Research Interests, Projects, Research Interests, Projects, Collaborations & OpportunitiesCollaborations & Opportunities

Priyank Kalla

Electrical & Computer Engineering

University of Utah

Typical Design Flow & CAD SupportTypical Design Flow & CAD Support

TLM Models

High-LevelAnalysis

HardwareDescription

DesignSpec

High-LevelSynthesis

Optimization

Testing for Defects

Fabrication

CircuitModels

Mask High-Level + LogicSynthesis & OptimizationArea, Speed, Power

Design VerificationDesign Verification

TLM Models

PropertyVerification

HardwareDescription

DesignSpec

High-LevelSynthesis

Optimization

Fabrication

CircuitModels

Mask “Core Engines” forCorrectness Reasoning &Constraint Solving

Equivalence Verification ?

My Research Projects at the UMy Research Projects at the U

Boolean Reasoning Engines (concluded)

Million+ variables, 10s Million Constraints

2 M.S. + 1 PhD

“Exploring” Design Automation for Photonic Devices

Collaboration w/ Steve Blair + his Colleagues

1 PhD student: Learning & Exploring the issues

Synthesis & Verification of Finite-Precision Arithmetic

Focus on Hardware – but applicable to Software

2 PhDs. + 2 more for sure…

Recognition + bread-earner [3 NSF Grants]

Datapath-Dominated ApplicationsDatapath-Dominated Applications

Floating-pointModel

AutomatedFixed-pointGeneration

Fixed-pointModel

Equivalence Verification ?

Real NumberSpecification

ConversionUtility

Optimization

SynthesisFPGA

HDL Model

HDL Model

MatlabXilinxAlteraSynplicitySynopsysCalyptoGalois

DSPCryptoEmbeddedASIC

Why Verify Finite-Precision Arithmetic Why Verify Finite-Precision Arithmetic

1996 Ariane Rocket Explosion64-bit floating to fixed point rounding error

Vancouver Stock-Exchange Index1982: Value initialized to 1000; 1984: value 520

Truncation error: Should have been 1098

Gulf War, Patriot Missile (28 dead, SCUD)

German Parliament: 5.0% versus 4.97% (2-bits!)

Bug in JPEG Decode Routine (discovered in ’05)Hacker Exploited – uploaded a virus: Microsoft website

Network Routers, Filter instability errors + ……

[Peymandoust et al, TCAD '03]

MP3 Decoder: Anti-Aliasing FunctionMP3 Decoder: Anti-Aliasing Function

MAC

x = a2 + b2

a b

x

F

DFF

coefficientscoefficients

Taylor series expansion

2 2

1 1

22F

xa b

6 5 4

3 2

1 9 115

64 32 6475 279 81 85

16 64 32 64

F x x x

x x x

Example: Anti-Aliasing FunctionExample: Anti-Aliasing Function

F[15:0] = 156x6 + 62724x5 +

17968x4 + 18661x3 + 43593 x2 +

40244x + 13281

G[15:0] = 156x6 + 5380x5 +

1584x4 + 10469x3 + 27209 x2 +

7456x + 13281

F ≠ G

F[15:0] = G[15:0]

Prove that F(x) % 216 ≡ G(x)% 216

Contemporary tools model the problem at (circuit) bit-level

Too many variables/constraints – infeasibleName of the game: “Abstraction” + model the “details”

Fixed-Size (m) Data-path: ModelingFixed-Size (m) Data-path: Modeling

Control the datapath size: Fixed size bit-vectors (Control the datapath size: Fixed size bit-vectors (mm))

* +

8-bit

8-bit

16-bit17-bit

Bit-vector of size Bit-vector of size mm: integer values in : integer values in 0,…, 20,…, 2mm-1-1

Fixed-size Fixed-size ((mm) bit-vector ) bit-vector

arithmeticarithmetic

Polynomials Polynomials reduced reduced %2%2mm

Algebra over Algebra over the ring the ring ZZ22

mm

* +

16-bit

16-bit

16-bit16-bit

Why is the Problem Difficult?Why is the Problem Difficult?

Z2m is a non-Unique Factorization Domain

F = x2 + 6x in Z8 (modulo 8) can be factorized as

Easy to do over Reals, Complex numbers, Integers (modulo p)

Textbook algebra solutions are not available over Z2m

Contacted a Mathematician: Prof. Florian Enescu

Partially unsolved Problems in “classical” mathematics

F

x+6

F

x+4 x+2x

““Zero” in Finite-PrecisionZero” in Finite-Precision

F[15:0] = 156x6 + 62724x5 +

17968x4 + 18661x3 + 43593 x2 +

40244x + 13281

G[15:0] = 156x6 + 5380x5 +

1584x4 + 10469x3 + 27209 x2 +

7456x + 13281

F ≠ G

F[15:0] = G[15:0]

F[15:0] - G[15:0] = 0?

F - G [15:0] = 57344 x5 + 16384 x4 + 8192 x3 + 16384 x2 + 32768 x

Ideals in Finite RingsIdeals in Finite Rings

Test for membership in the ideal of vanishing polynomials

Standard Problem formulation in Computer Algebra

But, how to “mathematically” generate this “ideal”?

Hilbert’s & Fermat’s results (mod p): Generalize to (mod pm)

Idealx

x % 2m

% 2m

0f

gf – g ? Z2m

Z2m[x]

ContributionsContributions

Abstraction of Arithmetic Datapaths Equivalence

Verification Problem

Equivalence of Polyfunction

Equivalence of Polynomial systems

Ideal Membership Testing

Canonical Forms

Simulation Vector Generation

ADD/MULT

Generalized to cover allBit-VectorArithmetic

Significance, Impact, InterestSignificance, Impact, Interest

Computer-algebra research group, Univ. Kaiserslautern

SINGULAR: Public domain computer algebra tool

Our approach implemented in their latest release

Univ. Kaiserslautern: Math + ECE + Infineon

Extension of my work to verify production-quality design

Univ. of Tokyo + Fujitsu

Use our work for Testing SoC (ATPG)

Bay-area Start-up: Calypto (NSF GOALI Partner)

GALOIS Inc., Cryptography applications (military funding)

Invited Talk: Intl. Joint Conferences on Automated Reasoning

Theoretical Computer Science Community