Formal Verification of Divider and Square-root Arithmetic Circuits Using Computer Algebra Methods

A considerable progress has been made in recent years in verification of arithmetic circuits such as multipliers, fused multiply-adders, multiply-accumulate, and other components of arithmetic datapaths, both in integer and finite field domain. However, the verification of hardware dividers and square-root functions have received only a limited attention from the verification community, with a notable exception for theorem provers and other inductive, non-automated systems. Division, square root, and transcendental functions are all tied to the basic Intel architecture and proving correctness of such algorithms is of grave importance. Although belonging to the same iterative-subtract class of architectures, they widely differ from each other. IEEE floating point standard specifies square-rooting and division as basic arithmetic operation alongside the usual three basic operations. The difficulty of formally verifying hardware implementation of a divider/square-root can be attributed mostly to the modeling of its characteristic function and the high memory complexity required by standard algebraic approach. The work proposed in this thesis discusses formal verification of combinational divider and square-root circuits. Specifically, it addresses the problem of formally verifying gate-level circuits using an algebraic model. In contrast to standard verification approaches using satisfiability (SAT) or equivalence checking, the proposed method verifies whether the gate-level circuit actually performs the intended function or not, without a need for a reference design. Firstly, we present a verification methodology for a constant divider, where the divisor value is fixed to a constant integer. Albeit simpler case of verification, it provides us with the basic understanding of verification techniques and the underlying issues applicable to divider verification. Secondly, a layered verification approach is proposed for the verification of generic array dividers. Finally, the work proposed in this thesis will further analyze the divider and square-root circuits and aim to curb the memory explosion issue experienced by computer algebra based verification methods in order to successfully verify large bit-width divider-type arithmetic circuits. More specifically, a novel idea of hardware rewriting is introduced, which avoids the high memory complexity. The mentioned technique verifies a 256-bit gate-level square-root circuit with around 260,000 gates in just under 18 minutes and 127-bit gate-level divider circuit in under one minute

Approximate Computing Survey, Part I: Terminology and Software & Hardware Approximation Techniques

The rapid growth of demanding applications in domains applying multimedia processing and machine learning has marked a new era for edge and cloud computing. These applications involve massive data and compute-intensive tasks, and thus, typical computing paradigms in embedded systems and data centers are stressed to meet the worldwide demand for high performance. Concurrently, the landscape of the semiconductor field in the last 15 years has constituted power as a first-class design concern. As a result, the community of computing systems is forced to find alternative design approaches to facilitate high-performance and/or power-efficient computing. Among the examined solutions, Approximate Computing has attracted an ever-increasing interest, with research works applying approximations across the entire traditional computing stack, i.e., at software, hardware, and architectural levels. Over the last decade, there is a plethora of approximation techniques in software (programs, frameworks, compilers, runtimes, languages), hardware (circuits, accelerators), and architectures (processors, memories). The current article is Part I of our comprehensive survey on Approximate Computing, and it reviews its motivation, terminology and principles, as well it classifies and presents the technical details of the state-of-the-art software and hardware approximation techniques.Comment: Under Review at ACM Computing Survey

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Reusable Reentry Satellite (RRS) system design study

The Reusable Reentry Satellite (RRS) is intended to provide investigators in several biological disciplines with a relatively inexpensive method to access space for up to 60 days with eventual recovery on Earth. The RRS will permit totally intact, relatively soft, recovery of the vehicle, system refurbishment, and reflight with new and varied payloads. The RRS is to be capable of three reflights per year over a 10-year program lifetime. The RRS vehicle will have a large and readily accessible volume near the vehicle center of gravity for the Payload Module (PM) containing the experiment hardware. The vehicle is configured to permit the experimenter late access to the PM prior to launch and rapid access following recovery. The RRS will operate in one of two modes: (1) as a free-flying spacecraft in orbit, and will be allowed to drift in attitude to provide an acceleration environment of less than 10(exp -5) g. the acceleration environment during orbital trim maneuvers will be less than 10(exp -3) g; and (2) as an artificial gravity system which spins at controlled rates to provide an artificial gravity of up to 1.5 Earth g. The RRS system will be designed to be rugged, easily maintained, and economically refurbishable for the next flight. Some systems may be designed to be replaced rather than refurbished, if cost effective and capable of meeting the specified turnaround time. The minimum time between recovery and reflight will be approximately 60 days. The PMs will be designed to be relatively autonomous, with experiments that require few commands and limited telemetry. Mass data storage will be accommodated in the PM. The hardware development and implementation phase is currently expected to start in 1991 with a first launch in late 1993

Optimisations arithmétiques et synthèse de haut niveau

High-level synthesis (HLS) tools offer increased productivity regarding FPGA programming.However, due to their relatively young nature, they still lack many arithmetic optimizations.This thesis proposes safe arithmetic optimizations that should always be applied.These optimizations are simple operator specializations, following the C semantic.Other require to a lift the semantic embedded in high-level input program languages, which are inherited from software programming, for an improved accuracy/cost/performance ratio.To demonstrate this claim, the sum-of-product of floating-point numbers is used as a case study. The sum is performed on a fixed-point format, which is tailored to the application, according to the context in which the operator is instantiated.In some cases, there is not enough information about the input data to tailor the fixed-point accumulator.The fall-back strategy used in this thesis is to generate an accumulator covering the entire floating-point range.This thesis explores different strategies for implementing such a large accumulator, including new ones.The use of a 2's complement representation instead of a sign+magnitude is demonstrated to save resources and to reduce the accumulation loop delay.Based on a tapered precision scheme and an exact accumulator, the posit number systems claims to be a candidate to replace the IEEE floating-point format.A throughout analysis of posit operators is performed, using the same level of hardware optimization as state-of-the-art floating-point operators.Their cost remains much higher that their floating-point counterparts in terms of resource usage and performance. Finally, this thesis presents a compatibility layer for HLS tools that allows one code to be deployed on multiple tools.This library implements a strongly typed custom size integer type along side a set of optimized custom operators.À cause de la nature relativement jeune des outils de synthèse de haut-niveau (HLS), de nombreuses optimisations arithmétiques n'y sont pas encore implémentées. Cette thèse propose des optimisations arithmétiques se servant du contexte spécifique dans lequel les opérateurs sont instanciés.Certaines optimisations sont de simples spécialisations d'opérateurs, respectant la sémantique du C.D'autres nécéssitent de s'éloigner de cette sémantique pour améliorer le compromis précision/coût/performance.Cette proposition est démontré sur des sommes de produits de nombres flottants.La somme est réalisée dans un format en virgule-fixe défini par son contexte.Quand trop peu d’informations sont disponibles pour définir ce format en virgule-fixe, une stratégie est de générer un accumulateur couvrant l'intégralité du format flottant.Cette thèse explore plusieurs implémentations d'un tel accumulateur.L'utilisation d'une représentation en complément à deux permet de réduire le chemin critique de la boucle d'accumulation, ainsi que la quantité de ressources utilisées. Un format alternatif aux nombres flottants, appelé posit, propose d'utiliser un encodage à précision variable.De plus, ce format est augmenté par un accumulateur exact.Pour évaluer précisément le coût matériel de ce format, cette thèse présente des architectures d'opérateurs posits, implémentés avec le même degré d'optimisation que celui de l'état de l'art des opérateurs flottants.Une analyse détaillée montre que le coût des opérateurs posits est malgré tout bien plus élevé que celui de leurs équivalents flottants.Enfin, cette thèse présente une couche de compatibilité entre outils de HLS, permettant de viser plusieurs outils avec un seul code. Cette bibliothèque implémente un type d'entiers de taille variable, avec de plus une sémantique strictement typée, ainsi qu'un ensemble d'opérateurs ad-hoc optimisés

Low-cost and efficient fault detection and diagnosis schemes for modern cores

Continuous improvements in transistor scaling together with microarchitectural advances have made possible the widespread adoption of high-performance processors across all market segments. However, the growing reliability threats induced by technology scaling and by the complexity of designs are challenging the production of cheap yet robust systems. Soft error trends are haunting, especially for combinational logic, and parity and ECC codes are therefore becoming insufficient as combinational logic turns into the dominant source of soft errors. Furthermore, experts are warning about the need to also address intermittent and permanent faults during processor runtime, as increasing temperatures and device variations will accelerate inherent aging phenomena. These challenges specially threaten the commodity segments, which impose requirements that existing fault tolerance mechanisms cannot offer. Current techniques based on redundant execution were devised in a time when high penalties were assumed for the sake of high reliability levels. Novel light-weight techniques are therefore needed to enable fault protection in the mass market segments. The complexity of designs is making post-silicon validation extremely expensive. Validation costs exceed design costs, and the number of discovered bugs is growing, both during validation and once products hit the market. Fault localization and diagnosis are the biggest bottlenecks, magnified by huge detection latencies, limited internal observability, and costly server farms to generate test outputs. This thesis explores two directions to address some of the critical challenges introduced by unreliable technologies and by the limitations of current validation approaches. We first explore mechanisms for comprehensively detecting multiple sources of failures in modern processors during their lifetime (including transient, intermittent, permanent and also design bugs). Our solutions embrace a paradigm where fault tolerance is built based on exploiting high-level microarchitectural invariants that are reusable across designs, rather than relying on re-execution or ad-hoc block-level protection. To do so, we decompose the basic functionalities of processors into high-level tasks and propose three novel runtime verification solutions that combined enable global error detection: a computation/register dataflow checker, a memory dataflow checker, and a control flow checker. The techniques use the concept of end-to-end signatures and allow designers to adjust the fault coverage to their needs, by trading-off area, power and performance. Our fault injection studies reveal that our methods provide high coverage levels while causing significantly lower performance, power and area costs than existing techniques. Then, this thesis extends the applicability of the proposed error detection schemes to the validation phases. We present a fault localization and diagnosis solution for the memory dataflow by combining our error detection mechanism, a new low-cost logging mechanism and a diagnosis program. Selected internal activity is continuously traced and kept in a memory-resident log whose capacity can be expanded to suite validation needs. The solution can catch undiscovered bugs, reducing the dependence on simulation farms that compute golden outputs. Upon error detection, the diagnosis algorithm analyzes the log to automatically locate the bug, and also to determine its root cause. Our evaluations show that very high localization coverage and diagnosis accuracy can be obtained at very low performance and area costs. The net result is a simplification of current debugging practices, which are extremely manual, time consuming and cumbersome. Altogether, the integrated solutions proposed in this thesis capacitate the industry to deliver more reliable and correct processors as technology evolves into more complex designs and more vulnerable transistors.El continuo escalado de los transistores junto con los avances microarquitectónicos han posibilitado la presencia de potentes procesadores en todos los segmentos de mercado. Sin embargo, varios problemas de fiabilidad están desafiando la producción de sistemas robustos. Las predicciones de "soft errors" son inquietantes, especialmente para la lógica combinacional: soluciones como ECC o paridad se están volviendo insuficientes a medida que dicha lógica se convierte en la fuente predominante de soft errors. Además, los expertos están alertando acerca de la necesidad de detectar otras fuentes de fallos (causantes de errores permanentes e intermitentes) durante el tiempo de vida de los procesadores. Los segmentos "commodity" son los más vulnerables, ya que imponen unos requisitos que las técnicas actuales de fiabilidad no ofrecen. Estas soluciones (generalmente basadas en re-ejecución) fueron ideadas en un tiempo en el que con tal de alcanzar altos nivel de fiabilidad se asumían grandes costes. Son por tanto necesarias nuevas técnicas que permitan la protección contra fallos en los segmentos más populares. La complejidad de los diseños está encareciendo la validación "post-silicon". Su coste excede el de diseño, y el número de errores descubiertos está aumentando durante la validación y ya en manos de los clientes. La localización y el diagnóstico de errores son los mayores problemas, empeorados por las altas latencias en la manifestación de errores, por la poca observabilidad interna y por el coste de generar las señales esperadas. Esta tesis explora dos direcciones para tratar algunos de los retos causados por la creciente vulnerabilidad hardware y por las limitaciones de los enfoques de validación. Primero exploramos mecanismos para detectar múltiples fuentes de fallos durante el tiempo de vida de los procesadores (errores transitorios, intermitentes, permanentes y de diseño). Nuestras soluciones son de un paradigma donde la fiabilidad se construye explotando invariantes microarquitectónicos genéricos, en lugar de basarse en re-ejecución o en protección ad-hoc. Para ello descomponemos las funcionalidades básicas de un procesador y proponemos tres soluciones de `runtime verification' que combinadas permiten una detección de errores a nivel global. Estas tres soluciones son: un verificador de flujo de datos de registro y de computación, un verificador de flujo de datos de memoria y un verificador de flujo de control. Nuestras técnicas usan el concepto de firmas y permiten a los diseñadores ajustar los niveles de protección a sus necesidades, mediante compensaciones en área, consumo energético y rendimiento. Nuestros estudios de inyección de errores revelan que los métodos propuestos obtienen altos niveles de protección, a la vez que causan menos costes que las soluciones existentes. A continuación, esta tesis explora la aplicabilidad de estos esquemas a las fases de validación. Proponemos una solución de localización y diagnóstico de errores para el flujo de datos de memoria que combina nuestro mecanismo de detección de errores, junto con un mecanismo de logging de bajo coste y un programa de diagnóstico. Cierta actividad interna es continuamente registrada en una zona de memoria cuya capacidad puede ser expandida para satisfacer las necesidades de validación. La solución permite descubrir bugs, reduciendo la necesidad de calcular los resultados esperados. Al detectar un error, el algoritmo de diagnóstico analiza el registro para automáticamente localizar el bug y determinar su causa. Nuestros estudios muestran un alto grado de localización y de precisión de diagnóstico a un coste muy bajo de rendimiento y área. El resultado es una simplificación de las prácticas actuales de depuración, que son enormemente manuales, incómodas y largas. En conjunto, las soluciones de esta tesis capacitan a la industria a producir procesadores más fiables, a medida que la tecnología evoluciona hacia diseños más complejos y más vulnerables

Proceedings of the 21st Conference on Formal Methods in Computer-Aided Design – FMCAD 2021

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

JUNO Conceptual Design Report

The Jiangmen Underground Neutrino Observatory (JUNO) is proposed to determine the neutrino mass hierarchy using an underground liquid scintillator detector. It is located 53 km away from both Yangjiang and Taishan Nuclear Power Plants in Guangdong, China. The experimental hall, spanning more than 50 meters, is under a granite mountain of over 700 m overburden. Within six years of running, the detection of reactor antineutrinos can resolve the neutrino mass hierarchy at a confidence level of 3-4$\sigma$, and determine neutrino oscillation parameters $\sin^2\theta_{12}$, $\Delta m^2_{21}$, and $|\Delta m^2_{ee}|$ to an accuracy of better than 1%. The JUNO detector can be also used to study terrestrial and extra-terrestrial neutrinos and new physics beyond the Standard Model. The central detector contains 20,000 tons liquid scintillator with an acrylic sphere of 35 m in diameter. $\sim$17,000 508-mm diameter PMTs with high quantum efficiency provide $\sim$75% optical coverage. The current choice of the liquid scintillator is: linear alkyl benzene (LAB) as the solvent, plus PPO as the scintillation fluor and a wavelength-shifter (Bis-MSB). The number of detected photoelectrons per MeV is larger than 1,100 and the energy resolution is expected to be 3% at 1 MeV. The calibration system is designed to deploy multiple sources to cover the entire energy range of reactor antineutrinos, and to achieve a full-volume position coverage inside the detector. The veto system is used for muon detection, muon induced background study and reduction. It consists of a Water Cherenkov detector and a Top Tracker system. The readout system, the detector control system and the offline system insure efficient and stable data acquisition and processing.Comment: 328 pages, 211 figure