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2.1 Business Processes and Patterns
Business processes are collections of structured ac-
tivities which allow for understanding of plans. They
are usually visualized with a flowchart of activities
which are work (or services) that must be performed
in a business model. The most popular business pro-
cesses notation is BPMN, Business Process Modeling
Notation, which was developed by the Business Pro-
cess Management Initiative (BPMI) as a standard for
business process modeling. The main requirement for
BPMN was the simplicity of business model creation.
BPMN notation consists of the following categories
of elements: flow objects, connecting objects, swim-
lanes, artifacts. Detailed information on BPMN ex-
ceeds the size of this work and can be found in many
works, e.g. (OMG, 2009).
The business modeling is related to the concept of
patterns which play an important role in the model-
ing of business processes. A pattern is “the abstrac-
tion form a concrete form which keeps reoccurring
in specific nonarbitrary contexts” (Riehle D., 1996).
Patterns are cataloged and documented in the 23 ob-
jects (der Aalst et al., 2003) which consist of the
following groups: Basic Control, Advanced Branch-
ing, Structural, Multiple Instances, State Based, and
Cancellation. Further considerations in this work
are limited to the five basic control patterns: Se-
quence, Parallel-Split, Synchronization, Exclusive-
Choice and Simple-Merge. In the latter part of the
paper, transformations of these patterns to the logic
formulas which constitute a logical specification of a
business model and which are processed using the se-
mantic tableaux methodology are introduced.
2.2 Logical Background
The logical framework for the presented approach are
temporal logic and reasoning using the method of
semantic tableaux. Temporal logic is a convenient
formalism for specification and verification of se-
quences of events without a strict timing, e.g. (Emer-
son, 1990). Temporal logic formulas can easily ex-
press liveliness and safety properties which play a key
role in proving the properties of a system. Considera-
tions in this paper are limited to axiomatic and deduc-
tive system for the smallest temporal logic, e.g. (Ben-
them, 95). This logic is also known as temporal logic
of the class K, and can be developed and expanded
through the introduction of more complex properties
of the time structure.
As already mentioned, this work focuses on for-
mal deduction as a method of verification, which is in
an opposition to the state exploration methods. The
deduction method of semantic tableaux for tempo-
ral logic (D’Agostino et al., 1999), which is based
on the formula decomposition, has some advantages
in comparison with traditional methods of inference.
Although the analysis starts from a long formula, at
each decomposition step it has fewer number of com-
ponents since logical connectives are removed and,
above all, the direction of inference is clearly stated
at all times. The method provides, through so-called
open branches of the semantic tree, the information
about the source of an error if one is found, which is
a very important advantage of this method. Tempo-
ral logic, which is used for the inference process and
for generation of the system specification is so-called
smallest temporal logic of class K. Work (Klimek
et al., 2010) contains an example of inference and
semantic tableaux for temporal logic in the context
of BPMN models. BPMN models are convenient in
this formal verification approach and in a process of
automatic or partly automatic extraction of formulas
which represent a logical specification. This follows
the nature of BPMN models, which constitute a kind
of a logical network. However, only BPMN design
patterns, c.f. (der Aalst et al., 2003), are considered,
since every business process might be modeled by
combining common design patterns. By providing
automatic transformation for these workflow patterns
to temporal logic formulae it is possible to automati-
cally build the logical specification for any givenbusi-
ness process.
3 DEDUCTION SYSTEM
The proposed system of inference, and its architec-
ture, using the semantic tableaux method for BPMN
models is presented in Fig. 1. The system works auto-
matically and consists of some software components.
The first component (the “TL Formulas Generator”
module) provides the functionality to produce a log-
ical specification. Logical specification is a set of
many temporal logic formulas. Formula generation
is performed by extracting directly from the design
patterns in the BPMN model. The extraction is fo-
cused on BPMN patterns and is shown later in this
work. Formulas are collected in a module (data ware-
house, i.e. file or database) that stores the specification
of the system (the “System’s Specification” module).
Properties of the system are treated as a conjunction
of formulas p
1
∧ . . . ∧ p
n
= P. The third component
(the second one is discussed later) provides the de-
sired properties of the system expressed in temporal
logic formulas (the “System’s Properties” module).
The easiest way to obtain such a formula is (manual)
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