1 Compositional Verification of Hybrid Systems Using Simulation Relations Doctorate Defense Goran Frehse Radboud Universiteit, Nijmegen, Oct. 10, 2005

2 Example 1: Überlingen, July 1, 2002 Boeing & Tupolew crossing 21:33:03 –Alarm from Collision Avoidance System (TCAS) B TU154M !

3 Example 1: Überlingen, July 1, 2002 Boeing & Tupolew crossing 21:33:03 –Alarm from Collision Avoidance System (TCAS) 21:34:49 –Human controller command B TU154M !

4 Example 1: Überlingen, July 1, 2002 Boeing & Tupolew crossing 21:33:03 –Alarm from Collision Avoidance System (TCAS) 21:34:49 –Human controller command 21:34:56 –TCAS recommendation B TU154M !

5 Example 1: Überlingen, July 1, 2002 Boeing & Tupolew crossing 21:33:03 –Alarm from Collision Avoidance System (TCAS) 21:34:49 –Human controller command 21:34:56 –TCAS recommendation 21:35:32 –Collision B TU154M !

6 ! Example 1: Überlingen, July 1, 2002 Boeing & Tupolew crossing 21:33:03 –Alarm from Collision Avoidance System (TCAS) 21:34:49 –Human controller command 21:34:56 –TCAS recommendation 21:35:32 –Collision Official Recommendation: “pilots are to obey and follow TCAS advisories, regardless of whether contrary instruction is given” Trust a computer!? Official Recommendation: “pilots are to obey and follow TCAS advisories, regardless of whether contrary instruction is given” Trust a computer!? B TU154M

7 Formal Verification Characteristics –mathematical rigour –sound proofs & algorithms Hybrid System –continuous environment –discrete software Problems –only computable for certain types of models –must check all possibilities  computational complexity Solution –abstraction –compositionality Model of Environment Model of Software Precise Specification Hybrid System

8 Formal Verification Characteristics –mathematical rigour –sound proofs & algorithms Hybrid System –continuous environment –discrete software Problems –only computable for certain types of models –must check all possibilities  computational complexity Solution –abstraction –compositionality Model of Environment Model of Software Proof (algorithmic) Precise Specification Hybrid System

9 Formal Verification Characteristics –mathematical rigour –sound proofs & algorithms Hybrid System –continuous environment –discrete software Problems –only computable for certain types of models –must check all possibilities  computational complexity Solution –abstraction –compositionality Model of Environment Model of Software Proof (algorithmic) Guaranteed Correctness Precise Specification Hybrid System TCAS verified in part Livadas, Lygeros, Lynch, ‘00

10 Example 2: Join Manoeuvre [Tomlin et al.] Traffic Coordination Problem –join paths at different speed Goals –avoid collision –join with sufficient separation

11 Example 2: Join Manoeuvre [Tomlin et al.] Traffic Coordination Problem –join paths at different speed Goals –avoid collision –join with sufficient separation Models –Environment: Planes –Software: Controller switches fast/slow Specification –keep min. distance

12 Abstraction and Simulation Relations Goal –check all possibilities Abstraction  simplified model –here: linear bounds on direction disturbances

13 Abstraction and Simulation Relations Goal –check all possibilities Abstraction  simplified model –here: linear bounds on direction –bounds on trajectories bounds on direction original trajectory bounds on trajectories of abstraction disturbances

14 Abstraction and Simulation Relations Goal –check all possibilities Abstraction  simplified model –here: linear bounds on direction –bounds on trajectories Simulation Relation  formal relationship between original and abstraction –everything possible in implementation is also possible in abstraction –specification = abstraction original trajectory bounds on direction bounds on trajectories of abstraction disturbances

15 Compositionality From Components to Systems –Simulation relations must hold after composition Original Plane Abstract Plane satisfies Original Controller Abstract Controller satisfies while active do if altitude > check distance else if speed >= 10 check heading check distance else warning end while while active do check distance end while

16 Compositionality From Components to Systems –Simulation relations must hold after composition Original Controller Original Plane Abstract Plane Abstract Controller Original Plane Abstract Plane satisfies sat. Original Controller Abstract Controller satisfies while active do if altitude > check distance else if speed >= 10 check heading check distance else warning end while while active do check distance end while composed system composed abstraction

17 Compositionality From Components to Systems –Simulation relations must hold after composition Benefits –modular verification –advanced deduction techniques possible Difficulty –formalisms must fit together hybrid system simulation relation composition Original Controller Original Plane Abstract Plane Abstract Controller Original Plane Abstract Plane satisfies sat. Original Controller Abstract Controller satisfies while active do if altitude > check distance else if speed >= 10 check heading check distance else warning end while while active do check distance end while composed system composed abstraction

18 Contribution of this Thesis Formal Framework for Compositional Verification –simulation relations for hybrid systems –semi-computable for linear bounds time collision possible! safety margin

19 Contribution of this Thesis Formal Framework for Compositional Verification –simulation relations for hybrid systems –semi-computable for linear bounds Verification Tool: PHAVer (Polyhedral Hybrid Automaton Verifier) –compute simulation relations and reachable states –most powerful verification tool for hybrid systems time collision possible! safety margin

20 Contribution of this Thesis Formal Framework for Compositional Verification –simulation relations for hybrid systems –semi-computable for linear bounds Verification Tool: PHAVer (Polyhedral Hybrid Automaton Verifier) –compute simulation relations and reachable states –most powerful verification tool for hybrid systems Future Work –compositional over- approximations (submitted) –efficiency & applications time collision possible! safety margin