1. Using Enumeration to Generate All Minimum Circuits for Boolean Functions of 4 and 5 Variables
Alan Mishchenko
UC Berkeley
(With many thanks to Donald Knuth,
TAOCP, Volume 4, Section “Boolean evaluation”)

2. Overview Small Boolean functions (n-functions where n < 6)
Minimum circuits (min-circuits)
Enumeration algorithms
Exhaustive enumeration of min-circuits
Experimental results
Conclusions and future work
Applications? 2

3. Two Different 12-Node Min-Circuits for The Most Complex 5-Function: 169ae443

4. Terminology Boolean value: 0 or 1 Boolean variable: x  {0,1}
abcd F
0000
0001
0010
0011 1
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Boolean value: 0 or 1
Boolean variable: x  {0,1}
A set of Boolean n variables, e.g. n=4, {a,b,c,d}
Minterm: a complete assignment of a set of Boolean variables, e.g. {a,b,c,d} = 0011
The total number of minterms = 2^n
Boolean function: a complete assignments of all minterms in terms of the set of variables
The total number of functions = 2^(2^n)
Truth table: a compact representation of Boolean function, e.g

5. NPN Classes
Two functions belong to the same NPN class if one of them can be derived from the other by complementing inputs (N), permuting inputs (P), and complementing the output (N)

6. Min-Circuit Types Using terminology introduced in TAOCP by Knuth
C(f): the minimum-node DAG (the most general type)
L(f): the minimum-node leaf-DAG (fanout only at inputs)
U(f): the minimum-node DAG with logic sharing (fanout) only at inputs and the first logic level
D(f): the minimum-level circuit
DN(f): the node count in the minimum-level circuit
Cm(f): the minimum-node width-n DAG (“minimum memory”)
Cm’(f): the same as Cm(f) with one extra generation rule
Cm1(f): the minimum-node width-(n+1) DAG

7. Min-Circuit Enumeration Methods
Bottom-up
Start with inputs, add gates – leads to L(f)
Top-down
Start with the output, add gates – leads to Cm(f)
Hybrid
Hard to define in an intuitive way – leads to C(f)

8. Bottom-Up Enumeration: L(f)
This is Algorithm L from TAOCP by Knuth, modified to work with NPN classes rather than individual functions
Maintain a hash table of reached NPN classes
Initialize it with constant 0 function and buffer function
Consider pairs of classes whose min-circuits have sizes n1 and n2
Add on top one two-input gate (AND/XOR) in five polarities:
The polarities are: f1 & f2, f1 & ~f2, ~f1 & f2, ~f1 & ~f2, f1  f2
(One of the classes should be exhaustively permuted/complemented)
Check each of the resulting functions among the reached classes
If it is there, skip it
If not, add the new class and set its min-circuit to have size n1+n2+1
Stop when all NPN classes have been reached
Runtime
Takes 0.01 sec for 4-input functions
Takes 200 sec for 5-input functions
Currently infeasible for 6-input functions

9. Other Enumeration Types
The most general one, C(f), leads to the best quality but is the hardest to implement
The top-down one, Cm1(f), is easy to implement but runs into problems with NPN classification (because it requires n+1 variable NPN classes)
L(f) is a good trade-off between quality, runtime, and the ease of implementation
Below we focus on L(f) is discussion and results

10. Min-Circuit Classes of 4-Functions

11. Min-Circuit Classes for 4-Functions (only AND, no XOR)
It is interesting to note that L(f) here has more classes with an odd number of nodes than with an even number of nodes!

12. Computing L(f) for 5-Functions

13. Computing D(f) for 5-Functions

14. Computing Cm(f) for 5-Functions

15. Min-Circuit Types for 5-Functions

16. Counting Min-Circuits
The focus of this presentation is on
counting the number of *different* min-circuits for each NPN class (finished)
generating all *different* min-circuits (not finished)
What min-circuits are considered different?
A min-circuit is characterized by the unordered pairs of NPN classes of fanins of the topmost node
Two circuits are different, if at least one fanin of the topmost node of the first circuit has NPN class that is not present among the fanins of the second circuit

17. Distribution of Min-Circuit Counts for 4-Functions

18. Distribution of Min-Circuit Counts for 5-Functions
It is interesting to note that among all 11-node NPN classes, there are 5450 classes with a unique min-circuit and one class with exactly 523 different min-circiuts. The next closest to it, is one class with exactly 314 different min-circuits.

19. Future Work Generate all min-circuits for 4- and 5-functions
Finish computation of Cm1(f) for 5-functions
This computation is based on Frontier-Based Search (Knuth)
Extend all results to (a subset of simple) 6-functions
Requires improving exact NPN classification for 6-functions
Find good practical applications

20. Conclusion Introduced the problem of min-circuit enumeration
Presented statistics for all min-circuit types for 4- and 5-functions (most of the info available in TAOCP by Knuth)
Showed distribution of min-circuits of L(f) (new result)

21. Four 4-Functions That Require 10 AND Nodes (XORs not used)
This slide is borrowed from presentation “Practical SAT” by Niklas Een