Combining Symbolic Simulation and Interval Arithmetic for the Verification of AMS Designs

1. Combining Symbolic Simulation and Interval Arithmetic for the Verification of AMS Designs 1 1 1 Mohamed Zaki, Ghiath Al Sammane, Sofiene Tahar, Guy Bois 2 1Hardware Verification Group, ECE Department, Concordia University 2Génie Informatique, Ecole Polytechnique de Montréal FMCAD'07 November 14th , 2007

2. Outline • Introduction • Related Work • Verification Methodology • Modelling AMS Designs • Symbolic Simulation • Verification Algorithm • Applications • ΔΣ Modulator • Analog Oscillator • Conclusion

3. Introduction A cornerstone in embedded systems are analog and mixed signal (AMS) designs, usually needed at the interface with the real world. • AMS applications • Front-end: sensors, amp., filters, A/D • Back-end: D/A, filters, oscillators, PLL • High performance digital circuits One important issue in the design process is verification. Used verification methods: Simulation and Symbolic Analysis. Formal Verification for AMS?

4. Problem in AMS Verification • Contains continuous components • Infinite continuous state space • Dense time • Strong nonlinear behavior with digital components Exhaustive simulation is out of reach The closed form solution of differential equations is only possible for specific cases

5. Motivation • Basic Idea: Approximate Analysis using (e.g.: interval, polyhedral). • Pros: guaranteeing theinclusion of the solution, hence soundness • Cons: computationally expensive, low dimension systems. Formal verification for AMS: Kurshan’91, Greenstreet ’98, Gupta’04, Dang’04, Hartong’05, Myers’05, Frehse’06 Verified Designs: - modulators, filters, oscillators, VCO… Used Tools: d/dt, PHAVer, Checkmate, Coho…

6. Motivation Proposed Methodology We propose a recurrence equations based bounded model checking approach for AMS systems. The idea is based on approximation by interval Taylor model forms Symbolic part Interval part

7. AMS System Discrete-Time Continuous- Time Temporal Property Digital Interval based Bounded Model Checking Verification Methodology Symbolic Simulation Recurrence Equations Taylor Approximation Combined SRE Property is False (Counterexample Generated) Property is Proved True for a Bounded Time

8. AMS System Discrete-Time Continuous- Time Temporal Property Digital AMS Modelling Symbolic Simulation Recurrence Equations Taylor Approximation Combined SRE Interval based Bounded Model Checking Property is False (Counterexample Generated) Property is Proved True for a Bounded Time

9. Differential Equations AMS designs are described using discrete time, continuous time analog behavior interacting with discrete digital components. The analog behavior is governed by the differential equations: A large class of AMS designs can be modeled using piecewise differential equations. • AMS exhibits piecewise behavior due to: • Abrupt change in input signal, parameters • Change in the analog behavior • Events generated by control logic, switching conditions

10. ► ► ► ► ► Logical, comparison or arithmetic formula If-Expression (If[Cond, y, z]) Differential Equations Extending System of ODEsusing Generalized Piecewise Formula A closed form solution is generally not available for ODE systems and discrete approximate models are used.

11. RE index ► Logical, comparison or arithmetic formula ► If-Expression (If[Cond, y, z]) ► Recurrence Equations Extending System of Recurrence Equations The generalized If-formula is a class of expressions that extend recurrence equations [Al Sammane’05] to describe digital and mixed signal designs

12. Behavior Mapping Requirement:- Discrete sampling that captures all the different states in the continuous evolution. :=: Approximation of the ODE as truncated Taylor series expanded about time instant with a remainder term :=: Map Piecewise ODE to SRE

13. Taylor Approximation The ODE system under certain assumptions, can be time descretized using Taylor Approximation Remainder Such representation allows an approximate polynomial description of the behavior of an ODE system using SRE.

14. AMS Example

15. AMS Example

16. Taylor Models Approximation To preserve the original behavior, the remainder term should not be discarded and instead bounds must be specified. Intervals are numerical domains that enclose the original states of a system of equations at each discrete step Symbolic part Interval part Taylor Model Approximation

17. Taylor Models Approximation A Taylor model for a given function f consists of a multivariate polynomial pn(x) of order n, and a remainder interval I, which encloses Lagrange remainder of the Taylor approximation • Taylor model arithmetic developed as an interval extension to Taylor approximations • Allowing the over- approximation of system reachable states using non-linear enclosure sets. • Preserve relationships between state variables. Symbolic Simulation

18. AMS System Discrete-Time Continuous- Time Temporal Property Digital Symbolic Simulation Recurrence Equations Taylor Approximation Combined SRE Interval based Bounded Model Checking Symbolic Rewriting Phase Next Interval States Verification Phase Property is False (Counterexample Generated) Property is Proved True for a Bounded Time Verification Methodology

19. Symbolic Simulation The symbolic simulation algorithm to obtain the generalized SRE is based on rewriting by substitution. Substitution rules Polynomial symbolic expressions Logical symbolic expressions If-formula expressions Interval expressions Interval-Logical expressions Taylor Models expressions

20. Symbolic Simulation Symbolic Simulation Algorithm Substitution Fixpoint Example Rewrites using two rules ► ►

21. Interval Rules To preserve the original behavior, the remainder term should not be discarded and instead bounds must be specified. Intervals are numerical domains that enclose the original states of a system of equations at each discrete step Basic interval arithmetic operators can be defined as follows:

22. Interval Rules Interval analysis provides methods for checking truth values of Boolean propositions over intervals by using the notion of inclusion test Inclusion test: Examples: ►

23. Taylor Models Rules • The evaluation of a function is transformed to symbolically computing the Taylor polynomial of the function. • Taylor polynomial will be propagated throughout the evaluation steps. • Only the interval remainder term and polynomial terms of high orders are bounded using intervals.

24. Vid Vid Example: id Arithmetic over Taylor Model

25. Example x, y bound

26. AMS System Discrete-Time Continuous- Time Temporal Property Digital Symbolic Simulation Recurrence Equations Taylor Approximation Combined SRE Interval based Bounded Model Checking Symbolic Rewriting Phase Next Interval States Verification Phase Property is False (Counterexample Generated) Property is Proved True for a Bounded Time Verification Methodology

27. Bounded Model Checking Bounded model checking (BMC) algorithm relying on symbolic and interval computational methods Properties

28. Bounded Model Checking Computing the (overapproximate) reachable states is based on image computation.

29. Bounded Model Checking Evaluation of the reachable states over interval domains Over-approximation guarantee: Every trajectory in the initial system, is included in the interval-based reachable states. Divergence problem in the interval based reachability calculation due to: 1) Dependency problem. 2) Wrapping effect Example: x - x = 0 for x in [1, 2], but X – X = [-1, 1] for X = [1, 2]

30. Bounded Model Checking Computing the (overapproximate) reachable states is based on image computation. is an interval evaluation of Taylor model form of the function Overapproximation guarantee: Every trajectory in the initial system, is included in the Taylor Model based reachable states.

31. Bounded Model Checking

32. Example 3rd Modulator

33. Application Not Verified with Counterexample Verified

34. Application Divergence

35. Conclusion • We presented a formal verification methodology for AMS designs. • Methodology based on symbolic rewriting and Interval methods • Continuous time is approximated using Taylor models • Avoiding conventional Interval arithmetic like wrapping effect. • Continuous state space is handled using symbolic-interval computations • Allowing the over- approximation of reachable states using non-linear enclosure sets. • Methodology implemented using the Mathematica computer • algebra system Future Work: • Automatic extraction of SREs form HDL-AMS designs. • Definition of an expressive property language for specifying • properties of AMS designs. • Explore more complex case studies.

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