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P versus NP

A researcher from HP Labs named Vinay Deolalikar announced a new proof that complexity classes P and NP are different. The paper made it to the Slashdot front-page (more on this below).

What constitutes a “proof”?

This is far from being the first claimed proof. There are about as many proofs that P is the same as NP than proofs of the opposite. With, for good measure, a few papers claiming that this is really undecidable… This just shows that the problem is not solved just because of one more paper. Indeed, the new “proof” takes more than 60 pages to explain, and it references a number of equally complex theorems and proofs.

This is interesting, because it means that very few, except some of the most advanced specialists in the field, will be able to understand the proof by themselves. Instead, the vast majority (including the vast majority of scientists) will accept the conclusion of a very small number of people regarding the validity of the proof. And since understanding the proof is so difficult, it may very well be that even the most experience mathematicians will be reluctant to draw very clear-cut conclusions.

Sometimes, clear-cut conclusions can be drawn. When I was a student, another student made the local news by announcing he had a proof of Fermat’s last theorem. We managed to get a copy of the paper, and shared that with our math teacher. He looked at it for about five minutes, and commented: “This is somewhat ridiculously wrong”.

However, In most cases, reaching such a definite conclusion is difficult. This puts us in the difficult position of having to trust someone else with better understanding than ours.

Understanding things by yourself

That being said, it’s always interesting to try and understand things by yourself. So I tried to read the summary of the proof. I don’t understand a tenth of it. However, the little I understood seemed really interesting.

If I can venture into totally bogus analogies, it looks to me like what Deolalikar did is build the mathematical equivalent of ice cube melting, and drew conclusions from it. Specifically, when ice freezes, phase change happens not globally, but in local clusters. You can infer some things about the cluster configuration (e.g. crystal structure) that were not there in the liquid configuration. In other words, the ice cube is “simpler” than water.

Now, replace atoms with mathematical variables, forces between atoms with some well-chosen Markov properties that happen to be local (like forces between atoms). The frozen cube corresponds to a P-class problem where you have some kind of strong proximity binding, so that you can deduce things locally. By contrast, liquid water corresponds to NP-class problems where you can’t deduce anything about a remote atom from what you learn about any number of atoms. Roughly speaking, Deolalikar’s proof is that if you can tell water from ice, then P and NP classes are distinct.

Of course, this is only an analogy, and it is very limited like any analogy, and I apologize in advance for totally misrepresenting Deolalikar’s subtle work. Nevertheless, I found the approach fascinating.

Crowds are stupid

Now, an alternative to personal understanding is to trust the crowd. Democracy. Unfortunately, if Slashdot is any indication, this doesn’t work at all.

Slashdot has a moderation system, where people vote for comments they find “Interesting” or “Insightful” or “Funny”. You’d think that this would let good comments rise to the top. But what really happens is that people with “moderation points” apparently feel an urge to moderate as quickly as they can. So the very first comments get moderated up very quickly, and drown any later comments in the noise.

Here are some of the comments on the P vs. NP announcement that Slashdot thought were “good”:

  • P is not NP when N is not 1 (“Funny”)
  • A random series of totally uninformed opinions on the cryptography impact (“Interesting” and “Funny”)

There are a few relevant gems in there, like the opinion of a professor at MIT that the proof is not valid. That leads to another more serious analysis. But there are a few redeeming gems even on Slashdot. Still, it’s too bad you have to sift through mud to find them.

Linus Torvalds on religion

Linus Torvalds, original creator of Linux, writes in his blog:

One of them starts to seriously talk about praying demons away, and then after the prayer has driven the demon out of the person, you have to support the person so that the demon doesn’t come back. And nobody laughs at him.

Seriously? What year is it again? I’m pretty sure they didn’t have Costco foodcourts in the middle ages, but maybe there was some time warping going on.

What the hell is wrong with people?

There is a large number of comments (169 to this date). Very few of them are critical of Linus at the beginning, but then a few start voicing a different opinion. And Linus Torvalds answers with gems like:

And yes, it is acceptable to ridicule people for their beliefs. If you meet a grown person who believes in the easter bunny, in santa claus, or in the tooth fairy, they’re damn well fair game.

Seriously, what’s wrong with Linus Torvalds?

If someone doesn’t know something, you educate them. You don’t run back home and cowardly post blog entries mocking your “opponent” to a friendly crowd! This is really not Linus Torvalds’ most glorious moment…

What I find more annoying is just how religiously illiterate Torvalds is, and how that doesn’t stop him from believing he knows everything about everything. He is not alone in his belief that to be religious, you have to be stupid.

Categories: Religion

What is your dangerous idea?

I’ve just finished reading John Brockman’s What is your Dangerous Idea?. This book is a collection of short essays (108 of them, I believe) by various authors selected by the Edge foundation. This makes it a book that is difficult to summarize or even analyze, but I’ll try. If you want to read the essays on-line, you can find them on the Edge site.

An impressionist painting of today’s intellectual landscape

One thing can be said upfront: this is a book worth reading, even if you are unlikely to like all of it. It’s worth reading, because it exposes the viewpoint of various smart thinkers on a large number of topics. You are unlikely to like all of it, because in this long list of idea, one is bound to trigger disagreement, angst, anger, or some other kind of negative sentiment.

The book covers a large range of topics, largely circling around science (physics, biology, neurophysiology, computer science), with peeks at sociology, religion or morality. Trying to talk about each and every individual idea would be quite pointless. It’s clear that the “Dangerous idea” question leaves things quite open, and so there is no real central theme. Some ideas appear quite insightful, others frightening, other depressing, some even seem clearly wrong or overly simplistic to me. But the general impression remains that of an impressionist painting of today’s intellectual landscape: it’s made of many little strokes, yet you can see trends and general shapes emerging from all these dots.

The dangerous uniformity of bland gray matter

Unfortunately, compared to real impressionists, the picture surprisingly lacks color and intensity, in my humble opinion. This was really the first impression I had after the few chapters of the book: “Hey, why are they all saying the same thing?” Now, to be honest, some of that may be a result of the subtle editorial work of John Brockman, who did a remarkable job of grouping writers who happened to be covering a similar theme together, which seems to give the book a kind of gentle flow.

But still, from such an honorable audience, I expected a lot more: more fire, more contrast, more debate, more disagreement. Oh, sure, many of the ideas are bound to shock a large fraction of the readers, in particular in the US, by claiming for example that we have no soul (John Horgan), that the probability of God is small (Philip W. Anderson), even that science must destroy religion (Sam Harris), that the fight against global warming is lost (Paul C.W. Davies), and so on.

However, while all these thinkers might liken themselves more or less consciously to Galileo or Darwin for their revolutionary ideas, for going against majority thinking, they are still nowhere close to these role models, because they largely agree among themselves and with their peers. The true honor of Galileo or Louis Pasteur was to go against the scientific establishment of the time. By contrast, What is your Dangerous Idea shows an unfortunate uniformity in the thinking of our present scientific establishment.

To illustrate this, it is sufficient to take the first three essays in the print edition: We Have No Souls (John Horgan), The Rejection of Soul (Paul Bloom), and The Evolution of Evil(David Buss). All three talk about a topic that is central to religious thinking, and all three go straight against conventional religious wisdom. Worse yet, they present their ideas under the authority of science, as if the ideas they present were scientific.

For example, John Horgan writes:

The dangerous (probably true) idea I’d like to dwell on is that we humans have no souls.

Paul Bloom confirms:

If what you mean by “soul” is something immaterial and immortal, something that exists independently of the brain, then souls do not exist.

and David Buss, on a different topic, similarly defines evil in a way that entirely ignores traditional morality:

At a rough approximation, we view as evil people who inflict massive evolutionary fitness costs on us, our families, or our allies.

Now, maybe you think that this is just the theme for essays at the beginning of the book? Not really. There is even an essay dedicated to that same ever-present idea, This Is All There Is, written by Robert R. Provine. And in one of the last essays on an unrelated topic, What We Know May Not Change Us (link apparently broken), Barry C. Smith finds it useful to begin his work by echoing exactly the same sentiment:

Human beings, like everything else, are part of the natural world. The natural world is all there is.

This would not be so dramatic if there was, in all this loud concert of consensual voices, at least one lone dissenter daring to proclaim a dangerous idea that would go against that flow, something like: Science has nothing to say about religion, or There may be something else after all. This would be the expected high-contrast picture of healthy scientific skepticism. Instead, what we have is bland gray matter, pun intended.

Somehow this all reminds me of a cartoon by French cartoonist Sempé, which unfortunately I can’t find on the web, where dozens of characters in a building were all saying I’m fortunate not to have seen that happen in my lifetime, but I’m afraid that you kids will live in a world where everybody believes the same thing. Hilarious, and so true…

How could you tell if you are in the Matrix?

The irony is that there is, actually, one essay that explains very well why the general consensus expressed by all these thinkers is, in reality, illogical, and why it should quickly appear untenable to such bright minds. The essay I’m referring to is: We Are All Virtual, by Clifford Pickover. In it, Pickover argues that there are more chances for us to be virtual than real. The idea is that we are building ever more realistic virtual worlds, and that at one point, we will certainly be able to make them appear as realistic to us as our own dreams. Since there will be more such virtual environments than real ones, chances are that you are currently in one.

I should know about virtualization. As the original designer of HP Integrity Virtual Machines, HP’s virtualization technology for servers, I am acutely aware that it’s possible to fool the most “reality-aware” form of software, the only form of software that routinely interacts with real, physical devices, namely operating systems. In other words, we can make an operating system such as Linux, Windows, HP-UX or OpenVMS believe that it’s running on some real hardware, with real disks, real network cards, a real console, and so on, when in reality, everything is pure software. This is why I’m often describing my own job by saying that I’m coding The Matrix.

Is there any reason the brain’s “software” can’t be fooled that way? If there is one, I don’t know it. It is certainly not widely known. Consequently, claims that there is nothing beyond natural reality, beyond the sum of our sense, are vacuous at best and fraudulent at worst. They are a belief, not science. Worse yet, from a scientific point of view, these claims fly in the face of simple technological and scientific evidence, from virtual machines to the psychology of dreams.

Ernst Pöppel also makes a similar point about the limits of what we can know in How Can I Trust, in the Face of So Many Unknownables (the on-line version differs somewhat from the print version). So that viewpoint is somewhat common, and as well it should. But to get back to my initial comment about uniformity of thought, what I find unfortunate is that both Pöppel and Pickover carefully wrap this simple point in a consensual, conflict-avoiding shroud of anything-but-religion. In today’s scientific community, it is apparently acceptable to write We are all virtual, but Gödel forbids that you’d write Chances are that the natural world is not all there is… even if the two sentences are essentially equivalent

My dangerous idea: there is a supernatural world

Now, from the reasoning above, you probably understand that what I truly believe is that we really can’t tell if there is something beyond the natural world or not. This is not exactly new philosophy. After all, it’s quite similar to Pascal’s Wager. But the obvious next question is: is there any way to tilt the balance one way or another?

In order to address this question, I need to point out that everything we “know” is really something we believe more strongly than the rest. Pöppel’s essay doesn’t say anything else, actually. In order to “know”, we must trust in a number of things, in general simply because we have been told about them, or because our senses report them. Sometimes, this approach leads us to hold as self-evident beliefs that are spectacularly false. But in general, it is a good working model we can use to drive our own actions. Trust in our own beliefs is the main driver that guides everyone of our actions.

Now, where do religions come from? They come from a testimony. According to the scriptures common to the majority of religious people on Earth today, someone once told a guy named Moses that reports of his nonexistence were vastly exaggerated. As I wrote earlier in this blog, I am that I am is actually quite profound: someone claims to exist without the need for anything else. This is definitely not the case for us: can we ascertain that we can exist without a brain or a physical body? Someone out there, allegedly, once said that He can.

Everything considered, monotheist religions are nothing but faith in such testimony. Unlike the characterization of religion given incorrectly at various places in Dangerous Idea, truly religious people do not claim that they know God, only that they know God spoke to us (Catholics often refer to God as “The Word” for that very reason.) In that respect, there is very little difference between believing that God exists and believing that George Bush exists, there is little difference between believing that God wants us to love one another and believing that George Bush wants to eradicate the axis of evil. In both cases, it’s mostly hearsay, testimony from someone who heard about something that someone else once heard, or indirect testimony through the Bible or newspapers.

Answering the Question Personally

There is one big difference, though. At least in the Catholic Faith (I’m not too sure about others), it is possible to speak to God directly, something called “prayer”. All things considered, this is probably much easier than speaking to George Bush directly. What the Church tells us is that we have some inner sense that allows us to listen to God. It is more subtle than hearing or seeing. It takes some faith to believe that it works. But thousands of people speak to God today.

To them, saying that God doesn’t exist is like saying that their parents don’t exist. Try this with your friends: “I’m sorry, Sir, but my computer has no trace of your father, so he doesn’t exist.” Of course, many believers will acknowledge that their faith may be a fluke in their brain, that the answers to their prayers may just be chance and random coincidence.

But in that respect, if we are all virtual, George Bush may also be a fluke in my brain for all I know, and the fact that my wife cooks a delicious dinner may also be a pure coincidence totally unrelated to her love for me. Still, in both cases, scientists beloved Ockam’s razor recommends that I act under the hypothesis that both George Bush and my wife’s love are “real”, whatever that word means. And believing in God is not more stupid than this belief, no matter what too many scientists in Dangerous Idea proclaim.

So here is why the idea that there is a supernatural reality is dangerous: if this person named God is real, and if He is who He claims to be, then we’d better listen carefully. After all, His message is not so bad, it’s all about love. Aren’t we lucky not to be in a world where the creator constantly yells “How can I make you miserable, you unworthy pieces of carbon-based junk”? Granted, the message is all garbled and difficult to comprehend. But then, my virtual machines probably also have a very poor understanding of my “reality”.

Other gems in the book

I can’t finish this review without mentioning that there are a number of essays in this book that stand out for a reason or another. The usual suspects talk about physics (e.g. Leonard Suskind, Lee Smolin, Brian Greene and Carlo Rovelli), and defend their now relatively well known core ideas with the usual brio.

As usual, I find Carlo Rovelli to be the most insightful of the bunch, pointing out that somehow, the two core discoveries of twentieth century physics have not yet been assimilated by most, including by physicists:

Many of today’s audacious scientific speculations – about extra dimensions, the multiverse and hte like – are not only unsupported experimentally, but also quite often formulated within a worldview that has not even fully digested quantum mechanics and relativity

Unfortunately, for most readers, it may not be entirely clear what he means by that, even if he does a good job at explaining it in the rest of the essay. Rovelli’s core idea, I believe, is that Relativity demonstrated that the notion of “fixed background” where events play out is a persistent illusion, one that is appropriate from our low-energy, low-speed environment, but totally wrong in general. As Brian Greene explains in one of his books, if you walk towards Andromeda or away from it, the “now” in the Andromeda galaxy shifts backward or forward by a few thousand years…

Yet, what Rovelli complains about is that some theories today are still written on some absolute background. This is certainly not the case for my own pet theory, which removes yet another “absolute background” (in my case, absolute variables such as x, t or m, as opposed to relative measurements of the corresponding physical entities, length, time or mass). But in general, I believe that he has a point. It’s a shame that relativity or quantum mechanics are still presented as dangerous ideas.

Computer geeks rule!

But the pearl of wisdom in the whole book, in my opinion, goes to Kai Krause for the strangely titled Anty Gravity: Chaos theory in an All-To-Practical Sense, probably in large part because it feels so “original” compared to all other essays. The Anty spelling is not a typo: it’s both in the title and in the contents table, and the essay does talk about ants. Kai’s essay is about what he calls “Super Individualism”, the need for every single one of us (myself included) to feel unique and special, and how this goes against the need of the many:

What if each ant suddenly wants to be the queen? What if soldiering and nest building and cleaning chores are just not cool enough anymore? If AntTV shows them, every day, nothing but un-Ant behavior…?

He concludes with:

Next year, let’s ask for good ideas, not dangerous ones. Really, practical, serious, good ideas, like: “What is the most immediate positive global impact of any kind that can be achieved within one year?”

And when I read that, I was proud to be a computer geek: in this large collection of essays from a large pool of bright minds, the one that really stood out from the rest was from a fellow geek…

Cyclic big-bang?

An interesting discussion about an alternative to the Big-Bang from the M-theory (check the link for audio). The discussion is pretty interesting, though I cannot help but react to a statement like the evidence at this point is mainly mathematical (4:21 into the talk). That’s not evidence (but I think he knows that :-) On the other hand, a good discussion of some key problems of cosmology, like why is the universe so large and so uniform? See also another analysis of the problems discussed in the talk.

Also, around 43:30, a claim that science is about experimental evidence. The problem is that many theories now are beyond the reach of experiments. So the speaker’s comment that [science] is not a faith based approach, while somewhat true in general, does not seem work for example in the debate between string theories and alternatives, where the evidence is not enough to allow the logic part of our brains to take over.

As an aside, I decided to remove Motl’s “Reference Frame” from my links of “physics blogs” because of the generally inflammatory tone both in the main text and the comments, but also because of some really bad science.

Categories: Physics, Religion, Sociology

Another Theory of Everything

Dr Max Tegmark looks like a very interesting person to discuss with. He is clearly a very credible scientist with a knack for explaining to the layman (that’s me), but he also indulges in what he calls “bananas” theories of everything.

I find his mathematical universe article fascinating. It is well researched and well argumented. However, I do not subscribe to most of the ideas presented there. Below are some of my objections to the reasoning in the article.

How do we find equivalent mathematical systems?

On page 4, Dr Tegmark writes a particular mathematical structure can be described as an equivalence class of descriptions, so that there is nothing arbitrary about the mathematical structure itself. If you read the article, however, you will realize that this is only illustrated with some very small, finite, mathematical structures. Mathematicians know that going from finite to infinite is usually fraught with peril.

On page 25, this is made more explicit, since Dr Tegmark writes that there is a simple halting algorithm for determining wether any two finite mathematical structures are equivalent (emphasis mine). Even ignoring the fact that finite mathematical structures are (in the present state of knowledge) less interesting as far as physics is concerned, I would have liked a reference here, because I’m not even sure what algorithm he is referring to.

How do we define mathematical systems without “baggage”?

Similarly, also on page 4, Dr Tegmark writes that the number 4 is well defined. I sent him email about this, and I am very curious to hear his response. In my opinion, the number 4 has a number of widely different definitions. It can be the result of counting, i.e. 1+1+1+1. It can be the product of 2 by 2. It can be the surface of a square of side 2. It can be the first composite number, the number of sides of any polygon with only square angles, or whatever else. Many of these definitions depend on some particular mathematical axioms. For example, the surface and square angles properties only hold in Euclidean geometry. On a sphere, you can build a triangle with three square angles.

This means in my opinion that “modding out the baggage”, to take Dr Tegmark’s expression, is in reality impossible, because the axioms of the mathematical theory have to be part of the baggage. In the case of a physics theory, there is one big central axiom, something like “this theory represents our universe”. To me, Dr Tegmark’s article does not prove at all that such an axiom can be modded out.

As an aside, in the theory of incomplete measurements (TIM), instead of trying to get rid of all axioms, I decided to see what could be considered axioms of physics. I proposed to define a measurement process as a) a physical process, b) with known input and output, c) giving consistent (repeatable) results, d) that depend only on their input, e) that impact only their output, and f) with a symbolic interpretation for changes in the output. These are more postulates than axioms, because it is a choice we make (a “baggage” as Dr Tegmark would call it) to isolate some physical processes among all possible ones and call them measurements. I believe that there is only one real axiom in the TIM, which is experimentally, there are measurements (at least to a degree of approximation that satisfies us). It is noteworthy that, if my reasoning is correct, the so-called axioms of quantum mechanics can be derived from this set of measurement postulates.

What does “defined” mean?

Even if I were to accept that “4 is well defined”, this does not mean that this applies for example to any real number. The evil spectre of Skolem’s paradox comes back to haunt us. Since the number of propositions you can write with a mathematical system made of finite sequences of symbols taken in a finite set is demonstrably countable, and since the set of real numbers is demonstrably not countable, there will be real numbers that are not the subject of any mathematical proposition. Of course, all real numbers are “subject” of a proposition like x=x, but this cannot in any way be interpreted as defining a particular real number. I can define 4 using a finite set of symbols, e.g. 1+1+1+1. I can even define some irrational numbers the same way, for example e=-1 is one possible definition of π. But there will be real numbers that will not be defined that way, in any meaningful sense of “defined”.

What does scientific (or falsifiable) mean?

Dr Tegmark is clearly aware of this problem. He notices on page 11 that an entire ensemble is often much simpler than one of its members. We can describe the set of real numbers, even if there are real numbers we cannot describe. This is relevant to the discussion of a “theory of everything”, however, in a way that he did not resolve in the article, at least, not in a way intelligible or satisfactory to me. Is a theory that predicts other universes than the observable one still a theory of physics? In particular, is a theory that predicts all possible universes or mathematical structures (which seems to be Dr Tegmark’s ultimate objective) still a scientific theory, since by construction it cannot be falsified by any physical experiment?

Is there a preferred length in the universe?

On page 6, the article states there appears to be no length scale “1″ of special significance in our physical space. I would personally have thought that the Planck length qualifies. Actually, as I wrote in section 3.5 of the TIM, the fact that hbar appears in physics equations at all is the best argument I can think of justifying Laurent Nottale’s scale relativity, leading to a “non-galilean” composition of scales law, as well as to impassable scales of nature playing with scale a role similar to c for translation. I do not agree with everything Nottale wrote, but the appearance of a non-dimensionless constant in a fundamental equation is worth explaining.

Did Dr Tegmark reformulate “God”?

In conclusion, I find that Dr Tegmark’s formulation for the “theory of everything” (TOE) is written in such a way that its existence itself belongs to belief more than to science. The TOE is “defined” largely by non-proven and, in my opinion, non-provable properties. It describes all possible universes (or multiverses), covers all possibilities, including identical copies of you if you look far enough (page 14), it leaves no room for initial conditions (page 10), it is eternal (page 18). And since we live within it, it is not unreasonable to state that it “creates” us at any instant.

I find it somewhat ironic that all these attributes are the very same attributes that the main monotheist religions gave to God. The theory of Dr Tegmark looks a lot like it could be described as I am that I am. One property is missing, however: these religions consider God as a person, a sentient being. Actually, the “self-aware” property is not entirely missing, since the article does mention “self-aware substructures”. But I did not see a discussion of whether the mathematical universe itself would be self-aware…

Is the mathematical universe self-aware?

That is an interesting topic. I am self-aware, but I cannot prove that you are. You being self-aware is a preferred hypothesis for me, based on what Dr Tegmark calls a “consensus view”, and recursively I believe that you believe that I am self-aware. None of this, however, addresses the question of whether, generally, a collection of people can be called self-aware. What about a collection of self-aware beings combined with some “inert” stuff? Further still, can you call “self-aware” a universe that contains self-aware beings capable of reflection about the universe?

Really, I can’t help but be amazed at how smart the guy was, who originally wrote “I am that I am” in the Bible. Thirty centuries or so later, it still takes us about 28 pages by a respected scientist to reach more or less the same conclusion. For some reason, I am ready to bet that this is not the way Dr Tegmark himself would put it :-)

Personal loss…

I lost my father, Gabriel de Dinechin, during the night of Good Friday. He was a truly remarkable figure. Brilliant yet humble, gentle yet true to himself and others, sometimes slightly out of touch with reality, but always ready to listen. He was considered the genius in the family, and he truly knew an amazing lot about a number of fields: mathematics, physics, biology, astronomy, history, languages (during his last days, he was trying to read Tintin in russian and studying the basque language)… Professionally, I had not realized how much he had done until an uncle of mine who had asked his peers told us about it. Whenever French drivers used to be caught in regular car jams and no longer are, notably in Paris and around Lyon, chances are he had a role in it…

I will truly miss him, and I know that those who knew him feel the same. For about 2 years, my family and myself knew that this sad moment would be coming. We had time to prepare, we had time to say goodbye. It helps somewhat. Religious beliefs also do help. But the pain is still there. Sometimes, you believe that you have it under control, you want to have it under control, and it just hits back.

Coincidentally, my previous blog entry was about a family who had lost their child tragically. The trial of the murderer is underway, and this family keeps giving a remarkable testimony. I now find myself mourning, and I may understand a little better how they feel, and how courageous these parents really are. After all, it’s in the normal order of things to lose one’s father, but losing a child is not. So I’d like anybody who is ready to give a prayer for me, our family, or my father, to also consider praying for the Kegelin family as well.

Categories: Religion

Sainthood…

The question of whether John-Paul II will be officially proclaimed a Saint by the Catholic church makes some headlines.

But there are less visible figures that might very well be saints without many knowing it. There was an extraordinary testimony in Famille Chrétienne by the parents of Jeanne-Marie Kegelin, a young girl who was savagely murdered. Two things make the testimony remarkable: the beautiful faith of this family despite their tragic loss, and the very simple, very short, yet very powerful prayers that this young girl had written shortly before her death, almost as if she was preparing for it.

As the trial of the murderer approaches, I invite all of you to pray for this family, for this young girl, and, as the family requested it, for the murderer and accomplices.

Categories: Christianity, Religion
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