1. Hybrid automata - Abstraction Anders P. Ravn Department of Computer Science, Aalborg University, Denmark Hybrid Systems – PhD School Aalborg University January 2007

2. Transition Semantics of HA X = {x 1, … x n } - variables (V, E) – control graph init: V  pred(X) inv: V  pred(X) flow: V  pred(X  X) jump: E  pred(X  X’) event: E   Q - states – {(v,x) | v  V and inv(v)[X := x]}. x’ = x-1   Q 0 – initial states - {(v,x)  Q | init(v)[X := x]} A - labels -   R  0 { (v,x) –  (v’,x’) | e  E(v,v’) and event(e) =  and jump(e) [X := x]} { (v,x) –  (v,x’) |   R  0 and f: (0,  )  R n s.t. f is diff. and f(0) = x and f(  ) = x’ and flow(v)[X := f(t), X:= f(t)], t  (0,  ) }.

3. Time Abstract Semantics X = {x 1, … x n } - variables (V, E) – control graph init: V  pred(X) inv: V  pred(X) flow: V  pred(X  X) jump: E  pred(X  X’) event: E   Q - states – {(v,x) | v  V and inv(v)[X := x]}. x’ = x-1   Q 0 – initial states - {(v,x)  Q | init(v)[X := x]} B - labels -   {  } - finite ! { (v,x) –  (v’,x’) | e  E(v,v’) and event(e) =  and jump(e) [X := x]} { (v,x) –   (v,x’) |   R  0 and f: (0,  )  R n s.t. f is diff. and f(0) = x and f(  ) = x’ and flow(v)[X := f(t), X:= f(t)], t  (0,  )}.

4. Tree Semantics Computation tree:  = q 00 a q 10 q 11... q 1n 1 … q 200 q 201 q 210 q 211 q 13 x’ = x-1   Q - states – {(v,x) | v  V and inv(v)[X := x]} Q 0 – initial states - {(v,x)  Q | init(v)[X := x]} B - labels -   {  } { (v,x) –  (v’,x’) | e  E(v,v’), event(e) = , jump(e) [X := x]} { (v,x) –   (v,x’) | f(0) = x, f(  ) = x’, flow(v)[X := f(t), X:= f(t)], t  (0,  )}

5. Trace Semantics Trajectory:  = where q 0  Q 0 and q i –a i  q i+1, i  0 Q - states – {(v,x) | v  V and inv(v)[X := x]} Q 0 – initial states - {(v,x)  Q | init(v)[X := x]} B - labels -   {  } { (v,x) –  (v’,x’) | e  E(v,v’), event(e) = , jump(e) [X := x]} { (v,x) –   (v,x’) | f(0) = x, f(  ) = x’, flow(v)[X := f(t), X:= f(t)], t  (0,  )} x’ = x-1  

6. Symbolic Analysis Q - states Q 0 – initial states, … A - labels, …  - transition relation, A  Q  Q a Theory: T = {p 1, … p n … }, p is a predicate, e.g. pred(X  V) Meaning of p: [p]  Q q 1  q 2 iff p(q 1 ) = r(q 2 ) for all p, r  T

7. Symbolic Bisimilarity Computation R R’ pre a

8. Timed Automaton X = {x 1, … x n } - variables (V, E) – control graph init: V  pred(X) inv: V  pred(X) flow: V  pred(X  X) jump: E  pred(X  X’). x’ = x-1   Init(v): v = v 0 and X = 0, where v 0  V inv(v): X <= C, where C is rational flow(v): X = 1 jump(e) : A boolean combination of X <= C, X < C and Y = 0, where Y  X.

9. Summary Abstraction: - subset of traces - subset of tree - simulation relation Predicates to describe trees and traces?