State-based event modeling

                        Alessander Botti Benevides1,2,3 and Claudio Masolo1
                       1
                      Laboratory for Applied Ontology, ISTC-CNR, Trento, Italy
     2
         Fondazione Bruno Kessler, Trento, Italy 3 ICT Doctoral School, University of Trento, Italy


             Abstract. The lack of formal semantics of traditional approaches to business
             process modeling negatively affects the integration and sharing of models. The
             approaches that provide a semantics often do not discuss the ontological foun-
             dation of the assumed primitives hindering the intelligibility of the models. We
             propose an ontological founded modeling framework in which synchronic prop-
             erties and relations among objects are reified into states. States represent the basic
             blocks on which events are built according to unity criteria. We discuss how our
             framework accommodates different philosophical standpoints on events and we
             illustrate how it can be used to ground event calculus, CLIMB, PSL, and BPMN.


1         Introduction
The ontological analysis of conceptual models in general, and business process mod-
els in particular, is still at the early stages. The lack of semantic transparency of the
used modeling languages often prevents models developed in different frameworks to
be compared and integrated (and understood). The representation of processes involve
general notions—e.g., object, state, event, participation—that have been analyzed in
philosophy and artificial intelligence (see, for instance, [4] and [8] for the concepts of
state and event) but, with few exceptions [20], relatively neglected in BPM formalisms.
     This work illustrates a preliminary effort towards the development of an ontolog-
ically well founded theory that, trying to accommodate different philosophical stand-
points about the nature and structure of events, can be used to ground different BPM
formalisms. The proposed framework in based on the reification into states of the (par-
tial) descriptions of world’s snapshots provided in terms of propositions that involve
objects (and their properties/relations). Complex events are built by collecting states
according to some unity criteria and types of events can be syntactically defined avoid-
ing the “philosophically unsound reification of types” [9]. Events and event-types serve
a compact and cognitively-oriented perspective on the world’s dynamic. Note that, both
the propositions and their reifications are present in our framework. This increases the
generality of our theory, an important feature for providing a common foundation to dif-
ferent modeling languages. To keep the framework simple, we consider a discrete and
linear time. This hypothesis can be relaxed without compromising the general frame-
work, even though we think that, in practical terms, it is not too limiting. We conclude
the paper sketching how our theory could be used to found CLIMB [15], Process Spec-
ification Language (PSL) 1 [2], event calculus (EC) [16], and Business Process Model
and Notation (BPMN) 2 [17].
 1
     http://www.mel.nist.gov/psl/ontology.html.
 2
     http://www.bpmn.org.
2        Botti Benevides, A. and Masolo, C.

2    The basic framework

We consider 3 disjoint and non-empty basic categories: time (tm), object (ob), and
state of affairs, or simply state, (st). Everything else will be built starting from these
categories. Time is considered here as linear, discrete, and atomic. Atoms are called
times, they are not composed by means of a mereological relation, and the precedence
relation defined on them is noted . We leave open if times are punctual or extended
entities. Objects—also called substances, endurants, or continuants—exist in time and
they are wholly present at every time they exist. E.g., tables, persons, bits of stuff. We
introduce the primitive of existence for objects: εt x stands for “the object x exists at time
t”. Events are usually understood as changes, in particular changes in objects, i.e., intu-
itively, events occur when objects acquire or loose some properties. Conversely, states
are static, nothing changes during them. States correspond to—using the terminology
of Kim [11]—exemplifications by objects of properties at a time, i.e., a state corre-
sponds to the fact that an object (several objects) has a given property (are in a given
relation) at a given time. E.g., sun’s being hot in 2013, Luca’s being 180cm high now,
Luca’s being enrolled in the University of Trento in 2010. More specifically, accord-
ing to Armstrong [1], states are linked to contingent properties, i.e., in temporal terms,
properties that objects do not necessarily have during their whole life.
     More technically, our idea is to reify into states, st-instances, temporally qualified
atomic propositions (on objects) of the FOL theory under development. In knowledge
representation, this technique in not new. It is explicitly addressed in [3], it is used in
conceptual modeling, both in ER and UML (see [10]), and it has been taken into account
in reified temporal logics as, for instance, situation and event calculi (see [16]).
     Following these approaches, let V be the extra-logical vocabulary of the FOL-
theory under consideration—we assume only closed formulas in this theory—and P ⊂
V be the set of temporally contingent predicates with one argument of type tm while
all the other arguments are existing entities of type ob (as better represented in (a1)).3
A univocally temporally qualified predicate P is temporally contingent if and only if
Pt x ∧ εt0 x → Pt0 x does not hold. This excludes kinds, e.g., ‘being a person’ or ‘being an
electron’, as well as ε, from P.4 In addition we assume P to be finite.
     P does not include attributes—to be understood as relations between objects and
concrete values, e.g., HAS COLORt (x, red). Our framework can be extended to account
for attributes, however here, to have a simpler framework, we just consider for each
attribute, e.g. color, a set of color-properties, e.g., REDt , BLUEt , etc.. We also exclude
from P diachronic predicates, e.g., HIGHERtt0 x standing for ‘x grew from t to t0 ’. Using
attributes, HIGHERtt0 x is reducible only to synchronic and atemporal predicates, e.g.,
∃hh0 (HEIGHTt (h, x) ∧ HEIGHTt0 (h0 , x) ∧ h < h0 ). Without attributes the reduction must
enumerate all the possible (finite, because P is finite) combinations of height-properties.
Assuming 1Mt , 2Mt , 3Mt are the only heigh-properties, HIGHERtt0 x can be reduced to
(1Mt x ∧ 2Mt0 x) ∨ (1Mt x ∧ 3Mt0 x) ∨ (2Mt x ∧ 3Mt0 x). It is not clear to us if this reduction
holds for all all the diachronic predicates but at least it is viable for attributes of objects.
 3
   We indicate the temporal argument as a subscript. Furthermore, xn stands for x1 , . . . , xn , ob xn
   stands for ob x1 ∧ · · · ∧ ob xn , εt xn stands for εt x1 ∧ · · · ∧ εt xn .
 4
   Existence is trivially rigid: εt x ∧ εt0 x → εt0 x.
                                                              State-based event modeling           3

Finally note that the requirement on the existence of objects in (a1) is quite problematic
for properties like FAMOUSt x because, intuitively, somebody can still be, or become,
famous after his/her death. At the same time, a sort of social recognition, at t, that
what x has done in the past is important seems necessary in order to be famous. These
predicates seem to have a double, both historical and actual, nature (see [13] for more
details). Leaving this delicate discussion apart, we adopt (a1).
    Ve is the extension of V by a set P̄ of unary predicates defined on states that are in
a 1-1 relation with the predicates in P. We indicate with P̄ ∈ P̄ the predicate associated
with P ∈ P. The primitives (i identify the ith object involved in the state where, by
convention, (0 identifies the time, i.e., existence can be extended to states as in (d1).

             xn → tmt ∧ ob xn ∧ εt xn
    V
 a1    P∈P PtW
 a2 st s ↔ P̄∈P̄ P̄s
 a3 P̄s ∧ P̄s0 ∧ εt s ∧ εt s0 i≤α ∀x(x (i s ↔ x (i s0 ) → s = s0
                             V
 a4 Pt x ↔ ∃s(P̄s ∧ εt s ∧ xn ( s)
         n
                                                   (P̄ ∈ P̄ is the predicate associated to P)
 a5 x (i s ∧ y (i s → x = y
 d1 εt s , t (0 s
 d2 pt xn , s(P̄s ∧ εt s ∧ xn ( s)
 d3 x ( s , x (1 s ∨ · · · ∨ x (α s          (α is the largest arity of the predicates in P)

(a2) guarantees that all states are covered, but not necessarily partitionated, by the pred-
icates in P̄ while (a3)5 enforces sufficient identity conditions for states. (a4)6 assures
that the reification of Pt xn exists only when Pt xn holds, i.e., states reify only true propo-
sitions. (a5) enforces (i to be injective. (a3) supports the definition (d2)—where is a
description operator à la Russell 7 —i.e., it is possible to introduce a set D of descrip-
tions p that are in a 1-1 relation with both the predicates P̄ ∈ P̄ and P ∈ P. Finally, (d3)
defines a participation relation ( in which the ith position is irrelevant.
     Our approach is in line with Kim’s theory; (a2)-(a4) closely correspond to Kim’s
existence condition— “the state [x, P, t] exists if and only if substance x has property
P at time t” [11] (where [x, P, t] corresponds to our pt x)—even though properties P
are here in the domain of quantification while we have a 1-1 meta-relation between
P and P̄. Differently from Kim’s identity condition—[x, P, t] = [y, Q, t0 ] if and only
if x = y, P = Q, and t = t0 —our framework allows for pt xn = qt xn with p dif-
ferent from q. In addition, while facts are usually considered to be in a 1-1 corre-
spondence, up to logical equivalence, with true propositions, this is not the case of
st-instances. Given two logically equivalent predicates, say Pt xn ↔ Qt xn , our theory
guarantees P̄-states and Q̄-states to be mutually existentially dependent but not identi-
cal. More specifically both pt xn , qt xn and pt xn = qt xn are consistent statements. We
can then partially account for the intension of properties. Let us suppose to organize
the P-predicates into a taxonomy by considering the isa relation. All the necessarily
disjoint taxonomical leaves can be safely reified into disjoint classes of states. Con-
sider now the leaves that can have common instances. One can distinguish a purely
 5
   Where α is the largest arity of the predicates in P.
 6
   xn ( s stands for x1 (1 s ∧ · · · ∧ xn (n s ∧ ¬∃y(y (n+1 s ∨ · · · ∨ y (α s), while xn = yn stands
   for x1 = y1 ∧ · · · ∧ xn = yn .
 7
   ψ( x(φx)) is equivalent to ∃x(φx ∧ ∀y(φy → y = x) ∧ ψ(x)).
4        Botti Benevides, A. and Masolo, C.

extensional overlap from an intensional one. For instance, ‘being tri-lateral’ (3L) and
‘being tri-angular’ (3A) are a typical example of extensionally coincident but inten-
sionally different properties. Vice versa, ‘being red’ and ‘being orange’ are often con-
sidered as non-intensionally disjoint because, for instance, vermilion objects are both
red and orange.8 We can represent this difference by imposing the disjointness of 3L-
and 3A-states or the REDt x ∧ ORANGEt x → redt x = oranget x constraint. Similarly,
given ELECTRONt x → [9.11×10−31 KG]t x and SCARLETt x → REDt x,9 we can impose
that ELECTRON-states are disjoint from [9.11×10−31 KG]-states while SCARLET-states
are RED-states.10 These distinctions are not logical, they require an ontological analysis
in charge of the user that, however, can now at least partially be represented. Our theory
is also compatible with Armstrong’s view on states of affairs intended as truth-makers
of propositions [1], even though we do not commit to the ontological primacy of states
with respect to (true) propositions.11
     States are not reducible to tuples because it is possible to have pt xn , qt xn (with
p, q ∈ D). Furthermore, states are completely determined, i.e., they reify atomic propo-
sitions with no variables; closed formulas with existentially or universally quantified
variables do not say anything of the actual configuration of the world, they do not
introduce states but only existential constraints on them. For instance, formulas like
∀txn (Pt xn → Qt xn ) do not state the existence of any state, they just introduce a depen-
dence of P-states12 on Q-states. Formulas like ∃txn (Pt xn )—or disjunctions and negations
of atomic propositions—introduce a sort of indeterminism because they are compatible
with different configurations of the world.
     Unless explicitly forced, our framework is quite conservative: the isa relation just
implies an existential dependence (see (a4)). This holds in general for implications with
form P1 ∧ · · · ∧ Pn → Q1 ∨ · · · ∨ Qn .13 The only exceptions are the equivalences with
form P1 ∧ · · · ∧ Pn ↔ Q that are taken into account in Section 3.

3    Events
We extend our framework by introducing a new kind of entities called event (ev)14
and an atomic classical extensional mereology (see [5] for the details) defined on them:
x v y stands for ‘the event x is part of the event y’. The usual mereological notions are
defined in (d4)-(d8). In this theory, events can always be decomposed into atoms in an
unique way, see (t1) and (t2). (a6) enforces atoms to be states, i.e., ev subsumes st and
events are sums of states. Thus, an event corresponds to the conjunction of the atomic
P-propositions corresponding to its atomic parts (states).
 8
   We assume here that vermillion is not in P.
 9
   The last isa relation is called determinate-determinable and has been deeply studied in philos-
   ophy (see [22]).
10
   One could also introduce a causation relation defined on states (or, more generally, on events)
   that does not reduce to material implication.
11
   In Armstrong’s theory, properties are in the domain and states of affairs are composed by
   substances and properties. His theory is close to Kim’s one but time is very marginally treated.
12
   A P-state is a state s such that ∃txn (s = pt xn ).
13
   Actually this is the implicative normal form every FOL-sentence can be converted in.
14
   Our events correspond to what Galton calls eventualities [7].
                                                             State-based event modeling          5

d4 x @ y , x v y ∧ x , y                                                (x is a proper part of y)
d5 x G y , ∃z(z v x ∧ z v y)                                                   (x and y overlap)
d6 x Σ yn , ∀w(w G x ↔ (w G y1 ∨ · · · ∨ w G yn ))                         (x is the sum of yn s)
d7 Λx , ev x ∧ ¬∃y(y @ x)                                                         (x is an atom)
d8 x Λv y , Λx ∧ x v y                                                (x is an atomic part of y)
t1 x v y ↔ ∀z(z Λv x → z Λv y)
t2 x Σ yn ∧ x0 Σ yn → x = x0
a6 st s ↔ Λs

Let us go back to the form P1 ∧ · · · ∧ Pn ↔ Q, i.e., Q is logically reducible to P1 , . . . , Pn .
By (a2) and (a4) there exist a state corresponding to Q and n-states corresponding to
the Pi s. According to the discussion at the end of the Section 2, the difference between
the Q-state and the sum of the n Pi -states is justified only in case Q and P1 ∧ · · · ∧ Pn
intensionally differ. Vice versa, Q is redundant and should be removed from P.
    With a slight abuse of notation, (d9) and (d10) extend ε and (, respectively, to
events. (d11) defines a temporally qualified version of ( while (d12) introduces the
usual notion of temporal slice.
d9 εt e , ∃s(s Λv e ∧ εt s)
d10 x ( e , ∃s(s Λv e ∧ x ( s)
d11 x (t e , ∃s(s Λv e ∧ x ( s ∧ εt s)
d12 x t y , ∀z(z v x ↔ εt z ∧ z v y)

     From a structuralist perspective, states can be seen as ‘sensory atoms’, as ‘temporal
dependent data’, as ‘pointlike observations’ on which complex entities can be built. In
a movie metaphor, states represent dynamic factual knowledge, i.e., factual knowledge
that concerns a single snapshot, that is ‘acquired’ at a given time. Vice versa, static
factual knowledge, once acquired, does not need to be re-considered. For instance, one
checks (at a given time) for the existence or some contingent properties of a person
but not for her personhood. Terminological knowledge can concern static-predicates,
e.g., PERSONx → MORTALx as well as dynamic ones. Both synchronic- and diachronic-
terminological knowledge, e.g., SCARLETt x → REDt x and 2Mt x ∧ εt0 x → (2Mt0 x ∨ 3Mt0 x),
introduce existential dependences among states that can be further specialized by means
of static-predicates, e.g., FERRARIx ∧ REDt x ∧ εt0 x → REDt0 x.
     In our movie metaphor, a movie—a narrative—can be seen as a sequence of snap-
shots (described in terms of our vocabulary) containing states that satisfy the existential
dependences introduced by the laws. Events offer an abstract and dynamically-oriented
point of view on narratives. Perception organizes stimuli by grouping them in unitary
objects that allow us to interact with the world in a quick and fruitful way (see [18] for
an introduction). Similarly, states can be organized by, synchronically or diachronically,
grouping them into events, entities that have a cognitive and/or practical relevance for
understanding the dynamic of the world. Furthermore, types of events can be used to
represent the laws that regulate the world in a cognitive-friendly fashion.
     It would be noted that nothing prevents the user to introduce predicates like STABt xy
in P. Differently from the usual conceptualisation, stab44bc (brutus, caesar) is here an
atomic state. This counter-intuitive classification can be explained in terms of granular-
ity. In temporal terms, STAB cannot be further analyzed. The user decided to consider
6         Botti Benevides, A. and Masolo, C.

stabbing-events as atomic, i.e., as ‘observable’ in a single snapshot, but no changes can
be observed in a snapshot. This does not contradict the foundations of our framework.
To intuitively explain this fact, one can think that the user considered a coarse temporal
granularity, i.e., a time corresponds to an interval in the actual movie. The FOL-theory
can then be seen as an (abstract) annotation of what happens during an interval. As-
sume now that STAB WITHt xyz is also in P. stab with44bc (brutus, caesar, k#1) and
stab44bc (brutus, caesar) can be dependent but they differ because they have differ-
ent participants. We loose the most relevant aspect of the approach of Davidson [6],
i.e., there is only one event, the stabbing, that is performed with a knife. By assum-
ing that STAB WITH is definable as, for instance, STAB WITHt xyz , STABt xy ∧ USEt xz,
STAB WITH-states could be considered as sums of STAB- and USE-states.15


3.1     Changes

Changes in objects and changes in general are often distinguished, compare (OC) and
(GC) below taken from [12]. (OC) commits to the survival of the object x: if something
does not exist it cannot lack a property ([12], p.82).16 (GC) commits to events (entities
that occur), it does not refer to objects, and even assuming that all the proposition S
concern only objects, still (GC) does not commit to the persistence of any object.
(OC) An object changes if and only if             (GC) A change occurs if and only if
   1. there is a property, P,                        1. there are distinct times, t and t0 ,
   2. there is an object, x,                         2. there is a proposition, S , and
   3. there are distinct times, t and t0 , and       3. S is true at t and false at t0 .
   4. x has P at t and fails to have P at t0 .
In our framework, (d13) and (d14) simulate, respectively, (GC) and the event version
of (OC). Note that (d14) just adds εt0 x to (d13), in both cases x could not participate to
e at t0 . One could strength (d14) as in (d15), however the clause x (t0 e is problematic
because it requires the existence of a state qt0 x with q ∈ D (and different from p)
                                                       W
but nothing guarantees that Pt x ∧ ¬Pt0 x ∧ εt0 x → Q∈P Qt0 x (more generally, nothing
                         W
guarantees that εt x → Q∈P Qt x). Lombard addresses this problem through the notion
of quality space—a set S of mutually exclusive (non-relational) properties such that if
Pit x, with Pi ∈ S , at every time t0 at which the object x exists there is a Pj ∈ S such
that Pjt0 x—and assuming that (basic) changes are movements of objects through quality
spaces. Assume P is partitioned in n-quality spaces that induce a partition S1 , . . . , Sn
in D. Basic changes can then be defined as in (d16) (where p is different from q).
d13 CNGe , p∈D ∃xtt0 (pt x v e ∧ ¬εt0 (pt0 x) ∧ εt0 e)
               W

d14 oCNGe , p∈D ∃xtt0 (pt x v e ∧ ¬εt0 (pt0 x) ∧ εt0 e ∧ εt0 x)
                W

d15 sCNGe , p∈D ∃xtt0 (pt x t e ∧ ¬εt0 (pt0 x) ∧ x (t0 e ∧ ¬∃y(y , x ∧ y (t0 e))
                W

d16 bCNGe , p,q∈Si ∃xtt0 (pt x t e ∧ qt0 x t0 e)
                W
                                                                             (for some Si )
                W         n 0    n                n             n
d17 gCNGe , p∈D ∃x tt (pt x v e ∧ ¬εt0 (pt0 x ) ∧ εt0 e ∧ εt0 x )

15
     However the ‘using’ and the ‘stabbing’ must be linked, x could do simultaneous actions.
16
     As observed by Kim, “[w]hether coming into being and passing away can be construed as
     changes in substances” ([11], p.310) is a question to be addressed.
                                                             State-based event modeling          7

(d14) (similarly for (d13)) can be generalized by allowing propositions that involve sev-
eral objects (d17). This generalization matches (GC) because there is no commitment
to the number (and nature) of entities involved in the proposition S . Still Pt xn ∧ ¬Pt0 xn
is an evidence of a change in the world delimitating this change to the objects xn , i.e., it
points out the part of the world involved in the change. However, Pt xn ∧ ¬Pt0 xn ∧ εt0 xn
does not say what objects change. For reason of space we cannot discuss in more details
this problem and the one (discussed in [12]) related to the cases where a parthood rela-
tion between objects is included in P. Here we assume a liberal approach that accepts
also ‘events’ where no object changes (e.g., homeomeric events like the sum of pt x and
pt0 x) and leaves to the user the possibility to filter the sums of states according to her
needs and the primitives of the FOL-theory under construction.
     Finally, Galton [7] defines instantaneous transitions as transitions from a state that
holds at t to a state of different kind that holds at t0 (the successor of t), e.g., the transi-
tion from redt x to bluet0 x. He claims that this transition does not occur at any time, it
occurs between times, between t and t0 in the example. Consequently he includes these
‘interfaces’ among the temporal entities. From a cognitive perspective, to be observed,
a transition requires the observation of two distinct states. According to (GC), we tend
then to see these transitions as non-instantaneous, i.e., as (specific) changes.


4    Comparison and discussion

We compare our framework with four approaches developed for representing and rea-
soning on events and processes: the event calculus (EC) [16], Computational Logic for
the verIfication and Modeling of Business constraints (CLIMB) [15], the PSL [2], and
BPMN [17]. Given the limited space and the preliminary nature of this work, we cannot
fully introduce these approaches nor provide a complete formal comparison. We focus
on some differences and similarities relevant from the ontological and representational
perspective outlining some strategies one could follow for a full comparison.
    The EC considers three sorts of entities: event, fluent, and timepoint.17 Events occur
in the world at times, Happens(e, t), while fluents are time-varying propositions that
hold, are true, at times, HoldsAt(f , t). Both events and fluents are terms individuated
by total functions. The user decides which functions identify events, e.g. stab(x, y),
and which functions identify fluents, e.g., on(x, y). While fluents can exist and hold at
different times, our states exist at single times. Consider the situation where x is on y
both at t1 and t2 . In the EC there is a single fluent, on(x, y), that holds both at t1 and
t2 while, in our framework, there is a complex event composed by two distinct states:
ont1 (x, y) and ont2 (x, y). In our framework, a fluent could be defined as the sum of all
the states of the same type (identified by the same description) that involve the same ob-
jects (in the same order), a notion quite similar to the one of homeomerous-perdurants
in the Descriptive Ontology for Linguistic and Cognitive Engineering (DOLCE) [14].18
However, note that in EC the representation of fluents through total functions f forces
17
   Differently from situation calculus [21], in EC time is usually considered as linear even though
   a branching version of EC exists.
18
   We would need to introduce infinite sums, see [5].
8          Botti Benevides, A. and Masolo, C.

the existence of f xn (n is the arity of f) whatever xn one considers. The fluent f xn ex-
ists in the domain of quantification even though it never holds. Fluents have then a
propositional nature.19 Vive versa, adopting the proposed reduction of fluents to sums
of states, fluents necessarily hold, they are sums of exemplifications of properties by
objects at a time. To avoid total functions and propositions, we pay the cost of introduc-
ing definite descriptions and (i primitives in our framework. As said, we see changes
and actions as intrinsically non atomic; it is not possible to observe changes or actions
in a single snapshot. The situation is different for ev-instances that persist in time. As
done for fluents, we can characterize a specific notion of event—disjoint from the one of
fluent—e.g., the one of change discussed in Section 3.1. Happens(x, t) and HoldsAt(x, t)
collapse then to εt x while all the EC’s primitives—Initiates(e, f , t), Terminates(e, f , t),
Releases(e, f , t), ReleasedAt(f , t)—are now defined on sums of states. Finally, while EC
embrace the unique name assumption—i.e., given two different descriptions p and q,
necessarily pxn , qym —we can force pt xn = qt xn as in the case of scarlet- and red-
states. This difference reminds us the one between identity and coincidence [19].
     Similarly to our approach, CLIMB does not make a distinction between events and
fluents. In addition to times, only events—represented as terms (usually identified by
functions)—are present. Both send(x, y, msg) and status(cable, off ) are reported as ex-
amples of events in [15].20 Similarly to EC, in CLIMB, the same event can happen at
different times; H(E, T ) stands for “event E happens at timepoint T ”, where E and T
are terms. A big difference concerns the fact that free variables are admitted in E and
T . In these cases events are sets of event-tokens (traces) and H(E, T ) means that some
of these event-tokens occur at one time in T . The variables can also be constrained
in scope; for instance one could consider only the messages sent by a given group of
persons. We prefer to clearly distinguish event-tokens from event-types. This is why
we excluded free variables from state-terms, associating to existentially and univer-
sally quantified P-formulas only existential constraints on states. Thus, we can reduce
H(E, T ) to εT E only in absence of free variables in E and T . In the other cases, H(E, T )
corresponds to the existential constraint ∃et(Et e ∧ Tt) where T is a temporal constraint
and E has the form Et e , ∃xn (e = pt xn ∧ Cxn ), where p ∈ D is a given description
(i.e., all the E-instances are linked to an unique description p) and C represents the con-
straints on the objects xn . The previous reduction can be generalized to allow H to apply
also to complex events. Integrity constraints are central in CLIMB. Roughly speaking,
integrity constraints represent the expected ‘outcomes’ of some events that happened at
given times, i.e., the possible events that would satisfy the system requirements. They
are represented by rules Body → Head where, in addition to temporal constraints, the
body is a conjunction of H(E, T ) clauses and the head is a disjunction of conjunctions
of E(E, T ) and EN(E, T ) clauses, where E(E, T ) (EN(E, T )) represents the positive
(negative) expectation that E happens at T . Since integrity constraints are seen as re-
quirements, they can be fulfilled or violated. Given a specific sequence s of (actual)
events, CLIMB is able to check if s satisfies (is compliant with) the integrity con-
straints. Our framework (as well as the EC) does not contemplate possible evolutions of
the world, it is narrative-based, hypothetical states are not represented. Our synchronic
19
     This maybe explains why EC uses the predicate HoldsAt instead of existence in time.
20
     One could discuss what is the ontological nature of off.
                                                              State-based event modeling            9

and diachronic (existential) constraints among (types of) states are not requirements;
like natural laws, they necessary hold, they cannot be violated, they represent how the
‘world works’. In our framework, integrity constraints can be (partially) represented by
introducing types of events that collect all the non-compliant sequences of states. E.g.,
given the requirement (t and t0 are constants) H(E, t) → E(E 0 , t0 ), one can introduce
R1 e , ∃s(s v e ∧ Et s ∧ εt0 e ∧ ¬∃s0 (E0t0 s0 )), where E and E0 are defined as before, while
for H(E, t) → EN(E 0 , t0 ) one can introduce R2 e , ∃ss0 (Et s ∧ E0t0 s0 ∧ s v e ∧ s0 v e).21
The compliance-check becomes then a sort of classification problem, we need to check
that there are no R1 - or R2 -instances.
     PSL theory encompasses a core theory (PSL-Core) and a number of extensions.
PSL-Core considers four kinds of entities: activity, activity occurrence, timepoint and
object. Every activity occurrence is an occurrence of —a primitive of PSL—a unique
activity and has a begin and an end timepoint. For instance, the activity (paint House#1
Paintcan#1) can have different occurrences: the House#1 (a specific object) can be (par-
tially) paint several times, during disjoint time intervals, using the same Paintcan#1.
Timepoints form a discrete infinite linear ordering with endpoints at infinity, while “[a]n
object is anything that is not a timepoint, nor an activity nor an activity-occurrence”.
An object can participate at a timepoint in an activity occurrence (participates in is a
ternary primitive of PSL) only when the object exists and the activity is occurring.22
     In our framework, an occurrence corresponds to a (non atomic) event23 while activi-
ties can be introduced as maximally specified event-types, i.e., leafs of the taxonomy of
(non-atomic) events. Classes of activities can then be reduced to isa-generalizations of
the leafs in the taxonomy of events. The PSL-primitive occurrence of becomes instan-
tiation while participates in can be mapped to (t . Note that occurrences are always
occurrences of an unique activity. We can introduce this constraint by assuming that the
leafs associated to activities partitionate the domain of (non atomic) events.
     From a general perspective, we can see BPMN-models as definitions of (our) event-
types. The core of BPMN provides a set of modeling constructs to specify how a pro-
cess (an event in our terminology) is structured in sub-activities. Activities, events, and
gateways (called Flow Objects) are used to specify this structure. Activities seem to
correspond to event-types—i.e., the whole process and its (sub-)activities have the same
ontological nature—while gateways, as well as sequence flows, introduce temporal con-
straints on (sub-)activities and add some indeterminism that, as said, can be represented
21
   Note that R1 and R2 are quite close to changes as discussed on Section 3.1.
22
   It is not clear whether and how the constants in the activity-term (paint House#1 Paintcan#1),
   i.e., House#1 and Paintcan#1, are linked by participates in to the occurrences of the activity. In
   addition, even though there are no axioms that guarantee that all the occurrences of an activity
   have the same participants, the examples reported in the PSL documentation consider activity
   terms with specific objects (constants) and no free variables.
23
   In PSL, activities are entities that can have occurrences happening at different intervals of time.
   The Outer Core extension called ‘Theory of Occurrence Trees’ defines occurrence tree as a
   poset representing all possible sequences of occurrences of all activities. It is not clear to us
   whether activity occurrences are actual or possible individuals (whether there is prescriptive
   or descriptive perspective on occurrences). In our framework we consider only actual events
   and we assume a descriptive attitude. Prescriptive laws can be enforced by means of axioms
   on types of events.
10       Botti Benevides, A. and Masolo, C.

by existential abstractions in our framework. BPMN-events seem to correspond to very
general types of states, however types of events cannot be introduced in BPMN pre-
venting, for instance, the representation of pre- or post-conditions of activities.
    Pools are intended to capture the notion of participant—the participant of a given
sub-process. Message Flows characterize the interchange of messages between partici-
pants. The exchanges of messages can be seen as (sending/receiving) events that involve
a document, a data object, or simply as a synchronization mechanism across pools that
can be reduced to some temporal constraints in our framework.
    The lack of semantics of BPMN prevents us to provide a safe semantics in terms of
our framework. Given the preliminary nature of this work, to stress our own view on
BPMN-constructs seems not appropriate.

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