**5.1 Definition of the interpretations of objective graphs**

For an objective graph �, we define a set [[�]], as follows.

**Case 1.** Let � be an objective graph defined in Case 1 of the definition of objective graphs. Then, [[�]]:={c∈C | *c.A1*=*a1* ⋀ …⋀ *c*.*An*=*an* },4 where *c.Ai* is the value of the attribute *Ai* on *c*.

<sup>2</sup> For sets *X* and *Y* with *Y X* and for a function *f* on *X*, *f*|*Y* denotes the function of *Y* that is defined by *f*|*Y*(*y*) := *f*(*y*) for all y*Y*. We often refer to *f*|*Y* as the restriction of *f* to *Y*.

<sup>3</sup> For sets *X*, *Y* with *YX*, *X*\*Y* denotes the set {*xX*| *x*Y}.

Representation System for Quality Indicators by Ontology 205

Moreover, consider Table 2 corresponding to [patient], which is defined in Figure 3 and

which has attributes ⟨result⟩ ��� ⟨����� ����⟩� ��� ��� �*1* be the set of tuples in Table 2.

Id Name (name) Sex (sex) Blood type (blood type)

Table 2. The table generated from the concept of a stakeholder [patient] with tuples �*2*.

Thus, the interpretation of �� ��s�� �� �*2* is { c∈�*2*|∃*x1*∈[[�*1*]] 〈subject (of a state)⟩(c, *x1*) },

A quantifying concept plays a role in a function that has an objective graph and optional parameters as input data and that outputs a numerical value. In general, one can classify quantifying concepts into three types. In the following, we explain each type of quantifying concept. We describe a quantifying concept by ⫷name of a quantifying concept⫸. Note that

For a finite set S, the summation of numbers obtained from elements of S is called the total number of S. For example, if each element is assigned to 1 as the existence of the element, then the total number is the same as the cardinality of S. The quantifying concept ⫷cardinality⫸ is regarded as a function that has an objective graph � as input data and that

For a concept S, attributes *A1*,…, *An* of S, and the real-valued function f on the set of values of instances of S with respect to *A1*,…, *An*, the summation Σ <sup>s</sup>∈<sup>S</sup> *f*(*s.A1*,..., *s.An*) is called the total attribute number of S with respect to *A1*,…, *An* and *f*, where *s.Ai* denotes the value of an

The quantifying concept ⫷total attribute number⫸ is regarded as a function that has the

⫷total attribute number⫸ outputs the total attribute number of [[�]] with respect to *A1*,...,

we often identify a concept with a set and that all sets are considered to be finite.

*<sup>3</sup>*}. Here, each tuple*2i* denotes the tuple in �*<sup>2</sup>*

*<sup>1</sup>*, tuple*<sup>2</sup>*

*<sup>2</sup>*, tuple*<sup>2</sup>*

*<sup>3</sup>*}.

*P1* Alice Johnson female A *P2* Richard Miller male O *P3* Robert Williams male AB *P4* William Brown male B *P5* Susan Wilson female O … … … …

[[��]] = { c∈�*2*|∃*x1*∈[[�*1*]] 〈subject (of a state)⟩(c, *x1*) } = {tuple*<sup>2</sup>*

instance s with respect to *Ai*, is an attribute quantifier function.

which is equivalent to {tuple*21*, tuple*22*, tuple*<sup>2</sup>*

whose id is *Pi*. That is,

**6.1 Total numbers** 

outputs the cardinality of [[�]].

following data as input data:

2. attributes *A1,..., An* of C(�), and

3. *f*: *C1*⨯...⨯*Cn*→*R*, where *Ci* := {*s.Ai*|s∈[[�]]}.

1. an objective graph �,

*An* and *f*.

**6. Quantifying concepts** 

**Case 2.** Let � be an objective graph defined in Case 2 of the definition of objective graphs. Then, [[�]]:={*x0*∈[[�*0*]]|∃*x1*∈[[�*1*]],…, ∃*xn*∈[[�*n*]]

> (⋀ *i=1,…,n Ri* (*xi*, *x0*)) ⋀ (⋀*i,j=0,…,n* (⋀ *k=1,…, n(i,j) R i,jk* (*xi, xj*)))}.

**Lemma.** For an objective graph � and a segment �\* of �, [[�]] ⊆ [[�\*]].

Proof. One can easily show the lemma above by induction on the structure of �.

## **5.2 Example of the interpretation of an objective graph**

In this subsection, we show a small example of the interpretation of an objective graph.

We first consider a concept of scheduled events denoted by [diagnosis] (cf. the definition of medical service ontology and Figure 5). Then, the concept has seven attributes (see the parenthetic names of columns of the table in Figure 5). Thus, one can obtain (the list of columns of) Table 1 corresponding to [diagnosis], whose attributes correspond to those of [diagnosis]. Let �*1* be data (a set of tuples) in Table 1 and assume that there is no tuple in �*<sup>1</sup>* whose value of the attribute "disease" is "stomach cancer" besides the tuples with id 1, 2, 3 and 5.


Table 1. The table generated from the concept of scheduled events [diagnosis] with tuples �*1*.

Let �1 be the objective graph in Section 4.2. Then, if the concept [diagnosis] is identified with �*1*, the interpretation of �1 ��s�� �� �*1* is { c∈�*1*| *c.* ⟨result⟩∋⟪stomach cancer⟫ }, which is equivalent to {tuple*11*, tuple*12*, tuple*13*, tuple*15*}. Here, each tuple*1i* denotes the tuple in �*<sup>1</sup>* whose id is *Ei*. That is,

 [[�1]] = { c∈�*1* | *c.* ⟨result⟩∋⟪stomach cancer⟫ } = {tuple*<sup>1</sup> <sup>1</sup>*, tuple*<sup>1</sup> <sup>2</sup>*, tuple*<sup>1</sup> <sup>3</sup>*, tuple*<sup>1</sup> 5,…*}.

Moreover, let �\* be the objective graph in Figure 7. Then, �\* = {{\**0*, \**1*}, \**0*, {*f 1*}, *L*, [patient]}, where *L* is the function satisfying the following properties.

(i) L(\*0) = �0 in Section 4.2, (ii) L(\*1) = �1 in Section 4.2, and (iii) L(f 1) = 〈subject (of a state)⟩ ( ⊆ [patient]×[diagnosis]) (cf. Section 3.2.2).

4 The symbol denotes the logical connective symbol of "and."

204 Semantics – Advances in Theories and Mathematical Models

**Case 2.** Let � be an objective graph defined in Case 2 of the definition of objective graphs.

(*xi*, *x0*)) ⋀ (⋀*i,j=0,…,n* (⋀ *k=1,…, n(i,j) R i,jk* (*xi, xj*)))}.

Set of Diseases (result)

ulcer}

*5,…*}.

(⋀ *i=1,…,n Ri*

In this subsection, we show a small example of the interpretation of an objective graph.

Term (content)

*E1 P1* 03-11-2011 *D1* - - - {stomach cancer} *E2 P2* 03-15-2011 *D1* - - - {stomach cancer} *E3 P3* 04-06-2011 *D2* - - - {stomach cancer} *E4 P4* 05-08-2011 *D2* - - - {gastric ulcer} *E5 P2* 06-09-2011 *D2* - - - {stomach cancer} *E6 P5* 07-06-2011 *D1* - - - {gastric varices, duodenal

… … … … … … … …

*<sup>2</sup>*, tuple*13*, tuple*<sup>1</sup>*

Table 1. The table generated from the concept of scheduled events [diagnosis] with tuples

]] = { c∈�*1* | *c.* ⟨result⟩∋⟪stomach cancer⟫ } = {tuple*11*, tuple*12*, tuple*13*, tuple*<sup>1</sup>*

Moreover, let �\* be the objective graph in Figure 7. Then, �\* = {{\**0*, \**1*}, \**0*, {*f 1*}, *L*, [patient]},

) = 〈subject (of a state)⟩ ( ⊆ [patient]×[diagnosis]) (cf. Section 3.2.2).

be the objective graph in Section 4.2. Then, if the concept [diagnosis] is identified with

Device (with what)

Method (how)

 ��s�� �� �*1* is { c∈�*1*| *c.* ⟨result⟩∋⟪stomach cancer⟫ }, which is

*<sup>5</sup>*}. Here, each tuple*1i* denotes the tuple in �*<sup>1</sup>*

We first consider a concept of scheduled events denoted by [diagnosis] (cf. the definition of medical service ontology and Figure 5). Then, the concept has seven attributes (see the parenthetic names of columns of the table in Figure 5). Thus, one can obtain (the list of columns of) Table 1 corresponding to [diagnosis], whose attributes correspond to those of [diagnosis]. Let �*1* be data (a set of tuples) in Table 1 and assume that there is no tuple in �*<sup>1</sup>* whose value of the attribute "disease" is "stomach cancer" besides the tuples with id 1, 2, 3

**Lemma.** For an objective graph � and a segment �\* of �, [[�]] ⊆ [[�\*]].

Staff (agent)

**5.2 Example of the interpretation of an objective graph** 

Date (occurring time point)

1

where *L* is the function satisfying the following properties.

4 The symbol denotes the logical connective symbol of "and."

Proof. One can easily show the lemma above by induction on the structure of �.

Then, [[�]]:={*x0*∈[[�*0*]]|∃*x1*∈[[�*1*]],…, ∃*xn*∈[[�*n*]]

and 5.

Patient (subject (of an event))

�*1*, the interpretation of �

whose id is *Ei*. That is,

equivalent to {tuple*11*, tuple*<sup>1</sup>*

in Section 4.2,

in Section 4.2, and

Id

�*1*.

Let �1

> [[�1

(i) L(\*0) = �0

(ii) L(\*1) = �1

(iii) L(f 1


Moreover, consider Table 2 corresponding to [patient], which is defined in Figure 3 and which has attributes ⟨result⟩ ��� ⟨����� ����⟩� ��� ��� �*1* be the set of tuples in Table 2.

Table 2. The table generated from the concept of a stakeholder [patient] with tuples �*2*.

Thus, the interpretation of �� ��s�� �� �*2* is { c∈�*2*|∃*x1*∈[[�*1*]] 〈subject (of a state)⟩(c, *x1*) }, which is equivalent to {tuple*21*, tuple*22*, tuple*23*}. Here, each tuple*<sup>2</sup> <sup>i</sup>* denotes the tuple in �*<sup>2</sup>* whose id is *Pi*. That is,

 [[��]] = { c∈�*2*|∃*x1*∈[[�*1*]] 〈subject (of a state)⟩(c, *x1*) } = {tuple*<sup>2</sup> <sup>1</sup>*, tuple*<sup>2</sup> <sup>2</sup>*, tuple*<sup>2</sup> <sup>3</sup>*}.
