2 Why We Need Model-Based Testing

Advanced Topics 241
14.3 Properties of model program composition
Recall from Section 7.3 that, under composition, model programs synchronize steps
for shared actions and interleave actions not found in their common action vocabulary.
We now describe in greater detail how actions are combined, taking into
account both action symbols and arguments, including the placeholder (underscore).
Recall that it is the actions (the terms) that are considered when combining
actions, not the action methods (the C# code).
It is sometimes easier to reason about properties of composition when the action
vocabularies of the composed model programs are the same. In order to do so we
use a simple transformation. The effect of interleaving for nonshared actions is
achieved by treating actions not in the vocabulary as self-loops. We transform a
model program M with respect to a set of action symbols a1,...,ak into a new
model program called a loop extension of M for a1,...,ak, as follows. For each
action symbol ai not in the action vocabulary of M, add a trivial guarded update rule
for ai whose enabling condition is true, and whose update rule produces no state
updates, that is, every action whose symbol is ai produces a self-loop in all states of
M. In reality, this transformation is not done explicitly but is built into the product
operation of model programs.
For example, consider the setup model program in Figure 14.15. The loop extension
of the setup model program for Cancel, say SetupC, has the additional
transitions
t("Inactive", Cancel(), "Inactive")
t("Activating", Cancel(), "Activating")
t("Active", Cancel(), "Active")
The arity of an action (symbol) is the number of arguments or parameters that
it takes. When describing a model program it is often convenient to omit trailing
parameters of an action that are not referenced by its guarded update rule. For
example, in the sample protocol model, the arity of a cancel action is one, the arity
of a request action is two, and the arity of a response action is three. Given an
action symbol f with arity k we often use the abbreviated form f (t1,...,ti) for
some i, 0≤ i ≤ k, to stand for the term f (t1,...,ti, _,...,_) with k − i trailing
underscores called placeholders. 1 For example, Cancel() stands for Cancel( ). An
instance of an action with placeholders is a replacement of some (or none) of the
placeholders with concrete values. A trace with placeholders is a sequence of actions
with placeholders and an instance of a trace with placeholders is an instantiation of
some of the placeholders in it. We often say trace to mean a trace that may contain
1 Formally, each occurrence of a placeholder represents a fresh logic variable. A term without
logic variables is called ground.
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