binomial coefficients
..binomial coefficients?! That’s right. I’ve found the web site of a Mr. Bob Jenkins with an entire page dedicated to a pairwise covering array generator named jenny.c. I’m fairly sure that only the most hardcore of the software testing weenies have some notion of what those are so for the sake of being succinct I’ll be providing my own explanation here: A pairwise covering array generator is a program for silicon computing machines that deduces sequences of input value possibilities for the purposes of software testing; and yes, I did say silicon computers–since testing their software is really a question of the great Mr. Turing’s halting problem, the existence of a practical, affordable, and efficient nano/molecular computing device such as a DNA computer, Feynman machine, universal quantum computer, etc. would essentially predicate a swift solution to the problem of testing contemporary computer software in non-deterministic polynomial time. The only problem we would have then is how to test those fantastic, futuristic, (seemingly science fictive) yet wondrous problem-solving inventions as they break through laborious barriers of algorithmic complexities that twentieth century computer scientists could have only dreamed about: PCP, #P, PSPACE-complete, 2-EXPTIME and beyond.. The stuff that dreams are made of.
Now, let’s return to Earth and learn about a few things that make Jenny so special. Computer scientists learned early on in their studies of software testing that pairwise or test cases with two input values were the most likely to uncover erroneous programming or “bugs.” Forget the luxury of automation for a minute, old school programmers typed input pairs manually to test their own software. Code tested in that manner was most likely some sort of special-purpose console mode utility. (Celsius to Fahrenheit, anyone?) As the computing power of the desktop PC increased according to Moore’s law, it became time-effective to write a simple program to generate these input pairs instead of toiling over it yourself–I suppose not testing at all was another option. Today, still some software is released to market after only very minor functional and/or quality assurance testing. Regression, stress, security, and other forms of testing cost money and reduce time to market, but in reality significant return on investment acts as a hedge against any losses incurred. Even ephemeral losses justify the absolute necessity of these expenditures.
A Jenny built in modern times undoubtedly has the power to deductively prove that a software product of the eighties decade is comprised of components (or units) that are fundamentally error-free. However, the paradox remains that improvements in automated software testers share a linear relationship with improvements of software in general. Thus, pairwise has become “n-way” which describes the process of utilizing greater multiples of input values in order to cover acceptable numbers of test cases. The number of covering arrays generated in this fashion grows exponentially and can be calculated as a binomial coefficient (see formula below.)
(n choose r) in factorial terms
According to Paul Black, former SAMATE (Software Assurance Metrics and Tool Evaluation) project leader, researchers at NIST have pegged 6-way as the magic number for optimal fault interaction coverage (notably Rick Kuhn and Dolores Wallace.) This conclusion is based on hard evidence from studies on real-world software scenarios including medical devices and the aerospace industry. However, it would not surprise me to see this approximation rise significantly in the coming decades, just as the paradoxical relationship between general-purpose software and automated software testing programs shifts itself in accordance with Moore’s law. If not by Moore, then by some other axiom of metric progression such as Rogers’ bell curve of technological adoption.
I’ve also got a hunch that the tiny percentage of bugs in that “n is arbitrarily greater than 6” range are some of the most critical, powerfully impacting software vulnerabilities known to man. They lie on an attack surface that’s almost non-existent; this makes them by definition, obscure, non-obvious, shadowy, and hidden. Vulnerabilities in this category are the most important by their very nature. Therefore, detecting vulnerabilities of this type will involve people and tools that are masters of marksmanship and artistic in their innovation. Research in this area is entering a steadfast beginning especially within the realms of dynamic instrumentation or binary steering, active analysis, fault propagation, higher-order preconditions/dependencies, concurrency issues, race conditions, etc. I believe that combining merits inherent in various analysis techniques will lead to perfection in software testing.
For perfection in hashing, check out GNU’s gperf, read how Bob used a perfect hashing technique to augment Jenny’s n-tuples; then get ready for our Big ßeta release of the BlockWatch client software (just in time for the holiday season!)
System of Systems 
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