Rolf Drechlser’s slides used Davio Gates and Kronecker Decision Diagrams Lecture 5 Rolf Drechlser’s slides used

Absolutely Minimum Background on Binary Decision Diagrams (BDD) and Kronecker Functional Decision Diagrams BDDs are based on recursive Shannon expansion F = x Fx + x’ Fx’ Compact data structure for Boolean logic can represents sets of objects (states) encoded as Boolean functions Canonical representation reduced ordered BDDs (ROBDD) are canonical essential for simulation, analysis, synthesis and verification

BDD Construction Typically done using APPLY operator Reduction rules remove duplicate terminals merge duplicate nodes (isomorphic subgraphs) remove redundant nodes Redundant nodes: nodes with identical children a b f 1 b c

BDD Construction – your first BDD Construction of a Reduced Ordered BDD 1 edge 0 edge f = ac + bc 1 a b c f a b c f 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 Truth table Decision tree

BDD Construction – cont’d 1 a b c f f 1 a b c f = (a+b)c 1 a b c 1. Remove duplicate terminals 2. Merge duplicate nodes 3. Remove redundant nodes

Decomposition types Decomposition types are associated to the variables in Xn with the help of a decomposition type list (DTL) d:=(d1,…,dn) where di { S, pD, nD}

KFDD Definition

Three different reductions types Type I : Each node in a DD is a candidate for the application g f g f xi xi xi

Three different reductions types (cont’d) xi xj xj g f g f

Three different reductions types (cont’d) Type D xi This is used in functional diagrams such as KFDDs xj xj f f g g

Example for OKFDD 1 x1 F=

Example for OKFDD (cont’d) This diagram below explains the expansions from previous slide X1 S-node X2 pD-node X3 nD-node

Example for OKFDD’s with different DTL’s You can save a lot of nodes by a good DTL

Complement Edges They can be applied for S-nodes They can be applied for D-nodes This is a powerful idea in more advanced diagrams – any operator on representation

Such recursive operations are a base of diagram creation and operation XOR-operation D-node S-node Such recursive operations are a base of diagram creation and operation

Such recursive operations are a base of diagram creation and operation

Restriction of Variables for S nodes is trivial Xi=0 Xi=1 xi S-node g f f g Cofactor for Xi=1 Cofactor for Xi=0

Restriction of Variables for D nodes is more complex, requires XORing Xi=1 Xi=0 xi pD-node g f f Please remember this

Restriction of Variables (cond’t) Xi=1 Xi=0 xi nD-node g f f

Optimization of OKFDD-Size Exchange of Neighboring Variables DTL Sifting

Experimental Results

Experimental Results (cont’d)

DIY PROBLEM Use simulated annealing algorithm for choice of a better variable ordering of the OKFDD

End of Lecture 5 Simple solution Solution space Neighborhood structure X1 X2 X3 X4 X3 X4 X1 X2 A solution A solution Neighborhood structure X1 X2 X3 X4 X1 X3 X2 X4 change End of Lecture 5