68 Y. Shi, Y. Mi and J. Li et al. / Information Sciences 496 (2019) 65–81
of all concepts under the cognitive operators F, H and
F ,
H are known as the conditional concept space G
C
FH
and decision
concept space G
D
F
H
, respectively.
Definition 5. [35] For any z
1
, z
2
∈ D , a quintuple ( G, M, I, D, J ) is referred to as a regular formal decision context when
H (z
1
) ∩
H (z
2
) = ∅ . In this case, we say that ( G, M, I ) and ( G, D, J ) are the conditional formal context and decision formal
context, respectively.
Here, the cognitive operators F , H and
F ,
H are known as the conditional cognitive and decision cognitive operators,
respectively. Note that a formal context with decision attributes is said to be a formal decision context [38,49] or training
context [16,17] . Furthermore, if each real-world object merely belongs to a decision attribute in a formal decision context,
the formal decision context is known as a regular formal decision context. This means that an instance is only connected
with a single label.
Property 1.
[35] Let ( G, M, I, D, J ) be a regular formal decision context. For any (A, B ) ∈ G
C
FH
and (Y, Z) ∈ G
D
F
H
, an object set A
is associated with a label set Z under the attribute set B when A ⊆Y, and A, B, Y and Z are nonempty. That is, if A = HF (g) and
Z =
F
H (z) , the object g is connected with a single label z.
According to Property 1 , we can fulfill the classification task by means of the interaction between the conditional and
decision concept spaces.
3. Proposed C3LM
In this section, we discuss the design of a new theoretical framework for concurrent computing, which comprises three
aspects: initial concurrent concept learning, the concurrent concept-cognitive process, and the concept generalization pro-
cess.
3.1. Initial concurrent concept learning in C3LM
In the real world, not all methods can be concurrent, as this often depends on their separability. In order to guarantee
concurrency for the C3LM in theory, we need to consider the following definitions and propositions.
Definition 6. Let ( G, M, I, D, J ) be a regular formal decision context. Suppose that D
1
, D
2
, . . . , D
K
is a partition of D by class
labels, and let G = G
D
1
∪ G
D
2
∪ ···∪ G
D
K
. Then, we say that G
D
k
(k ∈ { 1 , 2 , . . . , K} ) is a subclass-object set. For the sake of
brevity, hereinafter we write G
D
k
as G
k
.
Definition 6 indicates that an object set G can be decomposed into several subclass-object sets in a regular formal de-
cision context. Moreover, we only consider objects that are updated by newly input objects, as attributes can be taken as
relatively stable in real life. Therefore, in the following, we discuss the scenario of a subclass-object G
k
.
Let G
k
be a subclass-object set, and M and D be attribute sets. The set-valued mappings F
k
: 2
G
k
→ 2
M
, H
k
: 2
M
→ 2
G
k
and
F
k
: 2
G
k
→ 2
D
,
H
k
: 2
D
→ 2
G
k
are respectively referred to as the conditional and decision cognitive operators with a
subclass-object set G
k
when no confusion exists.
Definition 7. Let G
k
1
, G
k
2
, . . . , G
k
n
be a partition of an object set G
k
. If the following cognitive operators:
F
k
j
: 2
G
k
j
→ 2
M
, H
k
j
: 2
M
→ 2
G
k
j
, j = 1 , 2 , . . . , n,
F
k
: 2
G
k
→ 2
M
, H
k
: 2
M
→ 2
G
k
satisfy F
k
(g) = F
k
j
(g) , where g ∈ G
k
j
, we say that HS
F
k
H
k
= (F
k
1
, . . . , F
k
n
; H
k
1
, . . . , H
k
n
) is a conditional horizontal partition
state.
Proposition 1. Let HS
F
k
H
k
= (F
k
1
, . . . , F
k
n
; H
k
1
, . . . , H
k
n
) be a conditional horizontal partition state. For any g ∈ G
k
j
1
( j
1
∈
{ 1 , 2 , . . . , n } ) , if there exist objects g
1
, g
2
, . . . , g
n
∈ G
k
j
2
( j
2
∈ { 1 , 2 , . . . , n } ) such that F
k
j
1
(g) ⊆ F
k
j
2
(g
i
) (i = 1 , 2 , . . . , n ) , we have
(H
k
F
k
(g) , F
k
(g)) =
g ∪
n
∪
i =1
g
i
, F
k
j
1
(g)
; (6)
otherwise,
(H
k
F
k
(g) , F
k
(g)) = ({ g} , F
k
j
1
(g)) . (7)
Proof. See Appendix A for the proof of
Proposition 1 .
In
fact, from the perspective of objects, Definition 7 and Proposition 1 demonstrate that the separability holds for the
C3LM in the conditional formal context ( G, M, I ). Analogously, we can determine that the separability also holds for the
C3LM in the decision formal context ( G, D, J ) under the decision cognitive operators
F
k
and
H
k
.