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Tolerating permanent and transient value faults

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Abstract

Transmission faults allow us to reason about permanent and transient value faults in a uniform way. However, all existing solutions to consensus in this model are either in the synchronous system, or require strong conditions for termination, that exclude the case where all messages of a process can be corrupted. In this paper we introduce eventual consistency in order to overcome this limitation. Eventual consistency denotes the existence of rounds in which processes receive the same set of messages. We show how eventually consistent rounds can be simulated from eventually synchronous rounds, and how eventually consistent rounds can be used to solve consensus. Depending on the nature and number of permanent and transient transmission faults, we obtain different conditions on \(n\), the number of processes, in order to solve consensus in our weak model.

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Notes

  1. This assumption potentially allows corrupted messages on all links in a run; therefore it models dynamic faults.

  2. This assumption makes sense in the context of transient faults.

  3. We give three algorithms in order to show the generality of our approach.

  4. W.l.o.g., the same message is sent to all. Because of transmission faults, this does not prevent two processes \(p\) and \(q\) from receiving different messages from some process \(s\).

  5. The notion of a simulation differs from the notion of a translation of the HO model for benign faults. A translation establishes a relation purely based on connectivity, while with value faults, also some computation is involved. Because of this, we decided thus to use the term simulation instead.

  6. The sending function in a simulation algorithm is thus a function that maps \(states_p\) and the input from \(\mathcal {M}\) to a unique message from \(\mathcal {M}\); while the state-transition function \(T_{p}^r\) is a function that maps \(states_p\), the input from \(\mathcal {M}\), and a partial vector (indexed by \(\varPi \)) of elements of \(\mathcal {M}\) to \(states_p\).

  7. At line 16, the reception vector \(\varvec{\mu }_p^r\) is a vector of vectors: \(\varvec{\mu }_p^r[q']\) is the vector \(p\) has received from \(q'\), and \(\varvec{\mu }_p^r[q'][q]\) is element \(q\) of this vector.

  8. Similar as in the description of Algorithm 1, in case of messages that contain a vector of messages, we focus only on those elements of the vectors that are related to the message sent by process \(p_2\).

    Fig. 4
    figure 4

    Algorithm 2 from the point of view of \(v_2\) sent by \(p_2;\, p_1\) is the coordinator, \(n=4,\, f=1\)

    Full size image

  9. The two rounds of \({ BOTR}\) algorithm can be merged in a single round in which the code of both state-transition functions is executed at once. We have split them in two rounds to emphasize on the different communication predicates required.

  10. Consider two phases \(\phi _0\) and \(\phi _0+1\), such that a process has decided \(\bar{v}\) in phase \(\phi _0\). We consider the more general case in the presence of dynamic faults, and we assume that \(n = 5\), \(f=\alpha =1\) and \(T=4\). This means that at least \(T-\alpha =3\) processes have \(ts=\phi _0\) and \(vote=\bar{v}\). Consider in phase \(\phi _0+1\) that \((v,ts) \in possibleV_p\) at \(p\) with \(v\ne \bar{v}\). This means that \(p\), in round \(3(\phi _0\!+\!1)-2\), has received \(T=4\) messages with either \((v,ts,-)\), or \((-,ts',-)\) and \(ts'<ts\). Since \(n=5\) and \(T=4\), at least one of these messages is from a process \(c\) such that \(vote_c=\bar{v}\) and \(ts_c=\phi _0\). Since \(v\ne \bar{v}\), we must have \(\phi _0 < ts\). However, in phase \(\phi _0+1\), no process \(p\) can have \((v,ts)\) with \(ts> \phi _0\) in \(history_p\). Therefore, by line 31, we will not have \(v \in confirmedV\).

  11. Process \(p_1\) decided \(v_1\) by receiving correctly messages from processes \(p_1\),\(p_2\) and \(p_3\) and the corrupted message \(\langle v_1,\phi _1,-\rangle \) from \(p_4\).

  12. This observation was made already in [19] and [4], but without giving algorithms supporting the observation.

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Authors and Affiliations

  1. EPFL, 1015 , Lausanne, Switzerland

    Zarko Milosevic & André Schiper

  2. Fraunhofer AISEC, Garching, Germany

    Martin Hutle

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  1. Zarko Milosevic
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Correspondence to Zarko Milosevic.

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Milosevic, Z., Hutle, M. & Schiper, A. Tolerating permanent and transient value faults. Distrib. Comput. 27, 55–77 (2014). https://doi.org/10.1007/s00446-013-0199-7

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