1. Logic Synthesis Part II
Maciej Ciesielski
Univ. of Massachusetts
Amherst, MA

2. Outline Last time: Synthesis flow Two-level synthesis (PLA)
Exact methods (Quine McCluskey)
Heuristic methods (Espresso)
Multi-level synthesis (standard cells/FPGA)
Structural synthesis (SIS)
Today:
Functional decomposition
Traditional – disjoint decomposition (Ashenhurst Curtis)
BDD-based bidecomposition (BDS)
Logic Synthesis 2

3. Technology-independent
Synthesis Flow
HDL specification
Technology-independent
optimization
Technology
mapping
Cell library
Manufacturing
Front-end
parsing
Logic synthesis
Logic Synthesis 2

4. Logic Optimization methods
Two-level logic (PLA)
Exact (QM)
Heuristic
(espresso)
Multi-level logic
(standard cells)
Boolean
Structural
(SIS)
Functional
(AC, Karp)
(BDD-based)
algebraic
Boolean
Logic Synthesis 2

5. Two-level minimization - basic idea
Initial representation:
x y z
0 – 0
0 1 –
– 1 1
1 – 1
f1 f2
0 1
1 0
000
100
110
010
111
011 f1 f2
101
x y z
0 – 0
0 1 1
1 – 1
f1 f2
0 1
1 1
1 0
Minimized function: f1 f2
000
100
110
010
111
011
101 x y z
Logic Synthesis 2

6. Exact 2-Level Minimization (Quine-McCluskey)
(on set)
(don’t care set)
Primes of (F+D) = ( p p p3 )
Minterms of F y’ w
x’z’
x y z’ w
x’ y z w
x’ y z’ w
x’ y’ z’ w’
essential minterm
Solution: {p1,p2}  y + w is minimum prime cover (non-unique).
Possible approach (SAT): (p1+p3)(p2 +p3)(p1+p2)p2 = 1
Logic Synthesis 2

7. Heuristic 2-level logic minimization (espresso)
Consider F(a,b,c) initially specified as:
f = {abc, abc, abc} (on-set), and d ={abc, abc} (don’t care set)
abc is redundant
a is prime
F3= a+abc
Expand abc  bc
Expand abca
F2= a+abc + abc
F4= a+bc
F1= abc + abc+ abc
off on
don’t care a c b
Logic Synthesis 2

8. Multi-level Minimization: Boolean network6 1 5 3 4 7 8 9 2
Outputs
Inputs
Internal nodes,
single-output functions
Goal: minimize some measure of network complexity
number of 2-input gates
number of literals (variables), represe
Eventually, the nodes have to be mapped to standard cells
(technology mapping)
Logic Synthesis 2

9. Structural Operations (algebraic)
Basic Operations:
1. Decomposition (single function)
f = abc+abd+a’c’d’+b’c’d’ 
f = xy+x’y’, x = ab, y = c+d
2. Extraction (multiple functions)
f = (az+bz’)cd+e g = (az+bz’)e’ h = cde
f = xy+e, g = xe’, h = ye, x = az+bz’, y = cd
3. Factoring (series-parallel decomposition)
f = ac+ad+bc+bd+e
f = (a+b)(c+d)+e
Logic Synthesis 2

10. Structural Operations, cont’d.
4. Substitution
g = a+b f = ac+bc + d 
f = gc+d
5. Collapsing (elimination)
f = ga+g’b g = c+d
f = ac+ad+bc’d’
Note: algebraic division plays a key role in all these algorithms:
given function f, find g, such that
f = g h, where support(g)  support(h) = 
Logic Synthesis 2

11. Functional Decomposition (multi-level minimization)
Classical Asenhurst-Curtis decomposition, (based on decomposition charts)
A-C decomposition using BDDs
BDD-based bi-decomposition (BDS)

12. Overview The concept of functional decomposition
Two uses of BDDs in synthesis
as a computation engine to implement algorithms
as a representation that helps find decompositions
Two ways to direct decomposition using BDDs
bound set on top (Lai/Pedram/Vardhula, DAC’93)
free set on top (Stanion/Sechen, DAC’95)
other approaches
Disjoint and non-disjoint decomposition
Bi-decomposition
Logic Synthesis 2

13. Two-Level Curtis Decomposition
F(X) = H( G(B), A ), X = B  A
B = bound set A= free set F G H A B F X
if B  A = , this is disjoint decomposition
if B  A  , this is non-disjoint decomposition
Logic Synthesis 2

14. Decomposition Types F H A G B B F G H A F G H A B
Simple disjoint decomposition (Asenhurst) F H A G B B F G H A F G H A B
Disjoint decomposition (Curtis)
Non-disjoint decomposition
Logic Synthesis 2

15. Incompatibility Graph
Decomposition Chart
Bound Set = {a,b} 3 1 4 2
Incompatibility Graph
 =2 G G
Free Set = {c,d} 10 1 11 01 00 1 2 3 4
Definition 1: Column Compatibility
Two columns i and j are compatible if each element in i is equal to the corresponding element in j, or the element in either i or j is not specified
Definition 2: Column Multiplicity  = the number of compatible sets (distinct column patterns)
Logic Synthesis 2

16. Fundamental Decomposition Theorems
Theorem (Asenhurst)
Let k be the minimum number of compatible sets in the decomposition chart. Then function H must distinguish at least k values
Theorem (Curtis)
Let  (A | B) denote column multiplicity under decomposition into the bound set B and free set A. Then:
 (A | B)  2k  F(B,A) = H(G1(B), G2(B), …, Gk(B), A) F G H B A k
Logic Synthesis 2

17. Asenhurst-Curtis Decomposition1 2 10 11 01 00
Bound Set = {a,b}
Free Set = {c,d}
Here = 2, function H must
distinguish two values
need k=1 bit to encode outputs of G F G H a b c d
F(a,b,c,d) = (ab+ ab)c'+ (ab+ ab)(cd+cd)
G(a,b)= ab+ab H(G,c,d) = Gc+ G(cd+cd)
Logic Synthesis 2

18. Multi-Level Curtis Decomposition
Two-level decomposition is iteratively applied to new functions Hi and Gi, until smaller functions Gt and Ht are created, that are not further decomposable.
One of the possible cost functions is Decomposed Function Cardinality (DFC). It is the total cost of all blocks, where the cost of a binary block with n inputs and m outputs is m * 2n.
Logic Synthesis 2

19. Typical Decomposition Algorithm
Find a set of partitions (Bi, Ai) of input variables X into bound set variables Bi and free set variables Ai
For each partition, find decomposition F(X) = Hi (Gi(Bi ), Ai )
such that the column multiplicity is minimal, and compute DFC (disjoint function cardinality).
Repeat the process for all partitions until the decomposition with minimum DFC is found.
Logic Synthesis 2

20. Using BDD to direct Decomposition
Problem with classical AC decomposition:
Need one decomposition chart for each decomposition
Computationally expensive!
Alternatively, the algorithm may exploit the BDD structure of the function F to direct the decomposition:
in the bound set selection,
column multiplicity computation, and
deriving the decomposed functions G and H
Logic Synthesis 2

21. Binary Decision Diagram (BDD) - review
f = ab+a’c+a’bd
Graph representation of a Boolean function
- vertices represent decision nodes for variables
- two children represent the two subfunctions
f(x = 0) and f(x = 1) (cofactors)
- restrictions on ordering and reduction rules
can make a BDD representation canonical c a b d 1
c+bd
root
node
c+d 1
Logic Synthesis 2

22. BDD-Based Decomposition
Disjoint decomposition: based on BDD cut that separates
bound set from free set
Bound set on top (Lai/Pedram/Vardhula, DAC’93)
Free set on top (Stanion/Sechen, DAC’95)
Bi-decomposition using the concept of dominators
BDS system (Yang/Ciesielski, ICDD’99)
Recursive decomposition
(Bertacco/Damiani,ICCAD’97)
Implicit decomposition
(Wurth/Eckl/Legl,DAC’95)
Logic Synthesis 2

23. BDD-based Functional Decomposition
Functional decomposition (AC): F(X) = H (G(B ), A )
Find a BDD cut , dividing variables into disjoint sets:
bound set B
free set A F B A
BDD
cut
free set
bound set F G H B A
Logic Synthesis 2

24. Bound Set on Top Algorithm - Function G
Free Set 1 A B C
G={g0,g1}, A=g0g1, B=g0g1, C=g0g1
g0=abc+abc+abc, g1 = abc+ abc
= 1
= 0
Logic Synthesis 2

25. Bound Set on Top Algorithm - Function H
Free Set A B C 1
F(a,b,c,d,e) = H( g1(a,b,c), g2(a,b,c), d, e ) H=g0g1e + g0g1d + g0g1e
= 1
= 0
Logic Synthesis 2

26. AC Decomposition Algorithm (bound set on top)
Reorder variables in BDD for F and check column multiplicity for each bound set
For the bound set with the smallest column multiplicity, perform decomposition :
Encode the cut nodes with minimum number of bits (log )
derive functions G and H (both depend on encoding)
Iteratively repeat the process for functions G and H (typically, only H)
Note: This is an ALGEBRAIC decomposition
The algorithm work only for disjoint decompositions: the variables in the bound set and the free set are disjoint !
Logic Synthesis 2

27. Free Set on Top Algorithm - Function G
B=10
Free Set
Bound Set g1 g2 F 1
Bound Set
Is it the same function as for the “bound set on top” ? – NO, look at the truth table.
G={g1,g2}, g1=cde+cd, g2=d+e
= 1
= 0
Logic Synthesis 2

28. Free Set on Top Algorithm - Function H
Bound Set g1 g2
Bound Set
B=10
A=00
C=01 1
F(a,b,c,d,e) = H( a, b, g1(c,d,e), g2(c,d,e) )
H=(ab+ ab) g1 + (ab+ ab) g2
= 1
= 0
Logic Synthesis 2

29. Free Set on Top Algorithm
Find good variable order
Derive implicit representation of all feasible cuts on the BDD representing F
Use some cost function to find the best bound set and perform decomposition
Repeat the process for functions G and H
This algorithms is faster than “bound set on top” but it does not work for non-disjoint decompositions and incompletely specified functins (with DCs).
Logic Synthesis 2

30. Non-Disjoint Decomposition
Non-disjoint decomposition can be reduced to disjoint decomposition by adding variables
Bound Set = {a,b,c}, Free Set = {c,d}
Disjoint decomposition can be generated by introducing variables c1=c2=c instead of c
In terms of the Karnaugh map, it is equivalent to introducing two variables instead of one in such a way that c1c2 +c1c2 is a don’t care set.
Why: c1  c2  c1c2 +c1c2
Logic Synthesis 2

31. Non-Disjoint Decomposition ExampleB C B A A B A B B B A
There is no disjoint decomposition with any bound set.
But there exists a non-disjoint decomposition with bound set {a,b,c}
Logic Synthesis 2

32. Non-Disjoint Bi-Decomposition
There is a better,more systematic way to perform non-disjoint decomposition without adding variables
Bi-decomposition: F = D  Q
uses BDD cut , but there is no concept of bound set / free set
The top set defines a divisor D
The bottom set defines the quotient Q (sort of …) F D
F/D
BDD
cut F D  Q
Boolean operator
(AND, OR, XOR)
Logic Synthesis 2