Keywords

1 Introduction

Rough sets, constructed by Pawlak [12], are used as an effective method for data mining. The framework is usually applied to information tables with nominal attributes and creates fruitful results in various fields. However, we are frequently faced with attributes taking continuous values, when we describe properties of an object in our daily life. Therefore, we describe rough sets in information tables with continuous values.

Ways how to deal with attributes taking continuous values are broadly classified into two approaches. One is to discretize a continuous domain by dividing it into a collection of disjunctive intervals. Objects included in an interval are regarded as indistinguishable. From this indistinguishability the family of indiscernible classes is derived [1]. Results strongly depend on how discretization is made. Especially, objects that are located in the proximity of the boundary of intervals are strongly affected by discretization. This leads to that results abruptly change by a little alteration of discretization. The other is a way using neighborhood [7]. In this approach when the distance of an object to another one on an attribute is less than or equal to a given threshold, two objects are regarded as indistinguishable on the attribute. Results gradually change as the threshold changes. So, we use the latter approach.

Rules are induced from lower and upper approximations. Concretely speaking, when objects o and \(o'\) are included in the approximations, let single rules \(a_{i} = 3.60 \rightarrow a_{j} = v\) and \(a_{i} = 3.73 \rightarrow a_{j} = v\) be induced, where objects o and \(o'\) are characterized by values 3.60 and 3.73 of attribute \(a_{i}\) and the set approximated is specified by value v of attribute \(a_{j}\). For example, value 3.66 of attribute \(a_{i}\) is not indiscernible with 3.60 and 3.73 under the threshold 0.05. Therefore, we cannot say anything from these single rules for a rule supported by an object with value 3.66 of attribute \(a_{i}\). This means that the single rules are short of applicability. To improve such applicability, we consider a combined rule that is derived from a series of single rules supported by individual objects.

In addition, we are frequently confronted with incomplete information in daily life. We cannot sufficiently utilize information obtained from our daily life unless we deal with incomplete information. We express incomplete information in a partial value or an interval value. A missing value that means unknown in an attribute is expressed in all elements over the domain of the attribute. For example, the domain is given in the interval [1.23, 4.45], the missing value is expressed in [1.23, 4.45].

Most of authors fix the indiscernibility of an object with incomplete information with another object [3, 16,17,18], as was done by Kryszkiewicz [4]. However, object o characterized by a value with incomplete information has two possibilities. One possibility is that the object o may have the same value as another one \(o'\); namely, the two objects may be indiscernible. The other possibility is that o may have a different value from \(o'\); namely, the two objects may be discernible. To fix the indiscernibility is to take into account only one of the two possibilities. Therefore, this treatment creates poor results and induces information loss [9, 15]. We do not fix the indiscernibility of objects with incomplete information and simultaneously deal with both possibilities. This can be realized by dealing with objects having incomplete information from viewpoints of certainty and possibility [10], as was done by Lipski in the field of incomplete databases [5, 6].

We have an approach based on possible world from the viewpoints of certainty and possibility. This way creates possible tables. Unfortunately, infinite possible tables can be derived from an information table with continuous values. Another way uses possible classes of an object, in which the object is possibly indiscernible with anyone [8]. The number of possible classes grows exponentially, as the number of values with incomplete information increases. However, this difficulty can be avoided by using minimum and maximum possible classes in the case of nominal attributes [10]. In this work, we apply this approach to information tables with continuous values.

The paper is organized as follows. In Sect. 2, an approach using indiscernible classes is addressed in complete information tables. In Sect. 3, we develop the approach in incomplete information tables. This is described from two viewpoints of certainty and possibility. In Sect. 4, conclusions are addressed.

2 Rough Sets by Using Indiscernible Classes in Complete Information Systems with Continuous Values

A data set is represented as a two-dimensional table, called an information table. In the information table, each row and each column represent an object and an attribute, respectively. A mathematical model of an information table with complete information is called a complete information system. The complete information system is a triplet expressed by \((U,AT,\{D(a_{i}) \mid a_{i} \in AT \})\). U is a non-empty finite set of objects, which is called the universe. AT is a non-empty finite set of attributes such that \(a_{i} : U \rightarrow D(a_{i})\) for every \(a_{i} \in AT\) where \(D(a_{i})\) is the domain of attribute \(a_{i}\).

Indiscernible class \([o]_{a_{i}}\) for object o on \(a_{i}\) is:

$$\begin{aligned}{}[o]_{a_{i}} = \{o' \mid |a_{i}(o) - a_{i}(o')| \le \delta _{a_{i}} \}, \end{aligned}$$
(1)

where \(a_{i}(o)\) is the value for attribute \(a_{i}\) of object o and \(\delta _{a_{i}}\) is a threshold that denotes a range in which \(a_{i}(o)\) is indiscernible with \(a_{i}(o')\). The indiscernible class is a tolerance class. Using the tolerance class, rough sets are generalized [14]. And recently it is used in decision rule induction [13].

Family \(\mathcal{F}_{a_{i}}\) of indiscernible classes on \(a_{i}\) is:

$$\begin{aligned} \mathcal{F}_{a_{i}} = \{[o]_{a_{i}} \mid o \in U \}, \end{aligned}$$
(2)

where \(\cup _{i}[o]_{a_{i}} = U\). Using indiscernible classes, lower approximation \(\underline{apr}_{a_{i}}(\mathcal{O})\) and upper approximation \(\overline{apr}_{a_{i}}(\mathcal{O})\) of set \(\mathcal{O}\) of objects for \(a_{i}\) are:

$$\begin{aligned} \underline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid [o]_{a_{i}} \subseteq \mathcal{O}\}, \end{aligned}$$
(3)
$$\begin{aligned} \overline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid [o]_{a_{i}} \cap \mathcal{O} \ne \emptyset \}. \end{aligned}$$
(4)

Proposition 1

If \(\delta 1 \le \delta 2\), then \(\underline{apr}_{a_{i}}^{\delta 1}(\mathcal{O}) \supseteq \underline{apr}_{a_{i}}^{\delta 2}(\mathcal{O})\) and \(\overline{apr}_{a_{i}}^{\delta 1}(\mathcal{O}) \subseteq \overline{apr}_{a_{i}}^{\delta 2}(\mathcal{O})\), where \(\underline{apr}_{a_{i}}^{\delta 1}(\mathcal{O})\) and \(\overline{apr}_{a_{i}}^{\delta 1}(\mathcal{O})\) are lower and upper approximations under threshold \(\delta 1\) of attribute \(a_{i}\) and \(\underline{apr}_{a_{i}}^{\delta 2}(\mathcal{O})\) and \(\overline{apr}_{a_{i}}^{\delta 2}(\mathcal{O})\) are lower and upper approximations under threshold \(\delta 2 \) of attribute \(a_{i}\).

For object o in the lower approximation of \(\mathcal{O}\), all objects with which o is indiscernible are included in \(\mathcal{O}\); namely, \([o]_{a_{i}} \subseteq \mathcal{O}\). On the other hand, for an object o in the upper approximation of \(\mathcal{O}\), some objects with which o is indiscernible are in \(\mathcal{O}\); namely, \([o]_{a_{i}} \cap \mathcal{O} \ne \emptyset \). Thus, \(\underline{apr}_{a_{i}}(\mathcal{O}) \subseteq \overline{apr}_{a_{i}}(\mathcal{O})\).

Rules are induced from lower and upper approximations. Let \(\mathcal{O}\) be specified by restriction \(a_{j}=x\). Object \(o \in \underline{apr}_{a_{i}}(\mathcal{O})\) consistently supports a single rule \(a_{i} = a_{i}(o) \rightarrow a_{j} = x\). Object \(o \in \overline{apr}_{a_{i}}(\mathcal{O})\) inconsistently supports a single rule \(a_{i} = a_{i}(o) \rightarrow a_{j} = x\). The degree of consistency, called accuracy, is \(|[o]_{a_{i}} \cap \mathcal{O}|/|\mathcal{O}|\).

Since attribute \(a_{i}\) has the continuous domain, the antecedent part of single rules that individual objects support is usually different. We obtain lots of single rules, but they have a drawback for applicability. For example, let two values \(a_{i}(o)\) and \(a_{i}(o')\) be 3.65 and 3.75 for objects o and \(o'\) in \(\underline{apr}_{a_{i}}(\mathcal{O})\). When \(\mathcal{O}\) is specified by restriction \(a_{j}=x\), o and \(o'\) support single rules \(a_{i} = 3.65 \rightarrow a_{j} = x\) and \(a_{i} = 3.75 \rightarrow a_{j} = x\), respectively. By using these rules, we can say that a object having value 3.68 of \(a_{i}\), indiscernible with 3.65 under \(\delta _{a_{i}} = 0.03\), supports \(a_{i} = 3.68 \rightarrow a_{j} = x\). However, we cannot at all say anything for a rule supported by an object with value 3.70 discernible with 3.65 and 3.75. This shows that a single rule is short of applicability.

To improve the applicability of rules, we combine a series of single rules into one rule, which is called a combined rule. Let objects in U be aligned in ascending order of \(a_{i} (o)\) and be attached the serial superscript with 1 to \(N_{U}\) where \(|U| = N_{U}\). \(\underline{apr}_{a_{i}}(\mathcal{O})\) and \(\overline{apr}_{a_{i}}(\mathcal{O})\) consist of collections of objects with serial superscripts. For example, \(\underline{apr}_{a_{i}}(\mathcal{O}) = \{\cdots , o^{h}, o^{h+1}, \cdots , o^{k-1}, o^{k}, \cdots \}\) \((h \le k)\). Let \(o^{l}\) in \(\underline{apr}_{a_{i}}(\mathcal{O})\) support a single rule \(a_{i} = a_{i}(o{^l}) \rightarrow a_{j} = x\). Then, single rules derived from collection \((o^{h}, o^{h+1}, \cdots , o^{k-1}, o^{k})\) can be put into one combined rule \(a_{i} = [a_{i}(o^{h}), a_{i}(o^{k})] \rightarrow a_{j} = x\).

Next, when \(a_{j}\) is an attribute with the continuous domain, \(\mathcal{O}\) is specified by a restriction with an interval value. The interval value has the lower and the upper bounds that are existing values of attribute. Let the objects be aligned in ascending order of values of \(a_{j}\) and be attached the serial superscript with 1 to \(N_{U}\). For example, using the ordered objects, \(\mathcal{O}\) is specified like \(\mathcal{O} = \{o \mid a_{j}(o) \ge a_{j}(o^{m}) \wedge a_{j}(o) \le a_{j}(o^{n}) \}\) with \(m \le n\); in other words, \(\mathcal{O}\) is specified by restriction \(a_{j} = [a_{i}(o^{m}), a_{i}(o^{n})]\). In the case, the combined rule, derived from collection \((o^{h}, o^{h+1}, \cdots , o^{k-1}, o^{k})\), is expressed with \(a_{i} = [a_{i}(o^{h}), a_{i}(o^{k})] \rightarrow a_{j} = [a_{i}(o^{m}), a_{i}(o^{n})]\). The accuracy of the combined rule is \(\min _{h \le s \le k}|[o^{s}]_{a_{i}} \cap \mathcal{O}|/|\mathcal{O}|\).

Proposition 2

Let \(\underline{r}\) and \(\overline{r}\) be sets of combined rules obtained from \(\underline{apr}_{a_{i}}(\mathcal{O})\) and \(\overline{apr}_{a_{i}}(\mathcal{O})\), respectively. If \((a_{i} = [l, u] \rightarrow W) \in \underline{r}\), then \(\exists l' \le l, \exists u' \ge u \ (a_{i} = [l', u'] \rightarrow W) \in \overline{r}\), where \(\mathcal{O}\) is specified by restriction W.

Example 1

Information tables are depicted in Fig. 1. T0 is the original information table. U is \(\{o_{1}, o_{2}, \cdots , o_{18}, o_{19}\}\). T1, T2, and T3 are derived from T0, where some attributes are projected and objects are aligned in ascending order of values of attributes \(a_{1}\), \(a_{2}\), and \(a_{3}\), respectively.

Fig. 1.
figure 1

T0 is the original information table. T1, T2, and T3 are derived from T0.

Full size image

Let threshold \(\delta _{a_{1}}\) be 0.05. Indiscernible classes of objects are:

$$\begin{aligned}{}[o_{1}]_{a_{1}} = \{o_{1},o_{10},o_{14}\}, [o_{2}]_{a_{1}} = \{o_{2},o_{11},o_{16},o_{17}\}, [o_{3}]_{a_{1}} = \{o_{3}\}, [o_{4}]_{a_{1}} = \{o_{4}\}, \\ [o_{5}]_{a_{1}} = \{o_{5}\}, [o_{6}]_{a_{1}} = \{o_{6},o_{10},o_{15}\}, [o_{7}]_{a_{1}} = \{o_{7}\}, [o_{8}]_{a_{1}} = \{o_{8}\}, [o_{9}]_{a_{1}} = \{o_{9}\}, \\ [o_{10}]_{a_{1}} = \{o_{1},o_{6},o_{10},o_{14},o_{15}\}, [o_{11}]_{a_{1}} = \{o_{2},o_{11},o_{16}\}, [o_{12}]_{a_{1}} = \{o_{12}\}, \\ [o_{13}]_{a_{1}} = \{o_{13},o_{19}\}, [o_{14}]_{a_{1}} = \{o_{1},o_{10},o_{14}\}, [o_{15}]_{a_{1}} = \{o_{6},o_{10},o_{15}\}, \\ [o_{16}]_{a_{1}} = \{o_{2},o_{11},o_{16}\}, [o_{17}]_{a_{1}} = \{o_{2},o_{17}\}, [o_{18}]_{a_{1}} = \{o_{18}\}, [o_{19}]_{a_{1}} = \{o_{13},o_{19}\}. \end{aligned}$$

When \(\mathcal{O}\) is specified by restriction \(a_{4}=b\), \(\mathcal{O} = \{o_{1}, o_{2}, o_{5}, o_{9}, o_{11}, o_{14}, o_{16}, \) \(o_{19}\}\). Let \(\mathcal{O}\) be approximated by objects on attribute \(a_{1}\) with continuous values. Using formulas (3) and (4), lower and upper approximations are:

$$\begin{aligned} \underline{apr}_{a_{1}}(\mathcal{O})= & {} \{ o_{5}, o_{9}, o_{11}, o_{16} \}, \\ \overline{apr}_{a_{1}}(\mathcal{O})= & {} \{ o_{1}, o_{2}, o_{5}, o_{9}, o_{10}, o_{11}, o_{13}, o_{14}, o_{16}, o_{17}, o_{19} \}. \end{aligned}$$

Information table T1 is derived from information table T0, where objects are aligned in ascending order of values of attribute \(a_{1}\) and are attached the serial superscript from 1 to 19. The above approximations are described using the serial superscript as follows:

$$\begin{aligned} \underline{apr}_{a_{1}}(\mathcal{O})= & {} \{ o^{7}, o^{8}, o^{14}, o^{15} \}, \\ \overline{apr}_{a_{1}}(\mathcal{O})= & {} \{ o^{5}, o^{6}, o^{7}, o^{8}, o^{11}, o^{12}, o^{13}, o^{14}, o^{15}, o^{16},o^{17} \}, \end{aligned}$$

where

$$\begin{aligned} o^{5}= & {} o_{17}, \ o^{6} =o_{2}, \ o^{7} =o_{16}, \ o^{8} =o_{11}, \ o^{11}=o_{10}, \ o^{12}=o_{1}, \\ o^{13}= & {} o_{14}, \ o^{14} =o_{9}, \ o^{15} =o_{5}, \ o^{16}=o_{19}, \ o^{17} =o_{13}. \end{aligned}$$

From the lower approximation, consistent combined rules are

$$\begin{aligned} a_{1} = [2.95, 2.97] \rightarrow a_{4} = b, \ a_{1} = [3.22, 3.42] \rightarrow a_{4} = b, \end{aligned}$$

from collections \(\{o^{7}, o^{8}\}\) and \(\{o^{14}, o^{15}\}\), respectively, where \(a_{1}(o^{7}) = 2.95, \ a_{1}(o^{8}) = 2.97, \ a_{1}(o^{14}) = 3.22,\) and \(a_{1}(o^{15}) = 3.42\). From the upper approximation, inconsistent combined rules are

$$\begin{aligned} a_{1} = [2.89, 2.97] \rightarrow a_{4} = b, \ a_{1} = [3.07, 3.91] \rightarrow a_{4} = b, \end{aligned}$$

from collections \(\{o^{5}, o^{6},o^{7}, o^{8}\}\) and \(\{o^{11}, o^{12},o^{13},o^{14}, o^{15}, o^{16}, o^{17}\}\), respectively, where \(a_{1}(o^{5}) = 2.89\), \(a_{1}(o^{11}) = 3.07\), and \(a_{1}(o^{17}) = 3.91\).

Next, we consider the case where \(\mathcal{O}\) is specified by \(a_{3}\) with the continuous domain. Information table T3 is derived from T0, where the objects are aligned in ascending order of values of \(a_{3}\) and are attached the serial superscript from 1 to 19. Using lower bound \(a_{3}(o^{5}) = a_{3}(o_{15}) = 3.22\) and upper bound \(a_{3}(o^{10}) = a_{3}(o_{8}) = 3.49\), \(\mathcal{O} = \{o^{5}, o^{6}, o^{7}, o^{8}, o^{9}, o^{10}\} = \{o_{2}, o_{3}, o_{8}, o_{15}, o_{16}, o_{17} \}\). We approximate \(\mathcal{O}\) by attribute \(a_{2}\). Information table T2 where the objects are aligned in ascending order of values of \(a_{2}\) is derived from T0. Let \(\delta _{a_{2}}\) be 0.05. Indiscernible classes of objects are:

$$\begin{aligned}{}[o_{1}]_{a_{2}} = \{o_{1},o_{4},o_{7},o_{8}\}, [o_{2}]_{a_{2}} = \{o_{2},o_{3},o_{16}\}, [o_{3}]_{a_{2}} = \{o_{2},o_{3},o_{13},o_{16}\}, \\ [o_{4}]_{a_{2}} = \{o_{1},o_{4},o_{7},o_{8}\}, [o_{5}]_{a_{2}} = \{o_{5}\}, [o_{6}]_{a_{2}} = \{o_{6}\}, [o_{7}]_{a_{2}} = \{o_{1},o_{4},o_{7}\}, \\ [o_{8}]_{a_{2}} = \{o_{8}\}, [o_{9}]_{a_{2}} = \{o_{9}\}, [o_{10}]_{a_{2}} = \{o_{10}\}, [o_{11}]_{a_{2}} = \{o_{11},o_{18}\}, [o_{12}]_{a_{2}} = \{o_{12}\}, \\ [o_{13}]_{a_{2}} = \{o_{3},o_{13}\}, [o_{14}]_{a_{2}} = \{o_{14}\}, [o_{15}]_{a_{2}} = \{o_{15}\}, [o_{16}]_{a_{2}} = \{o_{2},o_{3},o_{16}\}, \\ [o_{17}]_{a_{2}} = \{o_{17}\}, [o_{18}]_{a_{2}} = \{o_{11},o_{18}\}, [o_{19}]_{a_{2}} = \{o_{19}\}. \end{aligned}$$

Using formulas (3) and (4), lower and upper approximations are:

$$\begin{aligned} \underline{apr}_{a_{2}}(\mathcal{O}) = \{ o_{2}, o_{8}, o_{15}, o_{16}, o_{17} \}, \ \overline{apr}_{a_{2}}(\mathcal{O}) = \{ o_{1}, o_{2}, o_{3}, o_{4}, o_{8}, o_{13}, o_{15}, o_{16}, o_{17} \}. \end{aligned}$$

Using information table T2 where objects are aligned in ascending order of values of attribute \(a_{2}\) and are attached the serial superscript from 1 to 19, the above approximations are described as follows:

$$\begin{aligned} \underline{apr}_{a_{2}}(\mathcal{O}) = \{ o^{6}, o^{7}, o^{9}, o^{10}, o^{11} \}, \ \overline{apr}_{a_{2}}(\mathcal{O}) = \{ o^{4}, o^{5}, o^{6}, o^{7}, o^{9}, o^{10}, o^{11}, o^{12}, o^{13} \}, \end{aligned}$$

From the lower approximation, consistent combined rules are

$$\begin{aligned} a_{2} = [3.11, 3.29] \rightarrow a_{3} = [3.22, 3.49], \ a_{2} = [3.51, 3.65] \rightarrow a_{3} = [3.22, 3.49], \end{aligned}$$

where \(a_{2}(o^{6}) = 3.11, \ a_{2}(o^{7}) = 3.29, \ a_{2}(o^{9}) = 3.51,\) and \(a_{2}(o^{11}) = 3.65\). From the upper approximation, inconsistent combined rules are

$$\begin{aligned} a_{2} = [2.98, 3.29] \rightarrow a_{3} = [3,22, 3.49], \ a_{2} = [3.51, 3.71] \rightarrow a_{3} = [3.22, 3.49], \end{aligned}$$

where \(a_{2}(o^{4}) = 2.98\) and \(a_{2}(o^{13}) = 3.71\).

This example shows that a combined rule is more applicable than single rules. For example, using the above consistent combined rule \(a_{2} = [3.11, 3.29] \rightarrow a_{3} = [3.22, 3.49]\), we can say that an object with 3.20 for a value of attribute \(a_{2}\) supports this rule, because 3.20 is included in interval [3.11, 3.29]. On the other hand, using single rules \(a_{2} = 3.11 \rightarrow a_{3} = [3.22, 3.49]\) and \(a_{2} = 3.29 \rightarrow a_{3} = [3.22, 3.49]\), we cannot say what rule the object supports under a threshold 0.05.

For formulas on sets A and B of attributes,

$$\begin{aligned}{}[o]_{A}= & {} \cap _{a_{i} \in A}[o]_{a_{i}}, \end{aligned}$$
(5)
$$\begin{aligned} \underline{apr}_{A}(\mathcal{O})= & {} \{o \mid [o]_{A} \subseteq \mathcal{O}\}, \end{aligned}$$
(6)
$$\begin{aligned} \overline{apr}_{A}(\mathcal{O})= & {} \{o \mid [o]_{A} \cap \mathcal{O} \ne \emptyset \}. \end{aligned}$$
(7)

3 Rough Sets by Indiscernible Classes in Incomplete Information Systems with Continuous Domains

An information table with incomplete information is called an incomplete information system. In incomplete information systems, \(a_{i} : U \rightarrow s_{a_{i}}\) for every \(a_{i} \in AT\) where \(s_{a_{i}}\) is a set of values over domain \(D(a_{i})\) of attribute \(a_{i}\) or an interval on \(D(a_{i})\). Single value v with \(v \in a_{i}(o)\) or \(v \subseteq a_{i}(o)\) is a possible value that may be the actual one as the value of attribute \(a_{i}\) in object o. The possible value is the actual one if \(a_{i}(o)\) is a single value.

In an incomplete information systemFootnote 1, an indiscernible class is a possible class that may be the actual indiscernible class. We have lots of indiscernible classes. Family \(\mathcal{F}[o]_{a_{i}}\) of indiscernible class is:

$$\begin{aligned} \mathcal{F}[o]_{a_{i}} = \{C[o]_{a_{i}} \cup e \mid e \in \mathcal{P}(P[o]_{a_{i}} \backslash C[o]_{a_{i}})\}, \end{aligned}$$
(8)

where \(\mathcal{P}(P[o]_{a_{i}} \backslash C[o]_{a_{i}})\) is the power set of \(P[o]_{a_{i}} \backslash C[o]_{a_{i}}\), and certainly indiscernible class \(C[o]_{a_{i}}\) and possibly one \(P[o]_{a_{i}}\) on attribute \(a_{i}\) of object o are:

$$\begin{aligned} C[o]_{a_{i}}= & {} \{o' \mid o' = o \vee (\forall u \in a_{i}(o) \forall v \in a_{i}(o') |u - v| \le \delta _{a_{i}}) \}, \end{aligned}$$
(9)
$$\begin{aligned} P[o]_{a_{i}}= & {} \{o' \mid o' = o \vee (\exists u \in a_{i}(o) \exists v \in a_{i}(o') |u - v| \le \delta _{a_{i}}) \}. \end{aligned}$$
(10)

The family of indiscernible classes has a lattice structure. The minimal element is the certainly indiscernible class and the maximal one is the possibly indiscernible class. In other words, \(C[o]_{a_{i}}\) is the minimum indiscernible class and \(P[o]_{a_{i}}\) is the maximum indiscernible class. Objects in the certainly indiscernible class of o are certainly indistinguishable with o. Objects in the possibly indiscernible class of o are possibly indistinguishable with o.

We can derive not the actual, but certain and possible approximations from the viewpoint of certainty and possibility, as Lipski obtained in query processing under incomplete information [5, 6]. We cannot definitely obtain whether or not an object belongs to the actual approximations, but we can know whether or not the object certainly and/or possibly belongs to approximations. Therefore, we show certain approximations (resp. possible approximations) whose object certainly (resp. possibly) belongs to the actual approximations.

Let \(\mathcal{O}\) be a set of objects. Using certainly and possibly indiscernible classes, certain lower approximation \(C\underline{apr}_{a_{i}}(\mathcal{O})\) and possible one \(P\underline{apr}_{a_{i}}(\mathcal{O})\) for \(a_{i}\) are:

$$\begin{aligned} C\underline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid P[o]_{a_{i}} \subseteq \mathcal{O}\}, \end{aligned}$$
(11)
$$\begin{aligned} P\underline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid C[o]_{a_{i}} \subseteq \mathcal{O}\}. \end{aligned}$$
(12)

Similarly, Certain upper approximation \(C\overline{apr}_{a_{i}}(\mathcal{O})\) and possible one \(P\overline{apr}_{a_{i}}(\mathcal{O})\) are:

$$\begin{aligned} C\overline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid C[o]_{a_{i}} \cap \mathcal{O} \not = \emptyset \}, \end{aligned}$$
(13)
$$\begin{aligned} P\overline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid P[o]_{a_{i}} \cap \mathcal{O} \not = \emptyset \}. \end{aligned}$$
(14)

As with the case of nominal attributes [10], the following proposition holds.

Proposition 3

\(C\underline{apr}_{a_{i}}(\mathcal{O})\) \(\subseteq \) \(P\underline{apr}_{a_{i}}(\mathcal{O})\) \(\subseteq \) \(\mathcal{O}\) \(\subseteq \) \(C\overline{apr}_{a_{i}}(\mathcal{O})\) \(\subseteq \) \(P\overline{apr}_{a_{i}}(\mathcal{O})\).

Using four approximations denoted by formulae (11)–(14), lower and upper approximations are expressed in interval sets, as is described in [11]Footnote 2, as follows:

$$\begin{aligned} \underline{apr}_{a_{i}}^{\bullet }(\mathcal{O}) = [C\underline{apr}_{a_{i}}(\mathcal{O}), P\underline{apr}_{a_{i}}(\mathcal{O}) ], \end{aligned}$$
(15)
$$\begin{aligned} \overline{apr}_{a_{i}}^{\bullet }(\mathcal{O}) = [C\overline{apr}_{a_{i}}(\mathcal{O}), P\overline{apr}_{a_{i}}(\mathcal{O}) ]. \end{aligned}$$
(16)

Certain and possible approximations are the lower and upper bounds of the actual approximation. The two approximations \(\underline{apr}_{a_{i}}^{\bullet }(\mathcal{O})\) and \(\underline{apr}_{a_{i}}^{\bullet }(\mathcal{O})\) depend on each other; namely, the complementarity property \(\underline{apr}_{a_{i}}^{\bullet }(\mathcal{O}) = U - \overline{apr}_{a_{i}}^{\bullet }(U - \mathcal{O})\) linked with them holds, as is so in complete information systems.

When objects in \(\mathcal{O}\) are specified by attribute \(a_{j}\) with incomplete information, \(\mathcal{O}\) is specified by using an element in domain \(D(a_{j})\). In the case where \(\mathcal{O}\) is specified by restriction \(a_{j}=x\) with \(x \in D(a_{j})\), four approximations: certain lower, possible lower, certain upper, and possible upper ones, are:

$$\begin{aligned} C\underline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid P[o]_{a_{i}} \subseteq C\mathcal{O}_{a_{j} = x}\}, \end{aligned}$$
(17)
$$\begin{aligned} P\underline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid C[o]_{a_{i}} \subseteq P\mathcal{O}_{a_{j} = x} \}, \end{aligned}$$
(18)
$$\begin{aligned} C\overline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid C[o]_{a_{i}} \cap C\mathcal{O}_{a_{j} = x} \not = \emptyset \}, \end{aligned}$$
(19)
$$\begin{aligned} P\overline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid P[o]_{a_{i}} \cap P\mathcal{O}_{a_{j} = x} \not = \emptyset \}, \end{aligned}$$
(20)

where

$$\begin{aligned} C\mathcal{O}_{a_{j} =x }= & {} \{o \in \mathcal{O}\mid a_{j}(o) = x \}, \end{aligned}$$
(21)
$$\begin{aligned} P\mathcal{O}_{a_{j} = x}= & {} \{o \in \mathcal{O} \mid a_{j}(o) \supseteq x \}. \end{aligned}$$
(22)

For rule induction, we can say as follows:

  • \(o \in C\underline{apr}_{a_{i}}(\mathcal{O})\) certainly and consistently supports rule \(a_{i} = a_{i}(o) \rightarrow a_{j}(o) = x\).

  • \(o \in C\overline{apr}_{a_{i}}(\mathcal{O})\) certainly and inconsistently supports rule \(a_{i} = a_{i}(o) \rightarrow a_{j}(o) = x\).

  • \(o \in P\underline{apr}_{a_{i}}(\mathcal{O})\) possibly and consistently supports \(a_{i} = a_{i}(o) \rightarrow a_{j}(o) = x\).

  • \(o \in P\overline{apr}_{a_{i}}(\mathcal{O})\) possibly and inconsistently supports \(a_{i} = a_{i}(o) \rightarrow a_{j}(o) = x\).

We create combined rules from them.

Let \(U^{C}_{a_{i}}\) and \(U^{I}_{a_{i}}\) be sets of objects having complete information and incomplete information for \(a_{i}\). \(o \in U^{C}_{a_{i}}\) is aligned in ascending order of \(a_{i}(o)\) and is attached the serial superscript with 1 to \(N_{_i}^{C}\) where \(|U^{C}_{a_{i}}| = N_{i}^{C}\). Objects \(o \in (C\underline{apr}_{a_{i}}(\mathcal{O}) \cap U^{C}_{a_{i}})\), \(o \in (C\overline{apr}_{a_{i}}(\mathcal{O}) \cap U^{C}_{a_{i}})\), \(o \in (P\underline{apr}_{a_{i}}(\mathcal{O}) \cap U^{C}_{a_{i}})\), and \(o \in (P\overline{apr}_{a_{i}}(\mathcal{O}) \cap U^{C}_{a_{i}})\) are aligned in ascending order of \(a_{i}(o)\). And then they are expressed by a sequence of collections of objects with a serial superscript like \(\{\cdots , o^{h}, o^{h+1}, \cdots , o^{k-1}, o^{k}, \cdots \}\) \((h \le k)\). From collection \((o^{h}, o^{h+1}, \cdots , o^{k-1}, o^{k})\), four types of combined rules expressed with \(a_{i} = [l, u] \rightarrow a_{j} = x\) are derived. For a certain and consistent combined rule,

$$\begin{aligned} l= \min (a_{i}(o^{h}), \min _{Y} e) \ \text{ and } \ u = \max (a_{i}(o^{k}), \max _{Y} e), \qquad \qquad \qquad \qquad \quad \nonumber \\ Y = \left\{ \begin{array}{ll} e< a_{i}(o^{k+1}), &{} \ \text{ for } h = 1 \wedge k \ne N_{i}^{C} \\ a_{i}(o^{h-1})< e< a_{i}(o^{k+1}), &{} \ \text{ for } h \ne 1 \wedge k \ne N_{i}^{C} \\ a_{i}(o^{h-1}) < e, &{} \ \text{ for } h \ne 1 \wedge k = N_{i}^{C} \nonumber \\ \end{array} \right. \\ \text{ with } \ e \in a_{i}(o') \wedge o' \in X, \end{aligned}$$
(23)

where X is \((C\underline{apr}_{a_{i}}(\mathcal{O}) \cap U^{I}_{a_{i}})\).

For certain and inconsistent, possible and consistent, possible and inconsistent combined rules, X is \((C\overline{apr}_{a_{i}}(\mathcal{O}) \cap U^{I}_{a_{i}})\), \((P\underline{apr}_{a_{i}}(\mathcal{O}) \cap U^{I}_{a_{i}})\), and \((P\overline{apr}_{a_{i}}(\mathcal{O}) \cap U^{I}_{a_{i}})\), respectively.

Proposition 4

Let \(C\underline{r}\) and \(P\underline{r}\) be sets of combined rules obtained from \(C\underline{apr}_{a_{i}}(\mathcal{O})\) and \(P\underline{apr}_{a_{i}}(\mathcal{O})\), respectively. When \(\mathcal{O}\) is specified by restriction W, if \((a_{i} = [l, u] \rightarrow W) \in C\underline{r}\), then \(\exists l' \le l, \exists u' \ge u \ (a_{i} = [l', u'] \rightarrow W) \in P\underline{r}\).

Proposition 5

Let \(C\overline{r}\) and \(P\overline{r}\) be sets of combined rules obtained from \(C\overline{apr}_{a_{i}}(\mathcal{O})\) and \(P\overline{apr}_{a_{i}}(\mathcal{O})\), respectively. When \(\mathcal{O}\) is specified by restriction W, if \((a_{i} = [l, u] \rightarrow W) \in C\overline{r}\), then \(\exists l' \le l, \exists u' \ge u \ (a_{i} = [l', u'] \rightarrow W) \in P\overline{r}\).

Proposition 6

Let \(C\underline{r}\) and \(C\overline{r}\) be sets of combined rules obtained from \(C\underline{apr}_{a_{i}}(\mathcal{O})\) and \(C\overline{apr}_{a_{i}}(\mathcal{O})\), respectively. When \(\mathcal{O}\) is specified by restriction W, if \((a_{i} = [l, u] \rightarrow W) \in C\underline{r}\), then \(\exists l' \le l, \exists u' \ge u \ (a_{i} = [l', u'] \rightarrow W) \in C\overline{r}\).

Proposition 7

Let \(P\underline{r}\) and \(P\overline{r}\) be sets of combined rules obtained from \(P\underline{apr}_{a_{i}}(\mathcal{O})\) and \(P\overline{apr}_{a_{i}}(\mathcal{O})\), respectively. When \(\mathcal{O}\) is specified by restriction W, if \((a_{i} = [l, u] \rightarrow W) \in P\underline{r}\), then \(\exists l' \le l, \exists u' \ge u \ (a_{i} = [l', u'] \rightarrow W) \in P\overline{r}\).

Example 2

Let \(\mathcal{O}\) be specified by restriction \(a_{4}=b\) in IT of Fig. 2.

Fig. 2.
figure 2

Information table IT with incomplete information

Full size image
$$\begin{aligned} C\mathcal{O}_{a_{4} =b}= & {} \{o_{2}, o_{5}, o_{9}, o_{11}, o_{14}, o_{16} \}, \\ P\mathcal{O}_{a_{4} = b}= & {} \{o_{1}, o_{2}, o_{5}, o_{9}, o_{11}, o_{14}, o_{16}, o_{17}, o_{19}\}. \end{aligned}$$

Each \(C[o_{i}]_{a_{1}}\) for \(i =1,\dots ,19\) is, respectively,

$$\begin{aligned} &C[o_{1}]_{a_{1}} = \{o_{1}, o_{10}\}, C[o_{2}]_{a_{1}} = \{o_{2}, o_{11}, o_{16}, o_{17}\}, C[o_{3}]_{a_{1}} = \{o_{3}\}, \\ &C[o_{4}]_{a_{1}} = \{o_{4}\}, C[o_{5}]_{a_{1}} = \{o_{5}\}, C[o_{6}]_{a_{1}} = \{o_{6}, o_{10}, o_{15}\}, C[o_{7}]_{a_{1}} = \{o_{7}\}, \\ &C[o_{8}]_{a_{1}} = \{o_{8}\}, C[o_{9}]_{a_{1}} = \{o_{9}\}, C[o_{10}]_{a_{1}} = \{o_{1}, o_{6}, o_{10}, o_{14}, o_{15} \}, \\ &C[o_{11}]_{a_{1}} = \{o_{2}, o_{11}, o_{16}\}, C[o_{12}]_{a_{1}} = \{o_{12}\}, C[o_{13}]_{a_{1}} = \{o_{13}, o_{19}\}, \\ &C[o_{14}]_{a_{1}} = \{o_{10}, o_{14}\}, C[o_{15}]_{a_{1}} = \{o_{6}, o_{10}, o_{15}\}, C[o_{16}]_{a_{1}} = \{o_{2}, o_{11}, o_{16}\}, \\ &C[o_{17}]_{a_{1}} = \{o_{2}, o_{17}\}, C[o_{18}]_{a_{1}} = \{o_{18}\}, C[o_{19}]_{a_{1}} = \{o_{13}, o_{19}\}. \end{aligned}$$

Each \(P[o_{i}]_{a_{1}}\) for \(i =1,\dots ,19\) is, respectively,

$$\begin{aligned} &P[o_{1}]_{a_{1}} = \{o_{1}, o_{6}, o_{10}, o_{14}, o_{15} \}, P[o_{2}]_{a_{1}} = \{o_{2}, o_{9}, o_{11}, o_{16}, o_{17}\}, P[o_{3}]_{a_{1}} = \{o_{3}\}, \\ &P[o_{4}]_{a_{1}} = \{o_{4}\}, P[o_{5}]_{a_{1}} = \{o_{5}\}, P[o_{6}]_{a_{1}} = \{o_{1}, o_{6}, o_{10}, o_{15}\}, P[o_{7}]_{a_{1}} = \{o_{7}\}, \\ &P[o_{8}]_{a_{1}} = \{o_{8}\}, P[o_{9}]_{a_{1}} = \{o_{2}, o_{9}, o_{11}, o_{16}, o_{17}\}, P[o_{10}]_{a_{1}} = \{o_{1}, o_{6}, o_{10}, o_{14}, o_{15} \}, \\ &P[o_{11}]_{a_{1}} = \{o_{2}, o_{9}, o_{11}, o_{16}, o_{17}\}, P[o_{12}]_{a_{1}} = \{o_{12}\}, P[o_{13}]_{a_{1}} = \{o_{13}, o_{19}\}, \\ &P[o_{14}]_{a_{1}} = \{o_{1}, o_{10}, o_{14}\}, P[o_{15}]_{a_{1}} = \{o_{1}, o_{6}, o_{10}, o_{15} \}, \\ &P[o_{16}]_{a_{1}} = \{o_{2}, o_{9}, o_{11}, o_{16}, o_{17}\}, P[o_{17}]_{a_{1}} = \{o_{2}, o_{9}, o_{11}, o_{16}, o_{17}\}, \\ &P[o_{18}]_{a_{1}} = \{o_{18}\}, P[o_{19}]_{a_{1}} = \{o_{13}, o_{19}\}. \end{aligned}$$

Four approximations are:

$$\begin{aligned} C\underline{apr}_{a_{1}}(\mathcal{O})= & {} \{o_{5}\}, \\ P\underline{apr}_{a_{1}}(\mathcal{O})= & {} \{o_{2}, o_{5}, o_{9}, o_{11}, o_{16}, o_{17}\}, \\ C\overline{apr}_{a_{1}}(\mathcal{O})= & {} \{o_{2}, o_{5}, o_{9}, o_{10}, o_{11}, o_{14}, o_{16}, o_{17}\}, \\ P\overline{apr}_{a_{1}}(\mathcal{O})= & {} \{o_{1}, o_{2}, o_{5}, o_{6}, o_{9}, o_{10}, o_{11}, o_{13}, o_{14}, o_{15}, o_{16}, o_{17},o_{19}\}. \end{aligned}$$
$$\begin{aligned} U_{a{1}}^{C}= & {} \{o_{2}, o_{3}, o_{4}, o_{5}, o_{6}, o_{7}, o_{8}, o_{10}, o_{12}, o_{13}, o_{14}, o_{15}, o_{16} \}, \\ U_{a{1}}^{I}= & {} \{o_{1}, o_{9}, o_{11}, o_{17}, o_{18}, o_{19} \} \end{aligned}$$

Objects in \(U_{a{1}}^{C}\) are aligned in ascending order of values of attribute \(a_{1}\) as follows:

$$\begin{aligned} o_{3}, o_{12}, o_{7}, o_{2}, o_{16}, o_{6}, o_{15}, o_{10}, o_{14}, o_{5}, o_{13}, o_{8}, o_{4} \end{aligned}$$

A series of superscripts is attached to these objects:

$$\begin{aligned} o^{1}, o^{2}, o^{3}, o^{4}, o^{5}, o^{6}, o^{7}, o^{8}, o^{9}, o^{10}, o^{11}, o^{12}, o^{13}, \end{aligned}$$

where \(o^{1}=o_{3}, o^{2}=o_{12}, \dots , o^{13}=o_{4}\). Using objects with the superscript, the four approximations are expressed as follows:

$$\begin{aligned} C\underline{apr}_{a_{1}}(\mathcal{O})= & {} \{o^{10} \}, \\ P\underline{apr}_{a_{1}}(\mathcal{O})= & {} \{o^{4}, o^{5}, o^{10}, o_{9}, o_{11}, o_{17}\}, \\ C\overline{apr}_{a_{1}}(\mathcal{O})= & {} \{o^{4}, o^{5}, o^{8}, o^{9}, o^{10}, o_{9}, o_{11}, o_{17}\}, \\ P\overline{apr}_{a_{1}}(\mathcal{O})= & {} \{o^{4}, o^{5}, o^{6}, o^{7}, o^{8}, o^{9}, o^{10}, o^{11}, o_{1}, o_{9}, o_{11}, o_{17},o_{19}\}. \end{aligned}$$

where objects with a superscript and with a subscript have complete and incomplete information for attribute \(a_{1}\), respectively; namely,

$$\begin{aligned} C\underline{apr}_{a_{1}}(\mathcal{O}) \cap U^{C}_{a_{1}}= & {} \{o^{10}\}, C\underline{apr}_{a_{1}}(\mathcal{O}) \cap U^{I}_{a_{1}} = \emptyset , \\ P\underline{apr}_{a_{1}}(\mathcal{O}) \cap U^{C}_{a_{1}}= & {} \{o^{4}, o^{5}, o^{10}\}, P\underline{apr}_{a_{1}}(\mathcal{O}) \cap U^{I}_{a_{1}} = \{o_{9}, o_{11}, o_{17}\}, \\ C\overline{apr}_{a_{1}}(\mathcal{O}) \cap U^{C}_{a_{1}}= & {} \{o^{4}, o^{5}, o^{8}, o^{9}, o^{10}\}, C\overline{apr}_{a_{1}}(\mathcal{O}) \cap U^{I}_{a_{1}} = \{o_{9}, o_{11}, o_{17}\}, \\ P\overline{apr}_{a_{1}}(\mathcal{O}) \cap U^{C}_{a_{1}}= & {} \{o^{4}, o^{5}, o^{6}, o^{7}, o^{8}, o^{9}, o^{10}, o^{11}\}, P\overline{apr}_{a_{1}}(\mathcal{O}) \cap U^{I}_{a_{1}} = \{o_{1}, o_{9}, o_{11}, o_{17}, o_{19}\}. \end{aligned}$$

From these expressions, four types combined rules are derived. For certain and consistent rules,

$$\begin{aligned} a_{1} = 3.42 \rightarrow a_{4} = b. \end{aligned}$$

For possible and consistent rules,

$$\begin{aligned} a_{1} = [2.89,2.97] \rightarrow a_{4} = b, \ a_{1} = [3.22,3.42] \rightarrow a_{4} = b. \end{aligned}$$

For certain and inconsistent rules,

$$\begin{aligned} a_{1} = [2.89,2.97] \rightarrow a_{4} = b, \ a_{1} = [3.07,3.42] \rightarrow a_{4} = b. \end{aligned}$$

For possible and inconsistent rules,

$$\begin{aligned} a_{1} = [2.89,3.92] \rightarrow a_{4} = b. \end{aligned}$$

Last, we describe the case where \(o \in \mathcal{O}\) is specified by numerical attribute \(a_{j}\) with incomplete information. \(o \in U^{C}_{a_{j}}\) is aligned in ascending order of \(a_{j}(o)\) and is attached with the serial superscript with 1 to \(N_{j}^{C}\) where \(|U^{C}_{a_{j}}| = N_{j}^{C}\). We specify \(\mathcal{O}\) by \(a_{j}(o^{m}) \in U^{C}_{a_{j}}\) and \(a_{j}(o^{n}) \in U^{C}_{a_{j}}\) with \(m \le n\).

$$\begin{aligned} C\underline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid P[o]_{a_{i}} \subseteq C\mathcal{O}_{[a_{j}(o^{m}),a_{j}(o^{n})]}\}, \end{aligned}$$
(24)
$$\begin{aligned} P\underline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid C[o]_{a_{i}} \subseteq P\mathcal{O}_{[a_{j}(o^{m}),a_{j}(o^{n})]} \}, \end{aligned}$$
(25)
$$\begin{aligned} C\overline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid C[o]_{a_{i}} \cap C\mathcal{O}_{[a_{j}(o^{m}),a_{j}(o^{n})]} \not = \emptyset \}, \end{aligned}$$
(26)
$$\begin{aligned} P\overline{apr}_{a_{i}}(\mathcal{O})= & {} \{o \mid P[o]_{a_{i}} \cap P\mathcal{O}_{[a_{j}(o^{m}),a_{j}(o^{n})]} \not = \emptyset \}, \end{aligned}$$
(27)

where

$$\begin{aligned} C\mathcal{O}_{[a_{j}(o^{m}),a_{j}(o^{n})]}= & {} \{o \in \mathcal{O}\mid a_{j}(o) \subseteq [a_{j}(o^{m}), a_{j}(o^{n})] \}, \end{aligned}$$
(28)
$$\begin{aligned} P\mathcal{O}_{[a_{j}(o^{m}),a_{j}(o^{n})]}= & {} \{o \in \mathcal{O} \mid a_{j}(o) \cap [a_{j}(o^{m}), a_{j}(o^{n})] \not = \emptyset \}. \end{aligned}$$
(29)

\(o \in U^{C}_{a_{j}}\) is aligned in ascending order of \(a_{j}(o)\) and is attached the serial superscript with 1 to \(N_{j}^{C}\). Now, \(\mathcal{O}\) is specified by attribute values \(a_{j}(o^{m})\) and \(a_{j}(o^{n})\) with \(o^{m} \in U^{C}_{a_{j}}\) and \(o^{n} \in U^{C}_{a_{j}}\). \(o \in U^{C}_{a_{i}}\) is aligned in ascending order of \(a_{i}(o)\) is attached the serial superscript with 1 to \(N_{i}^{C}\). Also, four types of combined rules with \(a_{i} = [l, u] \rightarrow a_{j} = [a_{j}(o^{m}), a_{j}(o^{n}) ]\) are obtained: certain and consistent, certain and inconsistent, possible and consistent, and possible and inconsistent combined rules.

These types of combined rules are obtained in incomplete information table IT in Fig. 2. For example, let \(\mathcal{O}\) be specified by numerical attribute \(a_{3}\) with incomplete information. When \(\mathcal{O}\) is approximated on numerical attribute \(a_{2}\) with incomplete information, the four types of combined rules are derived.

4 Conclusions

We have described rough sets and rule induction from them in information tables with continuous domains. First, we have dealt with complete information tables. Rough sets are obtained from indiscernible classes. Individual objects that belongs to the rough sets support single rules. The single rules are short of applicability. To improve the applicability of rules, we have put a series of single rules derived from the rough sets into one combined rule. The combined rule is expressed by using intervals.

Second, we have dealt with incomplete information tables. Incomplete information is depicted in a disjunctive set of values or an interval of values. We have dealt with it from viewpoints of certainty and possibility, as was introduced by Lipski in the field of incomplete databases. Lots of indiscernible classes are derived. The family of indiscernible classes is expressed by a lattice having the minimal and maximal elements. The number of indiscernible classes increases exponentially as the number of attribute values with incomplete information grows. However, approximations are obtained by using the minimal and the maximal indiscernible classes. Therefore, we have no difficulty of computational complexity. By using the minimal and the maximal indiscernible classes, four types approximations: certain lower, certain upper, possible lower, and possible upper approximations are obtained, as is so in incomplete information tables with nominal attributes. From these approximations, we have derived four types of combined rules that are expressed by using interval values: certain and consistent, certain and inconsistent, possible and consistent, and possible and inconsistent combined rules. The combined rules are more applicable than single ones.