2
In BRT, we will use set variables A
1
,A
2
,..., and function
variables f
1
,f
2
,... .
The BRT atoms consist of
∅
,U,A
i
, and f
i
A
j
, for any i,j
≥
1.
BRT expressions are defined as the least set containing the
BRT atoms, where if s,t are BRT expressions then so are s
t, s
«
t, s’.
The idea here is that U is the universal set, and s’ is U\s.
A BRT equation is s = t, where s,t are BRT expressions. A
Boolean inequation is s ≠ t, where s,t are BRT expressions.
Let V be a set of multivariate functions and K be a set of
sets. Let n,m
≥
1.
Equational (inequational, propositional) BRT on (V,K) of type
(n,m) analyzes statements of the form:
for all f
1
,...,f
n
V, there exists A
1
,...,A
m
K, such that a
given BRT equation (BRT inequation, propositional combination
of BRT equations) holds among the sets and their images under
the functions.
More formally,
for all f
1
,...,f
n
V, there exists A
1
,...,A
m
K, such that a
given BRT equation (BRT inequation, propositional combination
of BRT equations) holds,
where we require that in the given BRT equation (BRT
inequation, propositional combination of BRT equations), all
BRT atoms that appear are among the m(n+1) BRT atoms
A
1
,...,A
m
f
1
A
1
,...,f
1
A
m
...
f
n
A
1
,...,f
n
A
m
.
For this purpose, we take the universal set U to be the union
of the ranges of the elements of V and the elements of K.
Most typically, this will simply be the largest element of K
(which is not guaranteed to exist).