Exploiting Vanishing Polynomials for Equivalence Verification of Fixed-Size Arithmetic Datapaths

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Exploiting Vanishing Polynomials for Equivalence
Verification of Fixed-Size Arithmetic Datapaths

• Namrata Shekhar1, Priyank Kalla1, Florian
  Enescu2, Sivaram Gopalakrishnan1
• 1Department of Electrical and Computer
  Engineering,
• University of Utah, Salt Lake City, UT-84112.
• 2Department of Mathematics and Statistics,
• Georgia State University, Atlanta, GA-30303

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Outline

• Overall Verification Problem
• Our Focus Equivalence Verification of Fixed-size
  Arithmetic Datapaths
• Applications Polynomial Signal Processing, etc.
• Problem Modeling and Approach
• Polynomials over Finite Integer Rings
• Limitations of Previous Work
• Our Contributions
• Exploiting Vanishing Polynomials for Equivalence
  Test
• Related to Ideal Membership Testing
• Algorithm Design and Experimental Verification
  Runs
• Results
• Conclusions Future Work

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The Equivalence Verification Problem
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Fixed-Size (m) Data-path Modeling

• Control the datapath size Fixed size bit-vectors
  (m)

• Bit-vector of size m integer values in 0,, 2m-1

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Fixed-Size Data-path Implementation

• Signal Truncation
• Keep lower order m-bits, ignore higher bits
• f 2m g 2m
• Fractional Arithmetic with rounding
• Keep higher order m-bits, round lower order bits
• f - f 2m g - g2m
• 2m 2m
• Saturation Arithmetic
• Saturate at overflow
• Used in image-processing applications

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Example Anti-Aliasing Function

• F 1 1
• 2va2 b2 2vx
• Peymandoust et al, TCAD03
• Expand into Taylor series
• F 1 x6 9 x5 115 x4
• 64 32 64
• 75 x3 279 x2 81 x
• 16 64 32
• 85
• 64
• Scale coefficients Implement as bit-vectors

coefficients
coefficients
a
b
x a2 b2
x
MAC

F
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Example Anti-Aliasing Function

• F1150, F2150, x150
• F1 156x6 62724x5 17968x4 18661x3 43593
  x2
• 40244x 13281
• F2 156x6 5380x5 1584x4 10469x3 27209
  x2 7456x
• 13281
• F1 ? F2 F1150 F2150
• Transform the problem
• F1 - F2 57344x5 16384x4 8192x3 16384x2
  32768x
• 0 216
• F1 - F2 Vanishing polynomial

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Previous Work Function Representations

• Boolean Representations (f B ? B)
• BDDs, MTBDDs, ADDs etc.
• Moment Diagrams (f B ? Z)
• BMDs, KBMDs, HDDs etc.
• Canonical DAGs for Polynomials (f Z ? Z)
• Taylor Expansion Diagrams (TEDs)
• Required Representation for f Z2m ? Z2m

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Previous Work Others

• SAT and MILP-based techniques
• Suitable for linear/multi-linear forms
• Word-level ATPG, congruence closure based
  techniques, co-operative decision procedures
• Solve linear congruences under modulo arithmetic
• Theorem-Proving (HOL), term-rewriting
• Works when datapath size can be abstracted using
  data dependence, symmetry, and other abstractions
  
• Here, datapath size (m) defines ring cardinality
  (Z2m)

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Previous Work Symbolic Algebra

• MODDs Based on finite fields Pradhan et al,
  DATE 04
• Literal based decomposition
• Symbolic Algebra Tools Singular, Macaulay,
  Maple,
• Mathematica, Zen, etc.
• Polynomial equivalence over R, Q, C, Zp
• Unique Factorization Domains (UFDs) Uniquely
  factorize into irreducibles
• Match corresponding coefficients to prove
  equivalence

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Why is the Problem Difficult?

• Z2m is a non-UFD
• f x2 x in Z6 can be factorized as

f
f
x
x1
x3
x4

• Atypical approach required to prove equivalence

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Proposed Solution

• Equivalence can be proved by
• f (x) - g(x) 0 2m Zero Equivalence
• Exploit Vanishing polynomials
• An instance of Ideal Membership Testing
• Proposed solution based on
• Niven Warrens work Proc. Amer. Math.Soc,
  1957
• Singmasters work J. Num. Th, 1974

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Ideal Membership Testing
Q
P

• hP ?Q defined by 2m
• x in P maps to x 2m in Q
• Ideal members map to 0
• Derive Ideal of Vanishing Polynomials in Z2m
• Grobner's basis? Buchberger's algorithm?
• Known for UFDs, but for Z2m ?

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Ideal Membership Testing in Zp

• Fermats Little Theorem
• x p x (mod p) or
• x p x 0 (mod p)
• x p x generates the vanishing ideal in Zpx
• f(x) 0 p iff f(x) (xp-x)g(x)
• Zp Principal Ideal Domain
• This does not follow in Z2m

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Ideal Membership Testing
0
Q
P

• Generate the ideal of vanishing polynomials 2m
  ?
• Vanishing Ideal is finitely generated
• Niven et al, Am. Math. Soc., 57
• Need an algorithm for membership testing
• Representative expression for members of this
  ideal
• Singmaster, J. Num. Th 74

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Results From Number Theory

• n! divides a product of n consecutive numbers.
• 4! divides 99 X 100 X 101 X 102
• Find least n such that 2mn!
• Smarandache Function (SF).
• SF(23) 4, since 234!
• 2m divides the product of n SF(2m) consecutive
  numbers

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Results From Number Theory

• f g in Z23 or (f - g) 0 23
• 23(f - g) in Z23
• 234!
• 4! divides the product of 4 consecutive numbers

Write (f-g) as a product of SF(23) 4
consecutive numbers

• A polynomial as a product of 4 consecutive
  numbers?
• (x1)

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Results From Number Theory

• f g in Z23 or (f - g) 0 23
• 23(f - g) in Z23
• 234!
• 4! divides the product of 4 consecutive numbers

Write (f-g) as a product of SF(23) 4
consecutive numbers

• A polynomial as a product of 4 consecutive
  numbers?
• (x1)(x2)

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Results From Number Theory

• f g in Z23 or (f - g) 0 23
• 23(f - g) in Z23
• 234!
• 4! divides the product of 4 consecutive numbers

Write (f-g) as a product of SF(23) 4
consecutive numbers

• A polynomial as a product of 4 consecutive
  numbers?
• (x1)(x2)(x3)

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Results From Number Theory

• f g in Z23 or (f - g) 0 23
• 23(f - g) in Z23
• 234!
• 4! divides the product of 4 consecutive numbers

Write (f-g) as a product of SF(23) 4
consecutive numbers

• A polynomial as a product of 4 consecutive
  numbers?
• (x1)(x2)(x3)(x4)

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Basis for factorization

• S0(x) 1
• S1(x) (x 1)
• S2(x) (x 1)(x 2) Product of 2
  consecutive numbers
• S3(x) (x 1)(x 2)(x 3) Product of 3
  consecutive numbers
  
• Sn(x) (x n) Sn-1(x) Product of n
  consecutive numbers

Rule 1 Factorize into atleast Sn(x) to vanish,
where n SF(2m).
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Example 1 Vanishing polynomial

• 4th degree polynomial p in Z23 SF(23) 4
• p x4 2x3 3x2 2x
• p can be written as a product of 4 consecutive
  numbers.
• or p (x1)(x2)(x3)(x4) S4(x) in Z23.
• p is a vanishing polynomial.

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Example 2 Vanishing polynomial

• module fixed_bit_width (x, f, g)
• input 20 x
• output 20 f, g
• assign f20 x2 6x 3
• assign g20 5x2 2x 5

• h(x) f(x) g(x) 4x2 4x
• h(x) 0 for all values of x in 0,,7
• 4x24x not equal to (x1)(x2)(x3)(x4)
• Required To show that h(x) is a vanishing
  polynomial in Z23

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Constraints on the Coefficient

• h(x) 4x2 4x 4(x1)(x2)
• In Z23 , SF(23) 4. Product of 4 consecutive
  numbers
• S4(x) (x1) (x2) (x3) (x4)

Rule 2 Coefficient has to be a multiple of bk
2m/gcd(k!, 2m)

• Here, Coefficient of h(x) 4, Degree of h(x)
  2
• b2 23/gcd(2!, 23) 4 is a multiple of the
  coefficient
  

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Constraints on the Coefficient

• h(x) 4x2 4x 4(x1)(x2)
• compensated by constant
• In Z23 , SF(23) 4. Product of 4 consecutive
  numbers
• S4(x) (x1) (x2) (x3) (x4)
• missing factors

Rule 2 Coefficient has to be a multiple of bk
2m/gcd(k!, 2m)

• Here, Coefficient of h(x) 4, Degree of h(x)
  k 2
• b2 23/gcd(2!, 23) 4 is a multiple of the
  coefficient
  

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Deciding Vanishing Polynomials

• Polynomial F in Z2m vanishes if

F FnSn S n-1ak bk Sk
k0 Rule 1
Rule 2

• n SF(2m), i.e. the least n such that 2mn!
• Fn is an arbitrary polynomial in Z2mx
• ak is an arbitrary integer
• bk 2m/gcd(k!,2m)

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Algorithm
Example 1
F FnSn S n-1ak bk Sk k0

• Input poly, 2m
• Calculate n SF(2m)
• k n Reduce according to Rule 1
• Divide by Sn
• If remainder is zero, F FnSn, else Continue

• poly 4x2 4x in Z23
• n SF(23) 4
• k 4 Divide by S4
• Degree (poly) 2
• lt degree(S4) 4
• quo 0, rem 4x2 4x
• F4 0 Continue

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Algorithm
Example 1
F FnSn S n-1ak bk Sk k0

• Input poly, 2m
• Calculate n SF(2m)
• k n Reduce according to Rule 1
• Divide by Sn
• If remainder is zero, F FnSn, else Continue

• poly 4x2 4x in Z23
• n SF(23) 4
• k 4 Divide by S4
• Degree (poly) 2
• lt degree(S4) 4
• quo 0, rem 4x2 4x
• F4 0 Continue

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Algorithm
Example 1
F FnSn S n-1ak bk Sk k0

• Reduce according to Rule 2. Divide by Sn-1 to S0
• Check if quotient is a multiple of
• bk 2m/gcd(k!,2m)
• If remainder is zero, stop. Else, continue

• k 3 Divide by S3
• degree (poly) 2 lt degree(S3) 3
• quo 0, rem 4x2 4x continue
• k 2 Divide by S2
• quo 4 rem 0
• b2 23/gcd(2!,23) 4
• a2 quo/ b2 1 ? Z

poly a2.b2.S2 1.4.(x1)(x2) 0 in Z23
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Algorithm
Example 1
F FnSn S n-1ak bk Sk k0

• Reduce according to Rule 2. Divide by Sn-1 to S0
• Check if quotient is a multiple of
• bk 2m/gcd(k!,2m)
• If remainder is zero, stop. Else, continue

• k 3 Divide by S3
• degree (poly) 2 lt degree(S3) 3
• quo 0, rem 4x2 4x continue
• k 2 Divide by S2
• quo 4 rem 0
• b2 23/gcd(2!,23) 4
• a2 quo/ b2 1 ? Z

poly a2.b2.S2 1.4.(x1)(x2) 0 in Z23
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Example 2

• poly 5x2 3x 7 in Z23
• n SF(23) 4
• degree (poly) 2 lt n. Skip Rule 1 and goto Rule
  2
• Divide by S2
• quo 5 rem 5 4x
• b2 23/gcd(2!,23) 4
• a2 quo/ b2 is not in Z
• poly does not satisfy Rule 2
• poly is not a vanishing polynomial in Z23

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Experimental Setup

• Distinct RTL designs are input to GAUT U. de
  LESTER, 2004
• Extract data-flow graphs for RTL designs
• Construct the corresponding polynomial
  representations (f, g)
• Extract bit-vector size
• Find the difference (f-g) and invoke the
  algorithm
• Algorithm implemented in MAPLE
• Compare with BMD, SAT and MILP
• Complexity O(n1), n SF(Z2m)

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Results
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Applications to Synthesis

• Datapath size (m) 8 bits
• SF(28) 10
• Polynomial can be
• factorized into S10(x)

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Conclusions

• Technique to verify equivalence of univariate
  polynomial RTL computations
• Fixed-size bit-vector arithmetic is polynomial
  algebra over finite integer rings
• f(x) 2m g(x) 2m is transformed into f(x) -
  g(x) 0 2m
• Efficient algorithm to determine vanishing
  polynomials

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Future Work

• Future Work involves extensions for -
• Multivariate datapaths with fixed bit-widths
• To Appear, ICCAD 05
• Multiple Word-length Implementations
• In Review, DATE 06
• Verification of Rounding and Saturation
  Arithmetic
• Formal Error Analysis
• Applications to Synthesis
• Need cost models for low power, area, delay

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Questions?