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

1 Boolean functions
1.1 Fundamental concepts
1.1.1 Definition of Boolean functions
Boolean functions are functions that depend on a set of Boolean variables. The variables
and the function itself have their values in {0, 1}. We will also refer to the elements 0 and
1 as false and true respectively. Formally, we denote the infinite and numerable set of
Boolean variables as Vars. We also define a total order relation ≺ over Vars that allows
us to assign an index i ∈ N to each variable xi ∈ Vars. We can now give the formal
definition of Boolean valuation and Boolean function.
Definition 1. (Boolean valuations and Boolean functions.) The set of Boolean
valuations over Vars is defined as A def
= Vars → {0, 1}. The set of Boolean functions is
defined as F def
= A → {0, 1}.
Sometimes we may use variable assignment as a synonym for Boolean valuation.
1.1.2 Denoting specific Boolean functions
We will use a simple notation to refer to specific Boolean functions, exploiting the fact
that they can all be obtained by applying a few simple operators on a few elementary
functions. We can start by defining the Boolean functions ⊥ and ⊤ as the Boolean
functions which evaluate to false and true respectively in any given Boolean valuation.
For each variable x ∈ Vars, we also write x to denote the Boolean function f such that
∀a ∈ A : f(a) = 1 ⇐⇒ a(x) = 1. It should always be clear whether we are referring to
the variable or to the function in a specific context.
Now we define a few basic binary operations over F. For each pair of Boolean functions
f1, f2 ∈ F, we write f1 ∧ f2 to denote the Boolean function g ∈ F such that for each
a ∈ A, g(a) = 1 ⇐⇒ (f1(a) = 1 ∧ f2(a) = 1). We define all other common operations
analogously; here is a complete list:
Definition 2. (Binary operators on Boolean functions.) ∀f1, f2 ∈ F :
f1 ∧ f2 def
= g ∈ F . ∀a ∈ A : g(a) = 1 ⇐⇒ f1(a) = 1 ∧ f2(a) = 1 ,
f1 ∨ f2 def
= g ∈ F . ∀a ∈ A : g(a) = 1 ⇐⇒ f1(a) = 1 ∨ f2(a) = 1 ,
f1 ⊕ f2 def
= g ∈ F . ∀a ∈ A : g(a) = 1 ⇐⇒ f1(a) = f2(a) ,
f1 ⇔ f2 def
= g ∈ F . ∀a ∈ A : g(a) = 1 ⇐⇒ f1(a) = f2(a) ,
f1 ⇒ f2 def
= g ∈ F . ∀a ∈ A : g(a) = 1 ⇐⇒ f1(a) = 0 ∨ (f1(a) = 1 ∧ f2(a) = 1) .