The Usage of Stochastic Processes in Embedded System Specifications

1. The Usage of Stochastic Processes in Embedded System Specifications Axel Jantsch, Ingo Sander, Wenbiao Wu Royal Institute of Technology, Stockholm

2. Outline • Motivation • Nondeterminism • Constraining purpose • Descriptive purpose • ForSyDe Methodology •  Processes • Skeletons • (Refinement of stochastic processes) • Usage of stochastic processes A. Jantsch, KTH

3. Nondeterminism • Nondeterminism is used to model incomplete information • Not known • Not interesting B t3 t1 A D Communication time is not defined. C t2 t4 A. Jantsch, KTH

4. B t3 E t7 t1 t5 A D G C F t2 t4 t6 t8 Nondeterminism: Objectives System Objective: Specify and implement determinate behaviour. Constraining purpose Environment Objective: Describe relevant aspects of the environment. Descriptive purpose A. Jantsch, KTH

5. t1 t1 = c may be a constant in the implementation. A B Descriptive and Constraining Purpose • Descriptive: the real world object behaves like the model. • Constraining: the real world object instantiates one of the possibilities of the model. t1 t1 varies nondeterministically for each event. A B A. Jantsch, KTH

6. S t A B Nondeterminism in Specification and Design • System S behaves determinately and independently of the delay t. (E.g. data flow process networks) • System S behaves determinately if t T. (e.g. synchronous languages, synchronous circuits) • System S behaves differently for different values of t. (e.g. CCS, Erlang, SDL) A. Jantsch, KTH

7. Drawbacks with Nondeterminism • Pure nondeterminism is undesirable. • Fair nondeterminism is difficult to handle because we have to deal with many possible behaviours. • Simulation: It cannot be simulated. • Synthesis: Which particular possibility should be implemented? • Verification: Which behaviour is acceptable? Transaction Server Merge A. Jantsch, KTH

8. Probability versus Nondeterminism • Often we know a probability distribution. • Stochastic processes can be realized with pseudo random number generators. • Fair nondeterminism can be approximated well by stochastic processes. A. Jantsch, KTH

9. g2(f(e),state) e f f(e) g1,g2 map mealy ForSyDe - Formal System Design • System specification based on concurrent processes. • Computation and communication is perfectly synchronized. • ForSyDe models are determinate. • Timing, communication and computation is separated. • Skeletons are templates to instantiate processes. A. Jantsch, KTH

10. …, n3, n2, n1 …, 8, 2, 4  Processes • Bar-processes have to be implemented by selecting one or several possibilities. • Tilde-processes must be implemented such that the statistical properties are preserved. A. Jantsch, KTH

11. e f g f(e) or g(e) selMap (f, g, seed, type) Stochastic Skeletons • selMap is a stochastic skeleton which contains a  process with uniform distribution in the range [0,1]. • If  generates 0, f is applied, otherwise g is applied to the current input. A. Jantsch, KTH

12. Usage of Stochastic Skeletons • Descriptive purpose • Very useful for environment entities; • A stochastic description is superior to a nondeterministic description; • Not useful for system under design because we can control it if we want; • Constraining purpose • Stochastic processes allow for fine control over how behaviour and timing of the implementation is constrained. • Different probability distributions; • Bar and tilde processes; A. Jantsch, KTH

13. Summary • In system specifications nondeterminsm should be replaced by stochastic processes. • Stochastic processes can be used to fine tune the amount of information that shall be provided. • The descriptive usage is for environment entities. • The constraining usage is for entities under design. • Two types of stochastic processes are interpreted differently in design: • Tilde processes require the implementation of all probabilistic properties. • Bar processes allow the implementation of any of the possible behaviours. A. Jantsch, KTH