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

more. Formally, given m composite representations 〈n1, E1, D1, L1〉, . . . , 〈nm, Em, Dm, Lm〉,
their conjunction can be computed, for each i ∈ N : 1 ≤ i ≤ m, by:
〈n ′ i, E ′ i, D ′ i, L ′ i〉 = η ni,
m m m
Ej, Dj,
j=1
j=1
j=1
Lj ,
〈n1, E1, D1, L1〉 ∧ . . . ∧ 〈nm, Em, Dm, Lm〉 = η m
n ′ m
j, E ′ m
j, D ′ m
j,
j=1
j=1
j=1
j=1
L ′
j .
The actual implementation, in fact, does not utilize this exact expression: whenever we
perform normalization (except, of course, the final normalization) we actually avoid looking
for all entailed, disentailed and equivalent variables in the ROBDDs; instead we just
reduce the current ROBDD using the information on entailed, disentailed and equivalent
variables that we have collected so far by using algorithms elim_search_true_vars,
elim_search_false_vars and rename. If elim_search_true_vars or
elim_search_false_vars find new entailed or disentailed variables then they are added
to the sets of collected variables. We also conjunct each ROBDD immediately after this
partial reduction instead of reducing the remaining ones before.
4.2.4 Disjunction
Given the same arguments as conjunction, disjunction can be performed as:
where:
〈n1, E1, D1, L1〉 ∨ 〈n2, E2, D2, L2〉 = 〈n ′ 1 ∨ n ′ 2, E1 ∩ E2, D1 ∩ D2, L ′ 1 ∨ L ′ 2〉
L ′ 1 = L1 ∧
L ′ 2 = L2 ∧
(x,y)∈E1\E2
x≺y
(x,y)∈E2\E1
x≺y
(x, y) ∧
(x, y) ∧
(x,y)∈D1\D2
x≺y
(x,y)∈D2\D1
x≺y
(x, y) ,
(x, y) ,
E ′ 1 = λL ′(x)
′
x ∈ E1 \ E2 , E 2 = λL ′(x)
x ∈ E2 \ E1 ,
D ′ 1 = λL ′(x)
′
x ∈ D1 \ D2 , D 2 = λL ′(x)
x ∈ D2 \ D1 ,
L ′′
1 = (L ′ 1 \ L ′ 2) ∨ L1, L ′′
2 = (L ′ 2 \ L ′ 1) ∨ L2,
n ′ 1 = n1 ∧
x∈E ′ 1
n ′ 2 = n2 ∧
x∈E ′ 2
x ∧
x∈D ′ 1
x ∧
x∈D ′ 2
x ∧
(x,y)∈L ′′
1
x ∧
(x,y)∈L ′′
2
x ⇔ y,
x ⇔ y.
(recall that the ∨ operator on equivalence relations represents their intersection). If the
two arguments were already normalized, then it is proven that the result of the application
of this algorithm will be already normalized (otherwise a further call to the normalization
operator is required).
4.2 Algorithms with composite representations 41