In the mid 1990s Z. Pawlak relying on the rough set theory initiated the study of rough calculus in his many papers. He invented the investigation of its diferent subfields such as rough continuity, rough derivatives-integrals, rough diferential equations, etc. Some authors have systematically investigated the rough continuity of rough real functions in Pawlak's sense. The following reasonable step would be to define the derivative of rough functions. However, it does not seem clear how it could be carried through this important step. In the paper, a possible approach will be outlined.
basically, but not completely follows Pawlak’s method.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
domain is denoted by : → , ↦→ ( ). denotes the set of all functions with domain , in notation Dom = , and co-domain , in notation Im = .
If , ∈ , the operation ⊙ , ⊙ ∈ {+, −, ·, /} and the relation , ∈ {=, ̸=, ≤, <, ≥, >} are understood by pointwise.
For any ⊆ , ( ) = { ( ) | ∈ } ⊆ is the direct image of . Especially, ( ) ⊆ is the range of .
If , ∈ R ( ≤ ), [, ] = { ∈ R | ≤ ≤ } and ], [= { ∈ R | < < } denote closed and open intervals. [, ] = { } is identified with the real number ∈ R. It is easy to interpret the open-closed ], ] and closed-open [, [ intervals. (, ) means an ordered pair of real numbers and .
Let R≥0 denote the set of nonnegative real numbers. Let [ ] = {0, 1, . . . , } ⊆ N be a finite set of natural numbers. Accordingly, ] ] = 1, . . . , , [ [= 1, . . . , − 1, and ] [= 1, . . . , − 1.
Let denote a closed interval = [0, ] ( ∈ R≥0, > 0).
Definition 2.1. A categorization of is a sequence = { } ∈[ ] ⊆ R≥0, where ≥ 1 and 0 = 0 < 1 < . . . < = . is also called the discretization of .
, ∈ . if = = ∈ for some ∈ [ ], or , ∈ ] , +1[ for some ∈ [ [. Hence, the partition / associated with the equivalence relation is: / = {{ 0}, ] 0, 1[, { 1}, . . . , { −1}, ] −1, [, { }}.
a slightly diferent sense: Two compact real intervals nonoverlapping if either they are disjoint or their intersection contains at most one point, which necessarily an endpoint of both intervals ([2], p. 4). In the classical analysis context, a partition of is a collection of nonoverlapping closed intervals whose union is ([1], p. 149).
The block of the partition / containing ∈ is denoted by J K . In = [ , +1] is the particular, if ∈ , J K = { }. If ∈ J K = ] , +1[, J K closure of J K . Of course, when ∈ , J K = J K = { }.
naming of the following notions is consistent with the standard terminology of RST.
sets are referred to as definable sets. By definition, ∅ is definable. Their collection is denoted by / .
The principal notions of RST are the lower and upper approximation functions, l and u , respectively. Most commonly, their domain and co-domain are the power set of . In the following, however, the closed intervals of the form [0, ] ( ∈ ) will only be approximated. Therefore, l ([0, ])={ ′∈ | J ′K ⊆ [0, ]} = ∪{J ′K ∈/ | J ′K ⊆ [0, ]}, u ([0, ])={ ′∈ | J ′K ∩ [0, ] ̸= ∅} = ∪{J ′K ∈/ | J ′K ∩ [0, ] ̸= ∅}. PAS( ) = (, / , / , l , u ) is called Pawlak approximation space.
l ( ) = max{ ′ ∈ | ′ ≤ } and u ( ) = min{ ′ ∈ | ′ ≥ }. Of course, l( ) ≤ ≤ u ( ), and l ( ) = u ( ) = if ∈ . Moreover, • l ([0, ]) = [0, l ( )] = [0, ] and u ([0, ]) = [0, u ( )] = [0, ] (if ∈ ); • l ([0, ]) = [0, l ( )] $ [0, ] and u ([0, ]) = [0, u ( )[ % [0, ] (if /∈ ).
It is said that the number ∈ is exact with respect to PAS( ) if l ( ) = u ( ), otherwise is inexact or rough [13]. Of course, ∈ is exact if ∈ .
represented as J K =[l ( ), u ( )]={ } if ∈ , J K =]l ( ), u ( )[ if /∈ .
Let = [0, ], = [0, ] be two closed intervals with , ∈ R≥0, , > 0. Let , be categorizations of and , where ={ } ∈[ ], ={ } ∈[ ]⊆R≥0 with , ≥ 1, 0 = 0 < 1 < · · · < = and 0 = 0 < 1 < · · · < = .
as follows.
: / → [2 ], J K ↦→ ︂{ 22 +=1 =2, 2 + 1, iiff ∃∃ ∈∈ [[ ][ ((JJ KK =={] }, ⊂+ 1[)).,
︂{ { / 2}, if ≡ 0 (mod 2), −1 : [2 ] → / , ↦→ ] −1 , +1 [, if ≡ 1 (mod 2). 2 2
help of an enumeration function . Its values are referred to as ’s ( ∈ [2 ]). Example 3.1. In the running example, let =[0, 5] with ={ 0, 1, 2, 3, 4, 5}, and =[0, 4] with ={ 0, 1, 2, 3, 4}.
Figure 1 (a) shows the rough coordinate system with respect to PAS( ) and
real functions in this rough coordinate system.
sented taken into account the features of approximation spaces PAS( ) and PAS( ).
3.1. Pointwise approximation of rough real functions Definition 3.2 ([9, 13]). Let ∈ . The pointwise ( , )– lower and ( , )– upper approximations of are the functions
: → , ↦→ l ( ( )), : → , ↦→ u ( ( )). is exact at , if ( ) = ( ), otherwise is inexact (rough) at . is pointwise exact on ′ ⊆ , if ( ) = ( ) for all ∈ ′, otherwise is pointwise inexact (rough) on ′.
Remark 3.3. Rough real functions are ab ovo treated in ( , )–coordinate systems. Hence, it seems reasonable to define the pointwise approximation in this context, too. Nevertheless, a “pointwise” feature much better fits to the whole interval = [, ]. But with the choice = { 0 = 0, 1 = }, this case is also included in the above definition.
∈ is exact at ∈ if ( ) = ∈ for some ∈ [ ]. Geometrically it means that is exact at a point in if in this point touches or intersects a line segment = , where ∈ .
Example 3.4. Figure 2 (a) shows the pointwise ( , )–lower and ( , )–upper approximations, and , of a function ∈ . is exact at points , , 2, and rough at all other points. 3.2. Blockwise approximation of rough real functions Definition 3.5. Let ∈ . The blockwise ( , )– lower and ( , )– upper approximations of are the functions : / → , J K ↦→ l (inf (J K )), ←→ : / → , J K ↦→ u (sup (J K )). ←→ blockwise inexact (rough) on .
is blockwise exact on for some ∈ [2 ] if ( ) = ←→( ), otherwise is ←→
Let ∈ , ∈ [2 ]. is blockwise exact on if ( ) = { } ⊂ for some ∈ [ ]. Geometrically it means that is exact on ∈ / if touches or intersects a line segment = for some ∈ at the point / 2 if ≡ 0 (mod 2), or coincides with a line segment = for some ∈ on ] −1 , +1 [ when 2 2 blockwise rough o←→nall other blocks.
Example 3.6. Figure 2 (b) shows the blockwise ( , )–lower and ( , )–upper approximations, and ←→, of . is blockwise exact on 4 = { 2} only, and
above substantially difer from those considered in function approximation theory.
Im = { ∈ | ≤ ≤ ←→}, without any assignment rules. Certain functions in this family can a←→lso be interpreted as partially specified ones with unknown values.
3.3. Finite sequence approximations Definition 3.7. Let ∈ . The finite sequence ( , )– lower and ( , )– upper approximations of are the functions ∘ : [ ] → , ↦→ l ( ( )), ∘ : [ ] → , ↦→ u ( ( )).
points. However, it does not say anything about how a rough function behaves on the open intervals ] , +1[ ( ∈ [ [).
Definition 3.8.
Let
∈ . ( , )– upper approximations of are the functions
The extended finite sequence ( , )– lower and categorization points.
Of course, ∘ ( ) = ∙ (2 ), ∘ ( ) = ∙ (2 ) ( ∈ [ ]), i.e., they are equal at the Example 3.9. Figure 3 (a) and Figure 3 (b) illustrate the finite sequence and extended finite sequence approximations of the function (see Figure 2). (a) l u
(b) 3.4. Discrete sequence approximations
: → [ ], ↦→ max{ ∈ [ ] | ≤ }, : → [ ], ↦→ min{ ∈ [ ] | ≥ }. upper approximations of are the functions Definition 3.10.
Let ∈ . The discrete sequence ( , )– lower and ( , )– ⋆ : [ ] → [ ], ↦→ l ( ( )), ⋆ : [ ] → [ ], ↦→ u ( ( )).
points only. It will be extended on the whole partition / of . ( , )– upper approximations of are the functions Definition 3.11.
Let ∈ . The extended discrete sequence ( , )– lower and *: [2 ] → [ ], ↦→ l (inf ( )), *: [2 ] → [ ], ↦→ u (sup ( )).
Of course, ⋆( ) = *(2 ), ⋆( ) = *(2 ) ( ∈ [ ]), i.e., they are equal at the categorization points.
studied in the theory of “calculus of finite diferences”, see, e.g., [4], one of the classical treatments of this subject. Traditionally, the subject of the discrete calculus is the functions of the form : N → R, i.e., ’s are finite or infinite sequences.
whose both domain and co–domain are finite set of natural numbers which he called discrete functions. In the following, under the discrete functions it is meant a function of the form : N → Z. Obviously, : [ ] → [ ] is discrete function as well.
4.1. Pawlak’s approach
Definition 4.1. Let : [ ] → [ ] be a discrete function. The rough derivative ′ of is the function
′ : [ [= [ − 1] → Z, ↦→ ( + 1) − ( ).
• ( ± )′( ) = ′( ) ± ′( ); • ( · )′( ) = · ′( ) ( ∈ Z); where ( ± )′,( · )′,( · )′,( / )′ : [ − 1] → Z.
= (0), ′ = (
One can observe that Dom (0) = [ ], Dom (
direct interpretation of rough derivatives is simple. Let : [ ] → [ ] be a discrete function, and ′ : [ − 1] →
piecewise linear function with slopes ′( )’s ( ∈ [ − 1]). Of course, when ′( ) = 0, is constant between ( ) and ( + 1), i.e., ( ) = ( + 1). In this interpretation, the domain of can be considered as bounded compact real interval [0, ] ⊂ R. Example 4.3. Figures 5 (a) and 5 (b) show the discrete derivatives of ⋆ and ⋆.
calculus of rough functions. Pawlak used the rough calculus, which is also known as digital calculus or discrete calculus, to achieve this goal. However, Pawlak did not establish a connection between the rough derivatives and rough functions. Indeed, it is hard to interpret rough derivatives as rough functions.
defined above. In addition, let ∈ be a fixed rough function. Then, all diferent approximations of set up sets of rough functions. In regard to Pawlak’s approach, here we will focus on the discrete sequence approximation of rough functions only.
The discrete sequence approximation determines the following set of rough func tions: ℛ = { ∈ | ∀ ∈ [ ]( ⋆( ) ≤ ( ) ≤ ⋆( ))}. That is, ℛ on the open intervals formed by the categorization points of . of such the rough functions ’s which are bounded by ⋆( ) and ⋆( ) ( ∈ [ ], ⋆( ), ⋆( ) ∈ ) at the categorization points of , but they are not constrained consists Example 4.4. Figure 6 depicts the discrete approximation of (Figure 6 (a)), and the set of rough functions determined by it (Figure 6 (b)). One can observe that the discrete approximation constraints the rough functions at the categorizations points, but it does not say anything about how they behave on the open intervals. (a) (b)
Finding out rough derivatives of a set of rough functions, e.g., ℛ , may be
approached, for instance, by diferentiating both lower and upper discrete sequence
approximations. Some special dificulties of this approach are the following:
(i ) Of course, ⋆ ≤ ⋆ does not imply ⋆(
At an ∈ [ [ may occur ⋆
Their interpretations are simple. ⋆(
(
A possible solution is the following. If ⋆(
The paper, first, has presented four diferent rough approximation methods representing the rough real functions. The chances are that a definition of the rough diferentiation should rely on one of these representations. In this paper, basically but not completely following Pawlak’s method, the rough diferentiation based on discrete sequence approximation has been considered only. Of course, additional diferentiation definitions should also be studied based on the other representations which may be the subject of many subsequent papers in the future.