Abstract
We study foundational problems regarding the expressive power of parameterised systems. These (infinite-state) systems are composed of arbitrarily many finite-state processes that synchronise using a given communication primitive, i.e., broadcast, asynchronous rendezvous, broadcast with message loss, pairwise rendezvous, or disjunctive guards. With each communication primitive we associate the class of parameterised systems that use it. We study the relative expressive power of these classes (can systems in one class be simulated by systems in another?) and provide a complete picture with only a single question left open. Motivated by the question of separating these classes, we also study the absolute expressive power (e.g., is the set of traces of every parameterised system of a given class \(\omega \)-regular?). Our work gives insight into the verification and synthesis of parameterised systems, including new decidability and undecidability results for model checking parameterised systems using broadcast with message loss and asynchronous rendezvous.
Benjamin Aminof and Florian Zuleger are supported by the Austrian National Research Network S11403-N23 (RiSE) of the Austrian Science Fund (FWF) and by the Vienna Science and Technology Fund (WWTF) through grant ICT12-059. Sasha Rubin is a Marie Curie fellow of the Istituto Nazionale di Alta Matematica.
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Notes
- 1.
A cutoff is a maximal number of processes that needs to be model checked in order to guarantee correct behaviour of any number of processes. Our results show that, indeed, having a cutoff lowers the expressive power.
- 2.
Typically, each label is a set of atomic propositions.
- 3.
For lack of space, some proofs are missing or only sketched and can be found in the full version of this paper.
- 4.
We note that the transitivity of the relations \(\le \) gives rise to additional simulations that, for clarity, are not drawn in the figures.
- 5.
- 6.
In applications one typically takes \(\varOmega := 2^{\textsf {AP}}\) where \(\textsf {AP}\) is a set of atomic predicates.
- 7.
It is common to allow specifications (e.g., the LTL formula G \(_ p\)) to be satisfied by computations that loop forever in the same state. Thus, we don’t consider every transition in which the observables don’t change to be invisible. In particular, we can have both visible and invisible self loops. Using the CPU analogy, the former corresponds to a NOP in the instruction set, and the latter to a NOP in microcode.
- 8.
A slightly different version of \(\textsc {bc}\), in which R is also allowed to be empty, also appears in the literature [12]. Our results also hold for this version.
- 9.
If \(\textsc {cp}= \textsc {dg}\) then we also assume \(\varSigma = S\) (i.e., the synchronization alphabet is the set of local states), and for every local state \(s \in S\) there is a transition \(s \mathop {\longrightarrow }\limits ^{{t!,a}} r\) if and only if \(s = r = t\) and \(a = \mathbf {false}\) (i.e., the only transition \(\tau \) with \(\mathsf{{act}}(\tau ) = s!\) is an invisible self-loop on state s).
- 10.
Although distributed systems are routinely studied this way, one may also introduce final states to the local process and restrict to runs that end in final states [16].
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Aminof, B., Rubin, S., Zuleger, F. (2015). On the Expressive Power of Communication Primitives in Parameterised Systems. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_22
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