# An Ontology Design Pattern for representing Recurrent Situations

Valentina Anita Carriero
Department of Computer Science and Engineering
University of Bologna
Mura Anteo Zamboni 7, 40126 Bologna, Italy
<EMAIL_ADDRESS>&Aldo Gangemi
Digital Humanities Advanced Research Centre
Department of Classical Philology and Italian Studies
University of Bologna
Via Zamboni 32, 40126 Bologna, Italy
<EMAIL_ADDRESS>&Andrea Giovanni Nuzzolese
Semantic Technologies Laboratory
Institute of Cognitive Sciences and Technologies
Italian National Research Council
Via San Martino della Battaglia 44, 00185 Rome, Italy
<EMAIL_ADDRESS>&Valentina Presutti
Department of Modern Languages, Literatures, and Culture
University of Bologna
Via Cartoleria 5, 40124 Bologna, Italy
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present an Ontology Design Pattern for representing
situations that recur at regular periods and share some invariant factors,
which unify them conceptually: we refer to this set of recurring situations as
_recurrent situation series_. The proposed pattern appears to be foundational,
since it can be generalised for modelling the top-level domain-independent
concept of recurrence, which is strictly associated with invariance. The
pattern reuses other foundational patterns such as Collection, Description and
Situation, Classification, Sequence. Indeed, a recurrent situation series is
formalised as both a collection of situations occurring regularly over time
and unified according to some properties that are common to all the members,
and a situation itself, which provides a relational context to its members
that satisfy a reference description. Besides including some exemplifying
instances of this pattern, we show how it has been implemented and specialised
to model recurrent cultural events and ceremonies in ArCo, the Knowledge Graph
of Italian cultural heritage.

##  1 Introduction

In this paper, we investigate the concept of recurrence as applied to a series
of situations, which _recur_ , i.e. they occur periodically or repeatedly,
either by design (e.g. a festival), or by featuring similar attributes (e.g. a
bird migration pattern); hence the name _recurrent situations_. Recurrence is
associated with _invariance_. We typically recognise something as a known
entity when it is invariant under appropriate transformations (in time, space,
context, etc.). This is a basic biological competence, typically starting in
human babies at two months of age. As we recognise an invariant object, we can
also recognise an invariant situation, when multiple sets of related objects
feature similar patterns. While this ability in humans develops into
foundational concepts such as _permanence_ , _similarity_ and _difference_ ,
_type_ , etc., and is well investigated in cognitive science (see e.g. [1] for
links between current cognitive science theories and knowledge
representation), recurrence is less studied in ontology design. In particular,
recurrence under time transformation is not fully addressed and captured by
state-of-the-art patterns and ontologies.

In this paper, we deal with _constructed_ and _natural_ recurrence of
situations under time transformation. By _constructed_ we mean a recurrence
created by social norms (e.g. elections or festivals) or emerging and
established in local or even individual behavioral patterns (e.g. social
meetings or daily routines). By _natural_ we mean a recurrence observed in
nature (e.g. the raising of the sun, species migrations). By _situation_ we
mean any eventuality or structure that someone intends to talk about
(consider, refer to, mention), e.g. a conference, a system configuration, a
trial.

Existing ontologies mostly model recurrent situations as a special type of
event, which represents its (repeatable) occurrences as generic parts of a
more general event, neglecting their belonging to a _series_ , intended as a
collection, whose members are temporally ordered. This is relevant and
characterises the ontological nature of such a unifying entity.

The Recurrent Situation Series Ontology Design Pattern (ODP), explained in the
rest of this paper, addresses this issue. This pattern models collections of
situations, in which the member situations: (i) are ordered in a temporal
sequence, (ii) are separated by time intervals that obey some function (that
is in principle computable), and (iii) share at least one invariance, unifying
factor (e.g. a name, a topic, a purpose, etc.). Unifying factors make a
recurrent situation series distinct from others. This is a foundational
pattern, since it models the top-level, domain-independent concept of
recurrence, as applied to a series of situations.

The paper is organised as follows: Section 2 introduces the problem addressed;
Section 3 describes the pattern at an abstract level; Section 4 presents two
actual implementations of the pattern; Section 5 provides two usage examples.
Finally, Section 6 reports the state of the art and Section 7 concludes the
paper envisaging future work.

##  2 Problem addressed: recurrent situations

The _Palio di Siena_ is a horse race held twice each year on predefined dates
in Siena, Italy. Preceded by other events, e.g. an historical costume parade,
the race itself involves ten horses and riders and consists of three laps of
_Piazza del Campo_. The first horse that crosses the finish line, even without
its rider, is declared the winner and awarded a banner of painted silk
(_palio_). Indeed, by riding bareback, more than one rider is usually thrown
off his horse each year during the turns.

The Palio held on 2 July 2019 and the fall of the jockey _Brigante_ from his
horse during that Palio are both _situations_. Here, we refer to the notion of
situation as a generalisation over any relational concept, e.g. events,
states, actions. Both situations show a kind of repetition: indeed, every year
the Sienese horse race occurs, and, most likely, the same can be said for the
riders thrown off their horses. Nevertheless, it is possible, at both common-
sense and philosophical level, associate only the first situation with a
notion of recurrence that is strictly related to invariance: in recurrent
situations, we can recognise a pattern in the iteration, at periodic
intervals, of some properties that occur in each situation. While the Palio is
explicitly designed, based on a plan, to iteratively occur at specific dates
and following regulations (e.g. ten participants, crossbred horses), the falls
of the jockeys during the horse race have a high probability, which can be
empirically observed, to occur frequently, possibly at each race, but this
observation is statistical in nature and is based on neither a plan nor a rule
that regulates and assures its repetition with specific characteristics and at
regular time intervals. Therefore, the annual race is designed as part of a
_series_ with an inherent continuity, while the fall of a specific jockey is
perceived as an episode.

There are two main distinctions that can be made between different types of
recurrent situations, based on the origin of the regular recurrence: (i)
natural vs social, (ii) unplanned vs planned.

Natural recurrent situations are not constructed by social norms or practices:
humans can observe them in very different ways, but their occurrence is not
caused by the mere acceptance of social rules. For example, morning sunrise,
evening sunset, heartbeat, periodic migration of groups of animals from a
region to another for feeding or breeding, changing seasons during the
calendar year. We can be aware or not, we can provide alternative theories,
but those things are not happening because we decree they should exist.

Social situations instead are man-made situations that have a specific purpose
in a social setting, whose recurrence is set by humans: a world day, a train
schedule, festivals, religious holidays, awards, sporting events,
parliamentary elections, etc.

Planned situations are those defined by a clear plan, which regulates how the
things involved in that plan shall be carried out; on the contrary, unplanned
situations have not a predefined setting, and their properties are typically
observed after some occurrences.

When we informally (in conversations) refer to a regular repetition of a
situation (e.g. an edition of the Palio), the term _recurrent event_ is often
used, as we are talking about an event with invariant properties that occurs
repeatedly. The term _event_ however may cause confusion as it is attributed
in literature several different and very specific meanings. The concept
_situation_ generalises over them, addressing the large spectrum of possible
entities that can recur, either for construction or by nature.

Going back to our example, the _Palio di Siena_ is a series of consecutive
situations that belong, as members, to a collection characterised by some
unifying factors, i.e. properties that give those situations a unity.

##  3 The Recurrent Situation Series Ontology Design Pattern

In this section, we describe the abstract model of recurrent situation series,
which is represented as an Ontology Design Pattern, according to OWL
constructs.

#### Recurrent situation series.

A recurrent situation series is a set of situations repeating regularly over
time. It is intended as a _collection_ , since it can be seen as a social
object that depends on the collected items, i.e. its members [2]. A recurrent
situation series is a particular type of collection where all the members are
situations: entities flowing in time, either in the physical or social world,
located in place, and involving some objects. The members of a recurrent
situation series, i.e. the recurring situations, must all play at least the
role of ‘being a member of a recurrent situation series’, and share one or
more common properties.

At the same time, a recurrent situation series is a _situation_ itself,
intended as a context in which a set of entities is contextualized based on a
_description_ or “frame”. A recurrent situation series is similar to a plan,
which defines how the things involved in that plan (i.e. the specific
situations) shall be carried out, or follow some not completely predefined
rules, e.g. where the situations are located, in which time of the year, etc.

#### Unifying criteria.

The individual (re)occurring situations, while preserving their identity, are
kept together and form a new entity, i.e. a recurrent situation series. In
[2], collections of items are conceptually unified by _unity criteria_. A
unity criterion is a concept, a social object created for collecting existing
entities, representing a role that can be played by different entities.
Therefore, a recurrent situation series is a collection, whose members are
_unified_ by at least one unifying factor, i.e. organized according to one (or
more if it evolves through time) recognizable pattern shared by all members.
For example, unifying factors of Palio di Siena may be its name and the
holding of a horse race in Piazza del Campo. A set of situations can be
perceived as a unitary collection when at least one element or property occurs
in each situation member of the series, making all the situations ascribable
to a homogeneous collection. For instance, all the situations member of a
recurrent situation series can be placed in the same location, be about the
same topic, be inspired by the same theme, establish certain activities,
involve the same community, etc. A recurrent situation series must have at
least one unifying factor. Considering that, in theory, according to the
Description and Situation ODP [3], every situation satisfies some description.
We can expect that all members of a recurrent situation series satisfy the
same description. Therefore, we can assert that it exists a description that
plays the role of unifying factor for a recurrent situation series. This
description is the one satisfied by all situation members of the series.

A description is a socially-created object that provides a conceptualisation
or a view on a situation. A recurrent situation description (RSD hereafter)
either defines specific concepts, e.g. repetition interval, or (re)uses
existing ones, e.g. location, which are supposed to _classify_ the entities
involved in the recurrent situation, e.g. 6 months for repetition interval or
Siena for location.

A RSD looks like a plan for recurrent situations: it unifies the recurrent
situations series, describes its member situations, and is satisfied by these
situations: this implies that at least some concepts in the RSD classify the
entities (e.g. 6 months, Siena) involved in the situations.

These unifying factors may be more or less flexible in unifying the recurrent
situation series: properties that usually occur in all the situations member
of the same collection can undergo change under particular conditions. For
instance, yearly recurring situations may be usually held on a specific month
of the year, but they still remain members of the same series even if one
member situation takes place in another period of the year due e.g. to unusual
climatic conditions. E.g., the FIFA world cup has always been in May/June/July
since 1930, but in Qatar 2021 edition it will be held in December, since the
summer is too hot in Qatar. Moreover, a collection of situations recurring on
an annual basis remains the same entity even if one specific edition is
cancelled due to exceptional events.

#### Temporal sequence.

As opposed to other types of collections, a partly predictable temporal
interval holds between the members of a recurrent situation series. The
members of the collection are all consecutive situations, thus creating a
temporally-ordered sequence, so that each situation may have either previous
situations, or next situations, or both, unless the recurrent situation
series, planned to be a series of at least two situations, is interrupted
after the first situation, or is never instantiated.

In the context of this pattern, the temporal interval is relevant only between
members of the same recurrent situation series, while the sequence between a
member situation and any other situation is out of the scope of this pattern.

#### Regular time period.

A collection of situations can be defined recurrent if its members occurrence
are separated by regular time periods. As for natural situations, this time
period is observed in nature, while social situations usually define the
recurrence in a plan, explicitly. The time period concept works as a parameter
that classifies typical sets of values: for example, a “yearly” time period
classifies approximated time intervals, e.g. between 11 and 13 months instead
of exactly 12.

Even considering approximation, there is a difference between the estimated
time period, i.e. the time interval that is planned (for social and planned
situation series) or foreseen (for natural and unplanned situation series)
when the recurrent situation series starts, and the actual time period, i.e.
the time interval that can be observed after the series has been instantiated
and has multiple members. For instance, to a planned time period of
“approximately 2 years” corresponds a measured time interval that represents
the average of time intervals actually occurring between the situations, and
can vary significantly if e.g. some yearly situations are postponed or
canceled. This measurement could be especially relevant to recurrent
situations series, when the time period is a requirement to satisfy (e.g. a
periodical journal), or when a time period different from the expected is due
to negative external causes (e.g. natural climate changes).

##  4 Implementations of the Recurrent Situation Series ODP

The abstract pattern, as described in Section 3, can be formalised and
possibly implemented as an Ontology Design Pattern (ODP), in order to be
reused in different contexts. In this section, we report as an example two
implementations of this pattern: the ODP that has been published on the _ODP
portal_ 111http://ontologydesignpatterns.org [4], and an implementation
specific to the Cultural Heritage (CH) domain, in the context of the
ArCo222https://w3id.org/arco/ontology/arco ontology network [5].

###  4.1 ODP catalogue

The Recurrent Situation Series pattern333rss:
http://www.ontologydesignpatterns.org/cp/owl/recurrentsituationseries.owl#
reuses five existing Ontology Design Patterns (ODPs): _Collection_
444http://ontologydesignpatterns.org/wiki/Submissions:Collection, _Situation_
555http://ontologydesignpatterns.org/wiki/Submissions:Situation, _Description
and Situation_
666http://www.ontologydesignpatterns.org/cp/owl/descriptionandsituation.owl,
_Classification_
777http://ontologydesignpatterns.org/wiki/Submissions:Classification,
_Sequence_ 888http://ontologydesignpatterns.org/wiki/Submissions:Sequence.
These patterns have been extracted from DOLCE+DnS Ultra-lite999dul:
http://www.ontologydesignpatterns.org/ont/dul/DUL.owl# foundational ontology,
a lighter version of DOLCE+DnS [6], thus our ODP is also aligned to it.
Moreover, all reused ODPs are annotated with the OPLa ontology [7], which
facilitates future reuse of this pattern by supporting the process to
understand and explore the ODP. For instance, OPLa allows us to express that a
pattern in a local ontology is a specialisation or a generalisation of another
pattern.

Table 1: Competency questions answered by the Recurrent Situation Series ODP. ID | Competency question
---|---
CQ1 | Which are the situations of a recurrent situation series?
CQ2 | Which is the time period elapsing between two situations of a recurrent situation series?
CQ3 | When is the next situation of a recurrent situation series scheduled?
CQ4 | What are the unifying criteria of a recurrent situation series?
CQ5 | Which is the temporal validity of a unifying factor of a recurrent situation series?
CQ6 | Which is the description satisfied by all the situations of a recurrent situation series?
CQ7 | Which is the (immediate) next situation in a recurrent situation series?
CQ8 | Which is the (immediate) previous situation in a recurrent situation series?

The ODP, as submitted to the ODP portal, can be found at
http://ontologydesignpatterns.org/wiki/Submissions:RecurrentSituationSeries. A
diagram depicting the pattern is shown in Figure 1, and the Competency
Questions (CQs) this pattern can answer are reported in Table 1. The general
pattern provides a reusable template for modelling recurrent situation series.
For the concepts of situation and time period, which are not native to this
pattern, it reuses the semantics associated with: d0:Eventuality, from the
foundational ontology DOLCE-Zero101010d0:
http://www.ontologydesignpatterns.org/ont/d0.owl#, as a top-level concept for
any situation, event or activity, and the formalisation of time periods as
defined by the _Time Period_
ODP111111http://ontologydesignpatterns.org/wiki/Submissions:TimePeriod. In the
following, we describe classes and properties of the pattern.

Figure 1: Pattern for Recurrent Situation Series, in Graffoo notation
(https://essepuntato.it/graffoo/specification/). A recurrent situation series
is related to: its member situations, which are temporally ordered in
sequence; one or more unifying factors; one or more unifying situations, which
involve unifying factors and their temporal validity; the estimated and
measured time period between two consecutive situations. At least one unifying
factor of the series is a description that is satisfied by each situation
member of the series. Moreover, the series satisfies a description defining
the concepts characterising recurrent situation series.

#### RecurrentSituationSeries.

Recurrent situations are represented as individuals of the class
:RecurrentSituationSeries, which is modeled as the intersection of the classes
dul:Collection and :Situation (axiom 1): a recurrent situation series is both
a collection of situations, and a situation on its turn, extending along the
time in which its members occur (see below).

RecurrentSituationSeries $\displaystyle\sqsubseteq$
$\displaystyle\texttt{dul:Collection}\sqcap\texttt{Situation}$ (1)

#### hasMemberSituation.

The membership for a collection of recurring situations (a
:RecurrentSituationSeries) is restricted to situations only (axiom 2), and a
specific object property :hasMemberSituation is provided as a pattern
vocabulary item. Its inverse property is :isSituationMemberOf. A recurrent
situation series may also have 0 members, in order to envision series that are
never instantiated (axiom 3). As aforementioned, the concept of situation is
not native to this pattern, thus we introduce a generic class :Situation as a
subclass of d0:Eventuality (axiom 4).

RecurrentSituationSeries $\displaystyle\sqsubseteq$
$\displaystyle\forall\texttt{hasMemberSituation.Situation}$ (2)
RecurrentSituationSeries $\displaystyle\sqsubseteq$ $\displaystyle\geqslant
0~{}\texttt{hasMemberSituation.Situation}$ (3) Situation
$\displaystyle\sqsubseteq$ d0:Eventuality (4)

#### UnifyingFactor.

:UnifyingFactor is modelled as a subclass of dul:Concept, since it is a social
object for collecting situations. The :RecurrentSituationSeries is related to
the invariant unifying factors by means of the object property
:hasUnifyingFactor. An existential axiom (axiom 5) states that it exists a
dul:Description playing as unifying factor for a recurrent situation series.
This description defines concepts (dul:defines) that conceptualise all the
situations that are member of the series, thus it dul:isSatisfiedBy them.
Therefore, it works as a unifying criterion for the whole series. Moreover,
the recurrent situation series itself satisfies another description, which
defines all the concepts conceptualising a recurrent situation series (e.g.
the general concept of unifying factor, the concept of time period, etc.).

RecurrentSituationSeries $\displaystyle\sqsubseteq$
$\displaystyle\exists\texttt{hasUnifyingFactor.dul:Description}$ (5)

#### UnifyingSituation, isValidIn.

The validity of a unifying factor can be limited to a specific time interval,
for criteria that unify the series for a limited period of time, due to their
flexibility or external causes. For modelling the temporal validity of a
unifying factor, we need a temporally-indexed situation (:UnifyingSituation)
involving a unifying factor (axiom 6). The relation between this unifying
situation and the time interval of its validity is expressed through the
:isValidIn object property (axiom 7).

UnifyingSituation $\displaystyle\sqsubseteq$
$\displaystyle\exists\texttt{involvesUnifyingFactor.owl:Thing}$ (6)
UnifyingSituation $\displaystyle\sqsubseteq$
$\displaystyle\exists\texttt{isValidIn.dul:TimeInterval}$ (7)

#### hasNextSituation, hasPreviousSituation.

It is possible to represent which is the previous and the next situation of a
particular member of the series by means of the object properties
:hasPreviousSituation and :hasNextSituation. The two properties are further
specialised into :hasImmediatePreviousSituation and
:hasImmediateNextSituation, respectively, in order to associate the situations
with their immediate next and previous situations in the sequence.

These properties express a sequential relation between situations that are
members of the same recurrent situation series, while they do not relate a
situation to any other previous or next situation. This intended restriction
cannot be expressed in OWL 2 DL [8], because this requires a “diamond”
construction with explicit coreferent variables. However, it is possible to
check the local integrity of an instantiation of a :RecurrentSituationSeries
by using inference rules.

A situation (member of a recurrent situation series) can be linked by the
property :hasNextSituation to another situation, which is also member of a
recurrent situation series, if both situations are member of the same
collection. For instance, let us consider the following SPARQL query. A triple
asserting a local inconsistency between the two recurrent situation series
will be generated, if they are not the same or if there is not an identity
relation (owl:sameAs) between the two recurrent situation series. Similar
SPARQL queries can be executed for the related constraints (e.g. situations
connected by :hasPreviousSituation object property). These queries can be
embedded in the RDF model via SPIN121212https://spinrdf.org/ or
SHACL131313https://www.w3.org/TR/shacl/.

     PREFIX rss: <http://www.ontologydesignpatterns.org/cp/owl/
                                        recurrentsituationseries.owl#>
     CONSTRUCT {?re1 rss:isLocallyInconsistentWith ?re2}
     WHERE {
            ?rss1 rss:hasMemberSituation ?sit1 .
            ?rss2 rss:hasMemberSituation ?sit2 .
            ?sit1 rss:hasNextSituation ?sit2 .
            filter not exists {?rss1 owl:sameAs+ ?rss2}
            filter (?rss1 != ?rss2)
            filter (?sit1 != ?sit2)
           }

The datatype property :isTheLastSituation, with range xsd:boolean, expresses
whether a situation is the last of the series or not, while :situationNumber
represents the number of the situation within the sequence (e.g. “2” for the
2nd situation).

#### TimePeriod.

While the time period is a core element of this pattern, it can be modelled in
different ways, depending on the local requirements. Thus, we introduce a
generic class :TimePeriod. Nevertheless, we propose a possible solution for
implementing the pattern, by reusing the class tp:TimePeriod from the
_TimePeriod_ ODP141414tp:
http://www.ontologydesignpatterns.org/cp/owl/timeperiod.owl# in our usage
examples (see Section 5). A tp:TimePeriod is related to a measurement unit and
a measurement value. The collection of situations is related to the recurrent
time period with the object property :hasTimePeriod (axiom 8), which is
defined as a property chain [:hasMemberSituation $\circ$
:hasTimePeriodBeforeNextSituation]. The object property
:hasTimePeriodBeforeNextSituation relates a situation, member of a recurrent
situation series, to the approximate (e.g. yearly, monthly, etc.) time period
that typically elapses before the next situation, and is different from the
actual time interval between two specific situations of the series, which can
be derived from the time intervals measured between any two members.
:hasTimePeriod is indeed further specialised into :hasEstimatedTimePeriod and
:hasMeasuredTimePeriod.

RecurrentSituationSeries $\displaystyle\sqsubseteq$
$\displaystyle\exists\texttt{hasTimePeriod.TimePeriod}$ (8)

#### TimePeriodMeasurementUnit.

The tp:TimePeriod is related to the unit for measuring it,
tp:TimePeriodMeasurementUnit (e.g. year, month, week), by means of the object
property tp:hasTimePeriodMeasurementUnit, and to the time value through the
datatype property tp:timePeriodValue, with range xsd:integer.

###  4.2 ArCo: the Italian cultural heritage Knowledge Graph

A second implementation of the pattern is given in ArCo (Architecture of
Knowledge)151515https://w3id.org/arco, the Italian Cultural Heritage (CH)
knowledge graph [5]. We have been working on formally representing recurrent
situations while developing this KG, which consists of an ontology network and
facts on Italian cultural properties, based on the General Catalogue (GC)
maintained by the Italian Institute of the General Catalogue and Documentation
(ICCD-MiBAC). ArCo adopts an agile and iterative, pattern-based and test-
driven ontology development methodology, called eXtreme Design (XD) [9]. Its
requirements are provided in the form of user stories, as scenarios and real
use cases, by a growing community of customers, including ICCD. In particular,
two modelling issues led us to implement a solution for representing
situations recurring regularly over time.

In the GC catalogue records, which describe Italian cultural properties, it is
possible to find information about events (title, place, organizer) involving
cultural properties, such as exhibitions. While there is no explicit
information about possible recurrent cultural events, by analyzing the data,
it is clear that there are many cases of collections of recurrent events (e.g.
“third exhibition", “tenth painting award").

The second use case concerns a particular type of intangible cultural
heritage: ceremonies, customs and celebrations related to the _year cycle_
(e.g. Carnival, Ramadan) or to the _season cycle_ (e.g. popular belief,
science, myth, phenomena related to specific periods of the year), which recur
regularly and whose periodicity is explicitly referred to in the GC catalogue
records (e.g. annual, every two years, three times a year).

The Recurrent Situation Series ODP has been indirectly reused (i.e. has been
reused as a template) in the Cultural Event161616a-ce:
https://w3id.org/arco/ontology/cultural-event ontology module. It has been
specialised in order to represent collections of situations that are cultural
events (cis:CulturalEvent171717cis: http://dati.beniculturali.it/cis/) and
exhibitions (a-ce:Exhibition).

##  5 Use cases

In this section we introduce two use cases, and show how they can be addressed
by the Recurrent Situation Series ODP. The first example is a workshop,
planned to occur periodically; the second example concerns a collection of
recurrent situations observed in nature: annual animal migrations.

###  5.1 Artificial recurrent situation series: the Workshop on Ontology
Design and Patterns

The Workshop on Ontology Design and Patterns (WOP) is a series of workshops
that are co-located with ISWC (International Semantic Web Conference). ISWC
started in 2002, WOP in 2009. Both the conference and the workshop recur
yearly, between October and November, hence there is a time period of about
one year between two situations of the same series. Nevertheless, it can
happen that one edition is skipped: indeed, WOP has not been held in 2011. The
location of these events may change every year. Usually, 3 geographical
regions are involved: Europe, America, Asia/Oceania are sequentially
alternated. However, unforeseen conditions can alter this pattern: in 2020,
due to COVID-19 pandemic, they are organised as virtual events. The Workshop
on Ontology Design and Patterns can be seen as a unitary collection on account
of many (more or less) invariant factors: it is co-located with ISWC, its main
topic is modular and pattern-based design, it involves a community mostly
formed by computer scientists and ontologists, it is organised by the Ontology
Design and Patterns Association (ODPA), etc. Some factors are common to a
subset of the situations member of the series, e.g. the name of the workshop
changed 3 times: (i) _Workshop on Ontology Patterns_ from the first edition to
2012, (ii) _Workshop on Ontology and Semantic Patterns_ from 2013 to 2016, and
(iii) _Workshop on Ontology Design and Patterns_ since 2017. Moreover, the
workshop recurs regularly: its estimated time period is of about 1 year, but
the measured time period slightly differs, since it was not held in 2011, and
its dates are not predefined.

(a) The Workshop on Ontology Design and Patterns (WOP) is a recurrent
situation series and is related to its first three consecutive situations;
three unifying factors and one unifying situation with a limited temporal
validity; the estimated time period between two consecutive events.

(b) The situations member of the WOP series satisfy a description; this
description defines the concepts (topic, chair) characterising each workshop,
and provides a unifying factor to the series. The series, in turn, satisfies a
description defining the core concepts (time period, unifying factor, series
member) of recurrent situation series.

Figure 2: The Workshop on Ontology Design and Patterns (WOP) is a recurrent
situation series.

Figure 2 depicts how WOP can be modelled by reusing this ODP. In Subfigure
2(a), the series collecting the annual workshops is an instance of
:RecurrentSituationSeries, and is related to its members (e.g. WOP 2009, WOP
2010 and WOP 2012). The sequential relations between these (immediate)
consecutive situations are modelled by hasNextSituation and
hasImmediateNextSituation, and their inverses. Since the beginning of the
series, three unifying factors have been valid: the topic of the workshops
(ex:pattern-based-design), the organising committee (ex:wop-organisation), and
being held as joint event of ISWC (ex:co-location-iswc). So far, these
unifying factors do not necessarily need the specification of a temporal
validity. Differently, the name of the workshops can be considered a unifying
factor, however it has changed over time. This can be modelled by introducing
a :UnifyingSituation to specify the time interval in which e.g. the current
name of the series is valid (ex:2017-present). The series is related to its
estimated time period of ex:1year.

Subfigure 2(b) depicts the description satisfied by the members of the ex:wop-
series, i.e. ex:wop-description. This description defines a number of concepts
(the topic, the role of chair) that can be used for classifying the entities
involved in the member situations (each workshop edition). For instance, the
concept ex:topic classifies the entity ex:pattern-based-design, which is the
main topic of the workshop(s). Indeed, the factors unifying the whole series
may be classified by some or all the concepts defined in this description.
Finally, the WOP series satisfies a description that defines the core concepts
of recurrent situation series (ex:recurrent-situation-series-description),
such as the time period, the role of being a situation member of a series, the
unifying factor.

###  5.2 Natural recurrent situation series: the migrations of the Arctic
terns

Animal migration is the regular long-distance movement of animals, on an
annual or seasonal basis, that may be caused by local climate, the season of
the year, and food availability. The Arctic tern (_Sterna paradisaea_) is a
long-distance migrant seabird: it is famous for periodically flying from the
northern hemisphere, where it breeds, to the Southern Ocean and back again
each year. In this way, it lives two summers a year. It departs the breeding
site (Arctic) around August and starts its flight back from the Antarctic
around March. The migration of the Arctic tern is a unitary collection of
situations since its member situations involve all the members of a specific
bird species, happen along the North-South axis, depend on the season of the
year in a specific place, etc.

Figure 3: The Arctic tern migration is a recurrent situation series, with the
annual migrations of the Arctic terns as members. Each annual migration
consists in two parts, the migration from North to South and the migration
from South to North, which are situations members of two separate recurrent
situation series. Finally, these recurrent situation series are part of the
annual Arctic tern migration.

Figure 3, depicts how the migration of the Arctic ster is modelled with this
ODP. The ex:arctic-tern-migration is an instance of :RecurrentSituationSeries,
and represents the annual migration of the seabird. The series includes as
members all the annual migrations, e.g. the ex:arctic-tern-migration-2019 is
the member situation representing the migration from North to South and back
again happened in 2019. This situation actually consists in two parts
(dul:hasPart), which are also situations: the flight from North to South in
August 2019 (ex:arctic-tern-migration-ns-2019) and the flight from South to
North in March 2019 (ex:arctic-tern-migration-sn-2019). These situations are
in turn members of two separate recurrent situation series, also with an
annual time period: the collection of all the flights from North to South and
the collection of the ones from South to North. The ex:arctic-tern-ns-
migration and the ex:arctic-tern-sn-migration are series part of the annual
arctic tern migration series.

##  6 Related work

Most work about recurrence in situations focuses on different types of
recurrent situations and of constraints that bound a recurrent situation. [10]
and [11] investigate calendar-based periodic situations, e.g. repeating time
intervals such as “the first Friday each month”. [12] distinguishes between
strongly periodic events, intermittent events and nearly periodic events, with
respect to the strength and the regularity of intervals between different
occurrences. [13] defines a periodic pattern as the repetition of a component
(e.g. an event) over time, such as daily glucose measurements. Both local
(referring to a single component, e.g. the duration of one event of the
pattern) and periodic (referring to the set of repeating events, e.g. the
duration of the whole pattern) constraints can be applied to periodic
patterns, which can be also split into sub-patterns applied only to a subset
of the repeating events. Constraints related to single members of the periodic
series and to the series as a whole are also considered in [14].

The Recurrent Situation Series pattern follows the approach adopted by [14],
where repeated events are considered as collections of events of the same
type, and can be infinite in case no duration of the collection has been
defined. Here, the notion of recurrent situation is related to the idea of a
coherent series of events that become part of a pattern by means of their
iteration, at regular but not necessarily equal time intervals.

The concept of collection is studied in [2], but the focus of the authors is
on collections of _endurants_ , i.e. entities that are directly localised in
space such as objects and substances, while they do not deal with the
properties peculiar to a collection of situations, as entities directly
localised in time (_perdurants_).

There exist some examples of ontologies modelling recurrent situations, but in
most cases they are modeled as a particular type of event, rather than as a
collection of situations or events.

In DBpedia181818https://dbpedia.org/, recurrent situation series such as
annual film award ceremonies are represented as events, and DBpedia
ontology191919dbo: http://dbpedia.org/ontology/ models concepts such as
dbo:Olympics, dbo:Tournament and dbo:MusicFestival as subclasses of dbo:Event,
with an overlap between the series and its different editions. For instance,
the Palio di Siena202020http://dbpedia.org/resource/Palio_di_Siena is an
instance of dbo:SportsEvent, and there are no links to specific resources
representing its periodic editions. DBpedia ontology also derives classes from
YAGO ontology, which in turn are automatically derived from Wikipedia and
WordNet, such as yago:WikicatRecurringEventsEstablishedIn1895212121yago:
http://dbpedia.org/class/yago/, where the semantics of recurrence is only
implied by the name of the class. DBpedia instances of recurrent situations
are usually related to their frequency (e.g. “biennial, every two years”)
through the dbp:frequency222222dbp: http://dbpedia.org/property/ datatype
property.

Wikidata defines the concept of recurring event (wd:Q15275719232323wd:
http://www.wikidata.org/entity/), “event recurring at an interval”, as a
specific type of event with an associated time period (wdt:P2257242424wdt:
http://www.wikidata.org/prop/direct/) e.g. 4 years for the Olympic Games.
wd:Q15275719 is further specialised based on its frequency (e.g. biennial
event) or its category (e.g. recurring sporting events). Wikidata relates
different editions to their recurring event through the object property _part
of_ (wdt:P361). Being this property transitive, a sub-event that is part of a
specific edition (e.g. a talk by an author of a scientific paper that is part
of an edition of an annual conference), would also be considered part of the
annual conference as a recurring situation.

A mereological approach is less clear in distinguishing typical features of
recurrence, e.g. a recurrent situation may have no members at all and still be
a recurrent situation, but, in this case, a mereological whole would be empty,
i.e. without parts, which is both formally and intuitively impossible.
Moreover, the intermittent presence of the situation would make it a part of a
maximal continuous situation, which is not the same as a collection of
specifically identified situations. In other words, if we admit that
recurrence is a temporal mereological concept, we need also to admit maximal
mereological wholes for any set of situations that have some temporally
ordered similarity, which in the considered use cases would be an
overcommitment.

As part of the _pending section_ 252525Section with ontology terms still work-
in-progress, subject to possibly significant change:
https://schema.org/docs/pending.home.html schema.org defines the class
schema:EventSeries262626schema: http://schema.org/: “a collection of events
that share some unifying characteristic”, modelled as a subclass of
schema:Event and schema:Series. Included events are linked with the series
using the schema:superEvent property, which expresses a part-whole relation,
thus, as in Wikidata, both the series and the related events are modelled as
events, being the latter part of the former. Even if the comment of the class
cites the theme, location and organisers as typical examples of unifying
characteristics, these are not modelled in the ontology. schema:eventSchedule
associates a series of repeating events with a schema:Schedule, which is the
core class for defining the recurrence of an event series, i.e. its repeating
time period. schema:repeatFrequency defines the frequency (schema:Duration) at
which the events will occur according to a schedule, while schema:repeatCount
represents the number of times a recurring event will take place.

##  7 Conclusion

In this paper, we explained an abstract pattern to represent situations that
recur regularly over time, in a temporal sequence, and are members of a
unitary collection since they share some invariant properties. This pattern
can be generalised in order to represent the more general concept of
recurrence. We also discuss two implementations of this pattern as an Ontology
Design Pattern (ODP), a small ontology to be reused in different contexts.
This pattern has been created for addressing the requirement of modeling
cultural events, exhibitions, traditional ceremonies, festivals, etc. However,
it seems to be useful for representing all recurring situations, which can be
distinguished depending on the origin of their regular intermittence: (i) not
artificial situations whose regular recurrence can be observed in nature by
humans, but has a _natural origin_ (e.g. the sunrise every morning or periodic
migrations of animals); (ii) man-made situations where the regular repetition
and the unifying factors are _set by humans_ (e.g. a world day or a train
schedule); (iii) situations created by humans with a periodicity regulated by
a clear purpose, as part of a _workflow_ (e.g. a medical prescription defining
time intervals to take a medicine or periodical supply of services).

In our future work, we plan to study all types of recurrent situations, to
thoroughly investigate the general concept of recurrence, identifying possible
new features to be represented, and to experiment and evaluate other possible
patterns for recurrent situations, alternative to the one presented in this
paper.

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