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The Graph of Ancestors

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Is there an “Implex Method” in complexity theory?

Wikipedia src

Bill Wyman was the bass guitarist of the Rolling Stones until 1993. He married Mandy Smith in 1989. A few years later, in 1993, his son, Stephen Wyman, married Mandy Smith’s mother, Patsy Smith. Had Bill and Mandy not divorced by then, Wyman would have been his own step{\,^2}father. Which Wyman do we mean? Both of them.

Today we investigate the level of degeneracy in “family tree” type graphs.

We say “family tree” but it cannot be a tree. If your great{\,^k}-grandparents were all distinct for every value of {k}, then at {k = 30} you would have over one billion distinct ancestors at that level—and two billion in your whole tree—virtually all born after the year 1100 CE. The world population is estimated not to have passed one billion until after 1800.

At least we can say it cannot have cycles. You cannot be your own biological grandfather, for reasons more fundamental than not being able to go back in time to kill your grandfather. Thus the graph is a DAG—a directed acyclic graph. Is “DAG” a jargon term? It should be regarded as common—it is, after all, the stuff of us.

Implex and Amplex

The fact that you are descended by multiple paths from common ancestors is called pedigree collapse by Wikipedia, rather dramatically.

The degree of pedigree collapse, meaning the difference between {2^k} and your actual number {A} of level-{k} ancestors—is called the implex. This is not a legal word in official US Scrabble (TM), but does appear on the larger US-UK SOWPODS list.

The ratio {\frac{A}{2^k}} strikes me as more natural than the difference. To keep it representing “collapse,” one would have to do {1 - \frac{A}{2^k}}, but the original ratio is more convenient in what follows. I propose calling it the amplex, saying how ample one’s DAG of ancestors is. This term is not in SOWPODS but does appear (along with implex) in a downloadable dictionary of 466,551 meaningful words and names and abbreviations compiled by the organization dwyl for “Do What You Love.” The word amplex originally means to engage in a process by which some amphibians make ample trees—or rather DAGs—of descendants.

The ratio may be expected to stabilize once {k} becomes moderately large. We can say similar about the differences {k - \log_2 A} as {k} grows. Finally, the ratio {\frac{\log_2 A}{k}} is a technically different but similarly motivated notion of amplex. These logarithmic forms are most akin to information complexity measures.

An Example

For an example of implex and amplex, let Alice have a completely unique tree of ancestors, so zero collapse and amplex ration {1}. Then Alice’s full brother, named Bob of course, has the same. Suppose Alice and Bob incestuously marry and have a child C. Then for {k = 2} to {10}, C has implex {2^{k-1}} in the difference form, or {0.5} in the ratio form. The amplex ratio {\frac{A}{2^k}} is also {0.5}.

Now suppose instead that Alice and Bob come from different families and each have amplex {0.4}, i.e., {\frac{A}{2^k} = \frac{B}{2^k} = 0.4} (where {B} refers to Bob’s set of level-{k} ancestors) at level {k = 9}. Then their child D at level {k = 10} has amplex at most {0.4}. The maximum is achieved when Alice and Bob are unrelated to each other—i.e., the sets {S_A} and {S_B} of their respective ancestors (saying out to level {9}) are distinct. If they coincide, then D will have amplex only {0.2}.

It may seem intuitively wrong that the incestuous child C could have higher amplex—lower implex—than the child of unrelated parents, each with a reasonable ratio. This hints that some further modifications of these measures may be more effective, such as weighting differences of closer ancestors more. Insofar as different paths to the same level-{k} ancestor define an equivalence relation on binary strings of length {k}, there are notions of prefixes and prefix-freedom that are relevant.

I won’t try to sort such notions out now. I am only trying to be suggestive and invite reader suggestions. In the real world, genomics brings much more extensive notions of genetic diversity. What might help the world of computational complexity theory is more in mind here.

Analogy With Boolean Functions

There are various ways to set up a parenthood notion for Boolean functions. Given two Boolean functions {\alpha(x_1,\dots,x_n)} and {\beta(x_1,\dots,x_n)}, any of the following can be regarded as an offspring:

  • {\alpha(x_1,\dots,x_n) \wedge \beta(x_1,\dots,x_n)};

  • {\alpha(x_1,\dots,x_n) \vee \beta(x_1,\dots,x_n)};

  • {\alpha(x_1,\dots,x_n) \oplus \beta(x_1,\dots,x_n)} (etc.);

  • {(x_{n+1} \wedge \alpha(x_1,\dots,x_n)) \vee (\bar{x}_{n+1} \wedge \beta(x_1,\dots,x_n))}.

The idea that we are trying to tap into is the following: In complexity theory, we want to prove that a set {H} of functions are hard, meaning they do not have Boolean circuits of a given small size {k}. The sets of hard functions overall are large, but we mean sets {H} of functions that are known in some other way, such as belonging to families that define problems in {\mathsf{NP}}. For the most part, it suffices to prove that some member of such an {H} is hard.

The usual mindset is to prove that circuits of size {k} have insufficient “virility” to compute members of {H.} The idea I wish to suggest instead is to try to prove that they are too incestuous—that too many of them compute copies of the same function. Then show that they cannot encompass all of {H.}

To make this work in the above framework, we may need to invert the notion of Boolean “offspring” suggested above, so that the ultimate function computed is styled as an ancestor. In any event, the idea is to structure the counting so that size-{k} circuits cannot have all the members of {H} in their collective “gene pool”—so that some member of {H} must go uncomputed.

Open Problems

Can you suggest further notions in genealogy—without going into genetics—and can they help resolve impasses in complexity?

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15 days ago
OK, definitely got to study on this one. I've told the Bill Wyman being his own father-in-law story too many times. ;->
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Crossing the Pyrenees on foot

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Yesterday, I followed a more than thousand-year-old pilgrim route from France, across the Pyrenees mountains into Spain. A tough hike in normal circumstances, but even more challenging during a heatwave. It was an amazing experience and I hope that this video gives an impression.

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16 days ago
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Our national pandemic of 'sick individualism'

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Our national pandemic of 'sick individualism'

I couldn’t help a double-take when the rusty old truck passed me on a western Kentucky highway.

Painted on the side of its faded greenish hood was a bright red circle bisected by a similarly-hued diagonal bar that was superimposed over a white hypodermic needle.

Below the hand-painted “artwork” were the letters “F.T.M.,” which I took to mean “F--- the Mandate.” "FREEDOM" was also hand-painted in white letters on the hood.

I can imagine the reaction in World War II had the syringe been a ration book and the letters accompanying it were “F.R.R” — as in “F--- Rationing Regulations.”

Had the truck been so decorated, the driver doubtless would have suffered at least a verbal close encounter of the worst kind with a vet home on leave ,or with members of a family with a gold or blue star banner proudly hung in their window.

“What the coronavirus has revealed is the power of America’s cult of selfishness,” New York Times columnist Paul Krugman wrote going on two years ago. So does the old truck, which, too, was plastered with an obscene bumpersticker showing a stick man f-ing the word "LIBERALS."

The deadliest pandemic in history is still a worldwide killer. The virus has claimed the lives of more than a million Americans (including nearly 16,000 Kentuckians) and counting.

The truck driver is more proof that the “sick individualism” of anti-vaxxers/anti-maskers/covid deniers hasn’t gone away either, says Murray State University historian Brian Clardy. “It’s me, me, me — my ‘rights’ at the expense of everybody else’s.”

The angry, irrational — if not flat-out paranoid — opposition to government efforts to keep us out of the hospital and the cemetery “is sometimes portrayed as love of freedom,” Krugman added. “But people who insist on the right to pollute are notably unbothered by, say, federal agents tear-gassing peaceful protesters. What they call ‘freedom’ is actually absence of responsibility.

“Rational policy in a pandemic, however, is all about taking responsibility. The main reason you shouldn’t go to a bar and should wear a mask isn’t self-protection, although that’s part of it; the point is that congregating in noisy, crowded spaces or exhaling droplets into shared air puts others at risk. And that’s the kind of thing America’s right just hates, hates to hear.”

I don't know if the trucker is a churchgoer. But Jesus makes Krugman’s point in the Bible: “Therefore all things whatsoever ye would that men should do to you, do ye even so to them: for this is the law and the prophets.” (Matthew 7:12, King James Version, AKA “the Golden Rule.”)

The trucker’s up-yours, in-your-face signage and obscene sticker reminded me of this paragraph from Krugman’s column: “Indeed, it sometimes seems as if right-wingers actually make a point of behaving irresponsibly. Remember how Senator Rand Paul, who was worried that he might have Covid-19 (he did), wandered around the Senate and even used the gym while waiting for his test results?"

In The Deseret News last year, Jay Evenson proposed that “this generation of Americans has trouble with the concept of sacrifice.” (The trucker was a white guy who looked to be middle-aged.)

“The generation that understood it best — those who survived the Great Depression and World War II — are leaving us quickly, and much too soon. We could use their wisdom right now, as well as their memories of ration books, chocolate shortages, and having to drive on bald tires because rubber was needed for the war effort.”

In a column written after the Uvalde grade school gun massacre, Krugman suggested that “we should think of vehement opposition to gun regulations as a phenomenon closely linked to vehement (and highly partisan) opposition to mask mandates and vaccination in the face of a deadly pandemic, vehement opposition to environmental rules like the ban on phosphates in detergent, and more.”

Krugman asked, “Where does this hatred of the idea of civic duty come from? No doubt some of it, like almost everything in U.S. politics, is related to race.

"One thing it doesn’t reflect, however, is our national tradition. When you hear talk of home-schooling, remember that the United States basically invented universal public education. Environmental protection used to be a nonpartisan issue: The Clean Air Act of 1970 passed the Senate without a single nay. And Hollywood mythology aside, most towns in the Old West had stricter limits on the carrying of firearms than Gov. Greg Abbott’s Texas.”

In 1941-1945, “national tradition” included virtually the whole country uniting behind the war effort. Most Americans understood that the dire national emergency demanded measures like government-mandated rationing of gas, tires, sugar, meat, coffee, butter, shoes, and other consumer goods. John and Jane Q Citizen understood that sacrifice was a “civic duty” they owed to our servicemen and servicewomen who were risking — and sometimes losing — lives and limbs to defeat the deadly triple threats to democracy worldwide: Nazism and Fascism in Europe, and Japanese imperialism in the Pacific.

Almost nobody bellyached, at least out loud, that rationing infringed on their “personal freedom.” (If they did gripe, they usually got an unpleasant earful from their friends, neighbors, and even family members.)

“I don’t fully understand where this aversion to the basic rules of a civilized society is coming from,” Krugman confessed. “What’s clear, however, is that the very people who shout most about ‘freedom’ are doing their best to turn America into a ‘Hunger Games’-type dystopian nightmare, with checkpoints everywhere, loomed over by men with guns.”

This much is clear, too: If a majority of Americans today were infected with the “sick individualism” of the truck driver, the covid death toll would be considerably higher than a million plus, and if a “cult of selfishness” dominated the America of my parents (my mother worked for the local county ration board, and my father was a sailor on ships in the Pacific), the “The Man in the High Castle” would be dystopian reality and not just TV fantasy.


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19 days ago
Like yesterday:
Republicans want to tell you what to do.
Republicans don't want you to tell them what to do.
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This ex-Republican just tweeted the best thread to explain the GOP

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This ex-Republican just tweeted the best thread to explain the GOP

Cross-posted from Daily Kos; shared by user Bring the Lions.

I read this last week and was amazed at how well this Ethan Grey summed it up.  I asked him for permission to repost it here.  He said “sure!” So here it is.


Here is the Republican message on everything of importance:

1. They can tell people what to do.
2. You cannot tell them what to do.

This often gets mistaken for hypocrisy, there’s an additional layer of complexity to this (later in the thread), but this is the basic formula.

You've watched the Republican Party champion the idea of “freedom” while you have also watched the same party openly assault various freedoms, like the freedom to vote, freedom to choose, freedom to marry who you want, and so on.

If this has been a source of confusion, then your assessments of what Republicans mean by “freedom” were likely too generous. Here’s what they mean:

1. The freedom to tell people what to do.
2. Freedom from being told what to do.

When Republicans talk about valuing “freedom,” they’re speaking of it in the sense that only people like them should ultimately possess it.

So with this in mind, let’s examine some of our political issues with an emphasis on who is telling who what to do. And hopefully there will be no ambiguity about what the Republican Party message is ever again.

Let’s start with the COVID-19 pandemic. We were told by experts in infectious diseases that to control the spread of the pandemic, we had to socially distance, mask, and get vaccinated. So, in a general sense, we were being told what to do. Guess who had a big problem with that.

All Republicans saw were certain people trying to tell them what to do, which was enough of a reason to make it their chief priority to insist that they will not be told what to do. Even though what they were told to do could save lives, including their own.

As you can see, this is a very stunning commitment to refusing to be told what to do. So much so that it is not in fact “pro-life.” But Republicans will nevertheless claim to be the “pro-life” party. That is because they recognize “pro-life” can be used to tell people what to do.

The reason they say they are “pro-life” when they are trying to tell women what to do with their bodies is not out of genuine concern for human life, but because they recognize that in this position, they can tell women what to do with their bodies.

That’s why when you use that same appeal — “pro-life” — when you ask Republicans to do something about gun violence in schools, it doesn’t work. Because you are now in the position of telling Republicans what to do. That’s precisely why they don’t want to do anything about it.

Anyway, gun violence in schools is not a problem, but their children having to wear masks in schools is. Because somebody is telling their children what to do. Dead children don’t bother them, but telling their children what to do? Only *they* should do that.

They claim to be for “small government,” but that really means that a government that can tell them what to do should be as small as possible. But when the Republican Party recognizes it has an opportunity to tell people what to do, the government required for that tends to be large.

The reason Republicans are so focused on the border isn’t because they care about border security, it’s because they recognize it as the most glaring example of when they can tell other people what to do. That’s why it’s their favorite issue. You want in? Too bad. Get out.

If Republicans could do this in every social space tell the people who aren’t like them too bad, get the fuck out I’m here to assure that that would be something resembling their ideal society.

Now, there are economic policies that we’ve proposed that we can demonstrate would be of obvious benefit to even Republican voters. So how do Republicans leaders kill potential support for these policies? Make the issue about who is telling who what to do.

They focus on the fact that Democrats may raise taxes. Even when it’s painfully obvious that Democrats aren’t going to raise taxes on everyone (or on very few people), what’s important here is that Democrats are the people telling certain people what to do.

If you want to know why Republicans can easily be talked out of proposals from the Democratic Party that are shown to be of benefit to them, it is precisely because they have to entertain the idea of Democrats telling certain people what to do.

What you didn’t understand from the very beginning is that Democrats should not ultimately be in the position to tell anyone what to do. Only Republicans should be in the position to tell people what to do.

On the issue of climate change, a lot of them don’t regard it as a serious issue to the extent that they think it is a hoax. This is because when you tell Republicans to do something for the sake of the planet, you are still ultimately telling them to what to do.

Furthermore, you are conceiving the planet as a thing that all human beings should have to share. I am here to assure you that the GOP’s main concern with the planet is to ensure that they don’t have to share it.

Now here’s where things get interesting: when you explain to Republicans you want them to do something and explain it’s on the basis of benefitting other people. Now you have really crossed a line. Not only did you tell them what to do, you told them to consider others.

The whole point of an arrangement where you can tell people what to do, but you can’t be told what to do, is precisely to avoid having to consider others. This is why this is their ideal arrangement: so they don’t have to do that.

As you can see, this is a very toxic relationship with the idea of who can tell who what to do. So much so that it seems like the entire point is to conceive of a “right” kind of people who can tell other people what to do without being told what to do. Yep, that’s the point.

So let’s add one more component to the system for who tells who what to do:

1. There are “right” human beings and there are “wrong” ones.
2. The “right” ones get to tell the “wrong” ones what to do.
3. The “wrong” ones do not tell the “right” ones what to do.

As you can see, I've just been talking about white male supremacy and the accompanying caste system structure it enforces all along. And Im talking about this because the message of the Republican Party is that they quite like it.

But I realize that we are operating in an environment where white male supremacy is so entrenched that the press can’t even conceive of the Republican Party’s agenda of sorting the “right” human beings from the “wrong” ones as maybe presenting a “messaging problem.”

This is because the press has chosen to accommodate the Republican Party in a very specific way:

1. It normalizes the Republican agenda.
2. It normalizes framing the responsibility for stopping that agenda as ultimately being on Democrats.

Think about it: white supremacy is not allowed to be viewed as a “messaging problem.” Even when it’s a threat to democracy. Because if it’s a “messaging problem,” to Republicans, that sounds youre telling them thats a problem they have to solve.

Anyway, I made this thread mostly because I realize that the press has a “messaging problem.” Namely, in the sense that they seem extremely averse to explicitly identifying the message of the Republican Party. It’s called white male supremacy. Thanks for reading.


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20 days ago
Spot on!
Well done!
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The Virus Is Still Winning

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Despite the rapid development of COVID-19 vaccines and treatments to reduce mortality, the elderly, the immunocompromised, and other vulnerable populations remain at risk of severe infection. Worse, some SARS-CoV-2 strains are only a mutation or two away from a leap in lethality.

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24 days ago
True dis.
Our grandsons gave my wife COVID Mother's Day.
My oldest daughter just caught it in CA a few days ago.
I still haven't had it. I had my 2nd booster 4/13. Still masking up if indoors somewhere more than 1-2 minutes.
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Summer Camp for All

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It’s hard to imagine a more ideal escape for young people living through an extraordinary time of grief, loneliness and upheaval.
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37 days ago
Helping children should always be our highest concern.
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