Lecture Notes in Computer Science 3472

9 Testing Theory for Probabilistic Systems
Verena Wolf
University of Mannheim
Lehrstuhl für Praktische Informatik II
vwolf@pi2.informatik.uni-mannheim.de
9.1 Introduction
The aim of this chapter is to give a survey of testing relations for probabilistic
systems. We summarize the relevant material on probabilistic extensions of the
work of De Nicola and Hennessy [dNH84] who defined implementation relations
for nondeterministic processes based on a notion of testing (see also Chapter
5). We mainly concentrate on the relative expressive power of the different preorders
1 . All presented relations are primarily of theoretical interest and to the
best of our knowledge their usefulness in practical applications has not been
shown yet.
Testing can be described as recording the behavior of systems executing in a particularly
designed environment. In the classical setting a (testing) environment
of a process P is simulated by considering the parallel composition P�T of P
and a test process T (basically another nondeterministic process but equipped
with a set of success actions or states). De Nicola and Hennessy define PmayT
if a success state can (may) be reached by P�T and PmustTif P�T reaches
a success state on every run (execution). Two processes P, Q are related if P
may (must, respectively) T implies Qmay(must, respectively) T for all test
processes T .
In 1990, Ivan Christoff extended the classical testing theory to fully probabilistic
processes that are basically labeled transition systems enriched with probabilistic
information [Chr90]. He considered the parallel composition P�T of a fully
probabilistic process P and a nondeterministic process T and analyzed the trace
distribution of (the fully probabilistic result) P�T . Two fully probabilistic processes
P and Q are related if the trace distributions of P�T and Q�T coincide
for all possible test processes T . Additionally, Christoff constructed a characterization
by extended traces which are a denotational model simplifying the
development of algorithms for the computation of the preorder.
Two years later Cleaveland et al. presented a testing approach based on probabilistic
test processes [CSZ92]. They argued that the environment of a fully
probabilistic process P may also be probabilistic and accordingly, they applied
probabilistic test processes T , equipped with success states, and considered the
probability of reaching success in P�T . Furthermore, they lifted extended traces
1 We use the term ”preorder” as synonym for ”testing relation” although it might be
the case that the respective relation is not transitive or transitivity has not been
shown.
M. Broy et al. (Eds.): Model-Based Testing of Reactive Systems, LNCS 3472, pp. 233-275, 2005.
© Springer-Verlag Berlin Heidelberg 2005