XP2: A New Compact Representation
for Manipulating Arithmetic Circuits
Ajay K. Verma, Philip Brisk and Paolo Ienne
csda
csda
Processor Architecture Laboratory (LAP)
& Centre for Advanced Digital Systems (CSDA)
Ecole Polytechnique Fédérale de Lausanne (EPFL)
Logic Synthesis For Arithmetic Cicruits

Sum-of-Product (SoP) and Product-of-Sum (POS) Form
 Well-studied (e.g., Espresso), but…
 Arithmetic circuits are XOR-dominated

Reed-Muller Form (XOR of Products)
In principle, good for arithmetic circuits, but…
 Exponential growth in size compared to POS/SUM
 Parallel counter
 Leading Zero Anticipator (LZA)
 Arithmetic circuits with control logic at the periphery.


Contribution:

2
XP2: A Reed-Muller alternative without the exponential blowup
Motivational Example
z0 = Maj (a1, a3, a5, a7, a9, a11, a13)
z1 = Maj (a0, a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14)
0.35 ns
705 μm2
0.74 ns
2200 μm2
0.75 ns
2815 μm2
3
0.75 ns
2275 μm2
0.79 ns
Synthesis from SoP Form: 16397.2 μm2
Outline

Related work

XP2: a new compact representation


Results from abstract algebra


Null spaces and their use in factorization
Minimization algorithm for XP2 representation

4
Superiority over other representations
Iterative split-merge approach

Manipulation algorithms (CSE elimination)

Results

Conclusions and future work
Related Work

General circuits




Binary Decision Diagram [Lee59]
 Minimisation algorithm by partitioning [Yang02]
Representation of XOR-dominated circuits



5
SOP/POS Form
 Minimisation (e.g., Espresso) [Brayton84]
Factored form [Brayton82, Brayton87]
Generalized Reed-Muller form [Sasao90]
 Manipulation algorithms [Verma06, Verma07]
2-SPP form [Bernasconi06]
Three-level Boolean expressions [Ishikawa04]
Sum of Products and Reed-Muller Form
6

SP1 = SP – SOP Form

SPk – Sum of products of SPk-1 expressions

The SPk representation of the XOR of n variables is
exponentially large for any constant k.

Theorem 2. The Reed-Muller expansion of the OR of
n variables is exponentially large.

The n-bit parity function is exponentially large in any
SPk representation [Furst84]
XP2 Form
XP: Generalized Reed-Muller expression
PXP: Product of XP expressions
XP2: XOR of PXP expressions
XPk: XOR of PXPk-1 expressions
7
Why XP2?

8
Theorem 3. If the SoP/PoS representation of a circuit has size
k, then the size of the XP2 representation is O(k)

Linear growth!

Reed-Muller Form grows exponentially in k
f = ab + pqr + ac + xyr
SOP
f = 1  (1  ab) (1  pqr) (1  ac) (1  xyr)
XP2
f = (a + x) (p + y + r) (b + c) (a + r)
POS
f = (1  a x) (1  p y r) (1  b c) (1  a r)
XP2
Optimizing XP2 Expressions
1. 3.
Factorize
2.Merge
Split
(ay  bef)
q)
q)qd)
(cady
 d) bcef
(x (xz 
 z)
pc  (p
pd 
(p
qc
(ay
 bef)
(acy
((ay 
bef)(c
(p
d)

q)
bdef
x) x x)
AND
ANDAND
XOR
9
Not a Generalized
Reed-Muller Form
Expression!
AND
CSE Elimination in XP2 Representation
find_CSE (expr E1, expr E2)
{
Introduce new variables λ and μ;
E = λE1  μE2;
minimize (E);
S = set of PXP’s which have a product term of the form (λX  μY);
T = {p | p(λX  μY)  S, for some X, Y};
return T;
}
10
CSE Elimination: An Example
E1 = (ab  cd) (p  q)  pq (c  d)
E2 = (ab  pq) (c  d)  cd (a  b)
E = λE1  μE2 = λ(ab  cd) (p  q)  λpq (c  d) 
μ(ab  pq) (c  d)  μcd (a  b)
minimization
E = pq (c  d) (λ  μ)  ab (λ (p  q)  μ (c  d)) 
cd (λ (p  q)  μ (a  b))
CSE = {pq (c  d), ab, cd}
11
Experimental Setup
Manually designed
(CSE)
SOP form
Input circuit
RM and Generalized RM form
XP2 form
CSE elimination
Performance criteria: Literal Count
12
XP2 form
(CSE)
Results (1 of 4)
Benchmark
SOP form
Reed-Muller
Form
XP2 form
(no CSE)
XP2 form Manually
(CSE) Designed
16-bit LZD
220
1.81 x 106
332
154
66
64
16-bit LOD
220
602
332
154
66
64
16-bit Barrel Shifter
1280
4752
1280
896
288
192
16-bit Adder
5.24 x 106
9.18 x 105
9.18 x 105
1194
237
89
16-bit Comparator
1.05 x 106
4.81 x 108
1.05 x 106
246
85
63
15:4 Counter
5.19 x 105
5.72 x 104
5.72 x 104
1854
567
77
15-bit Majority
5.15 x 104
5.15 x 104
5.15 x 104
1479
543
71
12-bit CSA
Large
1.24 x 1012
1.24 x 1012
3336
250
149
16-bit LZA
Large
Large
Large
42602
3136
152
Performance criteria: Literal Count
13
Generalized
Reed-Muller
Form
Results (2 of 4)
Adder
14

SOP/GRM
Exponential growth as a function of bitwidth

XP2
Linear growth as a function of bitwidth
Results (3 of 4)
Barrel Shifter
15

Not XOR-dominated

XP2 has a similar literal count as SoP/GRM
Results (4 of 4)
Reed-Muller
XP2

16
Diminishing returns observed as k increases
Conclusions and Future Work

XP2

XOR-based representation for arithmetic circuits

Avoids exponential size complexity

Logic optimization fundamentals
 Factorization
 Split
 Merge
 CSE

17
Elimination
Develop a complete logic synthesis package using
XP2
Results from Abstract Algebra:
Null Space Factorization

Null space of X, N (X):

All expressions F, which satisfy FX = 0

ab  N (a  b)
f = (a  b) (cd  e)  (c  d) (ab  e)
ab  N (a  b)
f = (a  b) (cd  ab  e)  (c  d) (ab  e)
cd  N (c  d)
f = (a  b) (cd  ab  e)  (c  d) (ab  cd  e)
f = (a  b  c  d) (ab  cd  e)
18
Relative Compactness of SPk and XPk
XPk: XOR of products of XPk-1 expressions
SPk (n) = set of Boolean expressions whose representation size in SPk is n
XPk (n) = set of Boolean expressions whose representation size in XPk is n
19
Minimization of XP2 representation
Minimize (expr E)
{
do
{
Factorize (E)
Split (E);
Merge (E);
} while (there is a reduction in size)
output E;
}
Accepting only smaller expressions
might cause sub-optimality
20