# Notes

## Section 11: Traditional Mathematics and Mathematical Formulas

Binary decision diagrams

One can specify a Boolean function of n variables by giving a finite automaton (and thus a network) in which paths exist only for those lists of values for which the function yields True. The resulting so-called binary decision diagram (BDD) can be minimized using the methods of page 957. Out of all possible Boolean functions the number that require BDDs of sizes 1, 2, ... is for n = 2 : {1, 0, 6, 9} and for n = 3 : {1, 0, 0, 27, 36, 132, 60}; the absolute maximum grows roughly like 2n. For cellular automata with simple behavior, the minimal BDD typically grows linearly on successive steps. For rule 254, for example, it is 8t + 2, while for rule 90 it is 4t + 2. For cellular automata with more complex behavior, it typically grows roughly exponentially. Thus for rule 30 it is {7, 14, 29, 60, 129} and for rule 110 {7, 15, 27, 52, 88}. The size of the minimal BDD can depend on the order in which variables are specified; thus for example, just reflecting rule 30 to give rule 86 yields {6, 11, 20, 36, 63}.

In practical system design BDDs have become fairly popular in the past ten years, and by maintaining minimality when logical combinations of functions are formed, cases with millions of nodes have been studied. (Some practical systems are found to yield fairly small BDDs, while others are not.)

From Stephen Wolfram: A New Kind of Science [citation]