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Formal Verification of Analog Designs using MetiTarski William Denman , Behzad Akbarpour, Sofine Tahar 1 Mohamed H. Zaki 2 Lawrence C. Paulson 3 1 Concordia University, Montreal, Canada 2 University of British Columbia, Vancouver, Canada 3

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Formal Verification of Analog Designs using MetiTarski

William Denman, Behzad Akbarpour, Sofiène Tahar1 Mohamed H. Zaki2 Lawrence C. Paulson3

1Concordia University, Montreal, Canada 2University of British Columbia, Vancouver, Canada 3University of Cambridge, United Kingdom

FMCAD’09 November 17th, 2009

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2 / 36 FMCAD’09 William Denman

  • Should we care about formal verification for analog circuits?

Yes! Not really… Verifiers / Researchers Designers Common motivation

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3 / 36 FMCAD’09 William Denman

  • Some interesting statistics [IBS Corporation]

– Analog Circuitry 2% of the transistor count – 20% of the IC Area – 40% of the design Effort

  • Analog verification continues to be a

serious bottleneck

50% of the errors that require re-design are from analog circuitry

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4 / 36 FMCAD’09 William Denman

  • Challenges

– Infinite/Continuous state space – Infinite time – PVT : Sensitivity to process variation, voltage, temperature – Non-linear behaviour

  • We propose

– A time unbounded verification – Using MetiTarski : An Automated Theorem Prover

  • Formal Verification for Analog Circuits?
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5 / 36 FMCAD’09 William Denman

  • Motivation
  • Related Work
  • Proposed Methodology
  • Brief Introduction to MetiTarski
  • Illustrative Example
  • Conclusion
  • Future Plans
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  • Balivada [1995]

– Discretization of a circuit’s transfer function to the Z-domain – Apply digital based equivalence checking techniques

  • Hartong, Klausen and Hedrich [2004]

– From analog circuit transfer functions – Verify dynamic behaviour of the specification and implementation state spaces.

  • Model Checking/

Reachability Analysis Proof Based Equivalence Checking

Presence of tolerance margins

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7 / 36 FMCAD’09 William Denman

  • Kurshan and McMillan [1991]

– State space subdivision of transistor behaviour – Predict possible transitions between states

  • Gupta [2004] , Dang [2006], Frehse [2006], Little

[2006], Greenstreet [2007]

– Reachability relations using projection techniques – Over-approximation, but verification still sound Possible Time Bounded Verification

  • Model Checking/

Reachability Analysis Proof Based Equivalence Checking

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8 / 36 FMCAD’09 William Denman

  • Ghosh and Vemuri [1999]

– PVS used to prove functional equivalence between models – Specification built in VHDL-AMS – Approximated DC models

  • Hanna [2000]

– Predicates defining voltage and current behaviour – Theorem Proving used – Conservative approximation

  • Model Checking/

Reachability Analysis Proof Based Equivalence Checking

Manual/Heuristic steps

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9 / 36 FMCAD’09 William Denman

  • Motivation
  • Related Work
  • Proposed Methodology
  • Brief Introduction to MetiTarski
  • Illustrative Example
  • Conclusion
  • Future Plans
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SLIDE 10

10 / 36 FMCAD’09 William Denman

  • Analog

Circuit Closed Form Solution Specification Inequality MetiTarski Range Reduction

Property Verified True

Property of Interest Add Axioms Does not terminate Does not terminate Proof generated

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11 / 36 FMCAD’09 William Denman

  • Analog circuit specification

– Circuit must oscillate – Gain for certain frequency range

  • Isolate the property

– Oscillation : Is it present? – Gain : 3dB Bandwidth

  • Inequality

– Voltage < Upper threshold – Gain > Minimum Required Value

  • Specification

Inequality Property of Interest

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12 / 36 FMCAD’09 William Denman

  • Analog circuit

– Differential equations – Kirchoff law Equations

  • Closed Form Solution

– Bounded number of analytical functions – No differential operators – Not always easy to obtain

  • Analog

Circuit Closed Form Solution

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13 / 36 FMCAD’09 William Denman

  • Automated Theorem Proving

– The axioms are specific mathematical facts – Bounding properties – Definition of functions

  • Range Reduction

– Functions are not defined over all ranges – Large bounds cause proof to never end – Apply basic trigonometric identities

  • Range

Reduction Add Axioms

) 2 sin( ) sin( ) 2 cos( ) cos( π π + = + = x x x x

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14 / 36 FMCAD’09 William Denman

  • Motivation
  • Related Work
  • Proposed Methodology
  • Brief Introduction to MetiTarski
  • Illustrative Example
  • Conclusion
  • Future Plans
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15 / 36 FMCAD’09 William Denman

  • Developed by Akbarpour and Paulson [‘07]

– Automated Theorem Prover – Transcendental functions (sine, cosine, ln, exp, etc.) – Square Root

  • Theory behind the tool

– Resolution prover combined with a decision procedure – Decidability of real closed fields (RCF) by Tarski – Function families of upper and lower bounds by Daumas and others

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Metis QEPCAD-B Resolution Theorem Prover Decision Procedure MetiTarski

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  • QEPCAD-B

– Advanced implementation of cylindrical algebraic decomposition – Best available decision procedure for RCF – Eliminates quantifiers from a formula reduces to

  • .

2

= + + ∃ c bx ax x

) ( ) ( ) 4 (

2

= = = ∨ ≠ ∧ = ∨ ≥ − ∧ ≠ c b a b a ac b a

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18 / 36 FMCAD’09 William Denman

  • Assuming
  • We are given a function containing exp(x)

– Upper bound axiom is – Will usually need more than one axiom

  • 120

60 12 ) 120 60 12 (

2 3 2 3

− + − + + + − x x x x x x

4 ≤ ≤ x

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19 / 36 FMCAD’09 William Denman

  • Motivation
  • Related Work
  • Proposed Methodology
  • Brief Introduction to MetiTarski
  • Illustrative Example
  • Conclusion
  • Future Plans
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20 / 36 FMCAD’09 William Denman

  • PWL: Simplest class of nonlinear circuits
  • Behaviour can be reasonably approximated

. 1 723 . 723 . 276 . 276 . < ≤ ≤ < ≤ ≤

C C C

V V V

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21 / 36 FMCAD’09 William Denman

  • ODEs

Piecewise ODEs Transition Relations Initial Conditions MAPLE M1 MetiTarski M2 M3

Modes of operation

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  • Using a computer algebra system
  • Piecewise ODEs

– Separate behaviour of the component into modes

  • Transition relations

– Determined by the piecewise model

  • Initial Conditions

– Dependant on the system specification

  • Piecewise

ODEs Transition Relations Initial Conditions

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23 / 36 FMCAD’09 William Denman

  • Closed form solution

for each mode

  • Procedure followed

until each mode visited

ODEs Mode N Initial Conditions Maple Invlaplace Closed Form Solution Maple Fsolve Switching Time Maple Eval Initial Conditions Mode N+1

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24 / 36 FMCAD’09 William Denman

  • Starting with the ODEs of the system
  • ID(VC) is the current through the tunnel diode
  • Inverse Laplace transform taken to get closed

form solutions in each mode

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  • Using the produced solution

– Fsolve used to compute time when switches modes – Mode 1 -> Mode 2 : VD > 0.276

  • Initial conditions determined

– Take solution from Fsolve – Use Eval to evaluate function values

  • Continue until each mode visited
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  • Choose the property of interest

– Reason about oscillation – Reason about bounded behaviour

  • Turn into an inequality

– Non-oscillation : IL will never pass an upper bound – Bounded Behaviour : IL and VC will remain bounded

  • Input into MetiTarski

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27 / 36 FMCAD’09 William Denman

  • Transform inequality into the MetiTarski syntax
  • Remember: each mode must be checked
  • For All

Mode Switch Time Closed form solution Property inequality Time in a specific mode

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  • Property 1

– Non-Oscillation

  • In each mode upper threshold not passed

– IL : Current through the inductor

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29 / 36 FMCAD’09 William Denman

  • Property 2 – Bounded Behaviour
  • In each mode

the current and voltage are bounded

  • Necessary to

add axioms in 2 cases.

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  • Recall the property

!

IL will never pass an upper bound

Non Oscillation

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  • Applied methodology to a basic OP-AMP
  • Required additional method to obtain a closed

form solution.

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32 / 36 FMCAD’09 William Denman

  • Motivation
  • Related Work
  • Proposed Methodology
  • Brief Introduction to MetiTarski
  • Illustrative Example
  • Conclusion
  • Future Plans
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33 / 36 FMCAD’09 William Denman

  • Developed a methodology for the automated

verification of analog designs

– Algebra system steps are semi-automated, but mechanical in nature – MetiTarski completely automated – Most proofs complete quickly

  • Applied to several analog circuits

– Interesting and complex behaviour – Two different methods for closed form solutions

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  • Computing Closed Form Solutions

– Investigate methods for solving nonlinear ODEs

  • Scale to Larger Problems

– Efficient methods for calculating piecewise linear functions – Apply methodology to more precise models

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More details at: hvg.ece.concordia.ca