S-TEN D2.1: Temporal part and time


Ontology for self-describing networks	S-TEN D2.1

Cover page
Table of contents
1 Introduction
2 Terminology
3 Physical things

4 Temporal part and time
5 Observation and measurement
6 Information
7 Physical quantity, property and scale
8 What is possible

A SensorML
B Weather station example
C PML
D Computer interpretable listing
Index

4 Temporal part and time

4.1 What is a temporal part

A temporal part is all of a sten:classical_object for a part of its life. A temporal part is:

an instant in the life of a sten:classical_object;
a interval in the life of a sten:classical_object;
a combination of one or more instants and intervals.

A relationship with a sten:classical_object which exists for only a part of its life, is assigned to the appropriate temporal part.

NOTE 1 The assignment of relationships to temporal parts is the way in which time variation is addressed by the S-TEN ontology. This approach is sometimes refered to as " 4D modelling". This approach is used in ISO 15926-2.

NOTE 2 Some relationships are with instants rather than periods. Such relationships can be considered as a a function over a space of temporal instants. This is discussed in Section 4.4.

EXAMPLE Consider a network "my_network" with component instances of sten:facility "X-101" and "X-102". X-102 is added to the network at T₂, and X-101 is removed at T₃.

Instances of sten:physical_asset are installed to be the facilities as follows:

X-101 is asset 98-1234 from before T₀ until T₁. At T₁, 98-1234 is sent for refurbishment, and 03-2345 is taken from the stores and installed until the end of the life of X-101.
X-102 is asset 99-5678 from the beginning of its life until after T₄.

This information is shown diagramatically in Figure 3.

Figure 3: Temporal parts and installation

The recording of this information using temporal parts is as follows:

components of the network:

:my_network  a               sten:facility ;
             a               topology:directed_manifold_edge ;
             topology:end_1  [ sten:at_time [ sten:before :T0 ] ] ;
             topology:end_2  [ sten:at_time [ sten:after  :T4 ] ] .

:X-101       a               sten:facility ;
             a               topology:directed_manifold_edge ;
             topology:end_1  [ sten:at_time [ sten:before :T0 ] ] ;
             topology:end_2  [ sten:at_time :T3 ] ] .

:X-102       a               sten:facility ;
             a               topology:directed_manifold_edge ;
             topology:end_1  [ sten:at_time :T2 ] ;
             topology:end_2  [ sten:at_time [ sten:after :T4 ] ] .

[ sten:temporal_part_of :my_network ;
  topology:end_1        [ sten:at_time [ sten:before :T0 ] ] ;
  topology:end_2        [ sten:at_time :T3 ]
]
             sten:has_component [ sten:temporal_part_of :X_101 ] .

[ sten:temporal_part_of :my_network ;
  topology:end_1        [ sten:at_time :T2 ] ;
  topology:end_2        [ sten:at_time [ sten:after :T4 ] ]
]
             sten:has_component [ sten:temporal_part_of :X_102 ] .

installed assets:

:98-1234     a               sten:physical_asset ;
             a               topology:directed_manifold_edge ;
             topology:end_1  [ sten:at_time [ sten:before :T0 ] ] ;
             topology:end_2  [ sten:at_time [ sten:after  :T4 ] ] .

:99-5678     a               sten:physical_asset ;
             a               topology:directed_manifold_edge ;
             topology:end_1  [ sten:at_time [ sten:before :T0 ] ] ;
             topology:end_2  [ sten:at_time [ sten:after  :T4 ] ] .

:03-2345     a               sten:physical_asset ;
             a               topology:directed_manifold_edge ;
             topology:end_1  [ sten:at_time [ sten:before :T0 ] ] ;
             topology:end_2  [ sten:at_time [ sten:after  :T4 ] ] .

[ sten:temporal_part_of :X-101 ;
  topology:end_1      [ sten:at_time [ sten:before :T0 ] ] ;
  topology:end_2      [ sten:at_time :T1 ]
]
             =  [ sten:temporal_part_of :98-1234 ] .

[ sten:temporal_part_of :X-101 ;
  topology:end_1      [ sten:at_time :T1 ] ] ;
  topology:end_2      [ sten:at_time :T3 ]
]
             =  [ sten:temporal_part_of :03-2345 ] .

[ sten:temporal_part_of :X-102 ;
  topology:end_1      [ sten:at_time :T2 ] ;
  topology:end_2      [ sten:at_time [ sten:after :T4 ] ]
]
             =  [ sten:temporal_part_of :99-5678 ] .

The installation relationship between a temporal part of a sten:facility and a temporal part of a sten:physical_asset is expressed as an owl:sameAs relationship , which is abbreviated to "=" in N3. On one side of the "=", the beginning and end times of the temporal part are stated. They are not stated on the other side, because "=" requires that they are the same.

If the temporal parts are named object which are defined separately, then it may be better to specify the start and end times for each. This would enable software to check that the "=" is valid. This is shown as follows.

:X-101_T0_T1   sten:temporal_part_of :X-101 ;
               topology:end_1        [ sten:at_time [ sten:before :T0 ] ] ;
               topology:end_2        [ sten:at_time :T1 ] .

:98-1234_T0_T1 sten:temporal_part_of :X-101 ;
               topology:end_1        [ sten:at_time [ sten:before :T0 ] ] ;
               topology:end_2        [ sten:at_time :T1 ] .

:X-101_T0_T1   =  :98-1234_T0_T1 .

NOTE 3 The assumption that a sten:classical_object is a topology:directed_manifold_edge is convenient for this example. The assumption enables a temporal part to be defined by the properties topology:end_1 and topology:end_2. Each point within the 1D manifold is a sten:classical_object_at_instant.

The assumption of a structure for a temporal part, such as topology:directed_manifold_edge, is discussed in Section Physical thing and structure. The ontology used to defined the structure is contained in ISO/TS 15926-3. This standard has passed its CD ballot, but has not yet been published. Until publication, the S-TEN project is using a pre-publication OWL/XML serialisation of the ontology which is contained in http://www.s-ten.eu/iso15926-3_pre-publication/iso_ts_15926-3.htm. After publication, the OWL/XML serialisation will be downloadable from an ISO website.

Many different possible assumptions can be made about the structure of a sten:classical_object. These include:

A 4D manifold, within which each point is a material point at an instant.
A 2D manifold, within which each point is a cross-section at an instant. This approach can be used to represent a property of a beam as it changes with time.
A 1D manifold, within which each point is a "temporal plane". The example above is a special case of this in which the temporal plane corresponds to a sten:classical_object_at_instant. For some special applications, such as fluid flow through rotating machinery, it is also possible to consider "inclined temporal planes". This is shown in Figure 4.

Figure 4: "Vertical" and "inclined" temporal planes

4,2 Use of temporal part

Many types of relationship are necessarily between simultaneous temporal parts.

EXAMPLE 1 If Janet has a married_to relationship with John, then the relationship is between simultaneous temporal parts of each. This is shown as follows:

[ sten:temporal_part_of :Janet ;
  sten:during  [ topology:end_1 [ sten:utc_iso8601 "2007-03-10" ]  ]
]
    :married_to  [ sten:temporal_part_of :John ] .

The temporal part of Janet, that is married to John, has a beginnning on 2007-03-10, and an unspecified end. The temporal part of John, that is married to Janet, exists during the same sten:period.

If a relationship is necessarily between simultaneous temporal parts, then the sten:period for one need not be specified. That a type of relationship is necessarily between simultaneous temporal parts is specified by classification as a sten:simultaneous_mapping.

EXAMPLE 2: Information about married_to is as follows:

<owl:FunctionalProperty rdf:about="#married_to">
 <rdf:type rdf:resource="sten;simultaneous_mapping"/>
 <rdfs:domain>
  <owl:Restriction>
   <owl:onProperty rdf:resource="sten;temporal_part_of"/>
   <owl:someValuesFrom rdf:resource="#Person"/>
  </owl:Restriction>
 </rdfs:domain>
 <rdfs:range>
  <owl:Restriction>
   <owl:onProperty rdf:resource="sten;temporal_part_of"/>
   <owl:someValuesFrom rdf:resource="#Person"/>
  </owl:Restriction>
 </rdfs:range>
</owl:FunctionalProperty>

The domain and range of married_to is not Person, but temporal part of a Person.

An sten:activity is specifically defined to record relationships between instances of sten:persistent_object. Hence it is a sten:persistent_object which has a role in an sten:activity rather than a temporal part of it.

EXAMPLE 3 If Janet has a married_to relationship with John, then there is a marriage activity in which they both participate as spouse. This is shown as follows:

[ a            :marriage ;
  sten:during  [ topology:end_1 [ sten:utc_iso8601 "2007-03-10" ]  ]
]
  :spouse :Janet ;
  :spouse :John  .

4.3 Ontology for temporal part

4.3.1 has_temporal_part

An object is an sten:has_temporal_part relationship if and only if:

it is a sten:has_part relationship;
the part is all of the whole for a sten:part_of_time.

If the object in the domain is a basics:structure, then the object in the range consists of one or more members of basics:sub_structure or basics:structure_point.

If the object in the domain is a topology:directed_manifold_edge, then the object in the range consists of one or more members of topology:directed_manifold_edge or basics:structure_point.

The relationship sten:has_temporal_part includes equality. Hence a sten:classical_object has a sten:temporal_part_of relationship with itself.

OWL specification:

<owl:TransitiveProperty rdf:about="&sten;has_temporal_part">
 <rdfs:subPropertyOf rdf:resource="&sten;has_part"/>
 <rdfs:domain rdf:resource="&sten;classical_object"/>
 <rdfs:rangerdf:resource="&sten;physical_thing"/>
 <owl:sameAs rdf:resource="&sumo;temporal_part/">
 <rdfs:subPropertyOf rdf:resource="&iso15926-2;temporal_whole_part.part"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#has_temporal_part"/>
</owl:TransitiveProperty>

4.3.2 temporal_part_of

An object is a sten:temporal_part_of relationship if and only if:

it is a sten:part_of relationship;
it is the inverse of sten:has_temporal_part.

If the object in the range is a basics:structure, then the object in the domain consists of one or more members of basics:sub_structure or basics:structure_point.

If the object in the range is a topology:directed_manifold_edge, then the object in the domain consists of one or more members of topology:directed_manifold_edge or basics:structure_point.

The relationship sten:temporal_part_of includes equality. Hence a sten:classical_object has a sten:temporal_part_of relationship with itself.

NOTE sten:temporal_part_of is approximately equivalent to iso15926-2:temporal_whole_part. A difference is that a temporal_whole_part can relate instances of iso15926-2:possible_individual, where iso15926-2:possible_individual is equivalent to sten:physical_thing. In the STEN ontology a temporal part relationship is only valid for a whole sten:physical_thing which evolves through time, i.e. a sten:classical_object.

In the OWL ontology for ISO 15926-2, the bi-directional entity iso15926-2:temporal_whole_part corresponds to the two relationships iso15926-2:temporal_whole_part.part and iso15926-2:temporal_whole_part.whole.

OWL specification:

-->
<owl:TransitiveProperty rdf:about="&sten;temporal_part_of">
 <rdfs:subPropertyOf rdf:resource="&sten;part_of"/>
 <owl:inverseOf rdf:resource="&sten;has_temporal_part"/>
 <rdfs:domain rdf:resource="&sten;physical_thing"/>
 <rdfs:range rdf:resource="&sten;classical_object"/>
 <rdfs:subPropertyOf rdf:resource="&iso15926-2;temporal_whole_part.whole"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#temporal_part_of"/>
</owl:TransitiveProperty>
<!--

4.4 An approach to time

In this ontology, the universe is regarded as a 1D manifold of 3D manifolds, where there is a frame of reference such that in each 3D manifold all the points are simultaneous.

A "time" is such a 1D manifold. There are many possible times, of which one is International Atomic Time (TAI) experienced by a network of clocks on the surface of the earth. A point in TAI can be identified using Coordinated Universal Time (UTC).

The universe (in the neighbourhood of a set of clocks) is a sten:classical_object, which has temporal parts. Any smaller sten:classical_object can be regarded as a component of the appropriate temporal part of the universe. Similarly a smaller sten:classical_object_at_instant can be regarded as a component of the appropriate member of a "time". The two component relationships are given special names as follows:

sten:during which specifies that a sten:classical_object exists during a sten:part_of_time; and
sten:at_time which specifies that a sten:classical_object_at_instant exists at a time.

The two component relationships can be used to specify the duration of a sten:classical_object in two ways, as follows:

end points of sten:classical_object

:my_classical_object
     a               sten:classical_object ;
     a               topology:directed_manifold_edge ;
     topology:end_1  [ sten:at_time [ sten:utc_iso8601 "2007-02-10T11:30" ] ] ;
     topology:end_2  [ sten:at_time [ sten:utc_iso8601 "2007-02-10T15:45" ] ] .

end points of time interval

:my_classical_object
     a               sten:classical_object ;
     a               topology:directed_manifold_edge ;
     sten:during     [ topology:end_1  [ sten:utc_iso8601 "2007-02-10T11:30" ] ;
                       topology:end_2  [ sten:utc_iso8601 "2007-02-10T15:45" ]
                     ] .

4.5 Time ontology

4.5.1 time

An object is a sten:time if and only if:

it is a partition of a part of the universe;
there is an observer for which each item within the partition consists of things which are simultaneous.

OWL specification:

-->
<owl:Class rdf:about="&sten;time">
 <rdfs:subClassOf rdf:resource="&sten;classical_object"/>
 <rdfs:subClassOf rdf:resource="&topology;directed_manifold_edge"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#time"/>
</owl:Class>
<!--

4.5.2 instant

An object is a sten:instant if and only if:

it is a basics:structure_point within a sten:time.

OWL specification:

-->
<owl:Class rdf:about="&sten;instant">
 <owl:equivalentClass>
  <owl:Restriction>
   <owl:onProperty rdf:resource="&basics;point_has_containing_structure"/>
   <owl:allValuesFrom rdf:resource="&sten;time"/>
  </owl:Restriction>
 </owl:equivalentClass>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#instant"/>
</owl:Class>
<!--

4.5.3 part_of_time

An object is a sten:part_of_time if and only if:

it has a sten:temporal_part_of relationship with a sten:time.

OWL specification:

-->
<owl:Class rdf:about="&sten;part_of_time">
 <owl:equivalentClass>
  <owl:Restriction>
   <owl:onProperty rdf:resource="&sten;temporal_part_of"/>
   <owl:allValuesFrom rdf:resource="&sten;time"/>
  </owl:Restriction>
 </owl:equivalentClass>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#part_of_time"/>
</owl:Class>
<!--

4.5.4 period

An object is a sten:period if and only if:

it is a sten:part_of_time;
it is a topology:directed_manifold_edge.

OWL specification:

-->
<owl:Class rdf:about="&sten;part_of_time">
 <owl:equivalentClass>
  <owl:Class>
   <owl:intersectionOf rdf:parseType="Collection">
    <owl:Class rdf:resource="&sten;part_of_time"/>
    <owl:Class rdf:resource="&topology;directed_manifold_edge"/>
   </owl:intersectionOf>
  </owl:Class>
 </owl:equivalentClass>
 <owl:sameAs rdf:resource="&iso15926-2;period_in_time"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#period"/>
</owl:Class>
<!--

4.5.5 tai_time

An object is a sten:tai_time if and only if:

it is a sten:time;
it is the domain of the scale International Atomic Time (TAI).

NOTE This time is also the domain of the UTC scale. The UTC scale is defined with respect to the TAI scale.

OWL specification:

-->
<owl:Class rdf:about="&bipm;tai_time">
 <rdf:type rdf:resource="&sten;time"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#tai_time"/>
</owl:Class>
<!--

4.5.6 during

An object is a sten:during relationship if and only if:

it is a basics:function from sten:physical_thing to sten:part_of_time;
it specifies that the sten:physical_thing exists throughout the sten:part_of_time and does not exists outside the sten:part_of_time.

OWL specification:

-->
<owl:FunctionalProperty rdf:about="&sten;during">
 <rdf:type rdf:resource="&basics;function"/>
 <rdfs:domain rdf:resource="&sten;physical_thing"/>
 <rdfs:range rdf:resource="&sten;part_of_time"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#during"/>
</owl:FunctionalProperty>
<!--

4.5.7 at_time

An object is an sten:at_time relationship if and only if:

it is a basics:function from sten:classical_object_at_instant to sten:instant;
it specifies that the sten:classical_object_at_instant exists at the sten:instant.

OWL specification:

-->
<owl:FunctionalProperty rdf:about="&sten;at_time">
 <rdfs:domain rdf:resource="&sten;classical_object_at_instant"/>
 <rdfs:range rdf:resource="&sten;instant"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#at_time"/>
</owl:FunctionalProperty>
<!--

4.5.8 before

An object is a sten:before relationship if and only if:

it is a basics:mapping with domain and range sten:instant
it specifies that the domain object comes before the range object.

The relationship sten:before is a specialisation of the relationship sten:less_than for use with sten:time.

OWL specification:

-->
<owl:ObjectProperty rdf:about="&sten;before">
 <rdf:type rdf:resource="&basics;mapping"/>
 <rdfs:domain rdf:resource="&sten;instant"/>
 <rdfs:range rdf:resource="&sten;instant"/>
 <rdfs:subPropertyOf rdf:resource="&sten;less_than"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#before"/>
</owl:ObjectProperty>
<!--

4.5.9 after

An object is a sten:after relationship if and only if:

it is a basics:mapping with domain and range sten:instant
it specifies that the domain object comes after the range object.

The relationship sten:after is a specialisation of the relationship sten:greater_than for use with sten:time.

OWL specification:

-->
<owl:ObjectProperty rdf:about="&sten;after">
 <rdf:type rdf:resource="&basics;mapping"/>
 <rdfs:domain rdf:resource="&sten;instant"/>
 <rdfs:range rdf:resource="&sten;instant"/>
 <rdfs:subPropertyOf rdf:resource="&sten;greater_than"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#after"/>
</owl:ObjectProperty>
<!--

4.5.10 utc_iso8601

An object is an sten:utc_iso8601 relationship if and only if:

it is a basics:function from sten:tai_time to the space of text strings for the representation of UTC defined by ISO 8601:1988;
it identifies time points according to UTC.

NOTE An evaluation of sten:utc_iso8601 identifies a point within sten:tai_time and not a sten:period. This is true even if the text string could identify many different points. For example the text string "2007-03-14" could identify any point within the day.

By analogy, the use of "Janet" in the statement "Janet loves Fred Bloggs" identifies one person, even though there are many people called "Janet".

OWL specification:

-->
<owl:DatatypeProperty rdf:about="&sten;utc_iso8601">
 <rdfs:domain rdf:resource="&sten;tai_time"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#utc_iso8601"/>
</owl:DatatypeProperty>
<!--

4.5.11 utc_iec61850

An object is an sten:utc_iec61850 relationship if and only if:

it is a basics:function from sten:tai_time to the space of lists for the representation of UTC defined by IEC 61850-7-2:2003 (This link is to the first few pages of the standard only. The remainder has to be purchased from IEC);
it identifies time points according to UTC.

The items in a member of the space of lists have the semantics:

integer: SecondOfCentury;
integer pair: FractionOfSecond;
Boolean: LeapSecondPending;
Boolean: LeapSecondOccured;
Boolean: LeapSecondNegative,

where the definitions of SecondOfCentury, FractionOfSecond, LeapSecondPending, LeapSecondOccured and LeapSecondNegative are given in IEC 61850-7-2:2003.

The representation of the items in the list within a serialisation of OWL is as follows:

integer: encoded using basics:decimal;
integer pair: the first is the numerator and the second is the denominator, and each is encoded using basics:decimal; the numerator and the second ;
Boolean: encoded using sten:boolean.

EXAMPLE The time half a second after 2000-03-26T04:00:15 is represented as follows:

[ a                 sten:tai_time
  sten:utc_iec61850 ( [ basics:decimal "951537615" ]
                      ( [basics:decimal "5"] [basics:decimal "10"] )
                      [ sten:boolean "F" ]
                      [ sten:boolean "F" ]
                      [ sten:boolean "F" ]
                    )
]

OWL specification:

-->
<owl:DatatypeProperty rdf:about="&sten;utc_iso8601">
 <rdfs:domain rdf:resource="&sten;tai_time"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#utc_iso8601"/>
</owl:DatatypeProperty>
<!--

4.5.12 simultaneous_mapping

An object is a sten:simultaneous_mapping if and only if:

it is a owl:ObjectProperty;
each object in the domain maps to an object in the range which exists during the same sten:period or at the same sten:instant.

OWL specification:

-->
<owl:Class rdf:about="&sten;simultaneous_mapping">
 <rdfs:subClassOf rdf:resource="&owl;ObjectProperty"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#simultaneous_mapping"/>
</owl:Class>
<!--

4.6 An approach to property variation

Some properties are possessed by a sten:classical_object and others are possessed by a sten:classical_object_at_instant.

EXAMPLE 1 Consider the transmission line "my_line". The amount of energy transmitted between 10:00 and 11:00 on 2007-03-14 is recorded as follows:

[ sten:temporal_part_of :my_line ;
  a                     sten:classical_object ;
  a                     topology:directed_manifold_edge .
  sten:during           [ topology:end_1  [ sten:utc_iso8601 "2007-03-14T10:00" ] ;
                          topology:end_2  [ sten:utc_iso8601 "2007-03-14T11:00" ]
                        ]
]    sten:transmitted_energy  [ a            iso31:energy ;
                                joule_scale  [ basics:decimal  "360E6" ]
                              ] .

EXAMPLE 2 Consider the transmission line "my_line". The amount of power transmitted at 10:30 on 2007-03-14 is recorded as follows:

[ sten:temporal_part_of :my_line ;
  a                     sten:classical_object_at_instant ;
  sten:at_time          [ sten:utc_iso8601 "2007-03-14T10:30" ]
]    sten:transmitted_power  [ a            iso31:power ;
                               watt_scale  [ basics:decimal  "0.1E6" ]
                             ] .

A property, such as sten:transmitted_power, which is possessed by a sten:classical_object_at_instant cannot be possessed by a sten:classical_object. Instead a sten:transmitted_power can be associated with a sten:classical_object in two ways, as follows:

by a derived property such as "mean transmitted power" or "maximum transmitted power"; or
by a transmitted power function over time.

The first of these approaches is a variant of the first, because "mean" and "maximum" are properties of functions.

A property of a sten:classical_object_at_instant, such as sten:transmitted_power, cannot be possessed by a sten:classical_object, even if it remains unchanged. In this case the function over time is constant.

EXAMPLE 3 Consider the transmitted power function over time for "my_line", which is shown in Figure 5.

Figure 5: Transmitted power function over time

The information can be recorded in OWL as follows:

:my_line_from_2007-03-14T10:00_to_11:00
  sten:temporal_part_of :my_line ;
  a                     sten:classical_object ;
  a                     topology:directed_manifold_edge ;
  sten:during           [ topology:end_1  [ sten:utc_iso8601 "2007-03-14T10:00" ] ;
                          topology:end_2  [ sten:utc_iso8601 "2007-03-14T11:00" ] .

:my_line_from_2007-03-14T11:00_to_12:00
  sten:temporal_part_of :my_line ;
  a                     sten:classical_object ;
  a                     topology:directed_manifold_edge ;
  sten:during           [ topology:end_1  [ sten:utc_iso8601 "2007-03-14T11:00" ] ;
                          topology:end_2  [ sten:utc_iso8601 "2007-03-14T12:00" ] .

:my_line_from_2007-03-14T10:00_to_12:00
  sten:temporal_part_of :my_line ;
  a                     sten:classical_object ;
  a                     topology:directed_manifold_edge ;
  sten:during           [ topology:end_1  [ sten:utc_iso8601 "2007-03-14T10:00" ] ;
                          topology:end_2  [ sten:utc_iso8601 "2007-03-14T12:00" ] .

[ a                  basics:constant_function ;
  rdfs:domain        :my_line_from_2007-03-14T10:00_to_11:00 ;
  rdfs:range         iso31:power ;
  rdfs:subPropertyOf sten:transmitted_power ;
  basics:value_of_constant_function
                     [ a           iso31:power ;
		               watt_scale  [ basics:decimal  "0.1E6" ] ] .

[ a                  basics:constant_function ;
  rdfs:domain        :my_line_from_2007-03-14T11:00_to_12:00 ;
  rdfs:range         iso31:power ;
  rdfs:subPropertyOf sten:transmitted_power ;
  basics:value_of_constant_function
                     [ a           iso31:power ;
                       watt_scale  [ basics:decimal  "0.15E6" ] ] .

[ a                  basics:function ;
  rdfs:domain        :my_line_from_2007-03-14T10:00_to_12:00 ;
  rdfs:range         iso31:power ;
  rdfs:subPropertyOf sten:transmitted_power ;
  sten:function_mean [ a           iso31:power ;
                       watt_scale  [ basics:decimal  "0.125E6" ] ] .

4.7 Function property ontology

4.7.1 function_mean

An object is a sten:function_mean relationship if and only if:

it is a basics:function with domain basics:function;
for a function f:X → Y, it evaluates to give the member of Y as follows:
- as an integral:
  
  ∫_X f(x) dx
  
  ∫_X dx
- as a sum:
  
  ∑_{x ∈X} f(x)
  
  ∑_{x ∈X} 1

OWL specification:

-->
<owl:ObjectProperty rdf:about="&sten;function_mean">
 <rdf:type rdf:resource="&basics;function"/>
 <rdfs:domain rdf:resource="&basics;function"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#function_mean"/>
</owl:ObjectProperty>
<!--

4.7.2 function_maximum

An object is a sten:function_maximum relationship if and only if:

it is a basics:function with domain basics:function;
for a function f:X → Y, it evaluates to give the member of Y that is the maximum of f(x) for x ∈ X.

OWL specification:

-->
<owl:ObjectProperty rdf:about="&sten;function_maximum">
 <rdf:type rdf:resource="&basics;function"/>
 <rdfs:domain rdf:resource="&basics;function"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#function_maximum"/>
</owl:ObjectProperty>
<!--

4.7.3 function_minimum

An object is a sten:function_minimum relationship if and only if:

it is a basics:function with domain basics:function;
for a function f:X → Y, it evaluates to give the member of Y that is the minimum of f(x) for x ∈ X.

OWL specification:

-->
<owl:ObjectProperty rdf:about="&sten;function_minimum">
 <rdf:type rdf:resource="&basics;function"/>
 <rdfs:domain rdf:resource="&basics;function"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#function_minimum"/>
</owl:ObjectProperty>
<!--

4.8 Space ontology

4.8.1 Scope of the space ontology

Statements about how objects are placed in time and overlap in time are crucial to the S-TEN project. S-TEN is only interested in space in order to say where things are. The ontology in this section has been derived directly from the ontology for time in order to serve the purpose of the S-TEN project.

4.8.2 physical_space

An object is a sten:physical_space if and only if:

it is a partition of a part of the universe;
there is an observer for which each item within the partition consists of things which are at the same place.

OWL specification:

-->
<owl:Class rdf:about="&sten;physical_space">
 <rdfs:subClassOf rdf:resource="&sten;classical_object"/>
 <rdfs:subClassOf rdf:resource="&topology;manifold"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#physical_space"/>
</owl:Class>
<!--

4.8.3 point_in_space

An object is a sten:point_in_space if and only if:

it is a basics:structure_point within a sten:physical_space.

OWL specification:

-->
<owl:Class rdf:about="&sten;point_in_space">
 <owl:equivalentClass>
  <owl:Restriction>
   <owl:onProperty rdf:resource="&basics;point_has_containing_structure"/>
   <owl:allValuesFrom rdf:resource="&sten;physical_space"/>
  </owl:Restriction>
 </owl:equivalentClass>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#point_in_space"/>
</owl:Class>
<!--

4.8.4 part_of_space

An object is a sten:part_of_space if and only if:

it has a sten:part_of relationship with a sten:physical_space.

OWL specification:

-->
<owl:Class rdf:about="&sten;part_of_space">
 <owl:equivalentClass>
  <owl:Restriction>
   <owl:onProperty rdf:resource="&sten;part_of"/>
   <owl:allValuesFrom rdf:resource="&sten;physical_space"/>
  </owl:Restriction>
 </owl:equivalentClass>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#part_of_space"/>
</owl:Class>
<!--

4.8.4 region

An object is a sten:region if and only if:

it is a sten:part_of_space;
it is a topology:manifold.

OWL specification:

-->
<owl:Class rdf:about="&sten;region">
 <owl:equivalentClass>
  <owl:Class>
   <owl:intersectionOf rdf:parseType="Collection">
    <owl:Class rdf:resource="&sten;part_of_space"/>
    <owl:Class rdf:resource="&topology;manifold"/>
   </owl:intersectionOf>
  </owl:Class>
 </owl:equivalentClass>
 <owl:sameAs rdf:resource="&iso15926-2;spatial_location"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#region"/>
</owl:Class>
<!--

4.8.5 wgs84_space

An object is a sten:wgs84_space if and only if:

it is a sten:physical_space;
it is the domain of the scale Department of Defense, National Geophysical Agency (NGA) World Geodetic System 1984 (WGS 84).

OWL specification:

-->
<owl:Class rdf:about="&sten;wgs84_space">
 <rdf:type rdf:resource="&sten;physical_space"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#wgs84_space"/>
</owl:Class>
<!--

4.8.6 occupies

An object is a sten:occupies relationship if and only if:

it is a basics:function from sten:physical_thing to sten:part_of_space;
it specifies that the sten:physical_thing exists throughout the sten:part_of_space and does not exists outside the sten:part_of_space.

OWL specification:

-->
<owl:FunctionalProperty rdf:about="&sten;occupies">
 <rdf:type rdf:resource="&basics;function"/>
 <rdfs:domain rdf:resource="&sten;physical_thing"/>
 <rdfs:range rdf:resource="&sten;part_of_space"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#occupies"/>
</owl:FunctionalProperty>
<!--

4.8.7 at

An object is an sten:at relationship if and only if:

it is a basics:partial_function from sten:physical_thing to sten:point_in_space;
it specifies that the sten:physical_thing exists at the sten:point_in_space.

OWL specification:

-->
<owl:ObjectProperty rdf:about="&sten;at">
 <rdfs:domain rdf:resource="&sten;physical_thing"/>
 <rdfs:range rdf:resource="&sten;point_in_space"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#at"/>
</owl:ObjectProperty>
<!--

4.8.8 wgs84

An object is an nga:wgs84 relationship if and only if:

it is a basics:function from sten:wgs84_space to basics:list_of_reals;
it identifies points in space according to WGS84.

OWL specification:

-->
<owl:FunctionalProperty rdf:about="&nga;wgs84">
 <rdfs:domain rdf:resource="&sten;wgs84_space"/>
 <meta:defined_by rdf:resource="&D2.1;/temporal_part_and_time.htm#wgs84"/>
</owl:FunctionalProperty>
<!--

∫_X	f(x) dx

∫_X	dx

∑_{x ∈X}	f(x)

∑_{x ∈X}	1