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

Karchmer-Wigderson Games for Hazard-Free Computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity

Metastability Tolerant Computing

International audienceSynchronization using flip-flop chains imposes a latency of a few clock cycles when transferring data and control signals between clock domains. We propose a design scheme that avoids this latency by performing synchronization as part of state/data computations while guaranteeing that metastability is contained and its effects tolerated (with an acceptable failure probability). We present a theoretical framework for modeling synchronous state machines in the presence of metastability and use it to prove properties that guarantee some form of reliability. Specifically, we show that the inevitable state/data corruption resulting from propagating metastable states can be confined to a subset of computations. Applications that can tolerate certain failures can exploit this property to leverage low-latency and quasi-reliable operation simultaneously. We demonstrate the approach by designing a Network-on-Chip router with zero-latency asynchronous ports and show via simulation that it outperforms a variant with two flip-flop synchronizers at a negligible cost in packet transfer reliability

Metastability-Aware Memory-Efficient Time-to-Digital Converters

International audienceWe propose a novel method for transforming delay-line time-to-digital converters (TDCs) into TDCs that output Gray code without relying on synchronizers. We formally prove that the inevitable metastable memory upsets (Marino, TC'81) do not induce an additional time resolution error. Our modified design provides suitable inputs to the recent metastability-containing sorting networks by Lenzen and Medina (ASYNC'16) and Bund et al. (DATE'17). In contrast, employing existing TDCs would require using thermometer code at the TDC output (followed by conversion to Gray code) or resolving metastability inside the TDC. The former is too restrictive w.r.t. the dynamic range of the TDCs, while the latter loses the advantage of enabling (accordingly much faster) computation without having to first resolve metastability. Our all-digital designs are also of interest in their own right: they support high sample rates and large measuring ranges at nearly optimal bit-width of the output, yet maintain the original delay-line's time resolution. No previous approach unifies all these properties in a single device

Metastability-Containing Circuits

Communication across unsynchronized clock domains is inherently vulnerable to metastable upsets; no digital circuit can deterministically avoid, resolve, or detect metastability (Marino, 1981). Traditionally, a possibly metastable input is stored in synchronizers, decreasing the odds of maintained metastability over time. This approach costs time, and does not guarantee success. We propose a fundamentally different approach: It is possible to \emph{contain} metastability by logical masking, so that it cannot infect the entire circuit. This technique guarantees a limited degree of metastability in---and uncertainty about---the output. We present a synchronizer-free, fault-tolerant clock synchronization algorithm as application, synchronizing clock domains and thus enabling metastability-free communication. At the heart of our approach lies a 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 while at the same time reproducing established impossibility results regarding avoidance, resolution, and detection of metastability. Furthermore, we fully classify which functions can be computed by synchronous circuits with standard registers, and show that masking registers are computationally strictly more powerful

Karchmer-Wigderson Games for Hazard-free Computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth $2$ that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity.Comment: 34 pages, To appear in ITCS 202

Karchmer-Wigderson Games for Hazard-free Computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth $2$ that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity

Karchmer-Wigderson games for hazard-free computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity

On time, time synchronization and noise in time measurement systems

Time plays an important role in our modern lives. Especially having accurate time, which in turn depends on having clocks being synchronized to each other. This thesis is split into three distinct parts. The first part deals with the mathematical description of noise that is required to model clocks and electronics accurately. In particular we will address the problem that the generally used tools from signal theory fail for noise signals which are neither of finite energy nor periodic in nature. For this we will introduce a new function space based on the Pp-seminorm that is an extension of the Lp-norm for functions of potentially infinite energy but limited power. Using this new semi-norm we will modify the Fourier transform to work on signals from this P p-space. And last but not least, we will introduce, based on the above, a new mathematical model of noise that captures all the properties associated with 1/f -noise. In the second part, we will look at how noise propagates in a few classes of electronics, especially how the non-linear behavior of electronics leads to an amplification of noise and how it could be miti-gated. Lastly, in the third part we will look at one approach of fault-tolerant clock synchronization. After explaining its working principle and showing an implementation in an FPGA we will focus on meta-stability, the problems it can cause and how to handle them on two different circuit levels.Zeit spielt eine wichtige Rolle in unserem Leben. Insbesondere die Verfügbarkeit einer genauen Zeit. Welches wiederum davon abhängt, dass man Uhren hat die auf einander synchronisiert laufen. Diese Arbeit ist in drei Teile aufgeteilt: Im ersten Teil betrachten wir die mathematische Beschreibung von Rauschen um elektronische Systeme und Uhren korrekt beschreiben zu können. Im Besonderen betrachten wir die Probleme die die generell benutzten Methoden der Signalverarbeitung beim Umgang mit Rauschsignalen haben, die weder energiebegrenzt noch periodisch sind. Dafür erweitern wir den Funktionenraum der Lp-Norm auf leistungslimiterte Funktionene und führen die Pp-Halbnorm ein und modifizieren die Fouriertransformation zur Verwendung auf diesen Raum. Und letztlich führen wir ein neues mathematisches Model zur Beschreibung von Rauschen ein, welches alle üblicherweise angenommenen Eigenschaften gleichzeitig erfüllt. Im zweiten Teil analysieren wir wie sich einige Klassen von elektronischen Schaltungem im Bezug auf Rauschen verhalten. Insbesondere im Bezug auf das nicht-lineare Verhalten der elektronischen Elemente, welches zu einer Verstärkung des Rauschens führt. Im dritten Teil betrachten wir eine Möglichkeit um fehlertolerante Synchronization von Uhren zu erreichen. Nach einem Überblick über den verwendeten Algorithmus und wie dieser einem FPGA implementiert werden kann, schauen wir uns den Einfluss von Metastabilität an und wie dieser eingedämmt werden kann