1. A Boolean Paradigm in Multi-Valued Logic Synthesis
Alan Mishchenko
Electrical and Computer Engineering
Portland State University
Robert K. Brayton
Electrical Engineering and Computer Science
University of California, Berkeley
June 5, 2002

2. IWLS 2002, New Orleans, Louisiana
Overview
Motivation
Background
Optimization Algorithms
Node simplification
Partial encoding
Decomposition
Resubstitution
Common logic extraction
Case study: Decomposition and SAT
Conclusions
November 16, 2018
IWLS 2002, New Orleans, Louisiana

3. Algebraic vs. Boolean Algorithms
Most of the currently used logic synthesis algorithms are algebraic (do not use Boolean identities such as a  a = a and a  a’ = 0)
sub-optimal quality, fast
Boolean algorithms are also known but seldom used
better quality, slower
There is no unified picture of the Boolean algorithms for logic synthesis
November 16, 2018
IWLS 2002, New Orleans, Louisiana

4. Goals of the Present Work
Develop Boolean algorithms working on MV relations
Explore interdepen-dence and common computational core of the Boolean algorithms
This study is possible because of the vantage point of multi-valued logic optimization
November 16, 2018
IWLS 2002, New Orleans, Louisiana

5. Optimization Algorithms
Node simplification
Partial (value reducing) encoding
Decomposition
Resubstitution
Common logic extraction
November 16, 2018
IWLS 2002, New Orleans, Louisiana

6. IWLS 2002, New Orleans, Louisiana
Node Simplification
Compute and use complete flexibility (CF) to simplify the node:
CF in global space: R(X, yi) = Z [ R(X, yi, Z)  R(X, Z) ]
CF in local space: R(Y, yi) = X [ M(X, Y)  R(X, yi) ]
Use R(Y, yi) to optimize MV-SOP (heuristic, exact)
November 16, 2018
IWLS 2002, New Orleans, Louisiana

7. Partial (Value Reducing) Encoding
R2 is a wire, a cube, or a given function
Transformation is accepted if
log2 |v2| + log2 |v1|  log2 |v|
|v2|+ |v1| |v|
November 16, 2018
IWLS 2002, New Orleans, Louisiana

8. Decomposition and Encoding
Select bound set XB
Compute the CF of block B1 assuming v1 is the product of values in XB (column compatibility is not used)
Encode B1 using value-reducing encoding to get B2
Inputs in XC are shared, which leads to non-disjoint decomposition
November 16, 2018
IWLS 2002, New Orleans, Louisiana

9. IWLS 2002, New Orleans, Louisiana
Resubstitution
Compute R(Y, y), the complete flexibility of 
Find node R3(Y3, y3), Y3 Y, y3  Y
Substitute R3(Y3, y3) into R(Y, y): Rs(Ys, y) = R(Y, y)  R3(Y3, y3), Ys = Y  y3
Minimize Rs(Ys, y) to get Rr(Yr, y), |Yr| ≤ |Ys|
Accept transformation if Rr(Yr, y) reduces the cost of 
November 16, 2018
IWLS 2002, New Orleans, Louisiana

10. Common Logic Extraction
Get several functionally related nodes
Merge the nodes and derive complete flexibility, R, of the merged node
Look at cofactors appearing at a window of levels in the BDD of R
Find a set of compatible cofactors and express them as an MV relation
Try the MV relation as a Boolean divisor of the original nodes
November 16, 2018
IWLS 2002, New Orleans, Louisiana

11. Case Study: Decomposition and SAT
Ashenhurst-Curtis decomposition of multi-valued relations
R(A,B,v) = H( G(A,v), B, v )
disjoint-support: A  B = 
support-reducing: log2|range(G)|  log2|A|
Multi-valued SAT
Multi-valued input, binary output CNF formula
Each clause is composed of MV literals
An assignment of MV variables satisfies the CNF formula, if in each clause, there is at least one literal with a value belonging to the assignment
November 16, 2018
IWLS 2002, New Orleans, Louisiana

12. IWLS 2002, New Orleans, Louisiana
SAT Variables
n variables ci, 1  i  n, to encode the coloring of columns. Value j belongs to the value set of variable ci, if the corresponding column, i, can be colored with the color j. The range of ci is n/2.
m variables dij, one for each entry in the table that can take more than one value. These variables represent subsets of original values Sij, which are in agreement with the selected decomposition. The range is the range  of the relation.
November 16, 2018
IWLS 2002, New Orleans, Louisiana

13. IWLS 2002, New Orleans, Louisiana
SAT Clauses
Containment of the selected values in the original values of the cells in the decomposition chart
dij Sij, i,j: 0  i < n, 0  j < n
Coloring is compatible with the selected cell values
November 16, 2018
IWLS 2002, New Orleans, Louisiana

14. Simplifications of SAT Problem
No need to introduce variables dij and ci:
If a column is incompatible with other columns
If two columns can only be compatible with each other
No need to introduce variable dij:
If the corresponding cell in the decomposition chart has only one value.
Using an efficient SAT solver, it may be possible to solve the decomposition problem for nodes with 8-12 input variables
November 16, 2018
IWLS 2002, New Orleans, Louisiana

15. IWLS 2002, New Orleans, Louisiana
Conclusions
Boolean paradigm
Boolean operations are applied to the MV relation representing the complete flexibility of a node derived from the network structure
An efficient implementation may be possible due to common computational cores (BDDs and SAT) and recently improved algorithms
A more general point of view reveals underlying relationships among the procedures
November 16, 2018
IWLS 2002, New Orleans, Louisiana

16. Conclusions (continued)
A synthesis flow based of these methods may lead to the improvements in the optimization quality because
The use of multi-valued logic leads to searching a larger space of solutions.
Boolean (not only algebraic) properties of nodes are exploited.
The complete sets of don’t-cares (partial cares) give greater flexibility for optimizing the nodes of the network.
The SAT-based formulation is not limited to one particular encoding or coloring.
Functional decomposition can be performed concurrently with technology mapping (similar to Kravets and Sakallah, DAC’98).
November 16, 2018
IWLS 2002, New Orleans, Louisiana

17. IWLS 2002, New Orleans, Louisiana
A Boolean Paradigm in Multi-Valued Logic Synthesis Alan Mishchenko, ECE Dept, Portland State University Robert K. Brayton, EECS Dept, UC Berkeley
The goal of this work:
Develop Boolean algorithms working on MV relations
Explore interdependence and common computational core of Boolean algorithms
This study is possible because of the vantage point of multi-valued logic optimization
Optimization algorithms considered:
Node simplification
Partial encoding
Decomposition
Resubstitution
Common logic extraction
Case study: Decomposition and SAT
November 16, 2018
IWLS 2002, New Orleans, Louisiana