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model. Informal requirements, as in our example, mostly describe evolutions of
hybrid systems in a qualitative manner like “when something increases to a cer-
tain value another thing will start decreasing”. Finally, in order to transform the
informal requirements to the differential equation model, one needs experience
in physics and applied mathematics. To put a long story short: for our purposes
of model-based mutation testing, the full differential model - most of the time -
requires (and carries) too much detail. Hence, we abstract away these unneces-
sary details by using a technique called qualitative abstraction . After applying
qualitative abstraction to the continuous behavior of our hybrid system we ob-
tain a discrete model which we discuss in the next section. Having dealt with the
continuous parts of the hybrid system, we then use action systems to formalize
the discrete part. How action systems and qualitative evolutions can be joined
to a hybrid system model has already been described in [2] and yields a formal-
ism called Qualitative Action Systems (QAS) . This provides us a framework for
modeling and analyzing hybrid systems.
3 Environment Modeling with Qualitative Evolutions
In difference to most hybrid system models that use Ordinary Differential Equa-
tions (ODEs) to model continuous evolutions, a qualitative action system only
knows about discrete, qualitative evolutions. Each of these qualitative evolutions
forms a transition system that is constructed from Qualitative Differential Equa-
tions (QDEs) by applying a technique called Qualitative Reasoning (QR) [13].
Qualitative Reasoning originates from the area of Artificial Intelligence and is ap-
plied in common sense reasoning about physical systems with incomplete knowl-
edge. The technique is based on the well founded theory of QDEs which are
an abstraction of ODEs. Solutions to QDEs are usually found by inference sys-
tems like QSIM [13]. Qualitative reasoning relies on two abstractions: (1) value
abstraction and (2) time abstraction.
Value abstraction is a data abstraction mapping the continuous real-valued vari-
ables of a physical environment to discrete variables with symbolic values. These
symbolic variables are called quantities and have a finite domain of symbolic
values. This finite domain of a quantity variable, i.e., its type, is called quantity
space. There are two kinds of symbolic values in a quantity space: landmark
values and open intervals .
Landmark values are the “natural joints” that break a continuous set of values
into qualitatively distinct regions. A landmark value is a symbolic name for a
particular real number, whose numerical value may or may not be known. It
serves as a precise boundary for a qualitative region. For example, the landmark
values of our water level in tank T 2are Zero , Empty , Reserve and Full .These
names indicate the interesting points where a behavior changes from a qualitative
point of view. Hence, in our qualitative abstraction, the water level x 2 may
evaluate to the landmark values, e.g., x 2 = Empty . Furthermore, the landmark
values are defined to form a strict total order. In our example Zero < Empty <
Reserve < F ull holds.
 
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