Reactive Systems: Modelling, Specification and Verification - Cs.ioc.ee

3.3. STRONG BISIMILARITY 43
trace equivalent processes CTM and CTM ′ exhibited different deadlock behaviour
when made to interact with a third parallel process, namely CA. In hindsight, this
is not overly surprising. In fact, when looking purely at the (completed) traces of a
process, we focus only on the sequences of actions that the process may perform,
but do not take into account the communication capabilities of the intermediate
states that the process traverses as it computes. As the above example shows,
the communication potential of the intermediate states does matter when we may
interact with the process at all times. In particular, there is a crucial difference in
the capabilities of the states reached by CTM and CTM ′ after these processes have
received a coin as input. Indeed, after accepting a coin the machine CTM always
enters a state in which it is willing to output both coffee and tea, depending on
what its user wants, whereas the machine CTM ′ can only enter a state in which it
is willing to deliver either coffee or tea, but not both.
The lesson that we may learn from the above discussion is that a suitable notion
of behavioural relation between reactive systems should allow us to distinguish
processes that may have different deadlock potential when made to interact with
other processes. Such a notion of behavioural relation must take into account the
communication capabilities of the intermediate states that processes may reach as
they compute. One way to ensure that this holds is to require that in order for two
processes to be equivalent, not only should they afford the same traces, but, in some
formal sense, the states that they reach should still be equivalent. You can easily
convince yourselves that trace equivalence does not meet this latter requirement,
as the states that CTM and CTM ′ may reach after receiving a coin as input are not
trace equivalent.
The classic notion of strong bisimulation equivalence, introduced by David
Park in (Park, 1981) and widely popularized by Robin Milner in (Milner, 1989),
formalizes the informal requirements introduced above in a very elegant way.
Definition 3.2 [Strong bisimulation] A binary relation R over the set of states of
an LTS is a bisimulation iff whenever s1 R s2 and α is an action:
- if s1 α → s ′ 1 , then there is a transition s2 α → s ′ 2 such that s′ 1 R s′ 2 ;
- if s2 α → s ′ 2 , then there is a transition s1 α → s ′ 1 such that s′ 1 R s′ 2 .
Two states s and s ′ are bisimilar, written s ∼ s ′ , iff there is a bisimulation that
relates them. Henceforth the relation ∼ will be referred to as strong bisimulation
equivalence or strong bisimilarity.
Since the operational semantics of CCS is given in terms of an LTS whose states
are CCS process expressions, the above definition applies equally well to CCS