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

The ∧ operator has precedence over the ∨ operator.
For each f ∈ F, we define f (sometimes denoted as ¬f) as the Boolean function
such that ∀a ∈ A : f(a) = 1 ⇐⇒ f(a) = 0. For each i, t, e ∈ F we also define
if i then t else e def
= i ∧ t ∨ i ∧ e .
Finally, we need a notation for Boolean functions obtained by forcing some variables
in another Boolean function to a given truth value. For each valuation a ∈ A, each
x, y ∈ Vars, and each c ∈ {0, 1}, the valuation a[c/x] ∈ A is defined as
a[c/x](y) def
=
c, if x = y;
a(y), otherwise.
We extend this to a set of variables by defining, for each X = {v1, v2, . . . vn} ⊆ Vars,
the Boolean valuation a[c/X] def
= a[c/v1][c/v2] . . . [c/vn]. Now we can define, for each
f ∈ F, each x ∈ Vars, each a ∈ A and each c ∈ {0, 1}, the Boolean function
f[c/x] ∈ F (also denoted as f |x←c) as f[c/x](a) def
= f(a[c/x]). Like we did with
Boolean valuations, we define for each X = {v1, v2, . . . vn} ⊆ Vars the Boolean function
f[c/X] def
= f[c/v1][c/v2] . . . [c/vn]. We also define for each f ∈ F, each x, y ∈ Vars, and
each a ∈ A, the Boolean function f[x/y] ∈ F as f[x/y](a) def
= f(a[a(x)/y]).
1.1.3 Notable sets of variables
We now introduce, for any given Boolean function f, the set of variables on which f
depends, the sets of its entailed and disentailed variables, and the equivalence relation
representing its equivalent variables; calculating them is necessary for many applications.
Definition 3. (Variable dependency. Entailed, disentailed and equivalent variables.)
For each f ∈ F, the set of variables on which f depends, the set of entailed
variables for f, the set of disentailed variables for f, and the equivalence relation representing
the equivalent variables for f are given, respectively, by:
vars(f) def
= x ∈ Vars ∃a ∈ A . f a[0/x] = f a[1/x] ,
true(f) def
= x ∈ Vars ∀a ∈ A : f(a) = 1 =⇒ a(x) = 1 ,
false(f) def
= x ∈ Vars ∀a ∈ A : f(a) = 1 =⇒ a(x) = 0 ,
equiv(f) def
= (x, y) ∈ Vars 2 ∀a ∈ A : f(a) = 1 =⇒ a(x) = a(y) .
1.1.4 Equivalence relations over sets of variables
Let L be the set of all equivalence relations over a finite set of variables and consider a
particular equivalence relation L ∈ L. We define the leader of an equivalence class in L as
the minimum variable in the equivalence class; we also say that variable vl is the leader of
variable vi for L when vl is the leader of the equivalence class in L that contains vi (note
that each leader is the leader of itself), and define λL(vi) def
= vl. The set of all leaders for
1.1 Fundamental concepts 2