a modern C++ library for the manipulation of Boolean functions

4.2 Algorithms with composite representations
4.2.1 Normalization
All algorithms in this section will be described only for the case when entailed, disentailed
and equivalent variables are all represented explicitly.
The first and most important algorithm is the normalization algorithm, which allows
us to transform an internal representation where conditions 4.1, 4.2, 4.3 and 4.4 are not
necessarily met to one where they are all met without changing the semantics given by
Equation 4.5. Given this algorithm, all operations on composite representations (with
an arbitrary number of arguments) can be implemented, once we have a corresponding
implementation on ROBDDs, by this simple algorithm:
Denormalization Denormalize the representation, which means moving, for all the arguments
of the operator, all informations on entailed, disentailed and equivalent
variables that are stored in E, D and L into the ROBDD n.
Application Apply the corresponding operator on ROBDDs to the denormalized ROBDDs
computed in the previous step.
Normalization Normalize the ROBDD computed in the previous step, which means
moving all informations on entailed, disentailed and equivalent variables that are
stored in the ROBDD into three separate components as described previously.
Of course this trivial implementation is unacceptable, as it goes against the principle of
keeping the ROBDDs as small as possible when applying an operator on them in order to
improve efficiency. That is why we will present better implementations for these operators
right after describing our normalization algorithm.
The result of the application of the normalization function on the composite representation
〈n, E, D, L〉 will be denoted as η(〈n, E, D, L〉).
4.2.2 Implementing the normalization algorithm
CORAL’s generic normalization algorithm is an adaptation of an old normalization
algorithm that was a part of China and is described in [BS98]. It consists of these steps,
which must be applied repeatedly until a fixpoint is reached:
Step 1 If n = 0 the final result of the algorithm is ⊥.
Step 2 Look for all the equivalence classes in L that contain at least one variable in E
and add all their variables to E. Do the same for D.
Step 3 Remove all the equivalence classes that were found in the previous step from L.
Step 4 Apply the elim_true_vars algorithm on n with E as its argument, thus eliminating
all variables in E from n.
4.2 Algorithms with composite representations 37