1. Reachability Analysis for AMS Verification using Hybrid Support Function and SMT-based Method Honghuang Lin, Peng Li Dept. of ECE, Texas A&M University {linhh, pli} @neo.tamu.edu

2. Motivation Digital logic • Mixed-signal systems • Analog + Digital • Nonlinearity + Digital effects • Reachability Analysis • Formal method for AMS verification • Capable for PLL lock time checking TDC DCO Digitally-Intensive PLL [G. Yu et al JLPE’10]

3. Motivation • Challenges • Digital effects increase the complexity of the reachability analysis • Reachability analysis is expensive for nonlinear systems • Questions • Q1: How to model the two types of signals (especially digital) efficiently for verification? • Q2: How to accelerate reachability analysis?

4. Q1: Model • Linearization • TDC resolution effect • Complex transition • Digital  Linear analog • Staircase  Linear transition TDC

5. Q1: Model • Variable Reduction • IIR finite word length • Need state variables for internal nodes • Digital  Linear analog • Reduce # state variables by 8 Multiplication: round-off error Addition: overflow Second order IIR

6. Q2: Reachability Analysis Acceleration • Support function based method • [A. Girard et al IFAC2008] • Initial space to reachable space • Support function representation • Efficient for linear systems • Unable to solve nonlinear systems Support function Reachable space

7. Q2: Reachability Analysis Acceleration • Simulation-assisted SMT based method • [L.Yinet al ICCAD2012] • Generic method for nonlinear systems • Discretize SimulationApproximationSATConservative • Suffers from resolution and dimension explosion • Our goal: accelerate this generic method by leveraging support function

8. Proposed Method AMS Pure analog model Reachability analysis • Digital  Analog Support function based method Reachable space of the full system Linear subsystem Nonlinear subsystem SMT-based method

9. Outline • Motivation • Overview • Pure Analog Model with KRR • Hybrid Reachability Analysis • Experimental Results • Summary

10. Conservative Model Conservative? Upper bound Eu X Regression Eu Pure analog model Xa El Lower bound El Xd AMS System F Fa Fd Xa: Analog variables Xd: Digital variables Fa: Analog transition Fd: Digital transition X: Analog variables F: Pure analog transition function Eu: Upper bound of errors between F and Fa/d El: Lower bound of errors between F and Fa/d

11. Error Estimation with KRR • Kernel Ridge Regression (KRR) • [C. Saunders et al 1998; J.A.K. Suykenset al 2002] • Subject to • A.k.a Least Squares Support Vector Regression • Plenty of training dataaccurate prediction • Confidence interval computation • [K. De Brabanteret al 2011] • Error • Smootherbias and variance Min. structural risk

12. Error Estimation with KRR • Error Estimation of the Model • Next: Hybrid Reachability Analysis X(t) AMS system Ei(t+1) Error on the i-th state variable Pure analog model KRR Prediction + Confidence Intervals Features Targets

13. Partition the Pure Analog Model • Linear: • Nonlinear: • Variables on the boundary: Linearized

14. Hybrid Method Support function method NL-SMT with support function Combine the two reachable spaces with different dimension Reachable space with state variables , , Reachable space with state variables , ,

15. Support Function Based Method • Support function • Definition: • Intersection of half spaces: • Tight polyhedral over approximation of a reachable space • A list of vector li • Corresponding support function values ρ(li) • E.g. represent oval with pentagon

16. Support Function Based Method • Reachability analysis in linear subsystems • [A. Girard et al IFAC2008] • For the linear subsystem: • A useful property of support function: • The reachable space AU can be easily computed by the initial space U U AU A U AU AU+err Polyhedral over approximation

17. SMT Based Method • Nonlinear Satisfiability Modulo Theory(SMT) based method • [L.Yinet al ICCAD2012] • Convert verification problems to satisfiability problems composed of boolean combinations of multiple arithmetic constraints(can be nonlinear) • E.g. can be converted to • SAT solver • iSAT[http://isat.gforge.avacs.org/] • Davis-Putnam-Logemann-Loveland (DPLL) Algorithm • Produces an existing solution that satisfies all the constraints or “unsatisfiable”

18. NL-SMT with Support Function • Another Property of support functions: U V

19. Intersection of Reachable Subspaces • XY: Reachable space of linear subsystem • XZ: Nonlinear subsystem Linear Y Y Z X X Y Nonlinear Z X Z Y X Z X

20. Experiment Results • Error interval of modeling • KRR tool: Dlib-ml [Davis E. King, 2009] • Error intervals of phase difference and the output of loop filter for different word length • Reflects the error between digital implementation and ideal analog characteristic

21. Experiment Results • Speed up of the hybrid method • Compare single SAT solver running • Overhead in linear subsystem: 21.163 sec • At least 76X speedup

22. Hybrid Reachability Analysis of PLL • Lock time < 0.25 us • Resolution determined by the error interval prediction

23. Conclusions • Model with KRR • AMS  Pure Analog • KRR • Hybrid Reachability Analysis • Model partition • Respective reachability analysis in the linear and nonlinear subsystems • Reachable subspaces intersections • Experiment Results • Model error • Speedup • DI-PLL lock time verification

24. Thanks • Questions?