Reducing Multi-Valued Algebraic Operations to Binary

1
Reducing Multi-Valued Algebraic Operations to
Binary

• J.-H. Roland Jiang
• Alan Mishchenko
• Robert K. Brayton
• Dept. of EECS
• University of California, Berkeley

2
Outline

• Motivation
• Definitions
• Problem formulation
• Co-singleton transform
• EBD operations
• Experimental results
• Conclusions

3
Outline

• Motivation
• Definitions
• Problem formulation
• Co-singleton transform
• EBD operations
• Experimental results
• Conclusions

4
Motivation MV optimization

• In high-level design, system descriptions are
  inherently multi-valued.
• Common sub-expressions should be extracted and
  preserved at a higher level.
• Given an MV design, there are two paths of
  synthesis
• Encoding ? binary optimization (SIS)
• Optimizes design resting on the initial
  encoding
• Explores optimization only on restricted
  design space
• MV optimization ? encoding ? binary optimization
  (MVSIS)
• Preserves common sub-expressions at a higher
  level
• Explores larger design space
• Was thought to be computationally much harder
  and time consuming

5
Motivation Speed up MV operations

• Previous MV algebraic methods can be slow on
  large examples.
• Satisfiability matrix method
• Graph matching method (2-cube divisors only)
• Observations
• Many networks had many binary nodes.
• Good speed-up can be obtained by gathering all
  these together and applying SIS fast_extract ( fx
  ) method.
• Question
• Can we extend this to MV nodes besides binary
  ones ?

6
Outline

• Motivation
• Definitions
• Problem formulation
• Co-singleton transform
• EBD operations
• Experimental results
• Conclusions

7
Preliminaries

• Given a multi-valued function (a generalization
  of Boolean functions)
• f(a, b, ) A B ? F,
• we can express it in a sum-of-products (SOP)
  form.
• E.g.
• f0(a,b) a2,3b0,1 a0,3b1,2
  a1,2b0,3 a0,1b2,3
• Can think of this as a three level OR-AND-OR
  expression.

8
Definitions

• Purely algebraic operations (used in SIS)
• Operations which manipulate expressions like
  polynomials
• Semi-algebraic operations (used in MVSIS)
• Operations which include up to absorption rules
• Boolean operations
• Operations which include up to cube
  creation/annihilation
• A fact
• Algebraic optimization in Approach 2
  (semi-algebraic mv opt. ? enc. ? purely algebraic
  bin. opt.) may correspond to Boolean
  optimization in Approach 1 (enc. ? Boolean opt.).

9
Outline

• Motivation
• Definitions
• Problem formulation
• Co-singleton transform
• EBD operations
• Experimental results
• Conclusions

10
Problem formulation

• Given an arbitrary multi-valued SOP expression E
  and an oracle W which optimally factors a binary
  SOP input, how can we take advantage of W to
  factor E ?
• To do so, we have two criteria
• Transform E into E which looks like binary,
  and
• Transform the resultant output of W, say E,
  back to an expression that directly reflects an
  optimally factored form of E.

11
Outline

• Motivation
• Definitions
• Problem formulation
• Co-singleton transform
• EBD operations
• Experimental results
• Conclusions

12
Some unsuccessful naïve trials

• Apply binary encoding to the MV SOP expression
• For example, using 1-hot code, negative 1-hot
  code or any logarithmic code
• Satisfies the first criteria
• But NOT for the second (i.e. doesnt directly
  reflect a factored form)

13
The solution co-singleton transform

• Forward transformation
• Input an MV expression E
• Output a disguised binary expression E
• Begin
• For each literal xS ? E
• Replace it with ?i ? S xi
• End
• (xi ? x0,,i1, i1,,n is a co-singleton
  literal)
• Backward transformation
• Input a disguised binary expression E
• Output an MV expression E
• Begin
• For each ?i ? T xi ? E
• Replace it with x j j ?T
• End

14
Co-singleton transform

• An example Assume a, b are 4-valued variables
• E a2,3b0,1 a0,3b1,2 a1,2b0,3
  a0,1b2,3
• ? (forward co-singleton transform)
• E a0a1b2b3 a1a2b0b3 a0a3b1b2
  a2a3b0b1
• ? (factoring in the binary domain)
• (a1b3 a1b3) (a0b2 a2b0)
• ? (backward co-singleton transform)
• E (a0,2,3b0,1,2 a0,1,2b0,2,3)
  (a1,2,3b0,1,3 a0,1,3b1,2,3)

15
Co-singleton transform

• Co-singleton transform produces a bijection
  between an MV expression and a binary
  expression.
• Co-singleton transform is of time complexity
  linear in the size of the input expression.
• Co-singleton transform is optimally compact.
• Only n bits are used for an n-valued MV variable.
• Co-singleton transform vs. negative 1-hot
  encoding
• Co-singleton transform is NOT an encoding in the
  sense that its binary codes for the values of
  an MV variable are non-disjoint.
• Co-singleton transform negative 1-hot coding
  some minimization w.r.t unused codes

16
Closure properties

• Let F be a factored form of an MV SOP expression
  E, and F and E be the co-singleton transformed
  versions of F and E respectively.
• Theorem 1
• If F is a purely algebraic factorization of E,
  then F is a purely algebraic factorization of
  E.
• Theorem 2
• If F is a semi-algebraic factorization of E,
  then F can be derived from E using only
  semi-algebraic operations
• Theorem 3
• If F is a semi-algebraic factorization of E,
  then F is a semi-algebraic factorization of E.
• Corollary
• Semi-algebraic operations are closed under the
  co-singleton transform, but not for purely
  algebraic operations.

17
Outline

• Motivation
• Definitions
• Problem formulation
• Co-singleton transform
• EBD operations
• Experimental results
• Conclusions

18
EBD operations

• EBD Execution in Binary-in-Disguise
• Procedure
• Apply co-singleton transform
• Apply binary operations
• Apply inverse co-singleton transform
• Binary operations
• Factorization and decomposition
• Algebraic division
• Common divisor extraction
• (Non-algebraic operations)
• In the following discussion, binary operations
  are meant to be those used in SIS, and therefore
  are purely algebraic.

19
EBD operations vs. MV operations

• Different results
• In MV cases, results are maximally lowered. In
  EBD cases, results are maximally raised.
• Procedures exist to make them similar.
• Due to the implementation, there are results that
  can be obtained by one method, but not by the
  other.

20
Outline

• Motivation
• Definitions
• Problem formulation
• Co-singleton transform
• EBD operations
• Experimental results
• Conclusions

21
Experimental results

• Constructed a script (MV-script, similar to
  script.rugged in SIS).
• Replaced all MV algebraic operations with their
  equivalent EBD operation to obtain EBD-script.
• Measured total time and final number of literals
  in all factored forms of MV network.

22
Experimental results
circuit name EBD time EBD lits MV time MV lits
vg2 2.9 87 2.6 85
sse 2.1 128 2.2 120
b12 2.4 70 2.3 70
cht 1.8 163 1.9 164
sqrt8 1.1 67 1.2 56
clip 5.3 134 7.6 129
duke2 10.7 497 24.6 488
sand 23.6 545 47.5 525
f51m 1.8 108 2.4 97
sao2 2.4 109 4 110
circuit name EBD time EBD lits MV time MV lits
term1 5.2 147 6.2 142
9sym 3 72 4.6 120
alu2 12.5 266 19.4 278
sct 1.9 83 2 90
t481 14.2 36 63.9 40
ttt2 3.1 233 4.4 221
bw 3.2 194 4.4 194
rd84 5.3 87 9.5 106
squar5 1.4 58 1.4 58
z4ml 1.2 38 1.4 38
23
Experimental results
circuit name EBD time EBD lits MV time MV lits
C432 46.3 185 49.3 195
planet 24.7 605 63.5 611
vda 32.3 763 96 777
cps 93.3 1479 364.1 1524
dk16 7.0 248 9.3 238
S953 18.6 510 29.6 516
k2 269.2 1426 3351 1428
monk1t 0.9 7 0.8 7
sleep 36.6 33 63.7 37
car 1.2 43 1.3 44
circuit name EBD time EBD lits MV time MV lits
balance 8.1 182 84.1 217
conv35c 1.2 83 1 72
employ1 1.7 42 1.5 36
mm3 0.9 23 0.8 23
mm5 5.3 137 8.5 130
pal3x 4 114 4.5 100
aluack 1.4 91 1.4 76
iris 1.3 12 1.3 12
mm4 2 75 2.3 60
monks2t 1.2 51 1.2 43
24
Experimental results

• EBD method is significantly faster than MV
  method, especially for large examples.
• EBD results sometimes have little loss in quality
  due to the fact that there is no semi-algebraic
  capabilities in the binary domain.

25
Outline

• Motivation
• Definitions
• Problem formulation
• Co-singleton transform
• EBD operations
• Experimental results
• Conclusions

26
Conclusions

• EBD algebraic operations do not give same results
  as MV algebraic operations
• MV result is maximally lowered EBD result is
  maximally raised
• Due to the fact that binary operations are purely
  algebraic
• Binary semi-algebraic operations need to be
  revisited
• EBD operations are significantly faster,
  especially on large examples.
• When used in a full script, the quality of EBD
  results is comparable with that of MV results.