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 for TAOCP, Volume 4, Section 7.1.2 “Boolean evaluation”)

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

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

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 n Boolean variables, e.g. n=4, {a,b,c,d} Minterm: a complete assignment of a set of n Boolean variables, e.g. {a,b,c,d} = 0011 The total number of minterms = 2n Boolean function: a complete assignments of all 2n minterms in terms of the set of variables The total number of functions = 22^n Truth table: a simple representation of Boolean function, e.g. 0001 0001 0001 1111 = 0x111F

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)

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

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)

Bottom-Up Enumeration: L(f) This is the reformulated 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 the constant 0 function and the 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 all 4-input functions Takes 200 sec for all 5-input functions Currently infeasible for all 6-input functions

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, in discussion and results, we focus on L(f)

Min-Circuit Classes of 4-Functions

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

Computing L(f) for 5-Functions This table is generated using command “funenum –I 5 –l” in ABC.

Computing D(f) for 5-Functions This table is generated using command “funenum –I 5 –d” in ABC.

Computing Cm(f) for 5-Functions This table is generated using command “funenum –I 5 –S 5” in ABC.

Min-Circuit Types for 5-Functions

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 two 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

Distribution of Min-Circuit Counts for 4-Functions A column label is a range of counts of min-circuits. For example, [5;8] means that there are 5, 6, 7 or 8 different min-circuits.

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.

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) and call for improving exact NPN classification for 6-functions Extend all results to (a subset of simple) 6-functions Requires improving exact NPN classification for 6-functions Find good practical applications

Conclusion Introduced the problem of min-circuit enumeration Presented statistics for several types of min-circuits for 4- and 5-functions independently verified the information from TAOCP by Knuth Showed the distribution of min-counts of different L(f) min-circuits for 4- and 5-functions new result Plan to produce the actual implementation of all min-circuits for 4- and 5-functions, and a practical subset of those for 6-functions

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