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9.5 A Computational Interleaving Semantics 283
9.5.5 Transformation Equivalence and Semantic Function ¡ £¢¥¤
In the previous chapters we have informally discussed the notion of behaviourpreserving
transformation. A behaviour-preserving transformation takes a POOSL
description (during simulation) and transforms it into another, equivalent one. To
be able to prove the correctness of transformations, it is important to define the notion
of equivalence in a precise mathematical manner. Our notion of equivalence is called
transformation equivalence and is formally defined in this subsection.
Each POOSL description consists of a fixed number of process instances and cluster
instances. Instances communicate by exchanging messages over static channels. Some of
these channels are unobservable while others are observable. An unobservable channel
is used for internal communication between the different instances only. Observable
channels establish the link to the outer world. Transformation equivalence is based
upon a notion of observation. The intuitive idea is to only distinguish between two
specifications if the distinction can be detected by an external observer interacting with
each of them. The only way to interact with a specification is by means of message
passing over the specifications’ static channels.
Over the past years many different equivalence relations based upon the notion of
observation have been proposed. Examples include observation equivalence [Mil89],
trace equivalence (also called string equivalence), failures equivalence [Hoa85], testing
equivalence [NH84] and branching equivalence [vGW96]. These equivalence relations
differ in the way they discriminate among specifications. Next to equivalence relations,
a number of different so-called preorders have been proposed. Examples include, but
are not limited to, bisimulation divergence preorders [Wal88] and reduction preorders
[BSS87]. A preorder relation can be regarded as a specification-implementation relation
in the sense that it indicates that one specification is closer to an implementation than
another specification.
To be of practical use, equivalence relation as such is a notion that is not strong enough.
In general equivalence relations do not benefit from the property that replacing a component
of a specification by an equivalent one, yields a specification that is equivalent
to the original. An equivalence relation satisfying such a property is called a congruence.
In terms of languages, a congruence is an equivalence relation that is substitutive
under all language constructors. We will call an equivalence relation that is substitutive
under some, but not all, language constructors, a partial congruence 5 . All of the above
mentioned equivalence relations are (partial) congruences.
We have decided to base our notion of transformation equivalence upon one of the best
known behavioural equivalences called observation equivalence [Mil89]. An advantage
of observation equivalence is that it is quite strong. If two systems are known to be
observation equivalent, then they are also equivalent under many 6 other equivalence
5 An example of such a partial congruence is observation equivalence [Mil89]. Except for summation,
observation equivalence is preserved by all combinators.
6 An example of an equivalence relation that is (slightly) stronger than observation equivalence is