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Matthias Függer

Publications and source records attributed to Matthias Függer.

32 records · Page 2Linked to original sources
arXiv2020

PALS: Plesiochronous and Locally Synchronous Systems

Consider an arbitrary network of communicating modules on a chip, each requiring a local signal telling it when to execute a computational step. There are three common solutions to generating such a local clock signal: (i) by deriving it from a single, central clock source, (ii) by local, free-running oscillators, or (iii) by handshaking between neighboring modules. Conceptually, each of these solutions is the result of a perceived dichotomy in which (sub)systems are either clocked or fully asynchronous, suggesting that the designer's choice is limited to deciding where to draw the line between synchronous and asynchronous design. In contrast, we take the view that the better question to ask is how synchronous the system can and should be. Based on a distributed clock synchronization algorithm, we present a novel design providing modules with local clocks whose frequency bounds are almost as good as those of corresponding free-running oscillators, yet neighboring modules are guaranteed to have a phase offset substantially smaller than one clock cycle. Concretely, parameters obtained from a 15nm ASIC implementation running at 2GHz yield mathematical worst-case bounds of 30ps on phase offset for a 32x32 node grid network.

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arXiv2019

On the Radius of Nonsplit Graphs and Information Dissemination in Dynamic Networks

A nonsplit graph is a directed graph where each pair of nodes has a common incoming neighbor. We show that the radius of such graphs is in $O(\log \log n)$, where $n$ is the number of nodes. We then generalize the result to products of nonsplit graphs. The analysis of nonsplit graph products has direct implications in the context of distributed systems, where processes operate in rounds and communicate via message passing in each round: communication graphs in several distributed systems naturally relate to nonsplit graphs and the graph product concisely represents relaying messages in such networks. Applying our results, we obtain improved bounds on the dynamic radius of such networks, i.e., the maximum number of rounds until all processes have received a message from a common process, if all processes relay messages in each round. We finally connect the dynamic radius to lower bounds for achieving consensus in dynamic networks.

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arXiv2018

Tight Bounds for Asymptotic and Approximate Consensus

We study the performance of asymptotic and approximate consensus algorithms under harsh environmental conditions. The asymptotic consensus problem requires a set of agents to repeatedly set their outputs such that the outputs converge to a common value within the convex hull of initial values. This problem, and the related approximate consensus problem, are fundamental building blocks in distributed systems where exact consensus among agents is not required or possible, e.g., man-made distributed control systems, and have applications in the analysis of natural distributed systems, such as flocking and opinion dynamics. We prove tight lower bounds on the contraction rates of asymptotic consensus algorithms in dynamic networks, from which we deduce bounds on the time complexity of approximate consensus algorithms. In particular, the obtained bounds show optimality of asymptotic and approximate consensus algorithms presented in [Charron-Bost et al., ICALP'16] for certain dynamic networks, including the weakest dynamic network model in which asymptotic and approximate consensus are solvable. As a corollary we also obtain asymptotically tight bounds for asymptotic consensus in the classical asynchronous model with crashes. Central to our lower bound proofs is an extended notion of valency, the set of reachable limits of an asymptotic consensus algorithm starting from a given configuration. We further relate topological properties of valencies to the solvability of exact consensus, shedding some light on the relation of these three fundamental problems in dynamic networks.

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arXiv2018

Fast Multidimensional Asymptotic and Approximate Consensus

We study the problems of asymptotic and approximate consensus in which agents have to get their values arbitrarily close to each others' inside the convex hull of initial values, either without or with an explicit decision by the agents. In particular, we are concerned with the case of multidimensional data, i.e., the agents' values are $d$-dimensional vectors. We introduce two new algorithms for dynamic networks, subsuming classical failure models like asynchronous message passing systems with Byzantine agents. The algorithms are the first to have a contraction rate and time complexity independent of the dimension $d$. In particular, we improve the time complexity from the previously fastest approximate consensus algorithm in asynchronous message passing systems with Byzantine faults by Mendes et al. [Distrib. Comput. 28] from $Ω\!\left( d \log\frac{dΔ}{\varepsilon} \right)$ to $O\!\left( \log\fracΔ{\varepsilon} \right)$, where $Δ$ is the initial and $\varepsilon$ is the terminal diameter of the set of vectors of correct agents.

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arXiv2017

Metastability-Containing Circuits

In digital circuits, metastability can cause deteriorated signals that neither are logical 0 or logical 1, breaking the abstraction of Boolean logic. Unfortunately, any way of reading a signal from an unsynchronized clock domain or performing an analog-to-digital conversion incurs the risk of a metastable upset; no digital circuit can deterministically avoid, resolve, or detect metastability (Marino, 1981). Synchronizers, the only traditional countermeasure, exponentially decrease the odds of maintained metastability over time. Trading synchronization delay for an increased probability to resolve metastability to logical 0 or 1, they do not guarantee success. We propose a fundamentally different approach: It is possible to contain metastability by fine-grained logical masking so that it cannot infect the entire circuit. This technique guarantees a limited degree of metastability in---and uncertainty about---the output. At the heart of our approach lies a time- and value-discrete model for metastability in synchronous clocked digital circuits. Metastability is propagated in a worst-case fashion, allowing to derive deterministic guarantees, without and unlike synchronizers. The proposed model permits positive results and passes the test of reproducing Marino's impossibility results. We fully classify which functions can be computed by circuits with standard registers. Regarding masking registers, we show that they become computationally strictly more powerful with each clock cycle, resulting in a non-trivial hierarchy of computable functions.

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arXiv2016

Multidimensional Asymptotic Consensus in Dynamic Networks

We study the problem of asymptotic consensus as it occurs in a wide range of applications in both man-made and natural systems. In particular, we study systems with directed communication graphs that may change over time. We recently proposed a new family of convex combination algorithms in dimension one whose weights depend on the received values and not only on the communication topology. Here, we extend this approach to arbitrarily high dimensions by introducing two new algorithms: the ExtremePoint and the Centroid algorithm. Contrary to classical convex combination algorithms, both have component-wise contraction rates that are constant in the number of agents. Paired with a speed-up technique for convex combination algorithms, we get a convergence time linear in the number of agents, which is optimal. Besides their respective contraction rates, the two algorithms differ in the fact that the Centroid algorithm's update rule is independent of any coordinate system while the ExtremePoint algorithm implicitly assumes a common agreed-upon coordinate system among agents. The latter assumption may be realistic in some man-made multi-agent systems but is highly questionable in systems designed for the modelization of natural phenomena. Finally we prove that our new algorithms also achieve asymptotic consensus under very weak connectivity assumptions, provided that agent interactions are bidirectional.

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arXiv2015

Amortized Averaging Algorithms for Approximate Consensus

We introduce a new class of distributed algorithms for the approximate consensus problem in dynamic rooted networks, which we call amortized averaging algorithms. They are deduced from ordinary averaging algorithms by adding a value-gathering phase before each value update. This allows their decision time to drop from being exponential in the number $n$ of processes to being linear under the assumption that each process knows $n$. In particular, the amortized midpoint algorithm, which achieves a linear decision time, works in completely anonymous dynamic rooted networks where processes can exchange and store continuous values, and under the assumption that the number of processes is known to all processes. We then study the way amortized averaging algorithms degrade when communication graphs are from time to time non rooted, or with a wrong estimate of the number of processes. Finally, we analyze the amortized midpoint algorithm under the additional constraint that processes can only store and send quantized values, and get as a corollary that the 2-set consensus problem is solvable in linear time in any rooted dynamic network model when allowing all decision values to be in the range of initial values.

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arXiv2014

Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms

In this paper, we investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous systems, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree, i.e., there is a coordinator at each time. The striking point in this result is that the coordinator is not required to be unique and can change arbitrarily from round to round. Interestingly, the class of averaging algorithms, which are memoryless and require no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. Concerning the time complexity of averaging algorithms, we show that approximate consensus can be achieved with precision of $\varepsilon$ in a coordinated network model in $O(n^{n+1} \log\frac{1}{\varepsilon})$ synchronous rounds, and in $O(Δn^{nΔ+1} \log\frac{1}{\varepsilon})$ rounds when the maximum round delay for a message to be delivered is $Δ$. While in general, an upper bound on the time complexity of averaging algorithms has to be exponential, we investigate various network models in which this exponential bound in the number of nodes reduces to a polynomial bound. We apply our results to networked systems with a fixed topology and classical benign fault models, and deduce both known and new results for approximate consensus in these systems. In particular, we show that for solving approximate consensus, a complete network can tolerate up to 2n-3 arbitrarily located link faults at every round, in contrast with the impossibility result established by Santoro and Widmayer (STACS '89) showing that exact consensus is not solvable with n-1 link faults per round originating from the same node.

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arXiv2014

Faithful Glitch Propagation in Binary Circuit Models

Modern digital circuit design relies on fast digital timing simulation tools and, hence, on accurate binary-valued circuit models that faithfully model signal propagation, even throughout a complex design. Unfortunately, it was recently proved [Függer et al., ASYNC'13] that no existing binary-valued circuit model proposed so far, including the two most commonly used pure and inertial delay channels, faithfully captures glitch propagation: For the simple Short-Pulse Filtration (SPF) problem, which is related to a circuit's ability to suppress a single glitch, we showed that the quite broad class of bounded single-history channels either contradict the unsolvability of SPF in bounded time or the solvability of SPF in unbounded time in physical circuits. In this paper, we propose a class of binary circuit models that do not suffer from this deficiency: Like bounded single-history channels, our involution channels involve delays that may depend on the time of the previous output transition. Their characteristic property are delay functions which are based on involutions, i.e., functions that form their own inverse. A concrete example of such a delay function, which is derived from a generalized first-order analog circuit model, reveals that this is not an unrealistic assumption. We prove that, in sharp contrast to what is possible with bounded single-history channels, SPF cannot be solved in bounded time due to the nonexistence of a lower bound on the delay of involution channels, whereas it is easy to provide an unbounded SPF implementation. It hence follows that binary-valued circuit models based on involution channels allow to solve SPF precisely when this is possible in physical circuits. To the best of our knowledge, our model is hence the very first candidate for a model that indeed guarantees faithful glitch propagation.

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arXiv2013

Unfaithful Glitch Propagation in Existing Binary Circuit Models

We show that no existing continuous-time, binary value-domain model for digital circuits is able to correctly capture glitch propagation. Prominent examples of such models are based on pure delay channels (P), inertial delay channels (I), or the elaborate PID channels proposed by Bellido-Díaz et al. We accomplish our goal by considering the solvability/non-solvability border of a simple problem called Short-Pulse Filtration (SPF), which is closely related to arbitration and synchronization. On one hand, we prove that SPF is solvable in bounded time in any such model that provides channels with non-constant delay, like I and PID. This is in opposition to the impossibility of solving bounded SPF in real (physical) circuit models. On the other hand, for binary circuit models with constant-delay channels, we prove that SPF cannot be solved even in unbounded time; again in opposition to physical circuit models. Consequently, indeed none of the binary value-domain models proposed so far (and that we are aware of) faithfully captures glitch propagation of real circuits. We finally show that these modeling mismatches do not hold for the weaker eventual SPF problem.

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arXiv2012

New Transience Bounds for Long Walks

Linear max-plus systems describe the behavior of a large variety of complex systems. It is known that these systems show a periodic behavior after an initial transient phase. Assessment of the length of this transient phase provides important information on complexity measures of such systems, and so is crucial in system design. We identify relevant parameters in a graph representation of these systems and propose a modular strategy to derive new upper bounds on the length of the transient phase. By that we are the first to give asymptotically tight and potentially subquadratic transience bounds. We use our bounds to derive new complexity results, in particular in distributed computing.

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arXiv2012

FATAL+: A Self-Stabilizing Byzantine Fault-tolerant Clocking Scheme for SoCs

We present concept and implementation of a self-stabilizing Byzantine fault-tolerant distributed clock generation scheme for multi-synchronous GALS architectures in critical applications. It combines a variant of a recently introduced self-stabilizing algorithm for generating low-frequency, low-accuracy synchronized pulses with a simple non-stabilizing high-frequency, high-accuracy clock synchronization algorithm. We provide thorough correctness proofs and a performance analysis, which use methods from fault-tolerant distributed computing research but also addresses hardware-related issues like metastability. The algorithm, which consists of several concurrent communicating asynchronous state machines, has been implemented in VHDL using Petrify in conjunction with some extensions, and synthetisized for an Altera Cyclone FPGA. An experimental validation of this prototype has been carried out to confirm the skew and clock frequency bounds predicted by the theoretical analysis, as well as the very short stabilization times (required for recovering after excessively many transient failures) achievable in practice.

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arXiv2011

On the Transience of Linear Max-Plus Dynamical Systems

We study the transients of linear max-plus dynamical systems. For that, we consider for each irreducible max-plus matrix A, the weighted graph G(A) such that A is the adjacency matrix of G(A). Based on a novel graph-theoretic counterpart to the number-theoretic Brauer's theorem, we propose two new methods for the construction of arbitrarily long paths in G(A) with maximal weight. That leads to two new upper bounds on the transient of a linear max-plus system which both improve on the bounds previously given by Even and Rajsbaum (STOC 1990, Theory of Computing Systems 1997), by Bouillard and Gaujal (Research Report 2000), and by Soto y Koelemeijer (PhD Thesis 2003), and are, in general, incomparable with Hartmann and Arguelles' bound (Mathematics of Operations Research 1999). With our approach, we also show how to improve the latter bound by a factor of two. A significant benefit of our bounds is that each of them turns out to be linear in the size of the system in various classes of linear max-plus system whereas the bounds previously given are all at least quadratic. Our second result concerns the relationship between matrix and system transients: We prove that the transient of an NxN matrix A is, up to some constant, equal to the transient of an A-linear system with an initial vector whose norm is quadratic in N. Finally, we study the applicability of our results to the well-known Full Reversal algorithm whose behavior can be described as a min-plus linear system.

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