Lecture Notes in Computer Science 3472

5 Preorder Relations 119
power of refusal preorder over trace preorder comes thus at a cost which may or
may not be acceptable in practice.
One reason for which practical considerations are of interest is that preorders
are a key element in conformance testing [Tre94]. In such a framework we are
given a formal specification and a possible implementation. The implementation
is treated as a black box (perhaps somebody else wrote a poorly commented
piece of code) exhibiting some external behavior. The goal is then to determine
by means of testing whether the implementation implements correctly the
specification. Such a goal induces naturally an implementation relation, or a
preorder. Informally, the practical use of a preorder relation ⊑ consists then in
the algorithmic problem of determining whether s ⊑ i for two processes i (the
implementation) and s (the specification) by means of applying on the two processes
tests taken from the testing scenario associated with ⊑. Iftherelation
holds then i implements (or conforms to) s (according to the respective testing
scenario). The formal introduction of conformance testing is left to the end of
this chapter, namely to Section 5.9 on page 145 to which we direct the interested
reader for details. For now we get busy with defining preorders and analyzing
their properties.
Wherewegofromhere We present in the next section the necessary preliminaries
related to process representation and testing (including a first preorder to
compare things with). Sections 5.3 to 5.8 are then the main matter of this chapter;
we survey here the most prominent preorders and we compare them with
each other. We also include a presentation of conformance testing in Section 5.9.
5.1.1 Notations and Conventions
It is often the case that our definitions of various sets (and specifically inductive
definitions) should feature a final item containing a statement along the line that
“nothing else than the above constructions belong to the set being defined.” We
consider that the presence of such an item is understood and we shall not repeat
it over and over. “Iff” stands for “if and only if.” We denote the empty string,
and only the empty string by ε.
We present a number of concepts throughout this chapter based on one particular
paper [vG01] without citing it all the time, in order to avoid tiresome
repetitions.
Many figures show processes that are compared throughout the paper using
various preorders. We show parenthetically in the captions of such figures the
most relevant relations established between the depicted processes. Parts of these
parenthetical remarks do not make sense when the figures are first encountered,
but they will reveal themselves as the reader progresses through the chapter.
5.2 Process Representation and Testing
Many formal descriptions for processes have been developed in the past, most
notably under the form of process algebraic languages such as CCS [Mil80] and