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Combining Symbolic Simulation and Interval Arithmetic for the Verification of AMS Designs Mohamed Zaki, Ghiath Al Sammane, Sofiene Tahar, Guy Bois FMCAD'07 November 14 th, Hardware Verification Group, ECE Department, Concordia University 2 Génie Informatique, Ecole Polytechnique de Montréal 11 2

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

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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 Introduction One important issue in the design process is verification. Used verification methods: Simulation and Symbolic Analysis. One important issue in the design process is verification. Used verification methods: Simulation and Symbolic Analysis. Formal Verification for AMS?

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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

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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… Basic Idea: Approximate Analysis using (e.g.: interval, polyhedral). Pros: guaranteeing the inclusion of the solution, hence soundness Cons: computationally expensive, low dimension systems. Motivation

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

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

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

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

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

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RE index 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 If-Expression (If[Cond, y, z]) Logical, comparison or arithmetic formula ► ► ► Recurrence Equations

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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 Behavior Mapping :=: Map Piecewise ODE to SRE

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

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AMS Example

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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 Taylor Models Approximation Symbolic partInterval part Taylor Model Approximation

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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. Taylor Models Approximation A Taylor model for a given function f consists of a multivariate polynomial p n (x) of order n, and a remainder interval I, which encloses Lagrange remainder of the Taylor approximation Symbolic Simulation

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

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

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

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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:

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

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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. Taylor Models Rules

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Example: Arithmetic over Taylor Model id V id

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Example x, y bound

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

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

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

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Divergence problem in the interval based reachability calculation due to: 1) Dependency problem. 2) Wrapping effect 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. Example: x - x = 0 for x in [1, 2], but X – X = [-1, 1] for X = [1, 2] Bounded Model Checking

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

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3 rd Modulator Example

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Application Verified Not Verified with Counterexample

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Divergence Application

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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 Conclusion 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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THANKS ! More Info at hvg.ece.concordia.ca

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