Ordered Binary Decision Diagrams and Minimal Trellises

• Authors: John Lafferty;Alexander Vardy
• Venue: IEEE Transactions on Computers
• Year: 1999

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 ${\cal C}$ be a binary code of length $n$, and let $f_{\cal C}(x_1,\ldots,x_n)$ be the Boolean function that takes the value $0$ at $x_1,\ldots,x_n$ if and only if $(x_1,\ldots,x_n) \in {\cal C}$. Given this natural one-to-one correspondence between Boolean functions and binary codes, we prove that the minimal proper trellis for a code ${\cal C}$ with minimum distance $d 1$ is isomorphic to the single-terminal OBDD for its Boolean indicator function $f_{\cal C}(x_1,\ldots,x_n)$. Prior to this result, the extensive research during the past decade on binary decision diagrams驴in computer engineering驴and on minimal trellises驴in coding theory驴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.