Ordered binary decision diagrams and minimal trellises

Abstract

Ordered binary decision diagrams (OBDDs) are graph-based data structures for representing Boolean functions. They have found widespread use in computer-aided design and in formal verification of digital circuits. Minimal trellises are graphical representations of error-correcting codes that play a prominent role in coding theory. This paper establishes a close connection between these two graphical models, as follows. Let /spl Cscr/ be a binary code of length n, and let f/sub c/(x/sub 1/, ..., x/sub n/) be the Boolean function that takes the value 0 at x/sub 1/, ..., x/sub n/ if and only if (x/sub 1/, ..., x/sub n/)/spl isin//spl Cscr/. Given this natural one-to-one correspondence between Boolean functions and binary codes, we prove that the minimal proper trellis for a code /spl Cscr/ with minimum distance d>1 is isomorphic to the single-terminal OBDD for its Boolean indicator function f/sub c/(x/sub 1/, ..., x/sub n/). Prior to this result, the extensive research during the past decade on binary decision diagrams (in computer engineering) and on minimal trellises (in coding theory) has been carried out independently. As outlined in this work, the realization that binary decision diagrams and minimal trellises are essentially the same data structure opens up a range of promising possibilities for transfer of ideas between these disciplines.