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This question is about Model Checking for Software Formal Verification
How do you model the joint behavior of 2 independent and concurrent transition systems?

Specifically, given the two independent and concurrent Transition Systems below; TS1 and TS2 (left to right).
Left:TS1 Right:TS2

A tutor proposed that the resulting Interleaving Transition System ITS is enter image description here

I understand how all the shown states and transitions of this ITS were gotten, however, why is there no transition from state (l1,q1) to (l3,q2)--as would be expected if both TS1 and TS2 transitioned on action a?

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  • $\begingroup$ The definition of the interleaving operator (as opposed to e.g. the interface parallel operator in CSP) is that each component can take $a$- or $b$-transitions on its own - there is no synchronization. $\endgroup$ – Klaus Draeger Oct 21 '15 at 13:12
  • $\begingroup$ @KlausDraeger, thanks for your reply; I think I get it now: the interleaving operator is not synchronized by definition. In that case, can we say that the interleaving operator does not model the joint behavior of the two systems--since it does not account for synchronization? $\endgroup$ – eyeezzi Oct 22 '15 at 1:56
  • $\begingroup$ Only if you assume that systems are supposed to synchronize on all shared symbols. The point of having interleaving (and, as in CSP, parametrizing the parallel composition operator with a set of symbols on which to synchronize) is to allow you to represent a greater variety of models of interaction. $\endgroup$ – Klaus Draeger Oct 22 '15 at 8:52
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By definition, the interleaving operator is not synchronized, so any transition in $TS1\ |||\ TS2$ corresponds to a transition in a single component while the other one remains in whatever state it is.

More generally, in process algebras like CSP, the parallel composition operator $||$ can be parameterized with a set of symbols on which to synchronize - for example, you could consider $TS1\ |[\{a\}]|\ TS2$, which has the same states and $b$-transitions as $TS1\ |||\ TS2$, but only two (synchronized) $a$-transitions $(l_1,q_1)\to(l_3,q_2)$ and $(l_2,q_1)\to(l_1,q_2)$. In particular, $|||$ is the special case $|[\emptyset]|$.

The point is that (together with additional operations like hiding) this allows you to represent a far greater variety of models of interaction.

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