Diophantine triples in linear recurrence sequences of Pisot type

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness of Diophantine triples in such sequences. Whilst the case of binary recurrence sequences is almost completely solved, not much was known about recurrence sequences of larger order, except for very specialized generalizations of the Fibonacci sequence. Now, we will prove that any linear recurrence sequence with the Pisot property contains only finitely many Diophantine triples, whenever the order is large and a few more not very restrictive conditions are met.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1602.0823

Algorithms For Extracting Timeliness Graphs

We consider asynchronous message-passing systems in which some links are timely and processes may crash. Each run defines a timeliness graph among correct processes: (p; q) is an edge of the timeliness graph if the link from p to q is timely (that is, there is bound on communication delays from p to q). The main goal of this paper is to approximate this timeliness graph by graphs having some properties (such as being trees, rings, ...). Given a family S of graphs, for runs such that the timeliness graph contains at least one graph in S then using an extraction algorithm, each correct process has to converge to the same graph in S that is, in a precise sense, an approximation of the timeliness graph of the run. For example, if the timeliness graph contains a ring, then using an extraction algorithm, all correct processes eventually converge to the same ring and in this ring all nodes will be correct processes and all links will be timely. We first present a general extraction algorithm and then a more specific extraction algorithm that is communication efficient (i.e., eventually all the messages of the extraction algorithm use only links of the extracted graph)

Tolerating permanent and transient value faults

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 , the number of processes, in order to solve consensus in our weak model

Tolerating permanent and transient value faults

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 $$n$$ n , the number of processes, in order to solve consensus in our weak model

Quantitative Analysis of Consensus Algorithms

Consensus is one of the key problems in fault-tolerant distributed computing. Although the solvability of consensus is now a well-understood problem, comparing different algorithms in terms of efficiency is still an open problem. In this paper, we address this question for round-based consensus algorithm using communication predicates, on top of a partial synchronous system that alternates between good and bad periods (synchronous and non synchronous periods). Communication predicates together with the detailed timing information of the underlying partial-synchronous system provide a convenient and powerful framework for comparing different consensus algorithms and their implementations. This approach allows us to quantify the required length of a good period to solve a given number of consensus instances. With our results, we can observe several interesting issues, e.g., that the number of rounds of an algorithm is not necessarily a good metric for its performance

In Search of Lost Time

Dwork, Lynch, and Stockmeyer (1988) and Lamport (1998) showed that, in order to solve Consensus in a distributed system, it is sufficient that the system behaves well during a finite period of time. In sharp contrast, Chandra, Hadzilacos, and Toueg (1996) proved that a failure detector that, from some time on, provides "good" information forever is necessary. We explain that this apparent paradox is due to the two-layered structure of the failure detector model. This structure also has impacts on comparison relations between failure detectors. In particular, we make explicit why the classic relation is neither reflexive nor extends the natural history-wise inclusion. Although not technically difficult, the point we make helps understanding existing models like the failure detector model

Latency-aware Leader Election

Experimental studies have shown that electing a leader based on measurements of the underlying communication network can be beneficial. We use this approach to study the problem of electing a leader that is eventually not only correct (as aptured by the failure detector abstraction), but also optimal with respect to the transmission delays to its peers. We give the definitions of this problem and a suitable model, thus allowing us to make an analytical analysis of the problem, which is in contrast to previous work on that topic

Consensus when all processes may be Byzantine for some time

Among all classes of faults, Byzantine faults form the most general modeling of value faults. Traditionally, in the Byzantine fault model, faults are statically attributed to a set of up to t processes. This, however, implies that in this model a process at which a value fault occurs is forever "stigmatized" as being Byzantine, an assumption that might not be acceptable for long-lived systems, where processes need to be reintegrated after a fault. We thus consider a model where Byzantine processes can recover in a predefined recovery state, and show that consensus can be solved in such a model