Stochastic Formal Correctness of Numerical Algorithms

We provide a framework to bound the probability that accumulated errors were never above a given threshold on numerical algorithms. Such algorithms are used for example in aircraft and nuclear power plants. This report contains simple formulas based on Levy's and Markov's inequalities and it presents a formal theory of random variables with a special focus on producing concrete results. We selected four very common applications that fit in our framework and cover the common practices of systems that evolve for a long time. We compute the number of bits that remain continuously significant in the first two applications with a probability of failure around one out of a billion, where worst case analysis considers that no significant bit remains. We are using PVS as such formal tools force explicit statement of all hypotheses and prevent incorrect uses of theorems

Avoiding double roundings in scaled Newton-Raphson division

Abstract-When performing divisions using Newton-Raphson (or similar) iterations on a processor with a floating-point fused multiply-add instruction, one must sometimes scale the iterations, to avoid over/underflow and/or loss of accuracy. This may lead to double-roundings, resulting in output values that may not be correctly rounded when the quotient falls in the subnormal range. We show how to avoid this problem

Some issues related to double roundings

International audienceDouble rounding is a phenomenon that may occur when different floating- point precisions are available on the same system. Although double rounding is, in general, innocuous, it may change the behavior of some useful small floating-point algorithms. We analyze the potential influence of double rounding on the Fast2Sum and 2Sum algorithms, on some summation algorithms, and Veltkamp's splitting

Hardware Error Detection Using AN-Codes

Due to the continuously decreasing feature sizes and the increasing complexity of integrated circuits, commercial off-the-shelf (COTS) hardware is becoming less and less reliable. However, dedicated reliable hardware is expensive and usually slower than commodity hardware. Thus, economic pressure will most likely result in the usage of unreliable COTS hardware in safety-critical systems. The usage of unreliable, COTS hardware in safety-critical systems results in the need for software-implemented solutions for handling execution errors caused by this unreliable hardware. In this thesis, we provide techniques for detecting hardware errors that disturb the execution of a program. The detection provided facilitates handling of these errors, for example, by retry or graceful degradation. We realize the error detection by transforming unsafe programs that are not guaranteed to detect execution errors into safe programs that detect execution errors with a high probability. Therefore, we use arithmetic AN-, ANB-, ANBD-, and ANBDmem-codes. These codes detect errors that modify data during storage or transport and errors that disturb computations as well. Furthermore, the error detection provided is independent of the hardware used. We present the following novel encoding approaches: - Software Encoded Processing (SEP) that transforms an unsafe binary into a safe execution at runtime by applying an ANB-code, and - Compiler Encoded Processing (CEP) that applies encoding at compile time and provides different levels of safety by using different arithmetic codes. In contrast to existing encoding solutions, SEP and CEP allow to encode applications whose data and control flow is not completely predictable at compile time. For encoding, SEP and CEP use our set of encoded operations also presented in this thesis. To the best of our knowledge, we are the first ones that present the encoding of a complete RISC instruction set including boolean and bitwise logical operations, casts, unaligned loads and stores, shifts and arithmetic operations. Our evaluations show that encoding with SEP and CEP significantly reduces the amount of erroneous output caused by hardware errors. Furthermore, our evaluations show that, in contrast to replication-based approaches for detecting errors, arithmetic encoding facilitates the detection of permanent hardware errors. This increased reliability does not come for free. However, unexpectedly the runtime costs for the different arithmetic codes supported by CEP compared to redundancy increase only linearly, while the gained safety increases exponentially

Hardware Error Detection Using AN-Codes

Due to the continuously decreasing feature sizes and the increasing complexity of integrated circuits, commercial off-the-shelf (COTS) hardware is becoming less and less reliable. However, dedicated reliable hardware is expensive and usually slower than commodity hardware. Thus, economic pressure will most likely result in the usage of unreliable COTS hardware in safety-critical systems. The usage of unreliable, COTS hardware in safety-critical systems results in the need for software-implemented solutions for handling execution errors caused by this unreliable hardware. In this thesis, we provide techniques for detecting hardware errors that disturb the execution of a program. The detection provided facilitates handling of these errors, for example, by retry or graceful degradation. We realize the error detection by transforming unsafe programs that are not guaranteed to detect execution errors into safe programs that detect execution errors with a high probability. Therefore, we use arithmetic AN-, ANB-, ANBD-, and ANBDmem-codes. These codes detect errors that modify data during storage or transport and errors that disturb computations as well. Furthermore, the error detection provided is independent of the hardware used. We present the following novel encoding approaches: - Software Encoded Processing (SEP) that transforms an unsafe binary into a safe execution at runtime by applying an ANB-code, and - Compiler Encoded Processing (CEP) that applies encoding at compile time and provides different levels of safety by using different arithmetic codes. In contrast to existing encoding solutions, SEP and CEP allow to encode applications whose data and control flow is not completely predictable at compile time. For encoding, SEP and CEP use our set of encoded operations also presented in this thesis. To the best of our knowledge, we are the first ones that present the encoding of a complete RISC instruction set including boolean and bitwise logical operations, casts, unaligned loads and stores, shifts and arithmetic operations. Our evaluations show that encoding with SEP and CEP significantly reduces the amount of erroneous output caused by hardware errors. Furthermore, our evaluations show that, in contrast to replication-based approaches for detecting errors, arithmetic encoding facilitates the detection of permanent hardware errors. This increased reliability does not come for free. However, unexpectedly the runtime costs for the different arithmetic codes supported by CEP compared to redundancy increase only linearly, while the gained safety increases exponentially

Proceedings of the First NASA Formal Methods Symposium

Topics covered include: Model Checking - My 27-Year Quest to Overcome the State Explosion Problem; Applying Formal Methods to NASA Projects: Transition from Research to Practice; TLA+: Whence, Wherefore, and Whither; Formal Methods Applications in Air Transportation; Theorem Proving in Intel Hardware Design; Building a Formal Model of a Human-Interactive System: Insights into the Integration of Formal Methods and Human Factors Engineering; Model Checking for Autonomic Systems Specified with ASSL; A Game-Theoretic Approach to Branching Time Abstract-Check-Refine Process; Software Model Checking Without Source Code; Generalized Abstract Symbolic Summaries; A Comparative Study of Randomized Constraint Solvers for Random-Symbolic Testing; Component-Oriented Behavior Extraction for Autonomic System Design; Automated Verification of Design Patterns with LePUS3; A Module Language for Typing by Contracts; From Goal-Oriented Requirements to Event-B Specifications; Introduction of Virtualization Technology to Multi-Process Model Checking; Comparing Techniques for Certified Static Analysis; Towards a Framework for Generating Tests to Satisfy Complex Code Coverage in Java Pathfinder; jFuzz: A Concolic Whitebox Fuzzer for Java; Machine-Checkable Timed CSP; Stochastic Formal Correctness of Numerical Algorithms; Deductive Verification of Cryptographic Software; Coloured Petri Net Refinement Specification and Correctness Proof with Coq; Modeling Guidelines for Code Generation in the Railway Signaling Context; Tactical Synthesis Of Efficient Global Search Algorithms; Towards Co-Engineering Communicating Autonomous Cyber-Physical Systems; and Formal Methods for Automated Diagnosis of Autosub 6000