Automated Theorem Proving (ATP)

TMR (Triple Modular Redundancy) is a hardware fault-tolerance technique that uses three identical modules performing the same operation in parallel. Their outputs are fed into a majority-voting circuit. If one module fails and produces an incorrect output, the voter is still able to determine the correct output based on the other two modules, thus masking the fault and ensuring continuous operation.

Markov Chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from a probability distribution. A Markov chain is constructed that has the desired distribution as its equilibrium or stationary distribution. The state of the chain after a large number of steps is then used as a sample from the desired distribution, enabling computation of integrals and expectations.

G-code, formally known as RS-274, is the most prevalent programming language for controlling CNC machines. It consists of sequential commands that instruct the machine on positioning, speed, and specific actions. Commands begin with a letter address; 'G' denotes preparatory commands for motion (e.g., G01 for linear feed), while 'M' signifies miscellaneous functions (e.g., M03 for spindle start).

Verification techniques are broadly classified as static or dynamic. Static verification (or static analysis) examines the system's code or design without executing it. Examples include code reviews, inspections, and automated static analysis tools. Dynamic verification (or testing) involves executing the system with a set of inputs and observing its behavior to find defects. Both are complementary for comprehensive quality assurance.

The Risk Priority Number (RPN) is a quantitative measure used in FMEA to prioritize risks. It is calculated as the product of three ranked factors: Severity (S), Occurrence (O), and Detection (D). The formula is \(RPN = S times O times D\). Each factor is typically rated on a scale from 1 to 10, allowing teams to focus on the highest-scoring risks first.

A classic illustration of the Monte Carlo method is estimating the value of \(\pi\). By inscribing a circle of radius \(r\) within a square of side length \(2r\), the ratio of their areas is \(\frac{\pi r^2}{(2r)^2} = \frac{\pi}{4}\). Randomly scattering points within the square and counting the fraction \(p\) that fall inside the circle provides an estimate: \(\pi \approx 4p\).

A set of four decision rules for detecting non-random patterns on Shewhart control charts, indicating an out-of-control process even if no points are outside the 3-sigma limits. These rules identify unnatural runs, trends, or clustering of data points that signal the presence of a special cause of variation. They increase the sensitivity of control charts.

A regression model for a categorical, typically binary, dependent variable. Instead of modeling the outcome directly, it models the probability of the outcome using the logistic (sigmoid) function. The model predicts the log-odds of the event as a linear combination of the independent variables: \(\ln(\frac{p}{1-p}) = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p\), where p is the probability of the event.

The Metropolis-Hastings algorithm is a prominent MCMC method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. At each iteration, it generates a candidate for the next sample based on the current sample. This candidate is then accepted or rejected with a certain probability, ensuring the resulting chain converges to the desired distribution.

Abstraction in OOP is the concept of hiding complex implementation details and showing only the essential features of the object. It focuses on what an object does instead of how it does it. This is achieved through abstract classes and interfaces, which define a blueprint for other classes without providing a complete implementation, simplifying complex systems.