial points of trajectories which can cycle forever in σ Example: σ = e 1 e 2 . . . e 8 e 1 R 3 R 4 R 2 e 2 e 3 e 1 e 4 R 5 R 1 e 5 e 8 e 7 e 6 R 6 R 8 R 7 Theorem: Viab ( σ ) = Pre σ ( Dom ( Succ σ )) Algorithmic Analysis of Polygonal Hybrid Systems – p.31/66

Controllability Kernel Cntr ( σ ) : Is the greatest set of mutually reachable points via trajectories that remain in the cycle Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

Controllability Kernel Cntr ( σ ) : Is the greatest set of mutually reachable points via trajectories that remain in the cycle Example: σ = e 1 e 2 . . . e 8 e 1 R 3 R 4 R 2 e 3 e 2 e 4 e 1 l u R 5 R 1 e 8 e 5 e 6 e 7 R 6 R 8 R 7 Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

Controllability Kernel Cntr ( σ ) : Is the greatest set of mutually reachable points via trajectories that remain in the cycle Example: σ = e 1 e 2 . . . e 8 e 1 R 3 R 4 R 2 e 3 e 2 e 4 e 1 l u R 5 R 1 e 8 e 5 e 6 e 7 R 6 R 8 R 7 Theorem: Cntr ( σ ) = ( Succ σ ∩ Pre σ )( C D ( σ )) Algorithmic Analysis of Polygonal Hybrid Systems – p.32/66

Viability Kernel Algorithm: phase portrait for SPDIs for each simple cycle σ do Compute Viab ( σ ) ( viability kernel ) Compute Cntr ( σ ) ( controllability kernel ) Algorithmic Analysis of Polygonal Hybrid Systems – p.33/66

Viability Kernel Algorithm: phase portrait for SPDIs for each simple cycle σ do Compute Viab ( σ ) ( viability kernel ) Compute Cntr ( σ ) ( controllability kernel ) Both kernels are exactly computed by non-iterative algorithms! Algorithmic Analysis of Polygonal Hybrid Systems – p.33/66

Properties of the Kernels Theorem: Any viable trajectory in σ converges to Cntr ( K σ ) R 3 R 4 R 2 e 3 e 2 e 1 e 4 R 5 R 1 e 5 e 8 e 6 e 7 R 6 R 8 R 7 • Controllability Kernel: “ Weak”analog of limit cycle • Viability Kernel: Its “ local”attraction basin Algorithmic Analysis of Polygonal Hybrid Systems – p.34/66

Convergence Properties Every trajectory with infi nite signature without self-crossings converges to the controllability kernel of some simple edge-cycle R 11 R 3 R 4 R 2 R 12 e 3 e 11 e 2 e 10 e 1 e 4 e 12 R 5 R 13 e 5 e 13 e 8 e 15 R 1 e 14 e 6 e 7 R 6 R 8 R 15 R 14 R 7 Algorithmic Analysis of Polygonal Hybrid Systems – p.35/66

Between Decidable and Undecidable Algorithmic Analysis of Polygonal Hybrid Systems – p.36/66

More complex 2-dim systems What happens if ... • ...we allow jumps? • ...the PCD is on a 2-dim surface/manifold? • ...? Algorithmic Analysis of Polygonal Hybrid Systems – p.37/66

More complex 2-dim systems What happens if ... • ...we allow jumps? • ...the PCD is on a 2-dim surface/manifold? • ...? Answer: Reachability is equivalent to a well known open problem Algorithmic Analysis of Polygonal Hybrid Systems – p.37/66

Our Reference Model 1-dim Piecewise Affine Maps (PAMs): f : R → R , f ( x ) = a i x + b i for x ∈ I i Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Our Reference Model 1-dim Piecewise Affine Maps (PAMs): f : R → R , f ( x ) = a i x + b i for x ∈ I i a 1 x + b 1 R I 2 I 1 I 4 I 5 I 3 Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Our Reference Model 1-dim Piecewise Affine Maps (PAMs): f : R → R , f ( x ) = a i x + b i for x ∈ I i a 1 x + b 1 R I 2 I 1 I 4 I 5 I 3 a 5 x + b 5 Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Our Reference Model 1-dim Piecewise Affine Maps (PAMs): f : R → R , f ( x ) = a i x + b i for x ∈ I i a 1 x + b 1 a 4 x + b 4 R I 2 I 1 I 4 I 5 I 3 a 5 x + b 5 Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Our Reference Model 1-dim Piecewise Affine Maps (PAMs): f : R → R , f ( x ) = a i x + b i for x ∈ I i a 1 x + b 1 a 4 x + b 4 R I 2 I 1 I 4 I 5 I 3 a 2 x + b 2 a 5 x + b 5 Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Our Reference Model 1-dim Piecewise Affine Maps (PAMs): f : R → R , f ( x ) = a i x + b i for x ∈ I i a 1 x + b 1 a 4 x + b 4 R I 2 I 1 I 4 I 5 I 3 a 2 x + b 2 a 5 x + b 5 Reachability? Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

Our Reference Model 1-dim Piecewise Affine Maps (PAMs): f : R → R , f ( x ) = a i x + b i for x ∈ I i a 1 x + b 1 a 4 x + b 4 R I 2 I 1 I 4 I 5 I 3 a 2 x + b 2 a 5 x + b 5 Reachability? Open problem! Algorithmic Analysis of Polygonal Hybrid Systems – p.38/66

PCD on 2-dim manifolds ( PCD 2m ) Example: Torus R 1 R 2 R 3 R 4 Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

PCD on 2-dim manifolds ( PCD 2m ) Example: Torus R 1 R 2 R 3 R 4 Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

PCD on 2-dim manifolds ( PCD 2m ) Example: Torus R 1 R 2 R 3 R 4 Reachability? Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

PCD on 2-dim manifolds ( PCD 2m ) Example: Torus R 1 R 2 R 3 R 4 Reachability? Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

PCD on 2-dim manifolds ( PCD 2m ) Example: Torus R 1 R 2 R 3 R 4 Reachability? Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

PCD on 2-dim manifolds ( PCD 2m ) Example: Torus R 1 R 2 R 3 R 4 Reachability? Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66

PCD on 2-dim manifolds ( PCD 2m ) Example: Torus R 1 R 2 R 3 R 4 Reachability? Theorem: PCD 2m ≡ PAM Algorithmic Analysis of Polygonal Hybrid Systems – p.39/66