1300
B. Kulik, A. Fridman, A. Zuenko
In a space of properties S with attributes X
i
(i.e. S = X
1
× X
2
× . . . × X
n
),
the flexible universe will be comprised of different projections i.e. subspaces that use
a part of attributes from S . Every such subspace corresponds to a partial universe.
Definition 2. An elementary n-tuple is a sequence of elements each belonging to
the domain of the corresponding attribute in the relation diagram. An example of
an elementary n-tuple T [XY Z] is given above.
Definition 3. A C-n-tuple is an n-tuple of sets (components) defined in a certain
relation diagram; each of these sets is a subset of the domain of the corresponding
attribute.
A C-n-tuple is a set of elementary n-tuples; this set can be enumerated by
calculating the Cartesian product of the C-n-tuple’s components. C-n-tuples are
denoted with square brackets. For example, R[XY Z] = [ABC] means that A ⊆ X,
B ⊆ Y , C ⊆ Z and R[XY Z] = A × B × C.
Definition 4. A C-system is a set of homotypic C-n-tuples that are denoted as
a matrix in square brackets. The C-n-tuples that such a matrix contains are rows
of this matrix.
A C-system is a set of elementary n-tuples. This set equals to the union of sets
of elementary n-tuples that the corresponding C-n-tuples contain. For example,
a C-system Q[XY Z] =
A
1
B
1
C
1
A
2
B
2
C
2
can be represented as a set of elementary
n-tuples calculated by formula Q[XY Z] = (A
1
× B
1
× C
1
) ∪ (A
2
× B
2
× C
2
).
In order to combine relations defined on different projections within a single
algebraic system isomorphic to algebra of sets, NTA introduces dummy attributes
formed by using dummy components. There are two types of these components.
One of them called a complete component is used in C-n-tuples and is denoted
by “*”.
A dummy component “*” added in the i
th
place in a C-n-tuple or in
a C-system equals to the set corresponding to the whole range of values of the
attribute X
i
. In other words, the domain of this attribute is the value of the dummy
component. For example, if the domain of attribute X is given (here it equals to
the set {a, b, c, d}), the C-n-tuple Q[Y Z] = [{f, g} {a, c}] can be expressed in the
relation diagram [XY Z] as a C-n-tuple [∗{f, g} {a, c}]. Since the dummy component
of Q corresponds to an attribute with the domain X, the equality [∗{f, g} {a, c}] =
[{a, b, c, d} {f, g} {a, c}] is true. Another dummy component (Ø) called an empty
set is used in D-n-tuples.
A C-n-tuple that has at least one empty component is empty. In NTA, if we deal
with models of propositional or predicate calculuses, this statement is accepted as
an axiom which has an interpretation based on the properties of Cartesian products.
Below, we will show that usage of dummy components and attributes in NTA
allows to transform relations with different relation diagrams into ones of the same
type, and then to apply operations of theory of sets to these transformed relations.
Algebraic Approach to Logical Inference Implementation
1301
The proposed technique of defining dummy attributes differs from the known tech-
niques essentially due to the fact that new data are inputted into multiplace relations
as sets rather than elementwise, which significantly reduces both computational la-
boriousness and memory capacity for representation of the structures.
Operations (intersection, union, complement) and checks of relations of inclusion
or equality for these NTA objects are based on Theorems 1–6. Here they are given
without proof because their formulating in terms of NTA corresponds to the known
properties of Cartesian products. Let two homotypic C-n-tuples P = [P
1
P
2
. . . P
n
]
and Q = [Q
1
Q
2
. . . Q
n
] be given.
Theorem 1. P ∩ Q = [P
1
∩ Q
1
P
2
∩ Q
2
. . . P
n
∩ Q
n
].
Example 1. [{b, d} {f, h} {a, b}] ∩ [∗{f, g}{a, c}] = [{b, d} {f } {a}];
[{b, d} {f, h}{a, b}] ∩ [∗{g} {a, c}] = [{b, d} Ø {a}] = Ø.
Theorem 2. P ⊆ Q, if and only if P
i
⊆ Q
i
for all i = 1, 2, . . . , n.
Theorem 3. P ∪ Q ⊆ [P
1
∪ Q
1
P
2
∪ Q
2
. . . P
n
∪ Q
n
], equality being possible only in
two cases:
1. P ⊆ Q or Q ⊆ P ;
2. P
i
= Q
i
for all corresponding pairs of components except one pair.
Note that in NTA, according to Definition 4, equality P ∪Q =
P
1
P
2
. . .
P
n
Q
1
Q
2
. . .
Q
n
is true for all cases.
Theorem 4. Intersection of two homotypic C-systems equals to a C-system that
contains all non-empty intersections of each C-n-tuple of the first C-system with
each C-n-tuple of the second C-system.
Example 2. Let the following two C-systems be given in space S:
R
1
[XY Z] =
{a, b, d}
{f, h}
{b}
{b, c}
∗
{a, c}
,
R
2
[XY Z] =
{a, d}
∗
{b, c}
{b, d}
{f, h}
{a, c}
{b, c}
{g}
{b}
.
We need to calculate their intersection. First we calculate intersection of all the
pairs of C-n-tuples that the two different C-systems contain:
[{a, b, d} {f, h} {b}] ∩ [{a, d} ∗ {b, c}] = [{a, d} {f, h} {b}];
[{a, b, d} {f, h} {b}] ∩ [{b, d} {f, h} {a, c}] = Ø;
[{a, b, d} {f, h} {b}] ∩ [{b, c} {g} {b}] = Ø;
[{b, c} ∗ {a, c}] ∩ [{a, d} ∗ {b, c}] = Ø;