=Paper=
{{Paper
|id=Vol-160/paper-7
|storemode=property
|title=Meta-Modelling as a Means for Improved Communication and Interoperability ‐ The Case of Frisco
|pdfUrl=https://ceur-ws.org/Vol-160/paper5.pdf
|volume=Vol-160
|authors=P. Wohed,B. Andersson
|dblpUrl=https://dblp.org/rec/conf/caise/WohedA05a
}}
==Meta-Modelling as a Means for Improved Communication and Interoperability ‐ The Case of Frisco==
https://ceur-ws.org/Vol-160/paper5.pdf
Meta-Modelling as a Means for Improved
Communication and Interoperability
– The Case of Frisco
Petia Wohed1 and Birger Andersson2
1
Centre de Recherche en Automatique de Nancy
Université Henri Poincaré, Nancy 1/CNRS
BP239, 54506 Vandoeuvre les Nancy, France
petia.wohed@cran.uhp-nancy.fr
2
Department of Computer and Systems Sciences
Stockholm University/The Royal Institute of Technology
Forum 100, SE-164 40 Kista, Sweden
ba@dsv.su.se
Abstract. Unclear and disparate terminology are causing problems in
the area of information system development. To address these issues,
modelling frameworks and ontologies based on solid philosophical and
logical foundations have been developed. Hypothesizing that a diagram-
matic interface will be beneficial for comprehension and communication
resulting in wider dissemination, we propose a UML-based meta-model
of one of those frameworks–Frisco. The analysis done during the de-
velopment of the meta-model resulted in identification of a number of
ambiguities in the framework. Discussion of these as well as suggestions
of ways to solve them are also presented.
1 Introduction
Terminological fuzziness is characterizing the field of information systems creat-
ing all kinds of problems. To address this, during the last decades, frameworks
like Frisco (Framework of Information Systems Concepts) [1] and BWW (Bunge,
Wand and Weber models) [2] were developed. The goals of those frameworks co-
incides with the goals of ontology analysis and engineering [3], namely: to provide
a coherent system of concepts, the use of which would reduce misunderstandings
and ambiguities and would 1) support communication between stakeholder with
different needs and approaches 2) facilitate interoperability among systems and
3) be beneficial for requirement identification, reusability and reliability within
information system development process. Thus, the field of ontology engineering
has received a growing attention where ontologies like CYC [4] in the knowledge
representation area, and Toronto Virtual Enterprise (TOVE) and Enterprise On-
tology (EO) [5, 6] for enterprise modelling have been developed.
Characteristic for all these approaches is that they rely on solid theoretical
work, are developed by mathematicians, logicians and/or computer scientists,
are well formalised and their initial representation is in some formal notation.
While this last property is an absolute requirement for achieving interoperabil-
ity and clear semantics, it is hardly beneficial when it comes to the fulfilment
of the communication goal mentioned above. Not every person involved in an
information system development process can be assumed to be skilled in reading
and understanding formal notations.
To address the issue of comprehensibility, graphical formalisms has been de-
veloped. For instance, in [7] a graphical meta-model was developed for represen-
tation of BWW models. In this paper we report on using this approach when
developing such a meta-model for the Frisco framework. As in [7] we motivate
the approach by the following arguments.
– The development of a meta-model that visualises the concepts currently
existing and the framework and the relationships between them will increase
the communication and understanding of the framework and facilitate its
use.
– The development of a meta-model will support analysis of the framework
and facilitate a further development of it.
– The existence of a meta-model will further facilitate analysis of Frisco with
respect to other existing approaches e.g., BWW, Tove, EO, etc. During such
analysis overlapping and discrepancies between the ontologies shall be out-
lined which will result in a clearer positioning of Frisco according to the
other relevant works in the area.
– Meta-models of ontologies, in general, support the analysis of modelling lan-
guages in the sense that they can be used as reference frameworks through
which the considered languages are analysed.
– The experience of the work on the development and utilisation of the meta-
model will contribute to the overall knowledge within Ontology Engineering
field.
Unlike [7], where the benefits of the developed meta-model were mainly
demonstrated for the purposes of the analysis of two grammars, i.e., according
to the fourth point outlined above, and only briefly exemplifying the second
point above, we are in this paper in detail demonstrating how the development
of a meta-model can benefit for the further development of a framework. We
have selected Frisco to be the subject for our work because, 1) it was specifically
developed for addressing the conceptualization within the Information Systems
domain, 2) we are interested in an cross-analysis between FRISCO and BWW
(and later on with EO and TOVE). As an initial step in such a cross-analysis,
the creation and use of graphical meta-models, e.g. the one developed for BWW
or the one for Frisco partially presented here, will be beneficial.
As a graphical representation language, we are using UML class diagrams [8].
This choice is motivated by the wide dispersion and knowledge of UML, often
also referred to as a de-facto standard for information system analysis and devel-
opment. UML class diagrams limits our expressive power to what can be stated
using those diagrams, but for the purposes of this paper they are adequate.
The paper proceeds by presentation of Frisco and the development of meta-
model for it in the next section. In section 3, the results of the analysis of Frisco
provided during the development of the meta-model are presented and some
improvements suggested. Section 4, concludes the work and outlines directions
for further research.
2 The Frisco framework
As any other axiomatic system, Frisco starts by defining a small set of primitives
from which the rest of the concepts are gradually built up. It is divided into a
number of layers: a fundamental layer, an actor, actions and actands layer, a
system concepts layer, and an organizational and information system concepts
layer. Due to space limitation, in this paper we only present and work on the
the fundamental and actor, actions, and actands layers, which we discuss below.
2.1 The Fundamental Layer
The world is made up of Things. A thing is either an Elementary or a Composite
thing. Composite things are build up through Relationships. Relationships are sets
of binary tuples, the elements of which are things: the first element in a tuple
is called a Predicated thing and the second element a Predicator. Relationships
are themselves considered as things, which makes it possible to represent the
complex structures often existing in a domain.
The formal definitions are reprinted from [1] in table 1 in the Appendix. The
meta-model which we propose is drawn in figure 1. Generally, every concept is
represented by a class and in the lower right hand corner of every class the name
of the set as defined in table 1 is given.
The relationships between concepts are captured either by generalisation/
specialisation constraints (i.e., isa relationships) or by ordinary relationships to
which we have given names either appearing in the formal definitions, or just
close to the natural language comprehension. The cardinality constraints are
derived from the formal definitions. Some rules like that “A Thing is either Ele-
mentary or Composite” are directly captured in the graphical model (by joining
the ends of the corresponding isa relationships) while others like that “An Ele-
mentary Thing is defined as a Thing which is not a Relationship...” can rather
indirectly be seen in the graphical model through the implications they bring
e.g., “The sets of Elementary things and Relationships are disjoint” and through
the reasoning that Relationships are kinds of (isa) Composite Things which in
turns are disjoint with the Elementary Things. Furthermore, for the sake of
explicitness we have introduced the class Pair, which does not have a formal
definition of its own. However, the concept Pair is used for defining Relation-
ships and distinguishing between unary relationships (consisting of one pair),
binary (i.e., relationships composed of a couple of pairs), ternary relationships
(consisting three pairs), and so on.
According to the description in Frisco, Predicator and Predicated Things
do not need to be disjoint sets (so we have not joined their isa relationships
endings) but in most real life cases they are indeed disjoint. Furthermore, the
isa relationship between Relationship and Thing classes is derivable through the
Thing
Z
Composite Elementary Predicated
Thing CZ Thing EZ Predicator
P Thing Q
1 1
^ has ^ has
* *
Entity
E Pair
* ^ consists of
*
Set Member-
Relationship
R ship SM
Fig. 1. Frisco - basic concepts
transitivity of the isa relationships from Relationship to Composite Thing to
Thing, but we have chosen to explicitly draw it in the model to directly depict
the definition of Relationship concept stating that a relationships is a thing.
Finally for this part of the framework, a Set Membership is a binary relationship
with the predicators has-element and is-element-of within its first and second pair
correspondingly. As instances of the class Predicator has-element and is-element-
of are not directly depicted in the graphical model. Also the relationship between
Elementary Thing, as consisting of Things that are not Relationships and that
does not appear in pairs with the predicator has-element has been difficult to
capture graphically and is for the moment missing from the model.
In the remainder of the Fundamental layer the concept of Transition with sur-
rounding terminology is introduced. (The formal definitions are reprinted in ta-
ble 2 in the Appendix). A Transition is a kind of binary relationship in which the
predicators in the tuples are the primitives before and after and the predicated
things are composite things, so called States. Complex transitions can be built
up through Sequence, Choice and Concurrency to State Transitions Structures. A
coherent state transition structure, i.e., a structure with a unique input (before)
state and a unique output (after) state, is called a Composite Transition. Further-
more, Rules are used to define the set of permissible states and transitions in a
context. To capture these concepts we are gradually extending the meta-model
with the shaded classes as depicted in figure 2.
In this part of the framework the concept of Transition Occurrence for captur-
ing the different occurrences of the transitions is introduced. We have introduced
the corresponding class into the model, but we leave the discussion of it for the
next section. The concepts Type, Population and Instance, which are all part of
the formal definition of the framework, are for the moment left out of the model
but will be discussed in the next section.
2.2 The Layer of Actors, Actions and Actands
Transitions which are performed by someone are distinguished and called Ac-
tions. Actions are presented through a couple of tuples, the predicated thing of
the first of which shows the performing Actor and of the second one the transition
Thing
Z
Composite Elementary Predicated
Thing CZ Thing EZ Predicator
P Thing Q
1 1
^ has ^ has
* *
Entity
E Pair
* ^ consists of
*
Relationship Set Member-
R ship SM
1 < pre
* < of Transition
State 1 < post * Transition
S T 1 * occurrence
* defines permissable > 1 1
* ^ involves
^ consists of
permissable ^ involves Concurrency
* * * *
Composite StateTransition Choice
Rule
R Transition CT Structure ST
Sequence
Fig. 2. Frisco - Fundamental Layer
he/she is performing. The predicators used for describing this are the primitives
performing and performed-by, correspondingly. The things involved in the input
and the output states of an action, and which are not actors for that action, are
called Actands. The input actands for an action (i.e. the actands from the input
state) together with the actors are the Resources for that action. Also using the
primitive is-context, some of the input actands can be predicated, in order to
define the Action context. In a similar way the Goal of an action can be defined
by intentionally stating the desired output state.
The formal definitions for this layer are reprinted in table 3 in the Appendix.
The meta-model is drawn in figure 3. The transition part of the Fundamental
layer, as closely related to the concepts defined here, is reprinted in the bottom
of the figure. The isa relationships between the different layers are indicated by
dotted arrows. Furthermore, the basic concept from the fundamental layer on
which a concept is built on is indicated by giving the name of the set in the
upper right corner of the rectangle of the class.
3 Results from the analysis of Frisco during the
construction of the meta-model
During the process of building the meta-model we identified some minor prob-
lems within the formal definitions of Frisco. A discussion of them as well as
suggestions for how to solve them are provided below.
The idea of defining State Transition Structure and Composite Transition con-
cepts is to be able to build up complex transitions. However, the necessary
recursive element for achieving this and assumed in the natural language de-
scription of the framework is missing from the formal definitions. This makes
it unclear how complex transitions are actually built up. In the meta-model in
states desired >
Goal Output
G
Z
Actand
of v D Input
involves >
Composite CT T
Action involves >
Action CN N Q Z
Actor Resource
O RS
1 < pre * < of Transition
State 1 < post * Transition
S T 1 *
occurrence
* defines permissable > 1 1
* ^ involves
^ consists of
permissable ^ involves Concurrency
* * * *
Composite StateTransition Choice
Rule
R Transition CT Structure ST
Sequence
Fig. 3. Frisco - Actor, Action and Actands layer
figure 2, which is based on the formal definitions and not the textual description
of the framework, this defect can be seen through the absence of a recursive
relation. The relationship involves between State Transition Structure and Tran-
sition drawn twice in order to make explicit the fact that a State Transition
Structure is built on two Transitions, does not capture the necessary recursion.
One way to solve this and to introduce recursion would be to define a State
Transition Structure to be a Transition. This would not though work for Choice,
where a unique post-state is not defined. Besides, it would make the definition
of Composite Transition unnecessary. An alternative and more successful way
would then be to include recursion within the definition of State Transition
Structure. This is done by relaxing the existing assumption of tx and ty on
being just Transitions into being Transitions or State Transition Structures, i.e.
the following definition adjustment to the definition from table 2 is proposed.
Def adj 1 If tx : s1 → s2 , ty : s3 → s4 are transitions or state transition structures
then the following state transition structures exists
Sequence: sequ(tx , ty ) if s3 ⊆ s2
Choice: choice(tx , ty ) if s1 ∩ s3 = ∅
Concurrency: concur(tx , ty ) if s1 ∩ s3 = ∅
In the meta-model this change will be depicted by changing the range for one
of the involves relationship from the current Transition to be State Transition
Structure. The cardinality constraints of both involves relationship have to be
adjusted, as it is not longer known in advance how many of the involved in the
State Transition Structures elements are Transitions and how many are other
State Transition Structures. This is depicted in figure 4a. More complicated ways
for capturing this as well as a bit more of the semantics are available. However,
even the simple solution we are proposing here exemplifies well enough how a
meta-model can facilitate the analysis of a framework and how the model evolves
to capture its changes.
Furthermore, going back to the concept of Transition Occurrence (as we’ve
promised in the previous section), we note that interestingly, but not surprisingly,
Transition T Action Entity E < of Q
Transition Time Action < of
Actor
T occurrence Unit N 1 * occurrence Type ET * * O
0..2
^ involves involves ^ involves ^
< involves
Time
0..2 * * T
Interval < of Transition < of Action
StateTransition Transition Action
starts v
* *
ends v T 1 *
occurrence N 1 * occurrence
Structure ST 1 1
Time
a) Recursion b) Time Point c) Action Occurrence d) Entity Type
Fig. 4. Frisco - suggested improvements
after its formal definition in [1] an elaborated discussion on the formalisation of
the time concept was given and how a relative ordering between transition oc-
currences could be done. However, even if discussed, the concept of absolute time
was not formalised. Therefore, we propose the following formal definitions for
Time Point, Time Interval and Time Unit concepts (for a detailed analysis of the
time concept, please refer to [9]).
Def 2 Let T P = {tp|tp ∈ EZ} denote the set of time points for which the following
holds:
If tp1 ∈ T P and tp2 ∈ T P and tp1 = tp2 then tp1 < tp2 or tp2 < tp1
If tp1 < tp2 and tp2 < tp3 then tp1 < tp3 , where tp1 , tp2 , tp3 ∈ T P
Def 3 Let T I = {tps , tpe |tps , tpe ∈ T P ∧ tps < tpe } denote the set of time intervals,
where tps is called the start time point of an interval and tpe is called the end of the
interval.
Def 4 Let T U = T P ∪ T I denote the set of time units.
After have defined a set of Time Units, it is now natural to extend the definition
of Transition Occurrence with the notion of time. The following definition ad-
justment allows us to express not only that a transition has occurred, but also
the time unit at which it occurred. The changes in the meta-model are shown
in figure 4b. The distinction between a time point at which a transition occurs
versus a time interval spanning over the occurrence of a composite transition, in
the definition below, justifies the necessity of introducing both time points and
time intervals.
Def adj 5 A transition t : s1 → s2 is enabled to occur in state s if s1 ⊆ s.
If a transition t : s1 → s2 occurs in state s, then s is changed to the new state s =
(s \ s1 ) ∪ s2 .
The occurrence of a transition t : s1 ⇒ s2 in state s leading to state s’ at time unit tu
is denoted as occ(t, tu) : s → s . When the occurrence is of a composite transition the
time unit tu is a time interval, otherwise it is a time point.
We are now turning our attention into the Action concept of the Actor, Action
and Actands layer and how it builds on the Transaction concept. At the moment,
the Transition Occurrence concept from the Fundamental layer is used to capture
both the fact that a transition occurs, as well as the fact that an action is
performed. Graphically, it can be pointed out that the symmetry in figure 3
existing between the concepts Transition-Action and Composite Transition -
Composite Action (from the Fundamental and Actor, Action, Actand layers
correspondingly) is lacking for the concept Transition Occurrence. In order to
make explicit the occurrences of the actions we suggest the introduction of an
Action Occurrence concept (see figure 4c). Such a concept will naturally be based
on the Transition Occurrence concept, which is indicated by the intra-layers isa
relationship in the figure. In addition to bring explicitness and clarity to the
model, this change allows for an independent consideration of the Actor, Action
and Actands layer from the Fundamental layer. It also clearly establishes the
Fundamental layer as a supporting layer which we also believe was its initial
purpose.
Recall furthermore, that the difference between an Action and a Transition
is that an action is performed by an actor. According to the formal definitions
and to the meta-model in figure 3 when defining an action its actor is explicitly
specified. However, having now introduced the concept of Action Occurrence, it
would be far more natural to specify the actual performer of an action first when
it has occurred. Furthermore, instead of specifying an Actor for the performance
of an action it is rather natural to specify the potential performers for the action.
We also find the limitation that for the moment an action can only be specified
to be performed by an individual actor as quite unnecessary, especially as for
the most actions (e.g., composite actions) the opposite is valid. In order to make
the definition of an action more general, by allowing a whole group of actors
to be specified as potential performers of it, we introduce the concept of Entity
Type. This concept is actually used in some examples of the framework already.
However it is not formally separated, but occurs as a part of the Type concept
and even then it is not a part of its formal definition, but appears only in its
exemplification. Whether this name is the most suitable one can be discussed.
However, as we try to stay as close to the framework as possible we decide to
keep it for now.
Def 6 Let ET = 2E denote the set of entity types.
Now, the definition of action is changed to specify an Entity Type as the per-
former of an action, while the fact that a specific actor has performed an action
is captured in the definition for action occurrence adjusted below. The resulting
change in the meta-model is shown in figure 4d.
Def adj 7 Let N denote the set of all actions. N = {t ∈ T |∃q ∈ ET , ∃r ∈ R[r =
{q, perf orming, t, perf ormed by} ∧ q ∈ P restateof (t)]}
As argued above, we now give a separate definition for the concept of action
occurrence.
Def 8 An action n : s1 → s2 is enabled to be performed by an actor a in state s if
s1 ⊆ s and a ∈ Actorof (n). If an action n : s1 → s2 is performed in state s, then s is
The Actor, Action states desired > Output
and Actand layer Z
Actand
D Input
Goal G
EntityE Actor
Q
TypeET O Z
of > Resource
involves ^ RS
involves ^ performed by ^
Composite CT T < of Action
Action
Action CN N 1 * occurrence
1 < pre * < of Transition Time
State 1 < post * Transition
S T 1 * occurrence Unit
* defines permissable > 0..2
*
^ consists of ^ involves
Time
permissable < involves Concurrency
* * 0..2 * * Interval
Rule
Composite StateTransition Choice starts v* * ends v
Transition CT Structure ST 1 1
R
Time
Sequence Point
The Fundamental layer (partially)
Fig. 5. Frisco - an extended meta-model
changed to the new state s = (s \ s1 ) ∪ s2 .
The performance of an action n : s1 ⇒ s2 by actor a in state s leading to state s’ at
time unit tu is denoted as occ(n, a, tu) : s → s .
This definition invokes the function Actorof, which takes as input an action and
returns its performer, who is generally defined as a predicated thing. In order
to unify it with the adjusted definition of an action, where the performer is an
entity type, the following modification is suggested:
Def adj 9 Let Actorof : N → 2E be a function determining the entity types specified
as performers of an action, where Actorof (n) = {q ∈ 2E |q ∈ P restateof (n) ∧ ∃r ∈
R[r = {q, perf orming, n, perf ormed by}]}
The meta-model in figure 5 is drawn to summarize the results of the analysis
and present the suggested improvements. The introduced concepts are indicated
by shading the corresponding classes.
4 Conclusions and Further Work
In this work we focused on the construction of a meta-model for the Framework of
Information Systems Concepts (Frisco). The initial results, i.e., the development
of a meta-model for the first two layers of the framework was presented here
and as a modelling language the notation of UML class diagrams was used.
We believe this diagrammatical way of presenting a well formalised system of
concepts will serve as a communication aid facilitating the dissemination of it.
Furthermore, we have demonstrated how the analysis provided through the
meta-model development can be used in the further development of Frisco. This
was done by in detail presenting the ambiguities identified and the amendments
suggested to resolve them.
During the work Frisco was analysed in isolation. The next and even more in-
teresting step will be to use the proposed meta-model for analysing Frisco across
other alternative attempts existing in the area. In particular, the representatives
of two rather contrasting approaches are of interest:
– The closely related, i.e., top-down developed and well formalised, frameworks
or ontologies such as BWW, EO, TOVE are interesting to compare to Frisco
so that overlap and discrepancies are outlined. Such an analysis would not
only position the frameworks relative each other but it could also be ben-
eficial for the potential development of each one of them (according to the
technique of meta-model comparison as demonstrated in [7]).
– The pragmatic, bottom-up developed attempts like UEML [10] and the
framework proposed by Söderström et al. [11]. Such an analysis could ini-
tially be provided for surveying the similarities and differences between the
approaches (and be beneficial for works like the one presented in [12]). Fur-
thermore, mapping these non-formalised frameworks to the well-formalised
Frisco (in line with the work presented in [13]) would also give clear semantics
to them.
Common for all the attempts mentioned above is that they were all developed
addressing the interoperability issues and that they are all supposed to be used
for facilitating communication (between people, organisations and tools). Then
an interesting research direction is to investigate the “interoperability” of the
frameworks themselves.
References
1. Falkenberg, E.D., et al.: FRISCO - A Framework of Information System Concepts.
Technical Report ISBN 3-901882-01-4, IFIP (1998) http://www.wi.leidenuniv.
nl/∼verrynst/fri-full-7.pdf.
2. Wand, Y., Weber, R.: An ontological model on an information system. IEEE
Trans. Software Eng. 16 (1990) 1281–1291
3. Uschold, M., Gruninger, M.: Ontologies: Principles, methods and applications.
Knowledge Engineering Review 11 (1996)
4. OpenCyc.org: Opencyc 0.7b. Technical report, Cycorp (2004) http://www.
opencyc.org/doc/.
5. Enterprise Integration Laboratory, U.o.T.: TOVE (TOronto Virtual Enterprise)
Ontology Project. Accessed November 2004 from http://www.eil.utoronto.ca/
enterprise-modelling/tove/index.html (2004)
6. Uschold, M., King, M., Moralee, S., Zorgios, Y.: The Enterprise Ontology. Tech-
nical report, AIAI, The University of Edinburgh (1997) http://www.aiai.ed.ac.
uk/project/enterprise/enterprise/ontology.html.
7. Rosemann, M., Green, P.: Developing a meta model for the Bunge-Wand-Weber
ontological constructs. Information Systems 27 (2002) 75–91
8. OMG: UML 2.0 Superstructure Specification. OMG Draft Adopted Specification,
ptc/03-08-02 (2003) http://www.omg.org/cgi-bin/doc?ptc/2003-08-02.
9. Allen, J.: Towards a general theory of action and time. Artificial Intelligence 23
(1984) 123–154
10. Panetto, H., et al.: A Unified Enterprise Modelling Language for enhanced interop-
erability of Enterprise Models. In: Proc. of the 11th IFAC INCOM2004 Symposium,
Bahia, Brazil (2004)
11. Söderström, E., et al: Towards a framework for comparing process modelling lan-
guages. In: Proc. of the 14th Int. Conf. on Advanced Inf. Systems Engineering
(CAiSE). Volume 2348 of LNCS., Springer (2002)
12. Rukanova, B., Slooten, K., Stegwee, R.: Business Process Requirements, Modeling
Technique, and Standard: How to identify interoperability gaps on a process level.
In: Pre-proc. of the 1th Int. Conf. on Interoperability of Enterprise Sofware and
Applications (INTEROP-ESA2005). (2005) 13–25
13. Wohed, P., Andersson, B.: (Reconciliation of two Business Modelling Frameworks)
to be presented at CAiSE 2005 Forum, to be held in Porto, Portugal, June, 2005.
Appendix
Concept Set Definition Description
Thing Z primitive Any part of a conception of a
domain
RelationshipR R = {r ∈ Z|r ⊆ U ∧ U = A thing composed of several
{q, p|q, p ∈ Z} ∧ 1 = |r| < ∞} predicted things, each one
associated with one predictor
characterizing the role of the
predicted thing within the
relationship
Predictor P P = {p ∈ Z|∃u ∈ U, q ∈ Z[u = An atomic thing used to
q, p]} characterize or qualify other things
Predicted Q Q = {q ∈ Z|∃u ∈ U, p ∈ P[u = The thing being characterized or
thing q, p]} qualified by a predictor
Set mem- SM SM = {sm ∈ R|sm = A binary relationship between a
bership {q1 , has − element, q2 , is − thing (the set) characterized by a
element − of } ∧ q1 , q2 ∈ special predictor called
Q ∧ q1 = q2 } ’has-element’ and another thing
characterized by a special
predictor called ’is-element-of’
Elementary EZ EZ = {ez ∈ Z\R|¬∃u ∈ U[u = A thing not being a relationship
thing ez, has − element]} and not being characterized by the
predictor ’has-element’
Entity E E = EZ ∩ Q A predicted thing which is an
elementary thing
Composite CZ CZ = Z\EZ A non-elementary thing
thing
Table 1. Frisco - The Fundamental Layer (based on the primitives: Thing, has-element,
is-element-of)
Concept Set Definition Description
Transition T T = {t ∈ R|∃sb , sa ∈ CZ[t = A binary relationship between two
{sb , bef ore, sa , af ter}∧sb = different composite things, called
sa ]} pre-state and post-state of the
transition
State S S = {s ∈ CZ\T |∃t ∈ T [s ∈ A composite thing involved as a
P redthingin(t) 3 ]} pre-state or post-state in a
transition. No element of a state
may be a transition itself.
State- ST If tx : s1 → s2 , ty : s3 → s4 are A transition which is either a
transition transitions then Sequence, a Choice or a
structure Sequence: Concurrency
sequ(tx , ty ) if s3 ⊆ s2
Choice:
choice(tx , ty ) if s1 ∩ s3 = ∅
Concurrency:
concur(tx , ty ) if s1 ∩ s3 = ∅
Composite CT CT = ST ∩ T A state transition structure with
transition unique pre- and post- states
Transition A transition t : s1 → s2 is A specific occurrence of a
occurrence enabled to occur in state s if transition
s1 ⊆ s
If a transition t : s1 → s2
occurs in state s, then s is
changed to the new state
s = (s \ s1 ) ∪ s2
The occurrence of a transition
t : s1 ⇒ s2 in state s leading to
state s’ is denoted as
occ(t) : s → s
Type T Y T Y ⊆ 2(Q∪CZ) A specific characterization
applying to all things of that type
Population PO PO = {po ∈ 2(Q∪CZ) |∃ty ∈
T Y[po ⊆ ty]}
Instance IN IN = {in ∈ Q ∪ CZ|∃ty ∈
T Y[in ∈ ty]}
Rule R R = 2S ∪ 2T A set of permissible states and
transitions in a specific context
Table 2. Frisco - The Fundamental Layer, continuation (Primitives: before, after)
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3. Let Predthingin:R → 2Q be a function from relationships to sets of predicted things,
where P redthingin(r) = {q ∈ Q|∃p ∈ P[q, p ∈ r]}.
Concept Set Definition Description
Action N N = {t ∈ T |∃q ∈ Q, ∃r ∈ A transition involving a non-empty
R[r = {q, perf orming, set of actors
t, perf ormed − by} ∧ q ∈
P restateof (t) 4 ]}
Composite CN CN = ST ∩ N A composite transition with the
action same condition as applying for the
notion of action
Actor O O = {q ∈ Q|∃n ∈ N [q ∈ A thing able to cause transition
Actorof (n) 5 ]}
Actand D D = {z ∈ Z|∃n ∈ N [z ∈ A thing involved in the pre-state
Inputof (n) ∨ z ∈ or post-state of an action and not
Outputof (n) 6 ]} considered as an actor of that
action
Resource RS RS = {z ∈ Z|∃n ∈ N [z ∈ The union of the set of actors and
P restateof (n)]} the set of input actands of that
action
Action con- X X = {d ∈ D|∃n ∈ N , ∃r ∈ An optional part of the pre-state
text R[r = {d, is − context} ∧ d ∈ of an action qualifying the context
Inputof (n) 7 ]} or situation in which that action is
performed, and determining or
modifying at least one of its
output actands.
Goal G G = {d ∈ D|∃n ∈ N , ∃o ∈ A special input actand of that
O, ∃r ∈ R[r = {d, pursued − action, pursued by the actors of
by, o, pursuing} ∧ d ∈ that action and stating the desired
Inputof (n) ∧ o ∈ Actorof (n)]} output state intentionally
Table 3. Frisco - The Layer of Actors, Actions, and Actands (Primitives: performing,
performed-by, is-context, pursued-by, pursuing)
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4. Let Prestateof:T → S be a function from transitions to states,
where P restateof (t : sb ⇒ sa ) = sb denotes the (pre-)state before the transition t.
Let Poststateof:T → S be a function from transitions to states,
where P oststateof (t : sb ⇒ sa ) = sa denotes the (post-)state after the transition t.
5. Let Actorof:N → 2Q be a function determining the actors performing an action,
where Actorof (n) = {q ∈ Q|
q ∈ P restateof (n) ∧ ∃r ∈ R[r = {q, perf orming, n, perf ormed − by}]}.
6. Let Outputof:N → 2Z be a function determining the output actands of an action,
where Outputof (n) = {z ∈ P oststateof (n)|¬(z ∈ Actorof (n))}.
Let Inputof:N → 2Z be a function determining the input actands of an action,
where Inputof (n) = {z ∈ P restateof (n)|¬(z ∈ Actorof (n))}.