Online BDD Refinement

Binary Decision Diagrams are in widespread use in veriication systems for the canonical representation of Boolean functions. A BDD representing a function ' : B ! N can easily be reduced to its canonical form in linear time. In this paper, we consider a natural online BDD reenement problem and show that it can be solved in O(n log n) if n bounds the size of the BDD and the total size of update operations. We argue that BDDs in an algebraic framework should be understood as minimal xed points superimposed on maximal xed points. We propose a technique of controlled growth of equivalence classes to make the minimal xed point calculations be carried out eeciently. Our algorithm is based on a new understanding of the interplay between the splitting and growing of classes of nodes. We apply our algorithm to show that automata with exponentially large, but implicitly represented alphabets, can be minimized in time O(n log n), where n is the total number of BDD nodes representing the automaton.