Introduction to the OBDD algorithm for the ATP community

Abstract

We describe in terms familiar to the automated reasoning community the graph-based algorithm for deciding propositional equivalence published by R. E. Bryant in 1986. Such algorithms, based onordered binary decision diagrams orOBDDs, are among the fastest known ways to decide whether two propositional expressions are equivalent and are generally hundreds or thousands of times faster on such problems than most automatic theorem-proving systems. An OBDD is a normalized IF (‘if-then-else’) expression in which the tests down any branch are ascending in some previously chosen fixed order. Such IF expressions represent a canonical form for propositional expressions. Three coding tricks make it extremely efficient to manipulate canonical IF expressions. The first is that two canonicalized expressions can be rapidly combined to form the canonicalized form of their disjunction (conjunction, exclusive-or, etc.) by exploiting the fact that the tests are ordered. The second is that every distinct canonical IF expression should be assigned a unique integer index to enable fast recognition of identical forms. The third trick is that the operation in which one combines canonicalized subterms term should be ‘memo-ized’ or cached so that if the same operation is required in the future its result can be looked up rather than recomputed.