Equivalence Verification of Polynomial Datapaths with FixedSize BitVectors using Finite Ring Algebra

1
Equivalence Verification of Polynomial Datapaths
with Fixed-Size Bit-Vectors using Finite Ring
Algebra

• 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
• Problem Modeling
• Polynomial Functions over Finite Integer Rings
• Limitations of Previous Work
• Approach and Contributions
• Canonical form for Polynomials over Finite Rings
• Algorithm Design and Experimental Verification
  Runs
• Results, Conclusions Future Work

3
The Equivalence Verification Problem
4
Motivation

• Quadratic filter design for polynomial signal
  processing
• y a0 . x12 a1 . x1 b0 . x02 b1 . x0 c
  . x0 . x1

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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

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Example 1 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
• To Prove F1 216 F2 216
• F1 F2 in Z2mx1, , xd

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Previous Work Function Representations

• Boolean Representations (f B ? B) BDDs, ZBDDs
  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
• SAT, MILP, Word-level ATPG,
• Theorem-Proving (HOL), term-rewriting
• Works when datapath size can be abstracted away

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Previous Work Symbolic Algebra

• Symbolic Algebra Tools Singular, Macaulay,
  Maple,
• Mathematica, Zen, Dagwood etc.
• Polynomial representations Sparse, Dense,
  Recursive, Straight-line programs, DAGs, etc.
• Polynomial equivalence over R, Q, C, Zp
• Unique Factorization Domains (UFDs) Uniquely
  factorize into irreducibles
• Match corresponding irreducibles to prove
  equivalence

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Why is the Problem Difficult?

• Z2m is a non-UFD
• f x2 6x in Z8 can be factorized as

f
f
x6
x4
x2
x

• Atypical approach required to prove equivalence

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Proposed Solution

• f (x1, , xd) n g(x1, , xd) n
• Proving equivalence is NP-hard
• Vanishing polynomials ICCD 05
• f(x) g(x) 0 2m Zero Equivalence
• An instance of Ideal Membership Testing
• Efficient solutions over fields (Groebners
  bases) Z2mx1,, xd ?
• Canonical forms Current focus
• Unique representations for polyfunctions over Z2m
• Equivalence by coefficient matching
• Concepts from Hungerbuhler et al. To appear J.
  Sm. Not., 06

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Polyfunctions over Z2m
f
F2
g
G2
Z2mx1, , xd
Z2m

• Polynomials over Z2mx1, , xd
• Represented by polyfunctions from Z2mx1, , xd
  to Z2m
• F1 2m F2 2m gt they have the same
  underlying polyfunction (f )
• Use equivalence classes of polynomials
• Derive representative for each class Canonical
  form

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Motivating our Approach

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

• f (x) 5x2 6x - 3 (x2 2x 5) (4x2 4x)
  
• f (x) g(x) V (x) in Z23
• V (x) 4x2 4x 0 23 for x in 0,,7
• f (x) g (x) 0 in Z23
• Required To identify and eliminate such
  redundant sub-expressions

(vanishing)
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Vanishing polynomials Requirement
Set of all Vanishing polynomials
0
f
g
Z2m
Z2mx1, , xd

• Generate vanishing expressions V(x)
• Test if f (x) g (x) V(x)
• f (x) f (x) V(x)
• Challenge Infinite number of vanishing
  polynomials
• Required To generate V(x) specific to given f
  (x)

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Vanishing Polynomials for Reducibility

• In Z23, say f (x) 4x2
• f (x) f (x) - V(x)
• Generate V(x) of degree 2
• V(x) 4x2 4x 0 23
• Reduce by subtraction
• 4x2 f (x)
• 4x2 4x V(x)
• - 4x - 4x 8 4x
• 4x2 can be reduced to 4x
• Degree reduction

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Coefficient Reduction Example

• Degree is not always reducible
• In Z23, f (x) 6x2
• a 6
• k 2
• Divide and subtract
• 6x2 2x2 4x2 23
• 4x2 can be reduced to 4x
• f (x) 2x2 4x Lower Coefficient

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Our Approach

• Say f (x) akxk ak-1xk-1 a0
• In decreasing lexicographic order
• Required f (x) in reduced, minimal, unique form
• Check if degree can be reduced
• Check if coefficient can be reduced
• Perform corresponding reductions
• Repeat for all monomials

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Degree Reduction Requirement

• Generate appropriate vanishing polynomial , V(x)
  
• f (x) axk a1xk-1
• V(x) axk a2xk-1
• f (x) V(x) bxk-1
• V(x) axk is the leading term
• Identify constraints on
• Degree k
• Coefficient a
• Use concepts from number theory

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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) of 2m n
• SF(23) 4, since 234!
• 2m divides the product of n SF(2m) consecutive
  numbers
• Use SF(2m) to generate vanishing polynomial V(x)

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Results From Number Theory

• V (x) 0 23
• 23 V (x) in Z23
• 23 4! , since SF(23) 4
• 4! divides the product of 4 consecutive numbers

Write V(x) as a product of SF(23) 4 consecutive
numbers

• A polynomial as a product of 4 consecutive
  numbers?
• (x1)(x2)(x3)(x4) 4! x 4 0 23
• 
  4

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Constraints on the Coefficient

• In Z23 , SF(23) 4. Product of 4 consecutive
  numbers
• (x1) (x2)
• missing factors
• V (x) 4x2 4x (x1)(x2) 0 23
• compensated by constant
• 4(x1)(x2) 42! x 2
• 2

(x3) (x4)
4

• f (x) axk
  

Rule 1 If 2mak!, then V(x) ak! x k 0
2m k
axk a1xk-1..
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Example Vanishing Polynomial

• Consider f (x) 4x2 in Z23
• a 4
• k 2
• V (x) ?
• Rule 1 2mak! gt 23 42!
• V (x) ak! x k 4. 2! x 2 4. 2!
  (x2) (x1)
• k
  2 2!
• 4x2 4x 0 23
• f (x) 4x2
• V(x) - 4x2 4x
• - 4x - 4x 8 4x

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Coefficient Reduction Requirement

• Define v2(k!) max x ? N 2x k!
• number-of-factors-2 in k!
• v2(4!) v2(4 3 2 1) 3
• Rule 1 2m ak!
• Number-of-factors-2 in ak! m or
• v2(ak!) m
• If 2m does not divide ak!
• v2(ak!) lt m
• v2(k!) lt m and a lt 2m- v2(k!)

Constraint on degree
Constraint on coefficient
Rule 2 Coefficient (a) has to be in the range
0, , 2m- v2(k!)-1
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Notation Multivariate Polynomials

• Given d variables x ltx1, , xdgt
• with degrees k ltk1, , kdgt
• Replace in Rule 1 and Rule 2
• xk ?di1 xiki
• k! ?di1 ki!
• x ?di1 xi
• k ki
• v2(k!) ?d v2(ki!)
• Use lexicographic term ordering for variables

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Uniqueness

• Theorem Any polynomial F in Z2m can be uniquely
  written as

F S k?Nd ak xk

• d is the number of variables
• ak ? 0, , 2m- v2(k!)-1 is the coefficient
• v2(k!) lt m

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Reduction Procedure

• Given a monomial f (x) axk
• Degree reduction Determine if 2m ak!
• If yes, generate V(x) of with axk as the leading
  term
• f (x) f (x) V(x)
• If 2m ak!, check if coefficient (a) is in 0,
  , 2m- v2(k!)-1
• If not, perform division according to
• axk q. 2m- v2(k!)xk rxk

Degree reducible
Reduced form r lt 2m- v2(k!)
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Example Reduction
Given poly 6x2y 4xy in Z23

• Reducing 6x2y
• Degree Reduction
• a 6
• k! kx!.ky! 2!1! 2
• 23 does not divide (a k! ) 12
• Degree reduction not possible
• Perform coefficient reduction

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Example Reduction
Given poly 6x2y 4xy in Z23

• Coefficient Reduction
• v2(k!) v2(2!1!) 1
• Range 0, , 2m- v2(k!) 1 0,, 3
• a 6 gt 3 Can reduce coefficient
• 6x2y 4x2y 2x2y
• Degree reduction for 4x2y
• Rule 1 23 (4 2! 1!)
• 4x2y 4 . 2!1! x 2 y 1 4xy
• 2
  1
• poly (4xy 2x2y) 4xy 2x2y in Z23

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Experimental Setup

• Distinct RTL designs are input to GAUT U. de
  LESTER, 04
• Extract data-flow graphs for RTL designs
• Construct the corresponding polynomial
  representations (f, g)
• Extract bit-vector size
• Reduce f and g to canonical form
• Equivalence check by coefficient matching
• Algorithm implemented in MAPLE
• Complexity O(kd) where k is the total degree
  and d is the number of variables
• Compare with BDD, BMD and SAT

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Results
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Conclusions Future Work

• Technique to verify equivalence of multivariate
  polynomial RTL computations
• Fixed-size bit-vector arithmetic is polynomial
  algebra over the finite integer ring, Z2m
• f (x1,, xd) 2m g (x1, , xd) 2m is proved
  by reduction to canonical form
• Efficient algorithm to determine unique
  representations
• Future Work involves extensions for -
• Multiple Word-length Implementations DATE 06
• Verification of Rounding and Saturation
  Arithmetic

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Questions?
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Comparison