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Model checking parameterized asynchronous shared-memory systems

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Abstract

We characterize the complexity of liveness verification for parameterized systems consisting of a leader process and arbitrarily many anonymous and identical contributor processes. Processes communicate through a shared, bounded-value register. While each operation on the register is atomic, there is no synchronization primitive to execute a sequence of operations atomically. We analyze the case in which processes are modeled by finite-state machines or pushdown machines and the property is given by a Büchi automaton over the alphabet of read and write actions of the leader. We show that the problem is decidable, and has a surprisingly low complexity: it is NP-complete when all processes are finite-state machines, and is in NEXPTIME (and PSPACE-hard) when they are pushdown machines. This complexity is lower than for the non-parameterized case: liveness verification of finitely many finite-state machines is PSPACE-complete, and undecidable for two pushdown machines. For finite-state machines, our proofs characterize infinite behaviors using existential abstraction and semilinear constraints. For pushdown machines, we show how contributor computations of high stack height can be simulated by computations of many contributors, each with low stack height. Together, our results characterize the complexity of verification for parameterized systems under the assumptions of anonymity and asynchrony.

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Notes

  1. A finite-state automaton (FSA) is an FSM which decides languages of finite words. Therefore an FSA is an FSM with a set \(F\) of accepting states.

  2. In a nutshell, the Parikh image of a language \(L\) over a \(n\)-sized alphabet is a set of \(n\)-dimensional vectors of natural number. Each vector is the (Parikh) image of a word of \(L\) obtained by counting how many times each symbol occur and forgetting about the ordering, e.g. let \(L={abbb}\) over alphabet \(\Sigma =\{a,b\}\) the vector (1,3) is the image of \(abbb\) by counting 1 occurrence of \(a\) and 3 of \(b\).

  3. For readability, we write “configuration” for “PDM-configuration.”

  4. A pushdown automaton (PDA) is a PDM which decides languages of finite words. We define a PDA as a PDM with a set \(F\) of accepting states.

  5. See Theorem 4 for a definition of \(N\).

  6. Notice that the effective stack height of a configuration depends on the run it belongs to, and so \(c\,(\rho , i) = c_\psi (\rho ', i)\) does not necessarily imply that they have the same effective stack height.

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Acknowledgments

Pierre Ganty has been supported by the Madrid Regional Government project S2013/ICE-2731, N-Greens Software - Next-GeneRation Energy-EfficieNt Secure Software, and the Spanish Ministry of Economy and Competitiveness project No. TIN2015-71819-P, RISCO - RIgorous analysis of Sophisticated COncurrent and distributed systems.

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Authors and Affiliations

  1. TU Munich, Munich, Germany

    Antoine Durand-Gasselin & Javier Esparza

  2. IMDEA Software Institute, Madrid, Spain

    Pierre Ganty

  3. MPI-SWS, Kaiserslautern, Germany

    Rupak Majumdar

Authors
  1. Antoine Durand-Gasselin
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  2. Javier Esparza
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  3. Pierre Ganty
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  4. Rupak Majumdar
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Correspondence to Pierre Ganty.

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Durand-Gasselin, A., Esparza, J., Ganty, P. et al. Model checking parameterized asynchronous shared-memory systems. Form Methods Syst Des 50, 140–167 (2017). https://doi.org/10.1007/s10703-016-0258-3

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