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

Step 5 Apply the elim_false_vars algorithm on n with D as its argument, thus
eliminating all variables in D from n.
Step 6 Find all entailed variables in n using the algorithm entailed_vars and add
them to E.
Step 7 Find all disentailed variables in n using the algorithm disentailed_vars and
add them to D.
Step 8 If E ∩ D = ∅, the final result of the algorithm is ⊥.
Step 9 Find all equivalent variables in n by using the algorithm equivalent_vars, put
this information in an equivalence relation T and do the assignment L = L ∧ T .
Step 10 Eliminate all informations on equivalent variables from n by applying the
algorithm squeeze_equivalent_vars with T as its argument.
Step 11 Rename each variable in n to its leader in L by applying the rename algorithm.
This algorithm respects the loop invariant which states that after each iteration the
semantics given in Equation 4.5 is preserved. Pseudo-code for the normalization algorithm
can be found in Algorithm 22.
CORAL also implements a simpler, more efficient specialization of this algorithm for
positive functions; in this case, of course, we must not consider disentailed variables. A
pseudo-code for this specialized algorithm is shown in Algorithm 23.
4.2.3 Conjunction
The algorithms for conjunction and disjunction are an adaptation of the ones presented in
[BS98]. Given two (not necessarily normalized) composite representations 〈n1, E1, D1, L1〉
and 〈n2, E2, D2, L2〉, we compute their conjunction as:
〈n ′ 1, E ′ 1, D ′ 1, L ′ 1〉 = η(〈n1, E1 ∪ E2, D1 ∪ D2, L1 ∧ L2〉)
〈n ′ 2, E ′ 2, D ′ 2, L ′ 2〉 = η(〈n2, E1 ∪ E2, D1 ∪ D2, L1 ∧ L2〉)
〈n1, E1, D1, L1〉 ∧ 〈n2, E2, D2, L2〉 = η(〈n ′ 1 ∧ n ′ 2, E ′ 1 ∪ E ′ 2, D ′ 1 ∪ D ′ 2, L ′ 1 ∧ L ′ 2〉)
Notice that the initial collection of entailed, disentailed and equivalent variables allows
for greater reduction of the ROBDDs when normalization is called (and recall that the ∧
operator on equivalence relations represents the transitive closure of the union).
CORAL also provides a function meet_all_assign (and the corresponding
meet_all_up_assign that utilizes upward approximation) that conjuncts more than two
Boolean functions at the same time. In this case we start by collecting the entailed, disentailed
and equivalent variables of ALL functions in order to be able to reduce the ROBDDs
4.2 Algorithms with composite representations 38