Counter Examples to the Conjecture on the Complexity of BDD Binary Operations

(Abstract) In this article, we disprove the long-standing conjecture, proposed by R. E. Bryant in 1986, that any binary operation on two Boolean functions can be performed by his BDD algorithm in input-output linear time. We present Boolean functions for which his algorithm requires quadratic time in the input-output size for any non-trivial binary operation such as ∧, ∨, and ⊕. For the operations ∧ and ∨, we show a further stronger counterexample such that the output BDD size becomes only a constant but computation time is still quadratic in the input BDD size. We also present experimental results to support our theoretical observations.