Sunday, December 26, 2021

Simple Heuristic for High-School Algebra Students

Team. It seems that an unfortunate situation is occurring in modern history. Old books are slowly being destroyed and recycled.

And, in that language is in constant flux with slang terms from the urban dictionary like "throke" filling the airways of modern conversation.

Having a reference handy is quite wise.

For one, a term in modern computing which has been misused is "heuristic". It simply means a challenging problem in mathematics which might require that one recall some facts from early subjects which they should have learnt well and mastered. That lesson was learnt while scanning an old dictionary that was the property of the author's grandmother while he was in seventh or eighth grade.

At, which time he constructed a few "different" very simple ones.

The modern connotation for this term in computing is a "impossible" problem. Which might require an inordinate amount of computation resources in its resolution. And, it is often proposed before classrooms that an efficient solution in time and space resources is not feasible.

And, on the topic of "intractable" problems:


The following "simple" problem might be classed as a heuristic. It is a "simple" challenge question for algebra students who have just learnt the art of simple polynomial multiplication .

"How might one produce a pocket square from a 'bow-tie' or an 'handkerchief'?" provided the following binomials and trinomial. The diagrams are at the "base" of this www-history. And, one of the solutions might be somewhat shocking.

Actually, this is drawn from a classic lesson in algebra that can enter a segue that describes a complementary lesson in graph theory.

Although, this first question might be a simple one for many "highschool-aged" algebra students.

It would be a "splendid" conceptual basis for an "heuristic" that mixed principles from numerous subfields such as topology, abstract algebra, optimization, set theory, or others. When, first hidden under layers of postulates, theorems, corollaries, relationships, and axioms. This is all that mathematicians ever do. They often meld a challenging question around a simple mathematical construct.

One might spend an entire career building substantial and meaningful problems around a few such simple relational and logical building blocks.

Such is the case with ECT - elliptical curve technology - it was drafted up by the author. And, he was rather surprised that it reached the "realm of use" by "professional-grade" mathematicians apparently for ciphering information. The phrase near the original construction that was drawn in that notebook while lollygagging and daydreaming during a trigonometry and pre-calculus class be "could this be used for hiding information". This was an hint for the student who sat beside the author in class who was working on modules in upper division subjects, like undergraduate topology, while in high-school. He would often peer over and look at the recreational mathematics notes sketched by this loon.

Finally, a note for those taught otherwise. The mnemonic used for describing this procedure is often called based upon [ First, Outer, Inner, Last ] when addressing this question of finding the product of a couple of binomials such as ( A + B ) * ( C + D ) or our bow-tie. However, this is "best" remembered using the diagram below. It is more descriptive than the four-letter mathematical term drawn from the first capital letter of each word. The phrase is the equivalent of calculated profanity that permanently fixes a learning barrier in the mind of many students. One might decimate this barrier if he recognizes the relationships that occur between each term when forming the product. These fully-realized understanding makes the associated acronym as tenuous as the tin extracted from the hills of Tennessee drawn thin and spun around a cylinder of cardboard..




Saturday, December 11, 2021

Holiday Wishes : In Light

Team. In light of this author's last entry in this web history, a request for a harmonious and peaceful holiday season and upcoming year might seem somewhat strange. But life has taught that within the mire and abyss of abject hatred and disdain that this world is true beauty might be found. The author has found among men and woman who hail from various parts of this earth and practice a myriad of faiths. He has, at times, found a greater level of "humankindness" among those who do not practice his faith tradition than those who do. 

As said in earlier entries among these notes, most seem very disinterested in those lessons of true value which the author might give. After reading his religious canon of his chosen faith-walk, he can give very pithy and insightful lessons. And, some of these contradict traditional teachings. Although, these personal counterviewpoints are supportable by scripture. "He is a Rewarder of those who seek Him..." as it is written. Yet, such insight draw very little attention or interest when spoken of openly. They are teachings for which ears do not itch.

What seemingly enthralls folks is those tidbits of knowledge that might impress a manager in a technical interview. 

Such as, "Object-orientation is truly not a great departure from traditional structured programming techniques. In fact, it is a systematically-enforced method for imposing more structure through the bundling of related state and behavior and the utilization of principles such as encapsulation, information hiding, polymorphism, inheritence, and others. Those standard rules such as a "single point of modification", "modularity", "using well-known interfaces", and other still remain supreme. In fact, object-orientation is simply one form of concern partition and organization, a discipline that has also spawned aspect-orientation and might produce other novel "orientations" in future decades. And, concern partitioning and organization is simply part of a everyday task called problem-solving. Whose fancy name with Latin roots is 'parsology'....blah, blah, blah, yada, yada, yack, yack, yack..."

Everyone seeks sounding "impressive". We must have the most "illustrious" credentials from the most "storied" academic programs which teach the "grand principles" enunciated by the "great names" in history.

Truth be told, as simple-minded as the average human is, one can simply claim that a certain type of person made some great claim or discovery and few will question it. And, such a phenomenon is often that basis of a dissertation for doctoral students in social psychology and the fuel for many marketing and political campaigns.

Take a gander at this following video clip. Notice that men who it claims are doctoral-level researchers in mathematics. What is their demographic. All are male. Most have European ancestry. And, most have hair that is dark brown or lighter.

Would anyone dare question their abilities or the authenticity of their claims concerning the difficulty of this problem?

Yet, most middle and high school students who are "math-types" who truly earnt their honor roll accolades could tell anyone why a professional mathematician should not spend his career chasing after that solution of this problem and an amateur should not spend a "good" five seconds on it.

However, that is for those who spend much of their time honing their mathematics skills. For others, it can be a nice recreational diversion. And, it is "encouraging" uncovering something that the seemingly "smarty pants" cannot figure out.

In fact, most K-12 mathematics teachers who work with students at those grade levels could certainly say, "It looks like this one ( the Collatz sequence ) was thought up by some egghead junior that spends most of his afternoons watching Nova on PBS."

Yet, the "true" dilemma in this video is this, "How can men who have doctoral degrees not grasps this 'obvious' solution?" And, more troubling, "Why will so few among us question a European American male who presents himself as an authority?" Yes, such is the topic of many social psychology dissertations, sicherlich!

A search on YouTube for one of the most telling video clips concerning this phenomenon in the last fifty years turned up "nada, nichts, nothing". And, that footage was of Oprah Winfrey interviewing William Shockley. Who is a Nobel Prize winning scientist credited with discovering the semiconductor, was a Stanford physics professor, and was the well-known the eugenicist that ranked "social strength" of the earth's population groups. He used something called a "genetic" measure. And, during that hour long broadcast in the mid-1980s he presented a  model of "societal strength" which ranked the Asian community the highest followed by Europeans, Indo-Aryans, and then Africans. And, it was said that these ranking corresponded with "levels of intelligence" as measured averages on standardized examinations given globally. Now, who might argue with that. A Stanford professor in physics, Nobel Laureate, and European male who has proof concerning the superiority and inferiority of certain human groups.

The trouble is this, "In this grand scheme of societal rankings he left of a large portion of the human population, those who discovered the Americas. And, that is not the Spaniards. It is the many groups of indigenous Americans."

And, the question one must ask any eugenicist, "Are such men seen as human? If so, were they left off as a mere genocidal afterthought." The question mark is missing for a reason.

And, in the words quoted from this interview with Bertrand Russel in 1952. He was having trouble accepting the "self-assertion" of the Asiatic man. Apparently, he felt that they were getting out of their place. And, interestingly enough, Shockley placed this "conglomerate community" ahead of the European in terms of "human potential". Yet, that was based upon his "genetic" measure. And, that term is suggestive of describing inherited and innate biological factors; yet, it was defined in his work as a multivalued mathematical object.

So, Shockley's measure ranking his four "human" groupings based upon primarily on physical features was much like e-harmony's rating of personality traits and compatibility. In short, the results of the ranking could be altered drastically if it were based upon a different set of criteria. So, the questions are, "What criteria are important?", "Do any environmental or historical factors that have been left out of the measure play a role?", "Can statistics lie and liars use statistics?". In other words, "If proof establishes truth, what is proof?"

And, this, the fallacy in his research driven by personal motivations, Shockley implied without explicitly stating during his interview with Oprah. He quoted the percentage of individuals on the planet who meet the criteria sufficient for survival based upon the teachings of traditional eugenicists, such as the National Socialists in northern Europe. The number is about seventy million in a world of around seven billion. Many of whom have a dark hue. He also repeated that his genetic measure was defined as a mathematical measure and that he had stood beside the pyramids which were built over an eon ago. And, he emphatically said that he did not say that African were "stupid"! Recollections also suggests that he mentioned the low divorce rate among certain Indo-Aryan groups. However, he said that he had a sound reason for saying what he said based upon his research. And, that is certain groups simply have weaker societal structures. 

And, in doing so he did what many "well-educated" men will do, he called the masses ignorant in a way that they did not detect. Because, the majority of them liked what they heard and it served their purposes and itched their ears. And, no one would question who he said he was nor his claimed ability. He need not prove his competence or level of training. He was a European American man presented as well-educated and accomplished, much like those in the video on the Collatz sequence. Yet, they might only be actors as far as anyone knows. They are presented as mathematicians and give a convincing presentation which most non-math types would not question. Truth be told, anyone of them might work as a waiter at Perkins and hold a "hard-fought" GED. That does not mean that they lack intelligence; it just means that they are not who they are presented as.

And, as said in numerous earlier web entries, the "bell curve" argument presented by William Shockley was funneled through the international network of researcher most likely via a familial connection who had ties with the Warren Brown School of Social Work at Washington University. It was wagered with this person that the average American, most of whom which consider themselves European, would love what Shockley said, would never question it, and might very well become aggressive if someone dared contradict these findings. In fact, it is all what the author's Grandfather who was a Mason and Baptist Minister would call "hogwash". For, one could choose any set of "valid" measures that might rank these four groups in any order. It is all mathematical "smoke and mirrors" and "prestidigitation". And, albeit socially inflammatory, this "arbitrary" social order and a mathematical means for "proving" its infallibility was conjured up by a middle school student with woolen locks and dark brown skin.

And, it seem like many "great thinkers", many of whom are mathematicians. simply will not speak openly concerning this modern fallacy in social reasoning. Which they should see so clearly. It can quite convincingly be argued that "intelligence" and "phenotypical" traits are not correlated. And, one might make the same argument concerning "genotype". Considering that, the percentage of the human population across this globe that attempted a "solo" cloning experiment within the last twelve months greatly exceeds the of the percentage of the population of mules who did the same thing. It is a matter of "horse sense" folks! So, intelligence and mental acuity are relative.

And, how intelligent is the world's population when it will blindly accept "facts of science" from another person when he is male, European, and likely has a Germanic surname. If CNN reported that Helmut von Dummputz, a doctoral-level researcher at the University of Frankfurt, just proved the Reimann Hypothesis, would you spent all weekend wrapping your head around that unwieldy this postulate.  

Yet, the main premise of this note during this season of Channukka, Kiwanza, Advent, and other holiday lights is "seeking harmonious peace". And, we might all find such tranquility when we set our own egos aside and be not a respecter of personages, Why? The only Ego that merits any honor is the Force that created this existence and everything in it. We can spend between this moment and when this earth ceases its rotation about what we should call such a Force and who receives preferential treatment from Him based upon our "current" circumstances. Yet, one fact is true as seen in the balance that His Creation has, equitable measures are His delight. For every injustice, He eventually will bring about a meet justice. And, those very parts of His Creation which are seen as the least desirable often attain the greatest honor and are the most necessary for our longevity.

Also, stop and pause the next time someone is presented as an authority. They might only have a few more full days worth of training in a subject than you. Or, worse yet, like many a politician, they might have been briefed for thirty-minutes by a staffer before addressing the crowd in which you stand. Quietly questioning authority is wise. Openly arguing with it is foolhardy!

Fou Noir ou Feu Noir

Sunday, October 3, 2021

Algorithmic Memories – Reality Check

Algorithmic Memories – Reality Check

This note might be properly redacted for typographical errors in future days.

It is the case that a parent of this technical web journal writer worked in the physiology department of the local state medical center starting in the early-1970s. Seeing that physiology departments are not that common, he has been told that it has consistently ranked within the top ten nationally or globally. Whatever the case might, she was the third person of African-American descent hired in a research role in the history of the campus. And, they all had the rank of a research technologist. Actually, she was the first African American “officially” hired by the state seeing that the others joined the medical center along with a private cancer research center from Chicago that merged this facility around 1968. Those early years were difficult. Many of the other researchers were not “civil” enough that they would speak, and the work environment was very hostile. Also, it might be the case that her second pregnancy was aborted by the actions of one of her coworkers who put a substance in her morning coffee that resulted in a miscarriage between her third and fourth month between 1977 – 1978. It was the case that only a handful of researchers in the department could access the area of the hospital which had substances that would induce an abortion. Memories of conversations between his parents remind him of them discussing a claim made by an attorney for the medical center during a hearing on its premises that the only way that her claim could be established was by titrating the Pacific Ocean. And, it was the case that she said that whenever she was in the departmental restroom for more than five minutes her supervisor the department chairman would bang on the door and ask her if she were “auto-fornication” or “masturbating” in the American vernacular.

Yet, she often changes the subject when those issues are brought up these days. She says that she grew up in the Jim Crow era in the American South and Whites on a whole are much nicer now. When reminded of her childhood playmate whose entire family was lynched and their house torched simply because a European American farmer wanted the land and would not hand over a single red cent for it, she simply is not interested in discussing it. She shared with me a picture of the red hexagonal shape and Latin inscription that accompanies it which was a sign that some one should not enter their house when they came in from working in the fields, leave the county and the state, plus never return. Such was often placed on the homes of African American sharecroppers when someone decided that they wanted all that they had. If, they tried defending their homes. They would be brutalized and killed by a collection of European American Klansmen who were often held some “highly-respected” positions in the town such as membership in the Fraternal Order of the Mason along with roles on the police force, in the town governments, and as the local physician or attorney.

The only reason that this horrible memory is ever breached is this, a moment of conflict arose in the early-1970s when she made a suggestion in a research seminar. One of the male European American researchers lets her know that “quality, fundable” research ideas are very rare and that she should hold her peace among the doctoral-level researchers. Ironically, this was likely the one with a doctorate in mathematics from the University of Southern California. Who only was working in a physiology department because he could not find employment in his field with the skills that he had earnt and what he had truly learnt between smoking cannabis and daisy chaining in Southern California in the 1960s. Her quick retort was this, “My son can come up with a fundable research idea.” And, when she shared this inspiration with me that was coupled with a promise from the research staff that this online journal author would receive a full four-year scholarship at the local state school, that was sufficient motivation enough.

And, it often is said that it is not what you know but who you know that makes a difference. The opportunities that arose from the adoption and implementation of many successful research ideas became a burden whose full weight was felt working as a high school intern and undergraduate researcher in cancer biology under an NIH grant at the local hospital. For one, on more than one occasion, a European American graduate student or faculty person who was not happy with the presence of a few minority-interns placed us in harm’s way at different times. In particular, Kevin Stansbury and his shenanigans will never be forgotten.

Once, he asked that this fellow perform an act that could potentially had caused an explosion that easily could have leveled many of the buildings on the city block that house the main part of the medical. His reply was simply, “Oakey Doke!”. And, intrepidly compliance was given as Kevin slipped out of the laboratory and likely out the building heading for the safety of another part of the city. So, he could later report that he saw this negligent teenager conducting an unsanctioned experiment on the local evening news and in the local paper when they covered the imminent disaster. Truly, who would ever question a White man’s report? Yet, after about five minutes of mixing highly combustible substances, a faithful supervisor entered the laboratory and ask, “Why are you doing that?” The reply simply was, “Kevin Stansbury said it should be done.”

And, when Kevin was asked, “Why did you insisted that he do that.” He simply started laughing and guffawing as if life is a childish game which one might restart when he runs out of lives. Truth be told, had this chap dared question Kevin when he put him and many other of the members on the Institute and hospital in harm’s way, old Stansbury would become more aggressive, dangerous, and potentially violent. They always do. So, this web history author played the coon and went along with Kevin’s high shenanigans.

And, a knowledge of only a few of his contributions was enough that when he was “accidentally on purpose” placed on haloperidol in the VUMC after experiencing the central nervous system dysfunction associated with exertional heatstroke while a Vanderbilt Commodore football player during the Fall of 1988. This came in the wake of that football coaching staff driving a player named Sonny Bishop until his heart failed during the 1987 year. And, the fear of a civil suit from that player’s family likely loomed over the university. So, medicating a seventeen-year-old who did not know enough about medicine at that time that he would know the proper procedures and treatments for the temporary condition which he experienced was advantageous. Sadly, after contacting that university seeking support in the permanent cessation of the medication given years later, the inquiries were ignored. Interestingly enough, online records from the university’s past athletic rosters do not include the name of this author on the 1988-1989 team or Sonny Bishop during 1987 – 1988 and the previous years which he played for that team. Albeit, his unfortunate death was covered by the national news.

And, the medication administered became a convenient tool for inflicting harm by local physicians seeing that it lowers one level of cognitive functions and induces permanent neurological damage. It was given in large dosages and often by injections that this author could not escape. For one, it has a withdrawal profile that is worst than alcohol. And, its forced use has resulted in the author experiencing seizures. When these would occur in a hospital setting, the nursing staff would often say, “It looks like you are getting better.” And, this cold-hearted insult would often be followed a team of nurses and security guards tackling him on a cold concrete floor and forcibly injecting him with unordered dosages of haloperidol and combinations of other psychotropic drugs. And, history has shown that it is quite common the hospital records are written so they protect the facilities interest and the actual nature and amount of medications given can bury. And, seeing that this country is only about half a century from the supposed end of the Jim Crow era, what a vulnerable African-American experiences at the hands of the average European Americans, even when he is in a medical facility but when they are confident that they will never be held accountable for their actions, is the same. White hearts have not changed. One will rarely find any security films of takedowns and abuse of such wards in on-line forums. For one, it would damning for the facilities. And, they have the financial resources that are sufficient for guaranteeing that those video records are destroyed. One can find security films from a number of venues: stores, hotels, and even private in-home security camera footage. But, that of college campuses and hospitals will never be found in any on-line venue. In fact, these institutions truly only keep records that might potentially condemn their clients, if need be.

 Yet, the cold, hard reality check for those who see so little worth upon meeting this fellow with woolen hair and skin with a deep brown hue. He and his parent who reached the position grade of researcher at the local medical center by retirement are a academic tandem that “unofficially” hold Turing Awards, National Science Awards, and, possibly, one Field Medal, seeing that we were the source of these research vision which were subsequently implemented mostly by European Americans or a person that the masses of this world can accept as intelligent without having a color-struck conniption fit, rioting, plus lynching and burning in effigy.

His first contribution was as a child shortly after he learnt the basics of whole number division. And, as often said while in the local public school district by his teachers, “out of the mouth of babes…” You can certainly finish that phrase. Only knowing the rules of division for the composite of 0 through 10, he sat down in his small room in his family’s apartment and began dividing dividends by divisors producing a quotient and a remainder. Then, he would divide combinations of the dividend, the divisor, the quotient, and the remainder by each other. This produced a short irregular stream of numbers as output. He had seen something about Caesar’s cipher on public educational television and an idea dawned upon him. If the letters of the alphabet each had a single number associated with them, one could encode a secret message as follows. Simply, generate a stream of irregular (random) numbers, combine each letter in the message with an associated number in that stream: first with first, second with second, etcetera, and wrap the value around if it exceeds twenty-six. Beyond public television, this was inspired by a couple of his favorite afternoon shows. He loved watching episodes of Hogan Heroes and Black Sheep Squadron. Which were in syndication at the time.

Faint, misty water-colored memories of those moments recollect him calling his undecipherable message system the German Enigma, because Coronel Klink and that gang would never figure it out. And, fully aware that the world of early-1970s America and even during this day would never knowingly accept any advice from basically any child, especially an African American, he used a pseudonym as the author of the work.  The first name Alan was chosen randomly. The meaning of the second name had significance, “Touring”, intentionally misspelt as “Turing”. And, it was supposed at the time, that any African American male in that era who sought a career in computing would likely spend his whole life “touring” around the country looking for one having door after door slammed in his face. Only, a well-rounded mathematician who had sufficient training could establish whether “pseudorandom” number generators existed before roughly 1974. And, the reports of work done in the late-1930s and early 1940s is not simply governmental propaganda. Faint aural memories from that period remind him that his parent who working in research said that she asked a mother of a playmate for a picture of that child’s grandfather who resembled him and whose surname was Heinrich. Yet, only that family can verify such. Surely, the world can accept innovation from someone who has such an appearance.

Such a simplistic cryptosystem is truly undecipherable when used with privately exchanges keys (seeds) and unique, paired pseudorandom number generators on each communications channel.

If one had a “hidden” message that was one kilobyte in length, one must successfully find the combination of letters (A-Z) that spell the “secret” message using a circular traversal of the range [1..26]. For one, anything could be spelt from that sequence of ciphered values, and it would seem valid. And, it would be the easier finding the appropriate combination on a lock where each spinner had twenty-six possible values and a thousand and twenty-four such spinners existed. Why? The lock would open letting one know that he had the appropriate value. If, he struck upon it. One would never get any signal that he had the correct message with this ciphering system. It would only be discernible with the appropriate seed value and random number generator. Choosing a seed is easy. That number can be yanked out of thin air. And, spinning one’s own random number generator is not difficult. Many on-line references, papers, and books tell who such is done. The average junior high school student who is a computer programming hobbyist could do so. And, with a little knowledge of file formats, he can hide encoded messages in image, audio, and video files as temporary “static” in the multimedia’s signal stream. But, that is enough monkeying around.  Because, cracking messages simply becomes more challenging when one couples this approach with steganography, a vision for ciphering words and not letters, struck upon while a trigonometry pre-calculus student in high school during the mid-1980s. Yet, few in this world, especially those of European descent will accept such.

Yet, seeing that this fellow already has been administered an “apple” around 2015 and survived, based upon the medical knowledge he learnt in early years, and replenished the need chemicals that “apples” flushed from the body, while temporarily surviving on the street as a homeless person, after the local Charles Drew Health clinic would not provide him with any appropriate treatment, he really cares very little about the personal opinion that anyone has about him. The includes members of his own ethic or racial community who say he is an embarrassment for making such “grandiose” claims of accomplishment. Especially, Dr. Sherita Strong who received a promotion from the Director of Diversity and Inclusion at the local medical center became a Vice Chancellor after betraying this author. She, who was an athlete herself, stated that being driven on a practice field until confusion sets in is a complete impossibility and that the documented behavioral health condition must be accurate. Physicians simply do not make mistakes. They are well-trained professionals. She also implicitly stated that medical disparities based upon race and ethnicity simply do not occur in this era and that various genetic factors are the cause for the disproportionate number of African Americans, Latinos, Indigenous North Americans, and other minorities suffering and, at times, dying from certain ailments.

And upon the topic of the WWW, the greed of the business sector in the States United has placed this country in an operational security nightmare seeing that using such a simple ciphering system would allow for the exchange of clandestine messages freely across this global network. During early discussions concerning the development of a public portion of the Internet, called the World Wide Web, for commerce, communication, governmental surveillance, and the assessment of the social health of the world community, it was recommended that the delivered system be much like a read-only kiosk and the content made available by private citizens be approved. For one, considered the “foolish” words spoken by the staff of Charlie Hebdo in France. As the apostle Paul stated in the New Testament, “everything that is permissible is not beneficial”. And, such very acts of “free speech” might very well cost the world community that liberty. Yet, the European American leadership who adopted this initiative in a haste for the rapid accumulation of wealth like that seen by leaders of corporations like Google, Yahoo, Facebook, and others put the whole nation and its system of democracy at risk.

It was the late Rush Limbaugh who fondly told the account of African tribesmen who would catch monkeys by putting peanuts in an empty coconut with a small opening in one end. This hole was large enough that the hapless monkeys could fit their hand in it and grasp the nuts; however, it was small enough that the monkey could not remove his fist that was tightly holding the peanuts fast. And, despite cries of distress, hoots and hollers, as a tribesman reeled them in, the monkeys seemingly did not have enough sense that he would let go of those nuts even if it would save his life. Such bears itself out on this modern communication medium called the WWW “wholly-supported” and furnished the government of the States United in more ways than one. And, at this juncture in history, Uncle Sam simply cannot let them nuts go without causing total chaos in his world. The subsequent withdrawal systems of letting go are more than all of his military might could possibly handle.

And, the cold-hearted reality check is this, “someone has been lying so long about his ability and accomplishments and its nascent presence in his life and this existence that the daft fool believes it.” It is the case that this someone is either the author, who might be in need of more haloperidol, a neurotoxin, or the proverbial White man who, as an aggregate, has accepted the author’s research insights since his years as a toddler, implemented them, and predated their origination so his shallow sensibilities and false sense of superiority rooted in the insecurity of a feared imminent inferiority be pleasantly appeased. Whatever the case might be, one or both of us are a ninnyish nape.

By, the way, for those of you who read the entry on Gödel’s Halting Theorem, that was a mathematical heuristic drafted after learning some elementary BASIC in junior high. Gödel never existed. He is a mere legend. The product of government propaganda hoping that the social position of the European masses remains secure.

And, for those who do not teach; but, who are students, the term heuristic as defined in older dictionaries crafted before 1965 is a teaching tool given students that requires future investigation and the use of skills learnt throughout the educational process both old and new in the resolution of a challenging problem. In other words, heuristics are solvable. Open problems simply do not have a known solution. And, some closed problems are classified as not having any known solution and the discovery of one is unlikely. Yet, upon the topic of “Gödel’s work”, it was well-known in junior high that few students question an instructor or grand names. And, living in a community just fifty miles north of the international center for the distribution of National Socialist literature, most of the Europeans in this area of the country who can claim German-heritage do. So, the selection of such a pseudonym would place the work beyond the scrutiny of most. And, the convoluted proof that starts with the contrived construction of a contradiction and proceeds in a circle proving that a contradiction will arise after a series of deductive steps would ensure that most simply would not question the mathematical smoke and mirrors layered under a thick quilt of hocus pocus. The words written on the bottom of the notebook page on which this was drafted was simply curve ball.

And, nearly forty years later, that curve just hit the catcher’s mitt.

And, upon seeing the numerous inaccuracies in proof-work based upon this heuristic in textbooks like Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science by Martin Davis, Ron Sigal, Elaine J. Weyuker while working on a postbaccalaureate degree, the most mathematically merciful thing that this author and recreational mathematician could do is let his fishes of the hook plus give this advice, “[Jas 2:1 KJV] 1 My brethren, have ye not the faith of our Lord Jesus Christ, [the Lord] of glory, with respect of persons.” That is a rhetorical statement for those of you who were expecting a question mark at the end of that verse. And, for those who know scripture or have heard of Christ’s Legacy know that He would question and challenge any of the synagogue’s leadership without the least bit of trepidation.

Also, another small bit of advice, do not passively accept everything that instructors say. They might be wrong. Seek and discover the truth for yourself. Yet, be careful who you tell. This world’s ears itches for the soothing sweet nothingness of prevarications.




Monday, September 27, 2021

Effective Halt-ability - Gödel’s Halting Theorem

During the mid-1990s, while an undergraduate at the University of Nebraska at Omaha, a well-kept secret in the Midwest, a simple high-level language feature that might ameliorate the dilemma of Gödel’s Halting Theorem was struck upon. Its rudiments might have been realized during a first course in programming fundamentals in Pascal at Vanderbilt University circa the Spring of 1989. It was simply this, “the only programmatic structure which would ever produce an infinite iteration found within any code listing is a definite or indefinite repetition. Hence, it seemed reasonable that, if one places a limiting constraint before each instance of these classes of statements found within a program listing, one might guarantee halt-ability; however, one could not ensure the correctness of the program’s final state. When a limit on an iteration was exceeded, a program could gracefully exit and provide an exception report. Such a technique would not guarantee programmatic correctness. Yet, seeing the simplicity of this approach, it might be the case that Gödel’s Halting problem might have been left behind by him as a teaching heuristic, a challenging yet very solvable problem.

A simple, conceptual description of the Halting Problem follows the pattern of its formal mathematically construction; yet, it only uses abstractions.

Also, considering the fact that Gödel and Turing would have been working primarily in the context of machine language, this view of the Halting Problem is understandable. And, in the absence of structured programming or any formal software engineering principles and best practices, these early systems likely were very error prone and commonly entering states of infinite iteration. So, a reasonable first step in ensuring program correctness is guaranteeing halt-ability, the absence of any runaway conditions. Plus, crafting a program which could process any other program listing and detect whether it would eventually terminate or not would be a wise strategy for this first step. Then, the question becomes, “is such possible? Or, is such computable?” Hence, the Theorem of Halt-ability. 

Gödel’s proof is contradictory by construction. Hence, mathematical instinct says that the reasoning is somewhat circular. We start with a contradictory construction and the show that it produces a contradiction. Is such acceptable proof-work? Yet, the assertion of the proof and its claims has stood for decades.

And, if one might somehow show that a program shall eventually halt, the next goal which one must guarantee is that its beginning, intermediate, and final states are all correct. Such might be achievable through the use of formally verifiable programmatic contracts. Such as those developed by Bertrand Meyer.  

Guaranteeing halt-ability in the high-level language layer of a system description would require the use of a global plus numerous local constraints covering all of the iterative structures within the program specification. If none of these upper-bounds on repetition were met, the program would reach its final state and stop. If one of these constraints were violated, the program would raise an exception, print a report on the error stream, and gracefully halt execution. And, a language translation tool, like a compiler or interpreter, which inspected a program listing ensuring that every repetition structure was within the scope of such constraints, would be a counterexample contradicting Gödel’s claim. In other words, it would be the very class of program that he sought.

At the layer of the instruction set architecture, detecting the presence of a processing loop would be necessary. So, an architecture with commands that signal the beginning and ending of a loop are sufficient for its detection. A hardware implementation of such would place a value of high in a field upon entering a loop, a value of medium upon exiting, and a value of low when it is outside of any loop.

While this value is high, a loop counter would be incremented before the first instruction of any iteration. When a loop is exited, the state of any enclosing ones, which includes its counter, would be removed from a register stack maintaining these and made the active loop state. And, if an architecture does not support a loop-begin and loop-end imperatives, they might be simulated by placing values representing these events plus loop state variables within the register file.

It is the case that the “practical” nesting depth of loops has its bounds. It is the said that the goal of efficient programmatic design would use a level of nesting that would produce a quadratic running time or better. So, a hardware implementation that supported a depth of nesting around ten should sufficient. Yet, the placement of the loop frames within the lowest portion of their stack in the main memory file would support a practically unlimited level of nesting. Although, such is impractical from the standpoint of algorithmic efficiency. Seeing that, the development of algorithms with a quadratic time complexity be the goal.

 So, an approximate high-level language solution be this.

#GOEDEL’S HALTING PROBLEM RESOLUTION - EXAMPLE

@[ MAXIMUM-ALLOWABLE-CYCLES := ANY_RESOLVABLE_VALUE ]

WHILE NON-STOP:

                NON-STOP := METHODCALL( PARAMETER_LIST )

So, in modern computing, it might be the case that the question of halt-ability is solve-able. The battle for programmatically enforceable correctness remains. Although, the work of Bertrand Meyer and his programmatic (design) contracts show promise.

It also might be the case that the computing community already holds this insight; yet, it has not been deemed a worthwhile implementation as a hardware circuit or high-level language feature. It is certain that evidence of Gödel’s counterexample exists among current language translation tools. A Python process will gracefully exit and report an exception if a recursive function exceeds an allowable number of self-referential calls. This ensures “halt-ability” for any recursively defined program. Plus, recursion and iteration are processing equivalents, simply different sides of the same algorithmic coin. So, in a sense, Python’s modern hybrid compiler-interpreter is the very type of program that Gödel sought and proved was not computable.

In the case of this problem of halt-ability, it is likely that many students are very-well baffled, befuddled, and brain-bazzizalled by its proof-description, the inherent contradiction built in the proof-work, and the social dynamic involved in questioning a scientific holy-grail put in place by a respected name of Germanic heritage. The Nobel prize-winning physicist at Stanford, Professor William Shockley, helped measure the social weighting of racial cohort labels. His work was based upon what he called “genetic” measures. These were a collection of “carefully chosen” measures. Which he felt accurately and best described the relative strengths of four racial communities. Unfortunately, five primary ones exist on this earth. One of these, those who are descendants of the indigenous inhabitants of the Americas was completely left off the list. Considering the history of the European-led occupation of the Americas, they might simply have been deemed a genocidal afterthought. However, these rankings might shift and adopt an arbitrary ordering depending upon the nature, number, and type of measures chosen for the genetic tuple describing these relative strengths. So, as it oft has been said, “Statistics might lie; and, liars might use statistics.” Yet, Shockley’s work in the social sciences is rarely questioned. This is a function of his social demographic, earnt station in life, and “expert” power.

Most likely, Shockley’s sequential series of populations curves should coincide perfectly with aligned medians. The only difference would be the relative sizes of each normal curve based upon the corresponding population sizes of each group. Yet, that was a brief digression.

Few students in this generation will dare ask an instructor a question for fear of the coming wrath. Which might be dealt out by an insecure professor or peers who are hoping that they keep the average level of performance in the course exceedingly low. This, academic apathy, is the greatest factor limiting the progress made in the modern sciences.

This “insight” has applicability in language design, compiler construction, architecture, or theory. This observation is so incredibly trivial that it does not merit a formal publication. Although, it might potentially invalidate a number of instances of proof-work found in the modern theory of computation as described within Computability, Complexity, and Languages: Fundamentals of Theoretical Computer Science 2nd Edition by Martin Davis, Ron Sigal, and Elaine J. Weyuker. As such, a greater range of problem classes might be computable. Hence, a wider spectrum of problems might be solvable.

Homeboy From Heaven [JOD: Psalm 119:73 – 80 (KJV)]

Sunday, February 14, 2021

Certainty | A Diversion in Mathematics

 Team. Many among us are searching for an "absolute" truth. In an understanding of this lies certainty. From which, we can distill the reliable consequences of any action. Hence, many find solace in "faith-based" readings. It is their hope that they might further unravel and understand the  dynamic between spiritual actions and reactions. So, they know that certain behaviors elicit a predetermined set of responses. Such knowledge can potentially increase the "spiritual" stability in one's life.

It is this same desire for "predictability" and "stability" in the physical world that makes some pursue the field of mathematics. For seemingly, when one adds a pair of numbers "a" and "b", the result is consistently "c". This obviously apparent and timeless rule can never be violated.

And, it is through the historical investigation of such rules: +, -, *, and /, that a myriad of mathematical disciplines and subdisciplines have been established. This includes algebra, topology, calculus, and the theory of graphs. These subjects and their subfields hold "truths" which the modern mathematician sees as "self-evident" much like those inalienable "freedoms" of life, liberty, and the pursuit of happiness. Yet, are they? And, if not "self-evident", are they identically true.

And, if one might establish that a "fundamental" flaw exists in this "timeless" mathematical reasoning, how will the modern mathematician proceed? Will he persist in his "pollyannic" view of his discipline like a cogitating ostrich. Or, will he accept that centuries of reasoning performed by men which he deifies and holds as unquestionable, such a Descartes, Euler, Fermat, and Gauss, might be horribly flawed and incomplete?

One such observation concerning the sets of established numbers has been made on numerous occasions. And that is this, the set of "irrational" numbers simply is that, "irrational". The notion that another "whole" or "floating-point" number cannot be seen as a ratio of "whole" numbers is incredibly "counter-intuitive". In fact, it grates against "natural" mathematic instincts so much that one must consider such a mathematical proposition as "highly" suspect.

And, before one "raises" his mathematical hackles, consider this simple construction, an infinite "ten" or 100....0.  This one followed by an infinite expansion of zeroes, inf(10), is a "permissible" whole number. Although it has an abbreviated description, its complete expansion could never be rendered.

Now, take any "irrational" number with a "non-terminating" decimal such as pi or e and multiply them by inf(10).

    So, pi * inf(10) is an element of the set of natural numbers. Let us call it, PI. So, pi = PI/inf(10). And, pi is a "rationale" value formed from non-terminating whole numbers. The same is true of "e" or any other scalar which has previously been presented as "irrational". And, hence, any postulate, theorem, or worse yet, fundamental axiom, which has been established in the disciplines of mathematics since the establishment of the delusionary notion that number could actually be "irrational" is potentially flawed.

   So, "certainty" does not exist in mathematics, or for that fact, computation itself, in that it is built upon man's grossly limited, inherently flawed, and highly fallible capacity for reasoning. 

  And, it can be said that, with the introduction of the notion of the "irrational" number in the fifth century by the Greek mathematician Hippasus of Metapontum, mathematics itself has entered the realm of the "irrational". And, it should be said that as of 2021 A.D., it has not left.

irrational [ ih-rash-uh-nl ]

adjective

  1. without the faculty of reason; deprived of reason.
  2. without or deprived of normal mental clarity or sound judgment.
  3. not in accordance with reason; utterly illogical: irrational arguments.
  4. not endowed with the faculty of reason: irrational animals.

Mathematics.

  1. (of a number) not capable of being expressed exactly as a ratio of two integers.
  2. (of a function) not capable of being expressed exactly as a ratio of two polynomials.

Algebra. (of an equation) having an unknown under a radical sign or, alternately, with a fractional exponent.

Greek and Latin Prosody.

  1. of or relating to a substitution in the normal metrical pattern, especially a long syllable for a short one.
  2. noting a foot or meter containing such a substitution.

noun

Mathematics. irrational number.

Monday, January 25, 2021

Goofy Orientation

Team. Could you please help me find the right and left sides of the screen? Is the LHS and RHS, left-hand side and right-hand side, of a production in a context-free grammar inverted. In other words, what are the left and right sides of:

public static String aSimpleStringLiteral = "More backward than Aksarben."';

Could this be corrected? Or, is it the case that computing genius baffles "common sense".


When you "left-align" a paragraph in Microsoft Word or another online editor, what side of the page and screen does it align with? Are you saying, "Flush? On what side? Is that right-alignment?"

Just a reminder. Always keep this straight and remember that bolts tighten when they are turnt clockwise and loosen when they are turnt counter-clockwise. Common-sense is a nascent gift; but, it can be lost in this backwards world. Right? Or, is that Left?

And, if you think that it does not matter. Try this "natural" orientation for a year. And, see if life looks different.

Please Forgive the Rhetorical Nature of this Passage.

The Author Resolving Confusion. 

Free E-Book | A Synopsis of Programming Fundamentals

Team. Last week the author of this weblog drafted a short book covering some the concepts in computing which he has learnt within the past three decades of learning. He has a couple of post-baccalaureates in science and engineering. One is in computer science; the other is in engineering. And, the text does contain some religious content and references from the perspective of a Protestant Christian. So, if one would not find such offensive reading while seeing another man's perspective on computing, enjoy the reading. It is a "free" (*.pdf) that you can share with your friends.

Remember Hunt, Peck, and Think. It is the "Best" Programming Approach.