Verification

1
Verification Synthesis of Arithmetic Datapaths
using Finite Ring Algebra

• Priyank Kalla
• Electrical and Computer Engineering
• University of Utah
• Salt Lake City, UT-84112

2
Research Group Members

• Graduate Students
• Namrata Shekhar PhD
• RTL Verification of Arithmetic Datapaths
• Sivaram Gopalakrishnan PhD
• RTL Synthesis of Arithmetic Datapaths
• Vijay Durairaj PhD Chris Condrat BS/MS
• SAT SAT-based Decision Procedures
• Collaborators
• Prof. Florian Enescu Brandon
• Mathematics Statistics, Georgia State Univ.

3
Outline

• Introducing the Problems
• Equivalence Verification High-Level Synthesis
• Applications Fixed Point Arithmetic, Polynomial
  Signal Processing, Audio/Video/Multimedia DSP
• Modeling Poly-Functions over Finite Integer
  Rings
• Previous Work CAD Symbolic Computer Algebra
• Contributions
• Vanishing Polynomials, Canonical Forms for
  Verification
• Polynomial Reduction Decomposition for
  Synthesis
• Algorithm Design Experimental Results
• Open Problems Challenges CAD Algebra

4
The Verification Synthesis Problems
5
Polynomials over Bit-Vectors?

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

6
Bit-Vector Arithmetic 2m Algebra

• Represent integers as a vector of bits
• Bit x0 represents values 0 or 1
• Vector X10 x1, x0 represents integers
• 00, 01, 10, 11 (4 values from 0 to 3)
• Bit-vector of size m integer values in 0,, 2m-1
• Vector Xm-1 0 represents integers reduced
  2m

2-bit
2-bit


3-bit
4-bit
2-bit
2-bit
ADDER
MULTIPLIER
7
Fixed-Size (m) Data-path Modeling

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

16-bit
16-bit

16-bit

16-bit

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

8
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
• If( x70 gt 255 ) then x70 255
• Used in image-processing applications

9
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
10
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 F1 216 F2
  216
• Transform the problem
• F1 - F2 57344x5 16384x4 8192x3 16384x2
  32768x
• 0 216
• F1 - F2 Vanishing polynomial

11
Multiple Word-Length Operands

• Bit-vector operands with different word-lengths
• Input variables x1,, xd Output variables
  f, g
• Input bit-widths n1,, nd Output width
  m
• 
  
• Model as polynomial function
  

12
Example Digital Image Rejection Unit
input A110, B70 output Y1150, Y2150

• Y1 ? Y2
• Y1150 Y2150
• Y1 216 Y2 216

13
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

14
Previous Work Others

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

15
Previous Work Symbolic Algebra

• Galois Field Decomposition of Boolean Functions
  GF(2m) Pradhan, 1978 recent work
• Symbolic Algebra Tools Singular, Macaulay,
  Maple, Mathematica, ZEN, NTL, CoCoA
• Polynomial equivalence over R, Q, C, Zp
• Unique Factorization Domains (UFDs) Uniquely
  factorize into irreducibles
• Match corresponding coefficients to prove
  equivalence

16
Symbolic Algebra in CAD

• MODD DAG representation of polynomials over
  GF(2m) Pradhan, IWLS 05, DATE 04
• Guiding Synthesis engines using Groebners basis
  De Micheli, TCAD 02
• Given polynomial F and Library elements ltI1, ,
  Ingt
• F h1 I1 hn In
• Again, works over UFDs
• Approximate RTL as polynomials over Reals
• Theorem Proving Clarke et al.
• HOL Mathematica Analytica

17
Why is the Problem Difficult?

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

f
f
x
x6
x2
x4

• Atypical approach required to prove equivalence

18
Problem Formulations Solutions

• f (x1, , xd) n g(x1, , xd) n
• Proving equivalence is NP-hard Ibarra, J. ACM
  83
• Vanishing polynomials ICCD 05, DATE06
• f(x) g(x) 0 2m Zero Equivalence
• An instance of Ideal Membership Testing
• Efficient solutions over fields (Groebners
  bases) Z2mx1,, xd ?
• Canonical forms ICCAD 05
• Unique representations for polyfunctions over Z2m
• Equivalence by coefficient matching
• Concepts from Hungerbuhler et al. J. Sm. Not.,
  06

19
Ideal Membership Testing
Z2m
Z2mx1, , xd

• ( f g ) 2m 0 or ( f g ) vanishes 2m
• Membership in the Ideal of all Vanishing
  Polynomials in Z2m
• Grobner's basis? Buchberger's algorithm?
• Generate the Ideal!

20
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
• Generalize the result from
• p to pm to (any integer) n

21
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

22
Results From Number Theory

• (f-g) 2m 0 means that 2m (f-g)
• 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

23
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)

24
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)

25
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)

26
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)

27
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).
28
Example 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.

29
Example 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

30
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
  

31
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
  

32
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)

33
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

34
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

35
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
36
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
37
Example 2

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

38
Status of our Work - Extensions

• Multiple Variables Z2mx1, , xd
• Given polynomials (f, g) d variables
• x ltx1, , xdgt over Z2m
• Prove that (f-g) 0 2m
• What if word-lengths are different too?
• x1 ? Z2n1 , , xd ? Z2nd
• No problems!
• Straight-forward extensions of previous concepts
• Other approach Canonical forms of poly-functions

39
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

40
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)
41
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

42
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

• f (x) axk
  

If 2mak!, then V(x) ak! x k 0 2m
k
axk a1xk-1..
43
Coefficient Reduction Example

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

44
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

45
Experimental Setup

• Distinct RTL designs are input to GAUT U. de
  LESTER
• 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
  zero-testing algorithm
• Reduce f, g to canonical forms.
• Algorithm implemented in MAPLE
• Compare with BMD, SAT and MILP
• Complexity O(kd)

46
Results
47
Limitations Mathematics versus CAD

• Finite Ring Algebra Cannot DIVIDE!
• Right shift breaks the model
• Overflow arithmetic versus Saturation
• Comparators non-polynomial?
• Internal bit-vectors of arbitrary word-lengths
• Only Yes/No Equivalence? Find bugs too!
• Couple Verification with Simulation!
• Arithmetic interfaced with Boolean

48
Intermediate Signal Truncation
a70
b70
c70
a70
b70
c70


?
t170
t270


f180
f280
a 127 b 1 f1 383 c 255
a 127 b 1 f2 127 c 255
49
Proposed Solution
a70
b70
c70
a70
b70
c70


?
t170
t270


f180
f280

• Required Algorithm for reduction to canonical
  form over

50
Polynomial Abstraction from RTL

• If (x gt 2b10)
• then y x x x
• Else y xx

• Traditional modeling
• .
• Proposed modeling y as a polyfunction from
• Unique
  representation
• Issues with the proposed abstraction
• Scalability

51
Verification via Simulation?
52
Simulation Vector Generation

• Prove f g over Z8 via Simulation
• How many vectors to Simulate? Exhaustive
  Simulation?
• NO! Simulate only n SF(23) 4 CONSECUTIVE
  vectors

• f x4 x2 (specification)
• x0, f0
• x1, f2
• x2, f4
• x3, f2
• x4, f0
• x5, f2
• x6, f4
• x7, f2

• g 2x2 (implementation)
• x0, g0
• x1, g2
• x2, g0
• x3, g2
• x4, g0
• x5, g2
• x6, g0
• x7, g2

53
Applications to Synthesis

• Datapath size (m) 8 bits
• SF(28) 10
• Polynomial can be
• factorized into S10(x)
• F70 (x1)(x2)...(x10)

54
Polynomial Decomposition

• F 8xy2 11y3 9y2 1
• X, Y 30 are 4-bit vectors (16)
• Synthesize Area 320
• U 8x 3y
• F U3 U2 1
• Synthesize Area 18120 201

55
Conclusions

• Finite Word-Length Bit-Vector Arithmetic is
  Finite Ring Algebra
• Computations reduced integer power of 2
• NON-UFDs! Number Theory, Commutative Algebra
• Mathematicians should help us here.
• Techniques to verify equivalence of polynomial
  RTL computations
• f(x) 2m g(x) 2m is transformed into f(x) -
  g(x) 0 2m
• Ideal Membership Testing, Canonical Forms.
• Tremendous scope in high-level synthesis.

56
Questions?