Summary

Other BDD pictures for similar formulas:

• 
  BDD graph for the Boolean formula x1 * x2 + x3 * x4 + x5 * x6 + x7 * x8 using a bad variable ordering
• 
  Visualization of the BDD for the Boolean formula x1 * x2 + x3 * x4 + ... + x19 * x20 using a bad variable ordering

The following is the RML (Relational Manipulation Language) code that I fed to CrocoPat to produce the GraphViz dot files:

// RML program to generate BDD graphs for the formula
// x1 & x2 | x3 & x4 | x5 & x6 | x7 & x8,
// using two different variable orderings.
// "crocopat -e BDD_Variable_Ordering.rml" generates two files in dot format.
// "dot -Tsvg BDD_Variable_Ordering_Bad.dot -o BDD_Variable_Ordering_Bad.svg"
// generates a file in SVG format from the file in dot format.

// There are two ('Boolean') values for the variables x1, ..., x8.
DOM("0");
DOM("1");

// F is the name of the Boolean formula.
F(x1,x2,x3,x4,x5,x6,x7,x8) :=  (x1="1" & x2="1") 
                             | (x3="1" & x4="1")
                             | (x5="1" & x6="1")
                             | (x7="1" & x8="1");

// Prints the BDD as graph in GraphViz dot format,
// using a good variable ordering resulting in linear size of the graph.
PRINT GRAPH( F(x1,x2,x3,x4,x5,x6,x7,x8) ) 
      TO "BDD_Variable_Ordering_Good.dot";

// Prints the BDD as graph in GraphViz dot format,
// using a bad variable ordering resulting in exponential size of the graph.
// The first term of the conjunction sets the variable ordering.
PRINT GRAPH( TRUE(x1,x3,x5,x7,x2,x4,x6,x8) &
             F(x1,x2,x3,x4,x5,x6,x7,x8) ) 
      TO "BDD_Variable_Ordering_Bad.dot";;

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