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**Impossibility of Distributed Consensus with One Faulty Process**

Michael J. Fischer Nancy A. Lynch Michael S. Paterson

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Introduction Problem Reach agreement among remote processes Example: transaction commit problem Easy if the participating processes and network are completely reliable Real systems are subject to possible faults Process Crash Unreliable Communication Byzantine failure

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Result No completely asynchronous consensus protocol can tolerate even a single unannounced process death

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**Assumption Don’t consider Byzantine failure Reliable message system:**

messages are delivered correctly and exactly once Asynchronous: No assumption the relative speeds of processes or the delay time in delivering a message No synchronized clock Algorithms based on time-out can’t be used No ability to detect the death of a process

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**The weak consensus problem**

Initial state: 0 or 1 Decision state: Nonfaulty process decides on a value in {0, 1} Requirement: All nonfaulty processes that make a decision must choose the same value. Some processes eventually make a decision(for proof) Trivial solution is ruled out

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System model Processes are modeled as automata and communicate by messages One atomic step receive a message, perform local computation, send arbitrary but finite messages Atomic broadcast send a message to all other processes in one step all nonfaulty processes or none will receive it

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Consensus Protocols A consensus protocol P is an asynchronous system of N processes Each process p has a one-bit input register xp, an output register yp, with values in {b, 0, 1} Internal state: input/output register, pc, internal storage Initial state: all fixed starting values except the input register output register starts with b

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**p acts deterministically according to a transition function **

Decision state output register has the value 0 or 1 Once a decision is made, the value can’t be changed p acts deterministically according to a transition function Processes communicate by messages Message system send(p, m) receive(p) nondeterministically

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**A configuration consists of**

All internal state of each process, the contents of message buffer A step is a transition of one configuration C to another e(C), including 2 phases: First, receive(p) to get a message m Based on p’s internal state and m, p enters a new internal state and sends finite messages to other e = (p, m) is called an event and said e can be applied to C

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**Schedule, run, reachable and accessible**

A schedule from C a finite or infinite sequence σ of events that can be applied, in turn, starting from C The associated sequence of steps is called a run σ(C) denotes the reulting configuration and is said to be reachable from C An accessible configuration C If C is reachable from some initial configuration

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Lemma 1 Suppose that from some configuration C, the schedules σ1 and σ2 lead to configuration C1 ,C2 , respectively. If the sets of Processes taking steps in σ1 and σ2 respectively, are disjoint, then σ2 can be applied to C1 and σ1 can be applied to C2 , and both lead to the same configuration.

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**Decision value, partially correct, nonfaulty**

A configuration C has decision value v if some process p is in a decision state with yp =v. P is partially correct if No accessible configuration has more than one decision value For each v {0, 1}, some accessible configuration has decision value v. A process is nonfaulty If it takes infinitely many steps

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**Admissible and deciding run, totally correct**

Admissible run At most one process is faulty and all messages sent to nonfaulty processes are eventually received Deciding run Some process reaches a decision state A consensus protocol P is totally correct in spite of one fault if it is partially correct and every admissible run is a deciding run

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Theorem 1 No consensus protocol is totally correct in spite of one fault.

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**Bivalent, 0-valent/1-valent**

Let C be a configuration, V the set of decision values of configurations reachable from C. C is bivalent if |V| = 2. C is univalent if |V| = 1. 0-valent or 1-valent according to the corresponding decision value.

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Lemma 2 P has a bivalent initial configuration

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Lemma 3 Let C be a bivalent configuration of P, and let e = (p, m) be an event that is applicable to C. Let C be the set of configurations reachable from C without applying e, and let D= e(C) ={e(E) | E C and e is applicable to E}. Then, D contains a bivalent configuration.

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Theorem 2 There is a partially correct consensus protocol in which all nonfaulty processes always reach a decision, provided no processes die during its execution and a strict majority of the processes are alive initially

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Conclusion The problem of fault-tolerant cooperative computing cannot be solved in a totally asynchronous model of computation.. To solve this problem in practice, more refined models of distributed computing is needed

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