Michael J. Fischer, Nancy A. Lynch, and Michael Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, 1986.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....assumption like at most f nodes may appear Byzantine also implies at most f pairs of perceptions may appear Byzantine , for any pair of non faulty receiving nodes. A perception based fault model thus preserves the corresponding global one, which means that classic impossibility results like [4, 3] remain valid. Node 1 (send) recv) Node s . faulty node non faulty q ) p ) Figure 3. Effect of a link fault between sender s and receiver p upon the perception vectors of all receivers r. Only V p is faulty here. 3 Formal Framework In order to formalize our ....

M. Fischer, N. Lynch, and M. Merrit. Easy impossibility proofs for the distributed consensus problem. Distributed Computing, 1(1):26--39, 1986.

....like at most f nodes may appear Byzantine also implies at most f pairs of perceptions may appear Byzantine , for any pair of non faulty receiving nodes. A perception based fault model thus preserves the corresponding global one, which reveals that classic impossibility results like [4, 3] remain valid. Node 1 (send) recv) Node s . faulty node non faulty q ) p ) Figure 3. Effect of a link fault between sender s and receiver p upon the perception vectors of all receivers r. Only V p is faulty here. Some recent papers [21, 28] reveal that other ....

....assumption like at most f nodes may appear Byzantine also implies at most f pairs of perceptions may appear Byzantine , for any pair of non faulty receiving nodes. A perception based fault model thus preserves the corresponding global one, which means that classic impossibility results like [4, 3] remain valid. 3 Formal Framework In order to formalize our perception based fault model, we assume that all nodes s, 1 s n, are somehow provided with a (virtual) event V ] that occurs at some specified in any suitable global time scale. Besides its pure occurrence, V ] may or may ....

M. Fischer, N. Lynch, and M. Merrit. Easy impossibility proofs for the distributed consensus problem. Distributed Computing, 1(1):26--39, 1986.

....paper we present a more efficient protocol obtained by exploiting the properties of the graphs that characterize reliable communication. Historical Notes. The connectivity requirements for several distributed tasks in several models has been studied in many papers; for example Byzantine agreement [4, 8], approximate Byzantine agreement [6, 16] reliable message transmission [4, 5] and reliable and private message transmission [13, 5, 14] Simple impossibility results and references can be found in [8, 12] We mention that in Byzantine agreement all honest parties should agree on the same ....

.... tasks in several models has been studied in many papers; for example Byzantine agreement [4, 8] approximate Byzantine agreement [6, 16] reliable message transmission [4, 5] and reliable and private message transmission [13, 5, 14] Simple impossibility results and references can be found in [8, 12]. We mention that in Byzantine agreement all honest parties should agree on the same message while in reliable communication only the transmitter and the receiver agree on the message. Beimel and Franklin [1] considered the connectivity requirements in partially authenticated networks. In addition ....

M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, 1986.

....node is a node which never transmits a message, and every message transmitted to it is lost. For the general case where nodes can fail during the execution of an algorithm, no deterministic election protocol exists [FLP85, MW87] Other types of failures are also hard or impossible to cope with [F83, FLM85]. Fortunately, reliable hardware equipment makes failures of the most general type quite rare [G82] Thus, additional assumptions are needed. These include, for example, knowledge about synchrony in the network [G82] its topology [KW84, SG86] or its size [SG86] In our model, all faults are ....

Fisher, M.J., Lynch, N.A., and Merritt, M., Easy Impossibility Proofs for Distributed Consensus Problems, Proceedings of the 4-rd ACM Symposium on Principles Of Distributed Computing, Minaki, Canada, August 1985, pp. 59-70.

....relay messages since the primitive does this automatically. Since the primitive requires n 3f the nonauthenticated algorithm also has this limit on the number of faulty processes. It has been shown that, if authentication is not available, then synchronization is impossible unless n 3f[3, 5]. As in Section 2, we assume that clocks are initially synchronized such that, at ready, all correct processes are using C and these clocks are at most Dmax apart. The nonauthenticated algorithm is described in Figure 3. THEOREM 6. The nonauthenticated algorithm in Figure 3 achieves agreement ....

FISCHER, M., LYNCH, N., AND MERRITT, M. Easy impossibility proofs for distributed consensus problems. In Proceedings of the 4th Symposium on the Principles of Distributed Computing (Minaki, Canada, Aug.). ACM, New York, 1985, pp. 59-70.

.... graph map f : H G is a covering if both H and G are connected and for each vertex v 2 V (H) the map f maps bijectively the edges adjacent to v in H to the edges adjacent to f(v) 2 V (G) In Computer Science graph coverings were used to show some impossibility results in distributed computing [1, 6], they have also applications in the domain of interconnection networks [2] each covering of a network G by a network H gives an uniform emulation of H on G with dilatation 1. For a given covering f : H G, the number jf (v)j of vertices of H mapped to a vertex v 2 V (G) is the same for all ....

M. J. Fisher, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26--39, 1986.

....the original papers [89] 73] on Byzantine agreement. The idea is based on processes fooling other processes, making them believe they are in different systems. The most pleasing proof I know for this result is not the original, but the scenario proof I did with Mike Fischer and Mike Merritt [54]. The following argument is for the case of t 1, i.e. 3 processes and I fault. Suppose that p, q, and r comprise a 3 process solution that can tolerate 1 fault. Consider a system composed of two copies each of p, q and r joined into a ring, in order P0, q0, p, ql ,fl. Let a be an execution of ....

....[89] is basically the same as in this example, except that instead of describ ing the scenario as the execution that is generated by a certain system started with certain initial val ues, Lamport et al. construct the scenario explicitly. It seems to me that the higher level of abstraction of the [54] proof makes much clearer what is really going on. Perhaps there are other impossibility proofs con taining explicit constructions of bad executions that could be made more understandable by describing the bad executions implicitly, by a simple way of gener ating them. A related impossibility ....

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M. Fischer, N. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26-39, 1986.

....Recall that we are concerned with stateless composition. First, we show that it is impossible to construct an authenticated Byzantine Agreement protocol that composes in parallel (or concurrently) and is secure when n=3 or more parties are faulty. This result is analogous to the Fischer et al. FLM86] lower bound for Byzantine Agreement in the standard model (i.e. without authentication) We stress that our result does not merely show that authenticated Byzantine Agreement protocols do not necessarily compose; rather, we show that one cannot construct protocols that will compose. Since there ....

....seem to be achievable (and many others where it is undesirable) Theorem 7.1.1 No protocol for authenticated Byzantine Agreement that composes in parallel (even twice) can tolerate n=3 or more faulty parties. Proof. The proof of Theorem 7.1. 1 is based on some of the ideas used by Fischer et al. FLM86] in their proof that no unauthenticated Byzantine Agreement protocol can tolerate n=3 or more faulty parties. We begin by proving the following lemma: Lemma 7.3.1. There exists no protocol for authenticated Byzantine Agreement for three parties, that composes in parallel (even twice) and can ....

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Michael J. Fischer, Nancy A. Lynch, and Michael Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26-39, 1986.

....Recall that we are concerned with stateless composition. First, we show that it is impossible to construct an authenticated Byzantine Agreement protocol that composes in parallel (or concurrently) and is secure when n=3 or more parties are faulty. This result is analogous to the Fischer et al. [7] lower bound for Byzantine Agreement in the standard model (i.e. without authentication) We stress that our result does not merely show that authenticated Byzantine Agreement protocols do not necessarily compose; rather, we show that one cannot construct protocols that will compose. Since there ....

....mentioned, there are many scenarios where this does not seem to be achievable (and many others where it is undesirable) Agreement that composes in parallel (even twice) can tolerate n=3 or more faulty parties. Proof: The proof of Theorem 1 is based on some of the ideas used by Fischer et al. [7] in their proof that no unauthenticated Byzantine Agreement protocol can tolerate n=3 or more faulty parties. We begin by proving the following lemma: Lemma 3.1. There exists no protocol for authenticated Byzantine Agreement for three parties, that composes in parallel (even twice) and can ....

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M. Fischer, N. Lynch, and M. Merritt. Easy Impossibility Proofs for Distributed Consensus Problems. Distributed Computing, 1(1):26-39, 1986.

....that if players share no initial setup information beyond pairwise authenticated channels, then in fact broadcast is possible if and only if t n=3, where n is the number of players and t is the number of actively corrupted players to be tolerated by the protocol. By the impossibility proofs in [20, 10, 11], even additional resources (e.g. secret channels, private random coins, quantum channels and computers) cannot help to improve this bound unless some setup shared among more than just pairs of players is involved. On the other hand, the picture changes dramatically if some previous setup is ....

M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26-39, 1986.

....in fully connected asynchronous networks in which processes are reliable but some links may be arbitrarily faulty. One such a protocol, that is optimal with respect to the number of faulty links tolerated, can be found in [Sayeed et al. 1995] 10 Simpler impossibility proofs can be found in [Fischer et al. 1986]. An orthogonal approach to the impossibility of reaching consensus in time free asynchronous systems is the epistemic one proposed in [Halpern Moses 1990] Such an approach formally defines the concepts of knowledge and common knowledge (intuitively, a fact becomes common knowledge of a ....

M. J. Fischer, N. A. Lynch and M. Merritt, "Easy Impossibility Proofs for Distributed Consensus Problems", Distributed Computing, 1, pp.26-39, 1986.

....among n players in presence of an adversary that corrupts up to t n=3 players and make them misbehave arbitrarily. If a secure signature scheme can be used, they achieve broadcast even for any number t n of corruptions, consensus for up to t n=2. All these bounds are tight [LSP82, KY84, FLM86] but the proposed protocols are inecient. Ecient protocols were given in [DS83, DFF 82, TPS87, BDDS92, FM97, BGP89, CW92, GM98] The bounds t n for broadcast and t n=2 for consensus can also be achieved with unconditional security, when an unconditionally secure pseudo signature scheme ....

....extended consistency, it still needs to be proven that the bound t 2t n is optimal. Note that this impossibility result even holds for the ordinary variants without consistency detection, or validity detection, respectively. The proof proceeds along the lines of the impossibility proof in [FLM86] that broadcast is impossible if t n=3. Theorem 4. In Models M aut and M sec , neither broadcast with extended validity nor broadcast with extended consistency is achievable among a set of n players P if t 0 and t 2t n. For every protocol there exists a value x 0 2 f0; 1g such that, ....

M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26-39, 1986. 11

....n, the number of FCUs, satisfies n 3a 2s m. It is provably impossible (i.e. it can be proven that no algorithm can exist) to tolerate a arbitrary faults in clock synchronization with fewer than 3a 1 FCUs and 2a 1 disjoint communication paths (or a 1 disjoint broadcast channels) DHS86,FLM86] unless digital signatures are employed which is equivalent to reducing the severity of the arbitrary fault mode) Synchronization is approximate (i.e. the clocks of different FCUs need to be close together, not exactly the same) those problems that require exact agreement (e.g. group ....

Michael J. Fischer, Nancy A. Lynch, and Michael Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26--39, 1986.

....means that for any 3 players protocol, there is a choice of a bad player and a strategy for him that prevents BA. The original proofs of impossibility in this area were geared at supplying such a strategy for each possible protocol and tended to be quite cumbersome. Fischer, Lynch and Merrit [FLM] found a unified way to generate such a strategy, and their method provides short and elegant proofs for many impossibility results in the field, including the fact that t 1 rounds are required, and the necessity of high connectivity in general networks. Here, then, is the [FLM] proof that BA is ....

....Lynch and Merrit [FLM] found a unified way to generate such a strategy, and their method provides short and elegant proofs for many impossibility results in the field, including the fact that t 1 rounds are required, and the necessity of high connectivity in general networks. Here, then, is the [FLM] proof that BA is not achievable for n = 3; t = 1. The proof that in general, n 3t is a necessary condition for achieving BA, follows the same pattern, and will not be described here. A protocol Q which achieves BA for n = 3; t = 1 consists of six computer programs P i;j (i = 1; 2; 3; j = 0; 1) ....

M. J. Fischer, N. A. Lynch and M. Merritt, Easy impossibility proofs for distributed consensus problems, Distributed Computing 1(1986), 26-39.

....correct, then all correct recipients decide on the sender s input bit (v = b) We rst consider the special case of n = 3 and t 1, and then reduce the general case of n 3 and t dn=3e to this special case. The impossibility proof (for n = 3 and t 1) is based on the impossibility proof in [FLM86] where it is shown that broadcast for t dn=3e is not achievable in a model with pairwise authentic channels. In the new model, however, every pair of players can perform secure two party computation. The idea in the proof is to assume that there exists an unconditionally secure broadcast ....

M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26-39, 1986. 17

....is correct, then all correct recipients decide on the sender s input bit (v = b) We first consider the special case of n = 3 and t 1, and then reduce the general case of n 3 and t dn=3e to this special case. The impossibility proof (for n = 3 and t 1) is based on the impossibility proof in [FLM86] where it is shown that broadcast for t dn=3e is not achievable in a model with pairwise authentic channels. In the new model, however, every pair of players can perform secure two party computation. The idea in the proof is to assume that there exists an unconditionally secure broadcast ....

M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26--39, 1986.

.... in algebraic topology [10] and has also been studied in Graph Theory [11, 12] where it is in particular related to the notion of uniform emulation [13, 14] Concerning the theory of distributed computations, coverings of graphs have been used in particular for deriving impossibility results [2, 15]. In the rst subsection we introduce this notion of covering and give some basic properties. We then present some standard construction, the Kronecker product, which allows to build coverings of graphs. In order to be used within our framework this notion needs to be particularized to that of ....

M. J. Fisher, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distrib. Comput., 1:26-29, 1986.

....p 3 p 4 5 (a) b) c) Fig. 1. Rearrangement of processors in proof of Lemma 1 We first consider the special case of n = 3 and t 1, and then reduce the general case of n 3 and t dn=3e to this special case. The impossibility proof (for n = 3 and t 1) is based on the impossibility proof in [FLM86] where it is shown that broadcast for t dn=3e is not achievable in a model with pairwise authentic channels. In the new model, however, every pair of players can perform secure two party computation. The idea in the proof is to assume that there exists an unconditionally secure broadcast ....

M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26--39, 1986.

....way. The main question here is to nd some characterization of the classes of graphs that can be recognized by locally generated graph relabelling relations. Up to now, this question is still an open problem. 7 k coverings of graphs Inspired by techniques used by Angluin [1] and Fisher et al. [7], we de ne the notion of k covering, introduced in [13] which generalizes the classical notion of covering from graph theory. This notion is useful for proving negative results concerning locally generated graph relabelling relations. 12 Graph relabelling systems: a general overview Let k be a ....

Fisher, M.J.|Lynch, N.A.|Merrit, M.: Easy impossibility proofs for distributed consensus problem. Distrib. Comput. 1:26-39, 1986.

....can gain full control of a node. The behavior of a compromised node is then arbitrary and intrusions lead to Byzantine failures. Unfortunately, protecting against such failures require costly solutions and are difficult to implement in large distributed systems, when not theoretically impossible [8]. On the other hand, assuming that penetrated nodes are fail silent or only exhibit omission or timing failures is too optimistic. A better understanding and classification of the behavior of nodes after an intrusion is still needed. Building intrusion tolerant systems will require fast detection ....

M. Fischer, N. Lynch, and M. Merritt. Easy Impossibility Proofs for Distributed Consensus. Distributed Computing, 1(1):26--39, January 1986. 13

....of the network affects the number of processors and the number of rounds needed to achieve agreement. In general, as connectivity weakens, more processors and more rounds are required. The lower bound for the connectivity is 2t 1 in the non authenticated case and t 1 in the authenticated case [7, 17]. Message Complexity Of importance can be the number of messages sent or the size of the messages, i.e. the bit complexity. Message complexity and bounds are addressed in [10, 12, 4] Stated algorithms are mainly distinguished by the number of messages they require either exponential or ....

Fischer, M.J., Nancy, A.L., Merritt, M., "Easy Impossibility Proofs for Distributed Consensus Problems", Proceedings of the 4th Annual ACM Symposium on Principles of Distributed Computing , pp. 59-70, Minaki, Ontario, Canada, Aug. 1985.

....new clients from resolving to the orphaned address and minimizes the extra load on the new host because of the orphan. It also complements the automatic de installation scheme. Duplicate Addresses Since it is not possible to detect reliably host failures in asynchronous, unreliable networks [11], we may detect false failures. If we falsely conclude that a host has failed, we may reassign its service address(es) to other hosts. This will result in more than one hosts serving the same address. Vector messages (Sec. 3.2, expediting convergence) and larger timeouts help avoid duplicates. ....

M.J. Fischer, N.A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, 1986.

....is faulty, or can mislead other processors by sending 0 to one process and 1 to another. The protocol becomes far from obvious in this case. In fact, agreement is impossible if more than one third of the processors are faulty of if the network connecting the processors does not have enough links [7]. What does this have in common with the two examples given before First, the description defines an abstract space of possible programs. A program in this model is a finite set of processors connected by some fixed communication network, with some subset al..l executing the same instructions ....

....with some subset al..l executing the same instructions (the rest are faulty) and each processor having an input bit and an output bit. Execution of a program is also well defined. Second, the fact that agreement is impossible in certain situations can be proved by a simple combinatorial argument [7]. The proof does require some insight into the structure of the problem, but not much more than, say, the proof using logical relations that por is not definable. One thing that is different in this case is that the proofs of non computability do not seem to come from general principles. The lack ....

M. Fischer, N. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, 1985.

....we present a formalization of the local detection problem and we introduce some new methods for obtaining impossibility results. More precisely, we extend the notion of coverings, which is known from algebraic topology [13] and has been already used in distributed computing for negative results [1,5,7,9], to quasi coverings. Quasi coverings capture some topologies which fail to cope with the classical coverings. We show in this paper that one cannot detect locally the global termination for uniformly labelled graphs belonging to certain families of connected graphs C. More specifically, it ....

M. J. Fisher, N. A. Lynch and M. Merritt, Easy impossibility proofs for distributed consensus problems, Distrib. Comput. 1 (1986) 26--29.

.... map f : H G is a covering if both H and G are connected 1 and for each vertex v 2 V (H) the map f maps bijectively the edges adjacent to v in H to the edges adjacent to f(v) 2 V (G) In Computer Science graph coverings were used to show some impossibility results in distributed computing [1, 6], they have also applications in the domain of interconnection networks [2] each covering of a network G by a network H gives an uniform emulation of H on G with dilatation 1. For a given covering f : H G, the number jf Gamma1 (v)j of vertices of H mapped to a vertex v 2 V (G) is the same ....

M. J. Fisher, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26--39, 1986.

....does not receive a broadcast message. Proof Suppose that there exists an algorithm A which synchronizes the logical clocks to within fl, and in each execution of A, some process does not receive a broadcast message. We derive a contradiction, using the by now standard many scenarios technique [5, 6, 8, 12]. Choose D fl. Let s 1 be a scenario that corresponds to an execution of A in which the transmission time of every message (both broadcast and point to point) is 2D. Suppose without loss of generality that P 1 does not receive any broadcast messages in s 1 . Let s 2 be the scenario obtained ....

M. J. Fischer and N. A. Lynch and M. Merritt, "Easy impossibility proofs for distributed consensus problems," Distributed Computing 1:1, 1986, pp. 26--39.

.... tasks in several models has been studied in many papers; for example Byzantine agreement [4] approximate Byzantine agreement [6, 18] reliable message transmission [4, 5] and reliable and private message transmission [16, 5] Simple impossibility results and references can be found in [8]. We mention that in Byzantine agreement all honest parties should agree on the same message while in reliable communication only the transmitter and the receiver agree on the message. Digital signatures have been used for Byzantine agreement, see, e.g. 15] and discussion and references in [13] ....

M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, 1986.

....in Intern. J. Computer Math. Vol. 48, pp 21 29, 1993. 1 INTRODUCTION Since its introduction in 1980, 9] much attention has been given to the interactive consistency problem, also called the Byzantine Generals Problem, 7] Requirements for the existence of a solution have been explored, 3][4], 6] 9] and numerous algorithms for reaching agreement in the presence of malicious faults have been derived [1] 2] 6] 10] 12] As Pease, Shostak, and Lamport in their original paper pointed out, an algorithm for reaching agreement need not reveal which units are faulty; it matters only ....

M.Fischer, N.Lynch, and M.Merritt, Easy impossibility proofs for distributed consensus problems, Distributed Computing 1 (1986), 2639.

.... system it can learn by exchanging messages [29] The relationship between view and universal cover (a notion originating from algebraic topology) has been made explicit by Norris [22] Both notions are being used (sometimes under different names) in the distributed computing literature (e.g. see [1, 6, 2, 15, 22, 21, 26]) In the following, we shall refer to a node of a view by using the sequence of labels in the shortest path (in the view) from the root to that node. Since a view is a tree, such a naming in not ambiguous, and shall be called canonical. Thus, in a canonical naming, node x in view T is x = ff, ....

J. Fisher, N.A. Lynch, and Merritt. Easy impossibility proofs for distributed consensus. Distributed Computing, 1(1):26--39, 1986.

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Michael J. Fischer, Nancy A. Lynch, and Michael Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, Jan 1986. 12

....used in the design of algorithms for synchronizing clocks in distributed systems [11 ] For the synchronous case, it is not difficult to show that 3t I processes are necessary to solve the approximate agreement problem. The proof is an adaptation of the lower bound proof in [10] and appears in [8]. For the asynchronous case, our number of processes is not optimal. In fact, it appears possible to reduce the number of processes to as few as 3t 1. This reduction is obtained using a more complex algorithm, based on some of the interesting ideas of [2] This algorithm has a slower rate of ....

FISCHER, M., LYNCH, N. A., AND MERRITT, M. Easy impossibility proofs for distributed consensus problems. Distr. Cornput. 1, 1 (1986).

....bound for authenticated Byzantine failures, not presented in this thesis, is interesting (greater than Cd) only for the limited range of n 2f , and therefore says nothing interesting about unauthenticated Byzantine failures. The proof of this bound is similar to the shifting scenarios proofs of [FLM86]. Before suggesting other directions for further research, we first comment on the implications of our bounds for consensus in a closely related model. 6.1 Consensus in the related model of [HK89] Herzberg and Kutten [HK89] consider a model in which the actual worst case message delay in a ....

M. Fischer, N. Lynch and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, No. 1, 1986, pp. 26--39.

....these problems are unsolvable in network graphs having less than 2t 1 connectivity. These impossibility results do not apply if authentication is used. Since all of these bounds are tight, it is apparent that there must be a common reason for the many similar results. Fischer, Lynch, and Merritt [FLM86] tie together this large collection of impossibility results with a common proof technique. We illustrate this technique by proving the 3versus 1 impossibility result for reaching agreement on a value. Assume for the sake of obtaining a contradiction that there is such a solution for the system ....

M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, January 1986.

....to explore properties of faulty shared memory. The consensus problem is fundamental in distributed computing and is at the core of many algorithms for fault tolerant distributed applications. Much is known about the consensus problem in other fault models (see, e.g. Abr88, AH90, CIL87, Fis83, FLM85, FLP85, LAA87, SSW91] Sections 4 and 5 investigate the question of constructing reliable registers in an unreliable environment. This relates to the fundamental problem of implementing one type of shared object from another. Previous work on shared object implementations includes [Blo87, BP87, ....

M. Fischer, N. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, 1985. Computing with faulty shared objects \Delta 43

....in the design of algorithms for synchronizing clocks in distributed systems [LL 84] For the synchronous case, it is not difficult to show that 3t 1 processes are necessary to solve the approximate agreement problem. The proof is an adaptation of the lower bound proof in [LSP 82] and appears in [FLM 85] For the asynchronous case, our number of processes is not optimal. In fact, it appears possible to reduce the number of processes to as few as 3t 1. This reduction is obtained using a more complex algorithm, based on some of the interesting ideas of [B 84] This algorithm has a slower rate of ....

M. Fischer, N. A. Lynch, and M. Merritt, "Easy Impossibility Proofs for Distributed Consensus Problems," Distributed Computing, Vol. 1, No. 1, 1985.

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Michael J. Fischer, Nancy A. Lynch, and Michael Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, 1986.

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M. Fischer, N. Lynch, and M. Merritt. Easy Impossibility Proofs for Distributed Consensus Problems. Distributed Computing, 1(1):26--39, 1986.

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M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26-39, 1986.

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M. J. Fischer, N. A. Lynch and M. Merritt, " Easy impossibility proofs for distributed consensus problems", Distributed Computing, Vol. 1, pp. 26-39, 1986.

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M. Fischer, N. Lynch, and M. Merrit. Easy impossibility proofs for the distributed consensus problem. Distributed Computing, 1(1):26--39, 1986.

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M. J. Fischer, N. A. Lynch and M. Merritt, " Easy impossibility proofs for distributed consensus problems", Distributed Computing, Vol. 1, pp. 26-39, 1986.

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M. J. Fischer, N. A. Lynch and M. Merritt, Easy impossibility proofs for distributed consensus problems, Distributed Computing, Vol. 1, pp. 26-39, 1986.

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M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1):26--39, Jan. 1986.

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M. J. Fischer, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1:26-39, 1986.

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Michael J. Fischer, Nancy A. Lynch, and Michael Merritt. Easy impossibility proofs for distributed consensus problems. Distributed Computing, 1(1), pages 26--39, January 1986.

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M. J. Fischer, N. A. Lynch and M. Merritt, Easy impossibility proofs for distributed consensus problems, Distributed Computing, Vol. 1, pp. 26-39, 1986.

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M. J. Fisher, N. A. Lynch, and M. Merritt. Easy impossibility proofs for distributed consensus problems. Distrib. Comput., 1:2629, 1986.

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M. Fischer, N. Lynch, and M. Merritt. Easy Impossibility Proofs for Distributed Consensus Problems. Distributed Computing, 1(1):26--39, 1986. 26

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M. Fischer, N. Lynch, M. Merritt, "Easy impossibility proofs for distributed consensus problem", Information Processing Letters, vol.14, 1982, pp.183-186.

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M. J. Fischer, N. A. Lynch, and M. Merritt, "Easy impossibility proofs for distributed consensus problems," Distributed Computing 1, pp. 26-39, 1986.

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