In this section, we briefly review some fundamental conceptions of set-valued ordered decision systems [11] .

Definition 1 [11] . Let S = ( U , C ∪ { d } , V , f ) be an information decision system, where U = x 1 , x 2 , ⋯ , x | U | is a set of all objects, known as the universe. A = C ∪ d , where C is the collection of conditional attributes with preference relations, d is the decision attribute, and C ∩ d = ∅ . V A = V C ∪ V d , where V C is a collection of conditional attributes C for all values and V d is a collection of decision attributes d for all values. : U × C → 2 V C is an object attribute mapping, which is a set value mapping. If all conditional attributes are in increasing or decreasing preference, then S is said to be a set-valued ordered decision system.

Example 1. Table 1 shows a set-valued ordered decision system S = ( U , C ∪ { d } , V , f ) , where U = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 } , B = C = { a 1 , a 2 } = { Resding , Writing } , V B = { E , F , G } = { English , French , German } , V d = { poor , good } .

In Table 1, f ( x 2 , a 1 ) indicates that x₂ will be read in English, French, and German, and f ( x 1 , a 1 ) indicates that x₁ will be read in French only. It can be concluded that x₁ does not read as much language as x₂. That is, in the attribute a₁, object x₂ is at least dominant to object x₁. This type of set-valued ordered decision system is called conjunctive set-valued ordered decision system (C-SODS) [11] .

In C-SODS, the conjunctive dominance relation can be defined based on the set inclusion relation.

Definition 2 [11] . Let S = ( U , C ∪ { d } , V , f ) be a C-SODS. ∀ B ⊆ C , the

conjunctive dominance relation R_B on the domain U is defined as

R B = { ( y , x ) ∈ U × U | y ≽ B x } = { ( y , x ) ∈ U × U | f ( y , a ) ⊇ f ( x , a ) , ∀ a ∈ B } ,

It is clear that the conjunctive dominance relation satisfies reflexive, transitive, and not symmetry.

Definition 3 [11] . Let S = ( U , C ∪ { d } , V , f ) be a C-SODS. ∀ B ⊆ C , B ≠ ∅ two knowledge granules of x are called B-dominating set and B-dominated set, which are respectively defined as follows

[ x ] B + = { y ∈ U | y   R B   x } ,

[ x ] B − = { y ∈ U | x   R B   y } .

Definition 4 [11] . Let S = ( U , C ∪ { d } , V , f ) be a C-SODS. If the U can be divided into m equivalence sets, the equivalence sets are called decision sets. It can be expressed as:

C l = { C l n , n ∈ T } , where T = { 1 , ⋯ , m } .

Definition 5 [11] . Let S = ( U , C ∪ { d } , V , f ) be a C-SODS. If the U can be divided into m decision sets according to the decision attribute d, upward union C l n ≥ and downward union C l n ≤ can be defined as:

C l n ≥ = ∪ n ′ ≥ n C l n ′ ,

C l n ≤ = ∪ n ′ ≤ n C l n ′ , ∀ n , n ′ ∈ T .

where x ∈ C l n ≥ means “x belongs to at least C l n ”; x ∈ C l n ≤ means “x belongs to at most C l n ”.

Definition 6 [11] . Let S = ( U , C ∪ { d } , V , f ) be a C-SODS, for any B ⊆ C , the lower and upper approximations of C l n ≥ are respectively defined as follow:

R B _ ( C l n ≥ ) = { x ∈ U | [ x ] B + ⊆ C l n ≥ }

R B ¯ ( C l n ≥ ) = { x ∈ U | [ x ] B − ∩ C l n ≥ ≠ ∅ }

Definition 7 [11] . Let S = ( U , C ∪ { d } , V , f ) be a C-SODS, for any B ⊆ C , the lower and upper approximations of C l n ≤ are respectively defined as follow:

R B _ ( C l n ≤ ) = { x ∈ U | [ x ] B − ⊆ C l n ≤ }

R B ¯ ( C l n ≤ ) = { x ∈ U | [ x ] B + ∩ C l n ≤ ≠ ∅ }

Example 2. Continuing from Example 1, we can use the above definition to further compute the dominating and dominated sets, as well as the upper and lower approximations of the set-valued ordered decision system.

Dominating and dominated sets:

[ x 1 ] B + = [ x 6 ] B + = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 } , [ x 2 ] B + = [ x 3 ] B + = [ x 7 ] B + = { x 2 , x 3 , x 7 } , [ x 4 ] B + = { x 2 , x 3 , x 4 , x 5 , x 7 } , [ x 5 ] B + = { x 2 , x 3 , x 5 , x 7 } , [ x 8 ] B + = { x 2 , x 3 , x 5 , x 7 , x 8 } .

[ x 1 ] B − = [ x 6 ] B − = { x 1 , x 6 } , [ x 2 ] B − = [ x 3 ] B − = [ x 7 ] B − = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 } , [ x 4 ] B − = { x 1 , x 6 , x 4 } , [ x 5 ] B − = { x 1 , x 4 , x 5 , x 6 , x 8 } , [ x 8 ] B − = { x 1 , x 6 , x 8 } .

Decision classes:

C l 1 = { x 1 , x 6 , x 8 } , C l 2 = { x 2 , x 3 , x 4 , x 5 , x 7 }

Upward and downward unions of decision classes:

C l 1 ≥ = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 } , C l 2 ≥ = { x 2 , x 3 , x 4 , x 5 , x 7 }

C l 1 ≤ = { x 1 , x 6 , x 8 } , C l 2 ≤ = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 }

Approximations:

R B _ ( C l 1 ≥ ) = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 }

R B _ ( C l 2 ≥ ) = { x 2 , x 3 , x 4 , x 5 , x 7 }

R B ¯ ( C l 1 ≥ ) = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 }

R B ¯ ( C l 2 ≥ ) = { x 2 , x 3 , x 4 , x 5 , x 7 }

R B _ ( C l 1 ≤ ) = { x 1 , x 6 , x 8 }

R B _ ( C l 2 ≤ ) = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 }

R B ¯ ( C l 1 ≤ ) = { x 1 , x 6 , x 8 }

R B ¯ ( C l 2 ≤ ) = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 }

With the variation of an information system, the structure of information granules in the information system may vary over time which leads to the change of knowledge induced by RST. In a dynamic decision system, the attribute value domain can be updated due to different requirements, such as adding new conditional values, removing old conditional values, or correcting errors. Changes in the attribute value domain can directly lead to changes in the preference relation between any pair of attributes of two objects, resulting in corresponding changes in knowledge granules.

Example 3. From Table 1, we consider V B = { E , F , G } = { English , French , German } . If we add Chinese to V_B, we will also use Chinese as a language criterion to judge Reading and Writing. Then, the attribute value domain will be updated to V B = { E , F , G , C } = { English , French , German , Chinese } . At this point, Table 1 is updated to Table 2. x₁ is updated to {F, C} under a₁ in Table 2, so after updating, x₁ and x₂ are not comparable in Reading.

When the preference relation changes, the discovered knowledge may become invalid, or some new implicit information may emerge in the whole updated information system. Rather than restarting from scratch by the non-incremental

or batch learning algorithm for each update, developing an efficient incremental algorithm to avoid unnecessary computations by utilizing the previous data structures or results is thus desired.

In this section, we discuss the variation of approximations in the dynamic C-SODS when the conditional value evolves over time while the object set and attribute set remain constant. For convenience, we assume the incremental learning process lasts two periods from time t to time t + 1. ⋇ ( t ) and ⋇ ( t + 1 ) represent the set-valued ordered decision system at time t and time t + 1, respectively. ⋇ ( + ) denotes adding a conditional value.

In the following, V a ( t + 1 ) is denoted the updated attribute value domain of attribute a at t + 1, and v a + is the conditional value that is added to V a ( t ) , where V a ( t + 1 ) = V a ( t ) ∪ v a + .

Definition 8. For any x ∈ U , a ∈ B , where U + denotes the set of all objects with condition values added, and U ¯ = U − U + denotes the set of objects without condition values added. If v a + ∈ f ( t + 1 ) ( y , a ) , then y ∈ U + ; if v a + ∉ f ( t + 1 ) ( x , a ) , then x ∈ U ¯ .

When V a ( t ) changes, the dominating sets and dominated sets of y need to be updated. If x at time t dominates y, when y adds v a + at time t + 1, and x does not contain v a + . Thus, x no longer dominates y.

Property 1. If any y ∈ U + , then [ y ] B + , ( t + 1 ) = [ y ] B + , ( t ) − U ¯ .

Proof. If x ∈ [ y ] B + , ( t ) , f ( t + 1 ) ( y , a ) ⊆ f ( t + 1 ) ( x , a ) , then x ∉ [ y ] B + , ( t + 1 ) . If x ∉ [ y ] B + , ( t ) , x ∉ [ y ] B + , ( t + 1 ) .

The approximation sets are close to the dominating sets and dominated sets. According to Property 1, only the dominating sets have changed, thus we just need to update R _ B ( t + 1 ) ( C l n ≥ ) and R ¯ B ( t + 1 ) ( C l n ≤ ) .

Proposition 1. (1) If ∃ y ∉ R _ B ( t ) ( C l n ≥ ) , and U ¯ ⊆ C l n ≥ , [ y ] B + , ( t + 1 ) ⊆ C l n ≥ , then R _ B ( t + 1 ) ( C l n ≥ ) = R _ B ( t ) ( C l n ≥ ) ∪ { y } .

(2) If ∃ y ∈ R ¯ B ( t ) ( C l n ≤ ) , and U ¯ ⊆ C l n ≤ , [ y ] B + , ( t + 1 ) ∩ C l n ≤ = ∅ , then R ¯ B ( t + 1 ) ( C l n ≤ ) = R ¯ B ( t ) ( C l n ≤ ) − { y } .

Proof. (1) According to the Property 1, let Δ − = [ y ] B + , ( t ) − C l n ≥ . Clearly, if U ¯ ∩ Δ − = ∅ , and [ y ] B + , ( t + 1 ) ⊆ C l n ≥ , then R _ B ( t + 1 ) ( C l n ≥ ) = R ¯ B ( t ) ( C l n ≥ ) ∪ { y } .

(2) According to Property 1, if [ y ] B + , ( t + 1 ) ∩ C l n ≤ = ∅ , then R ¯ B ( t + 1 ) ( C l n ≤ ) = R ¯ B ( t ) ( C l n ≤ ) − { y } .

Similarly to y, if y is dominated by x at time t, then it is no longer dominated by x when y adds v a + at time t + 1 and x does not contain v a + . Thus, y is no longer dominated by x.

Property 2. If any x ∈ U ¯ , then [ x ] B − , ( t + 1 ) = [ x ] B − , ( t ) − U + .

Proof. This Property can be proved by the same method as employed in Property 1.

Similar to Proposition 1, only the dominated sets changed in Property 4. We just need to discuss the updating theories for the R _ B ( t + 1 ) ( C l n ≤ ) and R ¯ B ( t + 1 ) ( C l n ≥ ) .

Proposition 2. (1) If ∃ x ∉ R _ B ( t ) ( C l n ≤ ) , and U + ⊆ C l n ≤ , [ x ] B − , ( t + 1 ) ⊆ C l n ≤ , then R _ B ( t + 1 ) ( C l n ≤ ) = R _ B ( t ) ( C l n ≤ ) ∪ { x } .

(2) If ∃ x ∈ R ¯ B ( t ) ( C l n ≥ ) , and U + ⊆ C l n ≥ , [ x ] B − , ( t + 1 ) ∩ C l n ≤ = ∅ , then R ¯ B ( t + 1 ) ( C l n ≥ ) = R ¯ B ( t ) ( C l n ≥ ) − { x } .

Proof. This proposition can be proved by the same method as employed in Proposition 1.

Example 4. Continuing from Example 3, we can use Properties 1 - 4 and Propositions 1 - 4 to update the dominating sets and dominated sets, and approximations.

Updating dominating sets and dominated sets:

[ x 1 ] B + , ( t + 1 ) = [ x 1 ] B + , ( t ) − { x 1 , x 4 , x 5 , x 6 , x 8 } = { x 2 , x 3 , x 7 } ,

[ x 3 ] B + , ( t + 1 ) = [ x 3 ] B + , ( t ) − { x 2 , x 7 } = { x 3 } ,

[ x 4 ] B + , ( t + 1 ) = [ x 4 ] B + , ( t ) − { x 1 , x 4 , x 5 , x 6 , x 8 } = { x 2 , x 3 , x 7 } .

[ x 2 ] B − , ( t + 1 ) = [ x 2 ] B − , ( t ) − { x 1 , x 6 } = { x 2 , x 3 , x 4 , x 5 , x 7 , x 8 } ,

[ x 3 ] B − , ( t + 1 ) = [ x 3 ] B − , ( t ) − { x 1 , x 6 } = { x 2 , x 3 , x 4 , x 5 , x 7 , x 8 } ,

[ x 4 ] B − , ( t + 1 ) = [ x 4 ] B − , ( t ) − { x 1 , x 6 } = { x 4 } ,

[ x 5 ] B − , ( t + 1 ) = [ x 5 ] B − , ( t ) − { x 1 , x 6 } = { x 4 , x 5 , x 8 } ,

[ x 7 ] B − , ( t + 1 ) = [ x 7 ] B − , ( t ) − { x 1 , x 6 } = { x 2 , x 3 , x 4 , x 5 , x 7 , x 8 } ,

[ x 8 ] B − , ( t + 1 ) = [ x 8 ] B − , ( t ) − { x 1 , x 6 } = { x 8 } .

Updating approximations:

R _ B ( t + 1 ) ( C l 1 ≥ ) = R _ B ( t ) ( C l 1 ≥ ) , R _ B ( t + 1 ) ( C l 2 ≥ ) = R _ B ( t ) ( C l 2 ≥ ) , R ¯ B ( t + 1 ) ( C l 1 ≥ ) = R ¯ B ( t ) ( C l 1 ≥ ) ,

R ¯ B ( t + 1 ) ( C l 2 ≥ ) = R ¯ B ( t ) ( C l 2 ≥ ) .

R _ B ( t + 1 ) ( C l 1 ≤ ) = R _ B ( t ) ( C l 1 ≤ ) , R _ B ( t + 1 ) ( C l 2 ≤ ) = R _ B ( t ) ( C l 2 ≤ ) , R ¯ B ( t + 1 ) ( C l 1 ≤ ) = R ¯ B ( t ) ( C l 1 ≤ ) ,

R ¯ B ( t + 1 ) ( C l 2 ≤ ) = R ¯ B ( t ) ( C l 2 ≤ ) .