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

L is denoted as leaders(L) and its domain is denoted as dom(L). If E1, E2 ∈ L we define
E1 ∨ E2 as the intersection of E1 and E2 and E1 ∧ E2 as the transitive closure of the
union of E1 and E2. Formally it is defined inductively, for each i ∈ N, i > 0, by:
E1 ∧0 E2 def
= E1 ∪ E2,
E1 ∧i E2 def
= (x, z) ∈ Vars 2 ∃y ∈ Vars . ∃(x, y), (y, z) ∈ (E1 ∧i−1 E2) ,
E1 ∧ E2 def
=
(E1 ∧i E2).
i∈N
Finally, we define the domain of an equivalence relation L ∈ L as
dom(L) def
= x ∈ Vars . ∃(x, y) ∈ L .
1.1.5 Projections
Given a Boolean function f ∈ F and a variable x ∈ Vars, we define the Boolean function
∃xf def
= f[0/x] ∨ f[1/x]. (1.1)
This equation is also known as Schröder’s elimination principle and is often exploited by the
algorithms that compute projections. Given a finite set of variables {v1, v2, . . . vn} ⊆ Vars
we also define the Boolean function
∃ {v1,v2,...vn}f def
= ∃v1f ∧ ∃v2f ∧ . . . ∃vnf.
We finally define the projection of a Boolean function f onto a finite set of variables X as
the Boolean function
f | X
def
= ∃{Vars\X}f.
1.1.6 Positive Boolean functions
We define the set of positive Boolean functions Pos as the set of all Boolean functions
that evaluate to true in the variable assignment which assigns 1 to all variables in Vars.
Definition 4. (Positive Boolean functions.) If t ∈ A is the Boolean valuation such
that ∀v ∈ Vars : t(v) = 1, we define
Pos def
= f ∈ F f(t) = 1
Marriott and Søndergaard (see [MS93]) have found this class of Boolean functions to
be a good abstract domain for the problem of groundness analysis: that is why CORAL
provides some specialized algorithms for it.
1.1 Fundamental concepts 3