1. Fast Computation of Symmetries in Boolean Functions Alan Mishchenko
Electrical and Computer Engineering
Portland State University
February 8, 2002

2. Seminar at Cadence Berkeley Lab
Overview
Motivation
Symmetry types
Known symmetry detection algorithms
The new algorithm
Computational core
Specialized operators
Implementation
Experimental results
Conclusions
November 28, 2018
Seminar at Cadence Berkeley Lab

3. Seminar at Cadence Berkeley Lab
Motivation
In logic synthesis, if a variable group is symmetric, it can be easily decomposed out.
In technology mapping, symmetries allow for fast boolean matching.
In BDD minimization, SAT solving, etc, symmetries limit the search space.
In verification, symmetries can be used as invariants.
Any other applications?
November 28, 2018
Seminar at Cadence Berkeley Lab

4. Seminar at Cadence Berkeley Lab
Symmetry Types
Classical two-variable symmetry
Four basic two-variable symmetry types
Edward/Hurst (1978)
Tsai/Marek-Sadowska (1996)
Generalized two-variable symmetries
Research group at PSU ( )
Generalized multiple-variable symmetries
Mohnke/Molitor/Malik (1995)
Kravets/Sakallah (2000)
November 28, 2018
Seminar at Cadence Berkeley Lab

5. Classical Two-Variable Symmetry
The function F has a non-skew non-equivalence symmetry in variables xi and xj iff
November 28, 2018
Seminar at Cadence Berkeley Lab

6. Four Basic Two-Variable Symmetries
Non-skew non-equivalence symmetry
Non-skew equivalence symmetry
Skew non-equivalence symmetry
Skew equivalence symmetry
November 28, 2018
Seminar at Cadence Berkeley Lab

7. Detecting Classical Symmetries
Naïve approach: For each variable pair, derive the cofactors and compare them
November 28, 2018
Seminar at Cadence Berkeley Lab

8. Detecting Classical Symmetries
A better approach: Start by detecting as many non-symmetric pairs as possible
Idea 1: Check if and
Idea 2: Check if or depends on xj; and vice versa
Idea 3: Check if it is true that the cofactor sets for xi and xj in the shared BDD are the same
D.Moller et al. “Detection of Symmetry of Boolean Functions Represented by ROBDDs”. ICCAD’93
November 28, 2018
Seminar at Cadence Berkeley Lab

9. Naïve vs. Better Symmetry Detection
D.Moller et al. “Detection of Symmetry of Boolean Functions Represented by ROBDDs”. ICCAD’93
November 28, 2018
Seminar at Cadence Berkeley Lab

10. Seminar at Cadence Berkeley Lab
Disadvantages
Requires multiple BDD traversals
Uses naïve method to check a relatively large number of cofactors
Takes long time for functions that have no symmetries at all
Detects only one type of “classical” two-variable symmetry
November 28, 2018
Seminar at Cadence Berkeley Lab

11. New Symmetry Detection Algorithm
Overcomes the above disadvantages
Requires only one traversal
Does not use the naïve checking method
Quickly detects functions w/o symmetries
Detects four basic symmetry types
Achieves significant speed-up
Can be extended to other symmetry types
November 28, 2018
Seminar at Cadence Berkeley Lab

12. Seminar at Cadence Berkeley Lab
Symmetry Graphs
Symmetries form a symmetry graph
Vertices correspond to variables
Edges correspond to pair-wise symmetries
Graph operations
Union () and intersection () of graphs is the graph with the union or intersection of all edges
The product () of a variable by a variabe set is a graph with edges connecting the variable with all variables in the set
November 28, 2018
Seminar at Cadence Berkeley Lab

13. Generic Recursive Algorithm
solution RecursiveProcedure( problem P ) {
Step 1: If P is a trivial case, return the solution.
Step 2: Cofactor P w.r.t. a variable into P0 and P1.
Step 3: Get solutions, S0 and S1, of P0 and P1.
Step 4: Derive S, the solution of P, from S0 and S1.
Step 5: Return S. }
November 28, 2018
Seminar at Cadence Berkeley Lab

14. Symmetry Computation Core
symmgraph ComputeSymmetries( function F, varset V ) {
Step 1: if ( F is a constant function )
return CompleteGraph( V );
Step 2: x = Var( V );
( F0, F1 ) = Cofactors( F, x );
Step 3: RemainingVars = supp( F ) – x;
S0 = ComputeSymmetries( F0, RemainingVars );
if ( S0 =  ) S1 = ;
else S1 = ComputeSymmetries( F1, RemainingVars );
Step 4: Y = SymmetricVars( F0, F1, RemainingVars );
S2 = x  Y;
S3 = CompleteGraph( V – supp( F ) );
S = (S0  S1)  S2  S3;
Step 5: return R; }
November 28, 2018
Seminar at Cadence Berkeley Lab

15. Symmetry Computation Core
varset SymmetricVars( function G, function H, varset Y ) {
Step 1: if ( G = H ) return Y;
Step 2: x = Var( Y );
( G0, G1 ) = Cofactors( G, x );
( H0, H1 ) = Cofactors( H, x );
Step 3: R0 = SymmetricVars( G0, H0, Y – x );
if ( R0 =  ) R1 = ;
else R1 = SymmetricVars( G1, H1, Y – x );
Step 4: R = R0  R1;
if ( G1 = H0 )
R = R  x;
Step 5: return R; }
November 28, 2018
Seminar at Cadence Berkeley Lab

16. Seminar at Cadence Berkeley Lab
Computing Tuples
tuples Tuples( varset V, int i )
/* Tuples are the set of subsets of i elements out of V */ {
Step 1: if ( |V| < i )
return ;
if ( i = 0 )
return {};
Step 2: x = Var( V );
Step 3: S0 = Tuples( V – x, i );
S1 = Tuples( V – x, i –1);
Step 4: S = (add x to all subsets in S1)  S0;
Step 5: return S; }
November 28, 2018
Seminar at Cadence Berkeley Lab

17. Seminar at Cadence Berkeley Lab
Experimental Results
November 28, 2018
Seminar at Cadence Berkeley Lab

18. Notations Used in the Table
ins and outs – the number of input and outputs of the benchmarks
|BDD| – the number of nodes in the shared BDD after variable reordering
pairs – the number of pairs of symmetric variables, for all outputs
ratio – the ratio of the number of symmetric pairs to the number of all pairs
reading – time needed to read in the benchmark and build shared BDDs
naive – time of naïve method which derives and compares the cofactors
slow – time of our implementation of [1] based on BDD analysis
fast – time of the presented algorithm
[1], [2], [3] – time reported in the literature (see the next slide)
November 28, 2018
Seminar at Cadence Berkeley Lab

19. Seminar at Cadence Berkeley Lab
Methods Compared
[1] D. Möller, J. Mohnke, and M. Weber. Detection of Symmetry of Boolean Functions Represented by ROBDDs. Proc. Intl. Conf. Computer Aided Design, pp , November 1993.
[2] Ch. Scholl, D. Möller, P. Molitor, and R. Drechsler. BDD Minimization Using Symmetries. IEEE Trans. CAD, 18(2), pp , February 1999.
[3] C.-C. Tsai and M. Marek-Sadowska. Generalized Reed-Muller Forms as a Tool to Detect Symmetries. IEEE Trans. Computers, C-45(1), pp , Jan1996.
November 28, 2018
Seminar at Cadence Berkeley Lab

20. Seminar at Cadence Berkeley Lab
Conclusions
Defined two-variable symmetries
Reviews the known algorithms
Presented the new algorithm
Showed the new algorithm is at least 10 times faster than the known algorithms
November 28, 2018
Seminar at Cadence Berkeley Lab

21. Seminar at Cadence Berkeley Lab
Additional Slides
November 28, 2018
Seminar at Cadence Berkeley Lab

22. Generalized Symmetries (I)
Consider all cofactors of the function F w.r.t. two variables: F00, F01, F10, F11
Equal cofactor symmetries = classical symmetries
F01 = F10 (non-equivalent classical symmetry)
F00 = F11 (equivalent classical symmetry)
Other symmetries are derived from other relations between the cofactors
November 28, 2018
Seminar at Cadence Berkeley Lab

23. Generalized Symmetries (I)
November 28, 2018
Seminar at Cadence Berkeley Lab

24. Application: Regular Layout Synthesis
The upper part of the decision diagram of a function that has the classical non-equivalent symmetry (left) and equivalent symmetry (right).
November 28, 2018
Seminar at Cadence Berkeley Lab

25. Seminar at Cadence Berkeley Lab
W. Wang, M. Chrzanowska-Jeske. “Optimizing Pseudo-Symmetric Binary Decision Diagrams Using Multiple Symmetries”. IWLS'98.
November 28, 2018
Seminar at Cadence Berkeley Lab

26. Generalized Symmetries (II)
First-order symmetries = classical symmetries
Higher-order symmetries = symmetries that involve a hierarchy of variable groups
<group> means that the ordering of “group” is fixed
{group} means that the ordering of “group” is free
For example, function F = x(ab(c+d) + cd(a+b))
a second-order symmetry { <a,b>, <c,d> }
November 28, 2018
Seminar at Cadence Berkeley Lab

27. Seminar at Cadence Berkeley Lab
Computation
Theorem. Function F has symmetry {<s>,<t>} iff the cofactor matrix F<s>,<t> = [Fmi(s),mj(t) ] is symmetric: FT<s>,<t> = F<s>,<t>.
Example. Suppose F has symmetry {<a,b>,<c,d>}. Then FT<ab>,<cd>=F<ab>,<cd>
F<ab>,<cd> =
November 28, 2018
Seminar at Cadence Berkeley Lab

28. Higher-Order Symmetries in Benchmark Functions
V.Kravets, K.Sakallah. “Generalized Symmetries in Boolean Functions”. ICCAD’00.
November 28, 2018
Seminar at Cadence Berkeley Lab