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Binary decision diagrams (BDD’s)

a. E. T. b. b. T. T. E. E. c. c. E. E. d. T. T. T. E. 1. 0. Binary decision diagrams (BDD’s). Compact representation of a logic function ROBDD’s (reduced ordered BDD’s) are a canonical representation: equivalence of ROBDD’s implies that the functions are identical

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Binary decision diagrams (BDD’s)

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  1. a E T b b T T E E c c E E d T T T E 1 0 Binary decision diagrams (BDD’s) • Compact representation of a logic function • ROBDD’s (reduced ordered BDD’s) are a canonical representation: equivalence of ROBDD’s implies that the functions are identical • Example: f = abc + b’d + c’d • T = then edge, E = else edge • Same variable ordering on each path: a  b  c  d (Ordered BDD) Material taken mostly from G. Hachtel and F. Somenzi, “Logic Synthesis and Verification Algorithms,” Kluwer Academic Publishers, Boston, MA, 1996.

  2. Effect of variable ordering • Size of diagram varies with variable ordering a  d  b  c b  c  a  d b a E T E T d d c E T E T T b b E a E E E T T d c c T T E E T T E 1 0 1 0

  3. a E T b b T T E E c c E E d T T T E 1 0 Relation to the Shannon expansion • Each node is basically a Shannon expansion • f = a fa + a’fa’ f fa’ fa

  4. Building a BDD from a Shannon expansion f = abc + b’d + c’d b T E fb = ac + c’d c fbc’ = d T E fb’ = d fbc = a (… and so on …)

  5. a b d c d a a d c d d b d c c BDD as a compact truth table • Truth table: complete ordered binary tree • Reduce this by combining isomorphic parts and removing redundant nodes (T,E point to same node) to get ROBDD redundant T E T E T E T E T T E T E T E E T T redundant E T 1 0 T E T E T E T E 1 1 1 0 0 0 T E T E BDD shown earlier for the ordering b  c  a  d isomorphic 1 1 0 0 isomorphic

  6. Multioutput BDD’s • (Notation: solid line = Then edge; dashed line = Else edge) • Example: F1 = b+c, F2 = a+ b+c Combined Multioutput BDD Separate BDD’s F2 F1 F2 F1 a a b b b c c c 1 0 1 0 1 0

  7. Compactness of BDD’s • BDD’s are successful at compactly representing many common functions • XOR is an example of a function with a large SOP/POS representation, but a very compact BDD • Worst case: O(2n) nodes • Functions that require this many nodes do exist • Can use multilevel techniques to represent BDD’s more compactly (“partitioned ROBDD’s”)

  8. Operations on BDD’s • Given BDD’s for functions f and g, can use Shannon expansion to see how operations are performed • Assume variables v1, v2 … vn • f <op> g = v1 (f <op> g)v1 + v1’ (f <op> g)v1’ = v1 (fv1 <op> gv1) + v1’ (fv1’ <op> gv1’) • Can now do this recursively • Pictorially: • Identify identical subtrees as we come up the recursion tree using a hashing function v1 T E (BDD for fv1 <op> gv1) (BDD for fv1’ <op> gv1’)

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