Applications of Binary Decision Diagrams in Logic Synthesis, Verification, and Testing

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
Overview

• Motivation
• Background
• Optimization Algorithms
• Node simplification
• Partial encoding
• Decomposition
• Resubstitution
• Common logic extraction
• Case study Decomposition and SAT
• Conclusions

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

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

5
Optimization Algorithms

• Node simplification
• Partial (value reducing) encoding
• Decomposition
• Resubstitution
• Common logic extraction

6
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)

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

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

9
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 ?

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

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 log2range(G) ? log2A
• 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

12
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.

13
SAT Clauses

• Containment of the selected values in the
  original values of the cells in the decomposition
  chart
• dij ?Sij, ?i,j 0 ? i lt n, 0 ? j lt n
• Coloring is compatible with the selected cell
  values

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

15
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

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 dont-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, DAC98).

17
A Boolean Paradigm in Multi-Valued Logic
SynthesisAlan Mishchenko, ECE Dept, Portland
State University Robert K. Brayton, EECS Dept,
UC Berkeley

• Optimization algorithms considered
• Node simplification
• Partial encoding
• Decomposition
• Resubstitution
• Common logic extraction
• Case study Decomposition and SAT

• 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