Bit flipping

Another set of early studies came from the work of Judson and coworkers [35, 36], which emphasized using GAs for search problems on small molecules and peptides, especially cyclic peptides. A dihedral angle representation was used for the peptides with values encoded as binary strings, and the energy function used the standard CHARMM force field. Mutations were implemented as bit flips and crossovers were introduced by a cut-and-paste of the strings. The small size of the system enabled a detailed investigation of the various parameters and policies chosen. In Ref. [37], a comparison between a GA and a direct search minimization was performed and showed the advantages and weaknesses of each method. As many concepts are shared between search problems on small peptides and complete proteins, these studies have contributed to subsequent attempts on full proteins. 

The other observation is one that might be a little bit flip, but as you were referring to organizational models, you referred to the predatory nature of the industrial enterprise. I would refer you to an organizational model presented by microbial communities in nature. Although a population may be subject to predators, a good predator never completely destroys its host population. 

Let us consider the simple example, known as the bit flip code. The problem is the following we want to send one qubit of information through a noisy channel which flips the qubit with a probability p in other words, if the initial state of our qubit is a 0) + b 11), we get a 1) + b 0) with probability p, and a 0) + b 11) with probability (1 — p). The situation is much alike the classical case we have considered above, so we could be tempted to apply the same method. But if we try to do so, we rapidly run into major quantum trouble First, the no-cloning theorem forbids us to clone an arbitrary state. Moreover, even if cloning was possible, measurement of the qubits would completely destroy the information stored in the system. So, we have to find another way. 

The appropriate technique is the following. First, one encodes the original information a 10) + 6 1) on the two logical states 0/,) = 000) and 1 l) = 111) of a three qubit system. This is simply achieved by adding two physical qubits, initially prepared in the state 0), and by applying some well-chosen unitary transformation C to the compound system of three qubits this operation yields the state a 0/,) I 6 1/,) which is then submitted to the action of the noisy channel. Each of the three physical qubits of the system is likely to independently undergo a bit flip (with probability p). At the end of the channel, one performs the measurement associated with the four projectors... 

This procedure works perfectly provided bit flip occurs on one or fewer of the three qubits the probability that more than one bit flip occur is 3p2 — 2p3 which appears to be smaller than p, provided p < 1/2. In other words, when p < 1/2, the bit flip code decreases substantially the probability of error and thus makes the transmission through the channel sensitively more reliable. 

Other simple codes exist such as the phase flip code, which protects information against phase flip (see below for the definition of the phase flip) and can be simply derived from the bit flip code. Merging these two codes, Peter Shor proposed a code which protects one qubit of information against the action of arbitrary single qubit errors (bit and phase flips) this code involves nine physical qubits and shows the same schematic structure as the previous example. Its publication renewed the interest of physicists for the domain and gave hope that quantum errors are correctable. 

In the context of quantum informatics, the E/.. s are very dangerous, because they are likely to damage the information stored in the computer. They are often referred to as "quantum errors", and have to be corrected, in order to run a reliable computation. The main examples of quantum errors on a single qubit are bit-flip... 

The First idea of quantum error-correction, which we have already employed in the bit flip code, is to "give space" to the system by adding extra qubits, which play the role of ancillary qubits this ancilla adding procedure is highly related to the notion of redundancy in classical error-correction. Then, one encodes the information onto a well-chosen subspace C, the code space, of the extended Hilbert space of the system comprising the initial plus extra qubits. In other words, one applies a well-chosen unitary transformation C, the coding matrix, which "delocalizes" information on all the qubits of the system. That is exactly what we did in the bit flip code, when encoding information onto the subspace spanned by 0l) = 000), 1 l) = 111). ... 

In spite of impressive experimental demonstrations of basic quantum information effects in a number of different mesoscopic solid state systems, such as quantum dots in semiconductor microcavities, cold ions in traps, nuclear spin systems, Josephson junctions, etc., their concrete implementation is still at the proof-of-principle stage [1]. The development of materials that may host quantum coherent states with long coherence lifetimes is a critical research problem for the nearest future. There is a need for the fabrication of quantum bits (qubits) with coherence lifetimes at least three-four orders of magnitude longer than it takes to perform a bit flip. This would involve entangling operations, followed by the nearest neighbor interaction over short distances and quantum information transfer over longer distances. 

The separate runs have found a number of different solutions, although not necessarily local minima. For instance, population 5 has the correct values for the first two parameters and has values that differ from the correct values for the other two parameters by a single bit flip. A pair of correlated mutations would take this solution to the true minimum. In situations such as this, it is often useful to take the best solutions produced by the GA and apply a local minimization method to drive them to the bottom of the neatest local minimum. Population 2 is probably also in the basin of attraction of the solution and would drop in with the aid a local minimizer. 

In this book, we will refer fault as the SET pulse that may occur in the combinational logic and as the SEU that is the bit-flip that may occur in the memoiy element. 

We injected 2,944,640 faults in the AUT of the FPGA board running a 6 x 6 matrix multiplication protected with OCFCM, VAR and BRA. From those faults, 54,024 caused an error in the circuit s output when considering no farrlt tolerance detection. Since the fault injection was exhaustive, we can asstrme that, except for placement and routing differences, the microprocessor core has 54,024 sensitive bits, which represents 1.8% of the injected faults. This represents a proportion of 54 bit-flips in the configuration memory bits to cause a functional error in the design. 

FITS can emulate the existence of faults of different types (stuck-at, bit-flips, etc.) or even can emulate complex fault or error models. However, the most commonly used in the community is the single bit-flip fault as it well mimics the occurrence of Single Event Upsets (SEU [1, 3]). 

This paper presents new analysis functionality in the MODIFI tool that visualizes how errors propagate through the model, and identify sensitive blocks in the model. This can be used to reveal problems and to design more robust software. We exemplify the usefulness of our approach using a Brake-by-wire model and by using both bit-flip faults and sensor fault models. 

The objective of fault injection is to introduce artificial fault or errors in a system to test the system in presence of errors. The injection of faults using MODIFI is done by rerouting the connection between blocks in the model to also include a fault model. This is illustrated in Fig. 1 and Fig. 2, which show a Simulink model before and after MODIFI has inserted a fault injection block. The fault injection block will pass the input value to the output port unmodified unless a trigger is enabled. The trigger, which is based on the simulation time, will cause the block to apply the fault model, a bit-flip in this case, to the output. Fig. 2 also shows that MODIFI turns signals in the model into Simulink test points [10] which all have logging enabled. 

Fig. 2. Simulink model after insertion of a fault injection block for bit-flips...

MODM supports a wide range of fault models and can easily be extended with new fault models. In our current implementation, we provide fault models for single bit-flip faults and for sensor faults. The single bit-flip fault model is commonly used to emulate the effects of transient hardware faults, and we use this fault model to emulate the effects of hardware faults that affect registers and memory which are visible to software running on a microprocessor. 

To demonstrate sensitivity profiling and error propagation analysis, we apply two different failure modes mentioned in the IS026262 standard [14]. We apply the oscillation fault model on the signal representing the feedback from the Hall Effect, and the bit-flip fault model on the signals inside the behavior model of the wheel node. 

Fig. 10 shows the sensitivity profiling for about 20 000 injections of single bit-flip faults in the brake-by-wire model. We can see that the output of the last block LockDetect is the most sensitive part of the model part under analysis, where 40.6% of the injected faults lead to an incorrect output. This is reasonable since there are no fault handling mechanisms after this block, and faults here will therefore affect the output. The figure also shows that no faults were injected in the output of the comparison block Relational Operator. This is because there was no fault model implemented for Booleans in the fault library used for these experiments. 

Fig. 10. Sensitivity profiling of a campaign with the bit-flip fault model...

If A requires k bits and we encode values with a maximum size of n bits, we need n + k bits to store encoded values. Assuming a failure model with equally distributed bit flips and that the Hamming distance between all code words is constant the resulting probability p of not detecting an error is p = "TumbefoTpotTbi rdt = 2- = Thus, the error detection capability is... 

As the table shows, all the faults injected in the system during the simulations were detected by the voters. Since we configured all jobs in the system with a period of one macrotick (250ms), corruption and babbling faults were detected 250ms after their injection in the system, as expected. Bit-flips in signals and open circuits were also detected in the next macrotick. 

In contrast, the left half of Fig. 1 shows a visual representation of FI campaign results that were collected injecting faults into all possible fault locations of a particular benchmark application, requiring enormous computing power using the Fail [12] FI framework with an x86 simulator backend. The fault model used for Fig. 1 constitutes uniformly distributed transient bit flips in the main memory. The fault space spans all CPU cycles during a benchmark run, and all bits in the address space. Thus, each coordinate in Fig. 1 shows the outcome of one independent FI experiment after injecting a burst bit-flip at a specific point in time CPU cycles axis) and a specific byte in main... 

Symbolic Error Injection. Several projects try to provide more complete results than simple error injection provides by using symbolic error injection [11,16] They combine symbolic expressions of the analyzed software, error checks and possible errors into one model. Then, they check if all modeled errors are detected using model checking [16] or symbolic execution of the analyzed program [11], However, both projects demonstrate their approaches only for small examples and quite restricted error models such as bit flips. 

A Study of the Impact of Single Bit-Flip and Double Bit-Flip Errors on Program Execution... 

Keywords out-of-context dependability benchmarking, fault injection, single bit-flips, double bit-flips, error sensitivity. 

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