Thursday, November 10, 2011
First published Thursday, November 02, 2006
Pseudorandom thoughts on complexity
Draft 2This post supplements the previous post "Does math back 'intelligent design'?"
With respect to the general concept of evolution, or simply change over time, what do we mean by complexity?
Consider Stephen Wolfram's cellular automata graphs. We might think of complexity as a measure of the entropy of the graph, which evolves row by row from an initial rule whereby change occurs only locally, in minimal sets of contiguous cells. Taken in totality, or after some row n, the graphs register different quantities of entropy. That is, "more complex" graphs convey higher average information than "less complex" ones. Some graphs become all black or all white after some row n, corresponding to 0 information after that row. There exists a significant set of graphs that achieve neither maximum nor minimum entropy, of course.
How would we define information in a Wolfram cellular automaton graph? We can use several criteria. A row would have maximum entropy if the probability of the pattern of sequential cell colors is indistinguishable from random coloring. [To be fussy, we might use a double-slit single photon detector to create a random sequence whereby a color chosen for a cell is a function of the number of the quadrant where a photon is detected at time t.]
Similarly for a column.
Obviously, we can consider both column and row. And, we might also consider sets of rows and-or columns that occur as a simple period. Another possibility is to determine whether such sets recur in "smooth curve quasi-periods" such as every n^2. We may also want to know whether such sets zero out at some finite row.
Another consideration is the appearance of "structures" over a two-dimensional region. This effectively means the visual perception of at least one border, whether closed or open. The border can display various levels of fuzziness. A linear feature implies at least one coloration period (cycle) appearing in every mth row or every nth column. The brain, in a Gestalt effect, collates the information in these periods as a "noteworthy structure." Such a structure may be defined geometrically or topologically (with constraints). That is, the periodic behavior may yield a sequence of congruent forms (that proliferate either symmetrically or asymmetrically) or of similar forms (as in "nested structures"), or of a set of forms each of which differs from the next incrementally by interior angle, creating the illusion of morphological change, as in cartoon animation.
At this juncture we should point out that there are only 254 elementary cellular automata. However, the number of CA goes up exponentially with another color or two and when all possible initial conditions are considered.
So what we are describing, with the aid of Wolfram's graphs, is deterministic complexity, which differs from the concept of chaos more on a philosophical plane than a mathematical one.
We see that, depending on criteria chosen, CA graphs, after an evolution of n steps, differ in their maximum entropy and also differ at the infinite limit in their maximum entropy. Each graph is asymptotic toward some entropy quantity. By no means does every graph converge toward maximum entropy as defined by a truly random pattern.
So we may conclude that, as Wolfram argues, simple instructions can yield highly complex fields. The measure of complexity is simply the quantity of information in a a graph or subgraph defined by our basic criteria. And what do we mean in this context by information? If we went through all n steps of the rule and examined the sequence of colors in, for example, row n, the information content would be 0 because we have eliminated the uncertainty.
If, however, we don't examine how row n's sequence was formed, then we can check the probability of such a sequence with the resulting information value. At this point we must beware: Complete aperiodicity of cell colors in row n is NOT identical with maximum entropy of row n. Think of asking a high school student to simulate flipping of a coin by haphazardly writing down 0 or 1 in 100 steps. If one then submits the sequence to an analyst, he or she is very likely to discover that the sequence was not produced randomly because most people avoid typical sub-sequences such as 0 recurring six times consecutively.
So then, true randomness (again, we can use our quantum measuring device), which corresponds to maximum entropy, is very likely to differ significantly from computed chaos. This fact is easily seen if one realizes that the set of aperiodic computable irrational numbers is of a lower cardinality than the set of random digit sequences. Still, it must be said that the foregoing lemma doesn't mean there is always available a practical test to distinguish a pseudorandom sequence from a random sequence.
We might also think of deterministic complexity via curves over standard axes, with any number of orthogonal axes we like. Suppose we have a curve y = x. Because there is no difference between x and y, there is effectively no information in curve f(x). No work is required to determine f(x) from x. The information in y = 2x is low because minimal work (as counted by number of simple steps in the most efficient algorithm known) is required to determine g(x) from x. Somewhat more information is found for values of h(x) = x^2 because the computation is slightly slower.
A curve whose values hold maximum information -- implying the most work to arrive at an arbitrary value -- would be one whereby the best method of determining f(x+k) requires knowledge of the value f(x). Many recursive functions fit this category. In that case, we would say that a computed value whose computational work cannot be reduced from n steps of the recursive function or iterative algorithm holds maximum information (if we don't do the work).
So let's say we have the best-arranged sieve of Eratosthenes to produce the sequence of primes. On an xyz grid, we map this discrete curve z = f(y) over y = x^2, using only integer values of x. Now suppose we perceived this system in some other way. We might conclude that a chaotic system shows some underlying symmetry.
It is also possible to conceive of two maximally difficult functions mapped onto each other. But, there's a catch! There is no overall increase in complexity. That is, if f(x) is at maximum complexity, g(f(x)) cannot be more complex -- though it could conceivably be less so.
This conforms to Wolfram's observation that adding complexity to rules does little to increase the complexity of a CA.
Now what about the idea of "phase transitions" whereby order suddenly emerges from disorder? Various experiments with computer models of nonlinear differential equations seem to affirm such possibilities.
Wolfram's New Kind of Science shows several, as I call them, catastrophic phase transitions, whereby high entropy rapidly follows a "tipping point" as defined by a small number of rows. Obviously one's perspective is important. A (notional) graph with 10^100 iterations could have a "tipping point" composed of millions of rows.
Wolfram points out that minor aymmetries in a high entropy graph up to row n are very likely to amplify incrementally -- though the rate of change (which can be defined in several ways) can be quite rapid -- into a "complex" graph after row n. I estimate that these are low entropy graphs, again bearing in mind the difference between true randomness and deterministic chaos or complexity: the entropies in most cases differ.
What we arrive at is the strong suggestion -- that I have not completely verified -- that a high information content in a particular graph could easily be indicative of a simple local rule and does not necessarily imply an externally imposed design [or substitution by another rule] inserted at some row n.
However, as would be expected, the vast majority of Wolfram's graphs are high-entropy affairs -- no matter what criteria are used -- and this fact conforms to the anthropomorphic observation that the cosmos is en toto a low entropy configuration, in that most sets of the constants of physical law yield dull, lifeless universes.
I should note that New Kind of Science also analyzes the entropy issue, but with a different focus. In his discussion of entropy, Wolfram deploys graphs that are "reversible." That is, the rules are tweaked so that the graph mimics the behavior of reversible physical processes. He says that CA 37R shows that the trend of increasing entropy is not universal because the graph oscillates between higher and lower entropy eternally. However, one must be specific as to what information is being measured. If the entropy of the entire graph up to row n is measured, then the quantity can change with n. But the limiting value as n goes to infinity is a single number. It is true, of course, that this number can differ substantially from the limiting value entropy of another graph.
Also, even though the graphs display entropy, the entropy displayed by physical systems assumes energy conservation. But Wolfram's graphs do not model energy conservation, though I have toyed with ways in which they might.
The discussion above is all about classical models arranged discretely, an approach that appeals to the computer science crowd and to those who argue that quantum physics militates against continuous phenomena. However, I have deliberately avoided deep issues posed by the quantum measurement/interpretation problem that might raise questions as to the adequacy of any scientific theory for apprehending the deepest riddles of existence.
It should be noted that there is a wide range of literature on what the Santa Fe Institute calls "complexity science" and others sometimes call "emergence of order." I have not reviewed much of this material, though I am aware of some of the principle ideas.
A big hope for the spontaneous order faction is network theory, which shows some surprising features as to how orderly systems come about. However, I think that Wolfram graphs suffice to help elucidate important ideas, even though I have not concerned concerned myself here with New Kind of Science's points about networks and cellular automata.
This first appeared on Sunday, June 24, 2007
The Kalin cipher
Note: The Kalin cipher described below can of course be used in tandem with a public key system, or it can be done by hand with calculators. An appropriate software program for doing the specified operations would be helpful.The idea of the cipher is to counteract frequency analysis via matrix methods.
Choose a message of n characters and divide n by some integer square. The remainder can be padded out with dummy numbers.
Arrange the message into mxm matrices, as shown,
H I H x
O W A y
R E U z
We then augment the matrix with a fourth column. This column is the first key, which is a random number with no zeros. A second 4x3 matrix is filled with random integers. This is the second key. The keys needn't be random. Easily remembered combinations might do for some types of work because one would have to know the cipher method in order to use guessed keys.
We then matrix multiply the two as MK and put MK into row canonical form.
This results in the nine numbers in M being reduced to three in [I|b], where b is the final column.
8 9 8 x 7 9 2 1 0 0 a
8 15 1 y 5 5 4 0 1 0 b
18 5 21 z 3 2 3 = 0 0 1 c
5 2 3
where (a, b, c) is a set of three rationals.
The keys can be supplied by a pseudorandom number generator in step with a decoder program. A key can vary with each message matrix or remain constant for a period, depending on how complex one wishes to get. But, as said, if one is conducting an operation where it is advantageous for people to remember keys, this method might prove useful.
By the way, if a row of the original message matrix repeats or if one row is a multiple of another, a dummy character is inserted to make sure no row is a multiple of another so that we can obtain the canonical form. Likewise, the key matrix has no repeating rows.
In fact, on occasion other square matrices will not reduce to the I form. A case in point:
a+b b+c a+b
a b c
1 1 1
The determinant of this matrix is 0, meaning the I form can't be obtained.
In general, our software program should check the determinant of the draft message matrix to make sure it is non-zero. If so, a dummy letter should be inserted, or two inconsequential characters should be swapped as long as the meaning isn't lost.
But, if the program doesn't check the determinant it will give a null result for the compression attempt and hence would be instructed to vary the message slightly.
Notice that it is not necessary to transmit I. So the message is condensed into three numbers, tending to defeat frequency analysis.
Here is a simple example, where for my convenience, I have used the smallest square matrix:
We encrypt the word "abba" (Hebrew for "father") thus:
1 2 3
2 1 1
where the first two columns represent "abba" and the third column is the first key.
We now matrix multiply this 2x3 matrix by a 3x2 matrix which contains the second key.
1 2 3 x 2 1 = 10 16
2 1 1 1 3 7 8
2 3
We then apply key 1 (or another key if we like) and reduce to canonical form:
10 16 3
7 8 1
which is uniquely represented by
1 0 -1/4
0 1 11/32
By reversing the operations on (-1/4, 11/32), we can recover the word "abba." If we encode some other four characters, we can similarly reduce them to two numbers. Make a new matrix
-1/4 x 3
11/32 y 1
which we can fold into another two number set (u, v).
We see here an added countermeasure against frequency analysis, provided the message is long enough: gather the condensed column vectors into new matrices.
We can do this repeatedly. All we need do is divide by, in this case 4, and recombine. If we have 42n matrices to begin with, it is possible to transmit the entire message as a set of two numbers, though usually more than one condensed set would be necessary. Of course we can always pad the message so that we have a block of x2n. Still, the numbers tend to grow as the operations are done and may become unwieldy after too many condensations. On the other hand, frequency analysis faces a tough obstacle, I would guess.
So a third key is the number of enfoldments. Three enfoldments would mean three unfoldments. Suppose we use a 4x4 system on 64 characters with three enfoldments.
We write this on 16 matrices. After the transform, we throw away the I matrices and gather the remaining columns sequentially into a set of 4 matrices. We transform again and are left with a single column of four numbers. So if the adversary doesn't know the number of enfoldments, he must try them all, assuming he knows the method. Of course that number may be varied by some automated procedure linked to a public key system.
Just as it is always possible to put an nxn matrix without a zero determinant [and containing no zeros] into canonical form, it is always possible to recover the original matrix from that form. The methods of linear algebra are used.
Decryption is also hampered by the fact that in matrix multiplication AB does not usually equal BA.
The use of canonical form and the "refolding" of the matrices is what makes the Kalin cipher unique, I suppose. When I say unique, I mean that the Kalin cipher has not appeared in the various popular accounts of ciphers.
An additional possibility: when an nxn matrix is folded into an n column vector, we might use some "dummy" numbers to form a different dimension matrix. For example, suppose we end up with a 3-entry column vector. We add a dummy character to that string and form a 2x2 matrix, which can then be compressed into a 2-entry column vector. Of course, the receiver program would have to know that a 2-vector is compressed from a 3-vector. Also the key for the 2x2 matrix is so small that a decryption program could easily try all combinations.
However, if a 100x100 matrix folds into a 10-entry column vector, six dummy characters can be added and a 4x4 matrix can be constructed, leaving a 4-vector, which can again be folded into a 2-vector. All sorts of such systems can be devised.
Additionally, a "comma code" can be used to string the vectors together into one number. The decipherment program would read this bit string to mean a space between vector entries.
Clearly the method of using bases or representations as keys and then transforming offers all sorts of complexities -- perhaps not all useful -- but the emphasis on matrix condensation seems to offer a practical antidote to frequency analysis.
BTW, I have not bothered to decimalize the rational fractions. But presumably one would convert to the decimal equivalent in order to avoid drawing attention to the likelihood that the numbers represent canonical row vectors. And, of course, if one is using base 2, one would convert each rational to a linear digit string. Not all decimal fractions can be converted exactly into binary. However, supposing enough bits are used, the unfolded (deciphered) number will, with high probability, be very close to the correct integer representing the character.
Toward a signal model of perception
http://paulpages.blogspot.com/2013/03/toward-signal-model-of-perception.html
Synchronicity anecdotes
http://paulpages.blogspot.com/2013/04/appendix-anecdotal-accounts-of.html
Please send corrections and comments to Krypto78@gmail.com
The experiments cited here must be regarded as little more than anecdotes. Not only do my temperament and circumstances make a rigorous experimental approach unfeasible, replicability is a big issue because of the variability of the proposed reality signals. As with the anecdotes of Appendix A, we cannot say that statistical analysis would reveal a non-random covert correlation.
Hopefully, more talented investigators will devise experiments that will be, if not altogether convincing, at least more suggestive than the results I have obtained. In fact, I have obtained a number of remarkable "manufactured synchronicities" in times past but either took no notes or lost them. What remains is, alas, not the strongest stuff.
Techniques for achieving a degree of control over reality projection are centered on belief and focus. However, the unconscious strongly influences these actions and so we may often find somewhat goofy synchronicities, as with Jung's "fish story" related in my main article.
The experimenter will be able to devise endless varieties of "interference" experiments but will find that if the pattern mix is overly complicated, the resulting phenomenon signal may very well show no noteworthy signs of blending, interference or coherence.
I have found, and anecdotal accounts from history tend to confirm, that the use of a mirror or mirrors can have a major impact on setting off a burst of similarly constructed reality subsignals. Also, it seemed plausible that visual symmetry might have import. Yet I wished to limit the number of variables. So in one set of experiments, I chose two- or three-digit numbers by a pseudorandom process; drew the resulting number and its mirror symmetry; and then placed a mirror before this inscription such that I viewed the inscription and its reflected image simultaneously. I also rocked the mirror because previous tests had indicated that that might add to the effect.
Though I cannot say why mirrors might be useful, it is notable that quantum effects are often associated with photon interference. I know that sounds like hocus-pocus, and I admit to feeling my way along here.
I decided to use low integers because of their simplicity, which I believed might reduce variability and because they usually (but not always) had a low E-value. I also wondered what might happen using prime numbers of low E-value.
Later I would pair a prime with a pattern carrying a relatively high E-value, again using the mirror process.
I qualitatively determined that when low E-values are used for every element of an image, the aftermath is usually unremarkable.
Eventually, I used a method I'd developed a while back of studying mathematics while keeping some images -- such as books -- on my desk. The synchronicities are easily as strong or stronger than anything done with mirrors. My thinking is that math study, which requires sharp concentration and focus, is effective in bringing about "echoes" from the images.
A note of caution is in order. Tinkering with the reality stream could easily yield unpleasant results, analagous to the manner in which LSD can bring on a bad experience. It may help to include in the experiment some image associated in the experimenter's memories with goodness. However, I cannot guarantee experimenter safety. Such experiments are akin to Ben Franklin touching a key during a lightning storm; he just got a bit of a tingle whereas an imitator was electrocuted.
Another note: the system crashed while I was working on this page, killing about two hours worth of work. When I went to rewrite the material, the notes concerning the segment at which I left off went missing, and remain so. There is no reason to think those notes were dropped or pilfered. I have omitted further discussion of that experiment (which was about nothing sinister or spectacular).
In retrospect, many of the "echoes" are anything but spectacular, but often my subjective impression at the time was that the echo "jumped out" at me, as if there was something strange or unusual about it. Clearly, though, one can always find a routine psychological explanation for such a sensitive reaction.
As of June 30, 2009, the final experiment on this page is dated June 3, 2009. In future, I will no doubt perform more experiments and include them on this or a related page. I likely will try to increase the E-value and, at least for a few tests, simplify the symbolism.
February 2008; day missing
I focused on this pair [31, RED] for less than a minute at about 3 p.m.
About 45 minutes later, I was in a local library and noticed a 12-year-old neighbor busy with a school project. She was doing some sort of survey and asked a fellow pupil: "What is your favorite color?" (The reply was inaudible.)
I had a sense that this was an echo. Victoria's question was about color, even if not red. And she was working on something numerically oriented, even if not specifically about the number 31.
Later, I went to stay at a friend's apartment and upon reaching the doorway, I noticed that the building number was 131. Of course there are 10!/8! = 90 permutations of two digits, giving a 1/90 chance of encountering the number 31. But there is also a 1/90 chance for any other two digits, so in probability terms, this is unspectacular.
Also, there was no particular echo of the color red in this observation.
Feb. 8, 2008
At about 5:30 p.m., I obtained the number 978 from the last three calculator digits for 51/2 and flexed a mirror in front of a symmetrical image composed of these digits.
The image took this general appearance
9 7 9 7 8 7 9 7 9
except that the top two numbers were facing the other direction, as were the two right-most numbers.
As I was interested in checking a symmetrical design, I also drew the 8 as two circles and the 9's as a circle with a straight line segment tangent to it.
Later that evening I attended a seminar in which three very fine women reported on life-changing experiences. One speaker, who was one of seven siblings, described how her husband had died unexpectedly from three blood clots "in the lungs," though he had been taking Lipitor for high cholesterol.
Several echoes:
I tend to associate the numbers 9 and 78 with good things (a numerological idea I've picked up that I admit is hard to defend rationally). The women struck me as of especially fine character.
The two backward 9's were like P's, representing "pulmonary."
The 8 and the 9's looked like pills, representing the Lipitor.
Feb. 8, 2008
At 10:44 p.m. I chose the last two calculator digits for 110.71, which were 69.
This time I inscribed 6969 on a piece of paper, but had the second two digits facing the other direction.
I flexed the mirror about 10 times while looking at the figure and its reflected image.
After I had gone to bed, a teenage daughter of a close friend text-messaged me, asking that I fetch her some fast food. Her friend, a daughter of another close friend, also wanted food. (I had known both of them for years.)
I dropped off the food, and one of the girls, when she came downstairs, was a bit inappropriately attired (though not in her view). She told me her report card had two A's, three B's and three C's, a substantial improvement.
In our culture, of course, the number 69 is sexually suggestive. However, the incident was harmless.
I thought an echo had occurred because, though these girls were always asking little favors, the lateness of their call was a first.
There were other things I might report that could indicate echoes, but they are too nebulous to be worth detailing.
Feb. 11, 2008
At 10:02 a.m., the number 51 was chosen randomly from a book page. I inscribe 515 on a piece of paper with the last 5 facing the other direction and then rocked a mirror in front of the figure several times, using four rotations.
At about 11:30 a.m. I ran into a woman I know at a local library, a place I had never before seen her before. She had, unknown to me, recently moved into town with her husband and daughter.
I had used a book discarded by that library to select the "random" page number for my experiment.
Any other echoes, if any, are too nebulous to report.
Feb. 11, 2008
I first obtained 116, the last three calculator digits of the fifth root of pi. The prime decomposition of 116 is 2229 and so I used 29.
My notes record that a prime has an E-value for me because of my association of a prime with "hardness."
At 5:14 p.m., I inscribed on a piece of paper 2992, with the last two digits facing the other way and then put that paper in front of a large mirror. I then flexed a hand-held mirror in such a way that I could see the inscription in the hand-held mirror as reflected from the large mirror. I flexed the large mirror several times and also tried different positions for the hand-held.
During the mirror-flexing, I caught a glimpse of my abdomen and I wondered whether an echo concerning my belly or my body would occur.
After a nap, I went to the bathroom and found that a roommate had moved a sign reading "please wash out the shower, don't forget" to a spot under the mirror. The sign was intended for me.
I sense this as an echo though not a very pronounced one.
Later -- notes unclear about time -- I had to break an engagement with a friend set for the 19th because I had forgotten that I already had booked that date for another purpose. Apart from the number 9, there is little here.
Feb. 11, 2008
Unsatisfied with most results thus far, I tried another method in an attempt to get a more pronounced coincidence.
Using a pseudorandom process, I paired these two prime numbers thus:
(7)(19)
and then flexed the hand mirror about 15 times such that I could see the inscription doubly reflected, but stood aside so that very little of my image was seen.
The next day, there were several possible echoes but only one seems worthy of reporting. About 3 p.m., I read that Katherine Graham had been appointed publisher of the Washington Post. She was to report to her uncle, Post company CEO Donald Graham.
A "bit of me" is echoed here because I once served alongside Graham as an Army combat correspondent.
Weymouth and Graham are both "primes" in the Post company. Their family ties correspond to the pairing. Their independence is emphasized by the parentheses around the numbers. The ratio 7/19 approximates their age ratio.
Though these associations seem nebulous, at the time my subjective sense was that they were strong echoes.
Feb. 12, 2008
I flexed my hand mirror such that I could see reflections of this inscription:
E = ymc2 K = 1/2mv2 5,997
(The y was the Greek gamma.)
The number 5,997 was chosen by a pseudorandom process.
My thought was that perhaps the energy equations would have a higher E-value than mere numbers and hence yield a more pronounced coincidence.
I have notes concerning a possible echo in Abraham Pais's biography of Einstein, Subtle is the Lord, but I do not think it is worth including.
Feb. 12, 2008
I covered the back of the hand mirror with white paper, because I was concerned that its smoky design was muddying outcomes. At about 7:05 p.m., I flexed the mirror several times at an image I had taped to the wall. I am not sure what that image was, but was probably the inscription above.
I then held the mirror in front of the image and flexed it a few times in such a way that I couldn't see the reflection.
The following day, I spent most of the time in bed, sick with a cold. Checking my cell phone at one point, I found two text messages from one of the girls mentioned above. One was typical, asking that I bring her lunch to school (which I didn't, having been asleep). The other one was odd, asking if I had the number for Mr. Ice Cream.
I then realized that the white-backed mirror I had flexed looked rather like an ice cream cone.
I considered this coincidence to be a rather strong echo.
Feb. 18, 2008
By a pseudorandom process, I selected the prime 23 and performed a test with it. My notes of the test details are missing.
However, my notes record that sometime after 9 p.m. I was at a service organization meeting where various outreach opportunities were presented, including one for March 23. There was a short discussion about that date, as it was to fall on Easter.
This also seemed a rather pronounced emergence.
After this test, I ceased from experimenting with mirrors. Instead, I would put books with interesting titles on a table where I would study mathematics and check for rebounds. I have often done this in the past, getting numerous compelling echoes, but have no notes of those tests.
March 10, 2008 For an hour or two before 3:24 p.m. I did drill questions for second order differential equations. On the library table were these books: Just Six Numbers: The Deep Forces that Shape the Universe by Martin Reese and The Nine Numbers of the Cosmos by Michael Rowan Robinson. I only glanced at their content.
After finishing my study session, I checked the internet at saw that New York Governor Eliot Spitzer had been implicated in a prostitution ring. The numbers 6 and 9 (from the book titles) are, in American culture, considered sexually suggestive when written as 69. I recall thinking before checking the news that something sexy and maybe tawdry would show up as an echo. Interestingly, Spitzer was code-named "Client 9."
March 11, 2008
Did differential equations drills in the library between 11 a.m. and 1 p.m. or so. On the table were The Golden Ratio: The Story of the World's Most Astonishing Number by Mario Livio and The New Time Travelers: A Journey to the Frontiers of Physics by David Toomey. I glanced inside both books and sometime during my study session I noticed in Livio's book on the use of the number 666, which can be plugged in to a formula to get a close approximation of the negative of the golden ratio. I did some calculator manipulations but beyond that my notes are unclear.
In my mind, Middle Eastern events are often associated with the "number of the beast," because of the fact that much of the book of Revelation focuses on that region. The Iraq and Afghanistan wars, stemming from the insider atrocity known as the 9/11 attacks, make me think of the beast and the last days.
And so I saw an echo in the report that the top Middle Eastern commander had been fired for apparently emphasizing a moderate policy as opposed to the highly aggressive Bush-Cheney policy. The "666" is represented by the removal of a restraining hand.
As for any "time travel" or "golden ratio" echoes, my notes contain nothing worth reporting. This is not to say there are no echoes, but only that they are insufficiently pronounced to be useful.
March 12, 2008
At 1 p.m. I began to study differential equations with these books on the table: Infinite Ascent: A Short History of Mathematics by David Berlinski and A New Kind of Science by Stephen Wolfram.
I penciled in a cross inside NKS and took a brief look at Wolfram's cellular automata graphs (I had read most of the book some months earlier). Peeking inside Berlinski's book, I noticed he referred to NKS.
I noted a personal experience later that day but I think it unworthy to include, though I may reconsider in future.
However, the next day (time not noted) I read in the New York Times an obituary for the last French veteran of World War I. The cross in the "cellular automata" book is echoed by the fields of crosses in France and the low countries where the dead of World War I are buried. Armies reflect NKS's cellular automata and war reflects NKS's discussion of the mathematical topic of chaos theory.
French Premier Nikolas Sarkozy expressed "infinite sadness" in his tribute to Lazare Ponticelli, reflecting the "infinity" in the Berlinski book's title. That book also has the word "history" in its title, and the obituary concerns a momentous historical event.
The cross is also reflected in Ponticelli's comment on war. It is "stupid," he said. "You shoot at men who are fathers."
March 18, 2008
Sometime between 5 p.m. and 6:30 p.m. I was studying differential equations in a library. On the table were Extreme Waves by Craig B. Smith, showing on the cover a sailboat in peril; Blow the House Down, a novel by Robert Baer, a "bestselling N.Y. Times author"; Physics of Waves by William C. Elmore and Mark A. Head.
An orthodox Jewish man joined me at the table, where he placed three novels: The 5th Horseman, The Greatest Battle, and Sins of the Assassins. The first two were Apocalyptic fiction. He read from at least one of these books and also read the New York Times and the New York Sun.
About 10:45 p.m. the following day I read in the Times the obituary of Arthur C. Clarke, who wrote noted Apocalyptic science fiction novels. For example, Childhood's End features a race of aliens who look like devils.
Clarke was known for his efforts to steer humanity away from war, an echo of Sins of the Assassins. The writer had degrees in physics and mathematics, echoing the wave physics book and the math book. A scuba diving buff, he lived on Sri Lanka, which was struck by the devastating (almost Apocalyptic) tsunami of 2004. The Extreme Waves book discussed that tsunami, which killed 40,000 Sri Lankans.
Also, a very large rainstorm struck the central states today, causing massive flooding and killing 9 persons.
March 19 or 20, 2008
At about 6 p.m., I noted that I had been doing differential equation drills over the last 3 1/2 hours. I solved some easily but, because of mild fatigue, had difficulty with others.
On the table:
1. The Anchor Bible commentary on the gospel of John (I - XII), Volume 29, by Raymond E. Brown. I peeked at that and found it interesting.
2. General Relativity, an Einstein centenary collection of papers, S.W. Hawking and W. Israel, editors. The math looked difficult, I noted, but the book might be worth reading anyway.
3. Abstract Algebra by Lloyd R. Jaisingh and Frank Ayres Jr., a Schaum's Outline book (second edition).
Also on the table for a time was my folder full of math theorems, algorithms and formulae.
At any rate, during my study session, and after I had chosen the books, I got an email from a friend John telling me he was stranded in Atlantic City, several hours drive away, and needed a lift.
We fixed a time and place to meet the following morning.
The next day my breakfast and coffee was disrupted at a local eatery by a woman noisily complaining of having lost her wallet and wasting a trip. She then added that she'd been assaulted. I suspected that the high E-values imputed would have a negative affect during the day.
Echoes from the previous day: My friend "John" echoed in the book topic; perhaps my doing a Christian deed echoed in the book title; and "Atlantic City" echoed by the word "Anchor."
As it turned out, we missed each other at the casino and I wasted my trip. It later emerged that my friend had been assaulted and robbed of his wallet. So the woman's negativity echoed in these details.
My notes disclose nothing echoing relativity, other than the fact that we couldn't get our times and places to match up. Apparently, I was at one entrance and he at another, and when I went checking the other entrances, he was elsewhere.
My other notes concerning potential echoes concern things which are simply too nebulous to report.
March 26, 2008
Notes from March 25 are missing.
Mildly edited notes for March 26 have this to say:
"This morning I read on the New York Times front page a story about a new electronic fingerprint identification system introduced at a New York airport. Two Englishmen were interviewed, each with an opposing reaction. One, a businessman, had no problem with the testing; another, a writer, was offended by the intrusiveness.
"I remember thinking yesterday that an airplane echo was likely, since I wrote '7,127' and thought '727'."
The dualism of two prime numbers is reflected in the reporter's choice of two persons with contrasting views. Whether identification is also echoed is now unknown, my notes being lost.
April 22, 2008
I tried to study differential equations in the early afternoon, but was stymied by an allergy attack from accomplishing much. On the library table was the book Codes, Ciphers and Other Cryptic and Clandestine Communications by Fred B. Wrixson (Black Dog and Leventhal Publishers, 1998). Also on the table was a library survey form for April 12-27.
Later on I resumed study and placed on the table the book Star Wars Death Star by Michael Reaves and Steve Perry.
The next day, it was reported that a retired U.S. Army engineer had been accused of spying for Israel.
Espionage and counterespionage are strongly reflected in the title and the name Levanthol is Jewish, a link to the spy suspect, 87-year-old Ben-Ami Kadish, and the state of Israel. The spy case stemmed from the Star Wars era. There was a sudden supernova of exposure of a dark matter. The suspect couldn't have been far from death.
April 24, 2008
At 12:09 p.m., I noted that on the library table were The Invisible Man by H.G. Wells (a Tor Classic paperback); The Nomad of Time a three-novel collection by Michael Moorcock: The Warlord of the Air, The Land Leviathan and The Steel Tsar (Daw Books/Doubleday).
The Wells book concerns a man who takes a potion that makes him invisible and then into a psychotic killer. The Moorcock stories are about a former captain in the Royal Lancers who visits alternate realities.
I studied differential equations.
At 10:47 a.m. the following day, I noted: "re 'invisible man,' 'nomad of time' the echoes were either too personal or too nebulous to report." (I now don't recall, but I don't suppose they were anything terrible.)
April 25, 2008
At 11:38 a.m. I noted that I had leafed through Great American Deserts by Rowe Findley, photos by Wll? Meayers Edwards, National Geographic Society. I left that on the table while I was studying differential equations.
A note of 10:07 a.m. the next morning, recounts that the prior night was full of incidents of being deserted. By doing a favor for one of the teens mentioned above, I forsook an event that a friend had desired to attend with me. It was apparent that he felt deserted. The teenager, focused on her own agenda, deserted me without even a text message or phone call. Later, I realized I had told someone I would meet her at the event I had forsaken -- and hoped she didn't feel deserted. In sum, the evening turned out to be boring and lonely -- somewhat desert-like, that is.
Nov. 6 or 7, 2008
At about 3:30 p.m., I had on the library table the book The 23rd Cycle, which discusses solar flare activity.
I used my calculator to perform these operations:
(1 - 23-1 + 1 - 23-2)2 = 3.820583832 (1 + 23-1 + 1 + 23-2)2 = 4.18352732 ---------- 8.004116624
I repeated the calculations and noted that the interval for these exercises was about 10 minutes. I then busied myself with other matters.
Later that day I wrote a bit of fluff on my blog about "number 23" whimsies (I have since deleted it).
At between 8 p.m. and 9 p.m. that day, an acquaintance told of a man whose son, only in his twenties, had just died. The man had overcome cancer as a teen, but it had recurred a year ago and had hit full force.
Comments: "Son" is a homonym for "sun." The cancer was a sonspot; the cancer flareup echoes the surge in sunspot activity, which goes in cycles. There was an eight- or nine-year interval between the cancer remission and flareup, echoing the sum between 8 and 9 calculated.
Prior to that, at roughly 6 p.m. and after my calculator exercise, a teenage girl sent me a text message reading "SON!", something she'd never done before or since. I checked the internet for a possible meaning and came up with "signing off now." As I recall, she later told me she'd intended that message for someone else."
Also, it should be noted that my reality formulation may have been influenced by the alleged "badness" of number 23, even though that was something I attempted to refute in my little blog post.
There is a break in my notes of about a year. I had done few experiments and hadn't bothered to record anything. I resumed recording results once I began to prepare to publish online Toward a signal model of perception.
April 13, 2009
On the 14th, I recalled that on the 13th I had been using scrap paper found in a trash container. On the printed side was some sort of legal document.
I was doing exercises in complex numbers, specifically trying to calculate 120.5 correctly, eventually getting it right.
During a walk, I happened upon an expensive graphing calculator, which I turned over to the university police. These happenings echo, of course, the slightly advanced math and the legal papers.
Later that evening, I was working out the complex number (1 + i)i and then (3 + 5i)i. My erroneous answers were 20.5/e and (34)0.5e-5i. Moments after arriving at the wrong answers, my young friend called to say that she was lost on Route 34. Shortly thereafter she was pulled over by a policeman for driving too slowly on that road. On the 14th, I noted, she told me she had also been robbed of her wallet, which contained $50.
The clearest echoes are the number 34 and the things going wrong, associated with my wrong calculation.
April 14, 2009
At 2:01 p.m., I wrote that I would use the back of scrap legal documents to do complex analysis drills, and that on my apartment desk I had placed a Bible and Dostoyevsky's Crime and Punishment.
About 4 p.m., I quit working. For an exercise, I worked on the solution to (9 + 11i)i, choosing the numerals because of high E-value and not specifically out of any desire to wreak mayhem.
I then turned on the television, where I watched a news story about a woman Sunday school teacher charged with raping and killing an eight-year-old girl. The woman wept at her arraignment.
The story had been running for several days, but this was the first I knew of it.
So we have these associations: legal papers --> law court, 9 and 11 --> a heinous act, Bible --> Sunday school teacher, Crime and Punishment --> the whole scenario, including the remorse of the killer.
Curiously, I didn't actually compute the correct answer.
Later that evening, I watched a snippet of Law and Order and noticed some echoes. But as the echoes are typical of TV melodrama I have decided to omit them.
However, I wrote, at 8:59 p.m. I returned to trying to correctly compute that quantity with the Bible and the Dostoyevsky book still in place.
I also tried to compute (9 + 11i)i(11 + 9i)i.
As soon as I ceased studying and turned on the TV, there was this return: A melodrama about a 12-year-old girl and Satanic stuff with a cop saying about some beastly thing: "The spell is real." I turned the TV off.
May 12, 2009
On the library table as I studied complex analysis during the afternoon were the books Hitler's Scientists by John Cornwall, Jungles of Randomness by Ivars Peterson and Coincidences, Chaos, All That Math Jazz by Burger and Starbird, and Elementary Differential Equations by Lyman M. Kells. Also on the table was the newspaper The Jewish State, a fairly militant publication.
About 5 p.m., I picked up the day's Wall Street Journal at the library and scanned a story at the bottom of page one raising the question of whether amateur gene-splicers posed a threat to national security. The headline included this phrase: "Let out your inner Frankenstein." Concern was expressed over the idea that genetic technology might fall into the hands of terrorists who might unleash some plague virus, but the experts quoted didn't seem very troubled.
We see the echo of evil in science here, possibly muted by the fact that I found the Hitler's Scientists book to be only mildly interesting. And any evocation of international terrorists carries with it the association of pro-Israel militancy.
Amateurs getting involved in gene-splicing might be construed as an echo of randomness. The probability of a catastrophe becomes so unpredictable as to be essentially random.
May 16, 2009
During the morning and afternoon I studied complex analysis, and on the library table I had placed Essential Topology by Martin D. Crossley (Springer) and Explorations in Topology: Map Coloring, Surfaces, and Knits by David Gay, both of which I glanced through, and Teach Yourself Sanskrit, a Complete Course for Beginners, which I also scanned.
On the table for a brief period was A Beautiful Math, a book about John Nash, game theory and the "quest for a code of nature." Also for a short time there was a flyer about local jazz and arts, which I put away.
At 6:59, my notes say, three women sharing a train compartment with me struck up a conversation about a woman who, on seeing her deceased husband's soul enter another body, swiftly marries this man.
This "random" chatter concerns transmigration of souls, a subject of the Upanishads and Rig-Veda. The Sanskrit book's stated intention was to help people to read these Indian writings in the ancient language.
At 7:57 p.m., my notes say, I had watched a public television travelogue, "Adventures with a Purpose," about the marvels of travel high in the Swiss Alps. I was struck by the intensity and vividness of the scenery and discussion. It occurred to me that topology might be thought of in terms of the cool climes of abstract spaces, and space was a very strong theme of this travelogue.
Sometime after I got off the train but before I watched this show, I stopped into the supermarket. One cashier, referring to a promotional gimmick, said to another: "You have to tell the people about the game we have." The response: "I don't know anything about that game." Then the clerk asked whether I wanted a game sheet and, on learning what it was about, I replied, "I'm not too interested."
It was only an hour or two later that I suddenly recalled that I had initially included the game theory book on the table, but then, thinking better of it, put it on another table. (Other experiences, which I have not chronicled, suggest that game theory can bring unpleasant echoes; also the more one uses for interference, the more probable a muddy result.)
I had visited this store often but, for me, the game materialized only that evening. After I brushed off the game theory book, a clerk and I each gave the game the brushoff.
May 19, 2009
The two topology books and the Sanskrit book mentioned in the previous segment were on the library table as I studied complex analysis, beginning at about 2 p.m.
At 10:14 that evening, I began a note concerning a NOVA special I had just watched: "Parallel Worlds/Parallel Lives," concerning Mark Everett, son of the late physicist Hugh Everett. An independent producer had prodded Mark, an alternative rock musician, to learn more about his dad and his dad's many worlds interpretation of quantum physics.
A lack of emotional bonding troubled the family, and so Mark, though he had loved his father, felt as though he hadn't really known him.
I must say that I wasn't very interested in the day's experiment and its symbolism and so perhaps the result wasn't quite as vivid in my mind as in the previous episode. In fact, I didn't look at those books that day. My emotional vacuum may have been echoed by the family's emotional vacuum.
Anyway, the many worlds scenario echoes the strange realm of topological spaces and also the astonishing worlds of the Indian classics. The parallel lives and worlds of father and son are echoed by the two topology texts.
The special included a scene of Mark looking at a string of math symbols which was counterpointed with his slowly gaining new insight about his father's work. We may regard this as an echo of the study of the strange-looking symbols of Sanskrit. His sister, it was reported, had killed herself in order to go to be with her father, also a possible echo of Indian theology.
May 20, 2009
On my apartment desk are two Bibles -- a King James version and a New American Standard version -- and the chilling chronicle, The War Against the Jews by Lucy S. Dawidowicz (Holt, Rhinehart and Winston; Bantam edition June 1976), which I read decades ago. Inside that book was a Catholic devotional card with a prayer to Michael and a painting of the archangel slaying Satan. My notes give the time of this observation as about 3:45 p.m.
Feeling a bit unwell, I was able to do only a few routine complex analysis problems, quitting at 4:59 p.m. Shortly before putting down the math book, my roommate had called out to invite me to watch what he thought would be an interesting movie; it was about pedophilia in the Catholic church in the 1960s.
At 11:27 that evening, I noted that the news program World Focus included a brief item about pedophilia and child abuse in Catholic institutions in Ireland. (The show is prepared before 6 p.m. daily.)
The next morning's New York Times contained a major story on the child abuse report. Another page one story was headlined "4 Accused of Bombing Plot at Bronx Synagogues." Another, "Lawyer's Ways Spelled Murder..." concerned a former prosecutor and defense attorney accused of suggesting that witnesses be murdered.
Echoes: Michael image and the Bibles --> struggle against evil in the church; Michael image in The War Against the Jews --> the plot to bomb synagogues averted; Michael slays Satan --> murderous lawyer exposed; also, implicitly, Hitler defeated.
June 2, 2009
I returned to my mirror method on this day. At 1:28 p.m., I wrote that I had taped to the wall the page one segment of a New York Times story headlined, "Plane Vanishes Carrying 228; Cause a Puzzle." Next to the story I inscribed the numbers 22.57, 225.3.11 and 2352. The first is the prime decomposition of 228 and the latter two stem from the numbers found in A330-200, the model number of the Airbus that vanished off Brazil.
I flexed a folding double hand mirror in front of the report, with the two mirrors at about a 45 degree angle or less.
I initially caught a glimpse of myself, but then mainly noticed two blobs of light playing over the type for about 30 seconds.
An hour or so later, my roommate returned and showed me a set of photos taken at a rustic retreat that we'd visited a few weeks previously. The pictures were almost all of trees, the lake, cabins and so forth. A very few pictures had familiar faces in them.
There may have been some minor echoes, but the most notable one was the matter of the missing people. There had been some 200 people at the conference but hardly any were in the photos. It would be days before a few bodies were recovered from the Atlantic; all others remain missing as of June 29, 2009. If the numbers had any echoes, they were insufficient to be worthy of notice.
June 3, 2009
The following was written on a piece of paper:
8 71 + 17 8
Just after 2:14 p.m., I flashed the hand-held folding mirror in front of the inscription above, which I had taped to a wall in my apartment. I flexed the mirror in various ways, continually observing the reflection of the numbers.
Later, while sitting on a bench in a vest-pocket park I glanced at my cell phone and saw the time: 7:17; suddenly two girls emerged and were frolicking around the fountain of around a Civil War monument in front of me. Each appeared to be about 7 or 8 years old.
About a half hour later a friend gave me some phone numbers of some people I might need to be in touch with. I had trouble distinguishing her 8's from her 3's and had to ask her to verify an 8.
A bit later, someone else I know told me of newly discovered heart artery problems. I recall him mentioning that one artery had an 80 percent blockage.
A note of June 5, (may mean June 4) 2009, says that I checked the obituaries of David Carradine, who died June 4 in a Bangkok hotel. One story related that the 72-year-old had recently begun shooting a film (Stretch), which concerns a 17-year-old who publicly announced that he was going to kill himself before doing so.
Carradine's body was found hanging with a rope around his neck and genitals. Death apparently resulted from "auto-erotic asphyxiation."
Echoes: 71 --> age 72 (close but not exact); 17 --> age 17; two 8's symmetrically placed --> nooses about neck and genitals and suicide of 72-year-old and 17-year-old.
Now this may seem unpleasant, but let us recall the fine echoes of the two girls. The issue is not the creation of life or death, but the tuning in of reports. As an analogy, all sorts of things are broadcast on TV or via the internet, but one can't process all reports. Similarly, many "histories" are enfolded into reality, but only some are "energized" insofar as the observer is concerned.
Appendix A: anecdotal accounts of synchronicity
Toward a signal model of perception
Synchronicity 'experiments'
First posted June 25, 2009
Please send corrections and comments to Krypto78@gmail.comPresented here is a collection of anecdotal accounts of what some may believe are episodes of synchronicity, along with my comments. I have not sought to arrange these accounts in any particular order.
We must frankly admit that none of these incidents could be shown, using standard statistical tools, to result of non-random covert connections.
Pauli's pull
The physicist George Gamow gave a humorous example of the Pauli effect in his book Thirty Years that Shook Physics (Dover reprint 1985):
"It is well known that theoretical physicists cannot handle experimental equipment; it breaks whenever they touch it. Pauli was such a good theoretical physicist that something usually broke in the lab whenever he merely stepped across the threshold. A mysterious event that did not seem at first to be connected with Pauli's presence once occurred in Professor J. Franck's laboratory in Gottingen. Early one afternoon, without apparent cause, a complicated apparatus for the study of atomic phenomena collapsed. Franck wrote humorously about this to Pauli at his Zurich address and, after some delay, received an answer in an envelope with a Danish stamp. Pauli wrote that he had gone to visit Bohr and at the time of the mishap in Franck's laboratory his train was stopped for a few minutes at the Gottingen railroad station. You may believe this anecdote or not, but there are many other observations concerning the reality of the Pauli effect!"
Similarly, the physicist Abraham Pais relates in his book Niels Bohr's Times (Oxford 1991), that Pauli was very proud of the "Pauli effect" whereby, beginning in 1922, "something would go wrong whenever he entered a laboratory." Pauli "would tell with glee how his friend Otto Stern, experimentalist at Hamburg, would consult him only through the closed door leading to his working space."
Wolfgang Pauli, a founder of quantum mechanics, seems to have come to believe in a phenomenon that his former therapist and friend, Carl Jung, dubbed "synchronicity," though he was careful about his phrasing.
His number came up
The novelist William S. Burroughs was fascinated by the number 23, recounts Robert Anton Wilson, a writer and mystic. "According to Burroughs, he had known a certain Captain Clark, around 1960 in Tangier, who once bragged that he had been sailing 23 years without an accident. That very day, Clark's ship had an accident that killed him and everybody else aboard. Furthermore, while Burroughs was thinking about this crude example of the irony of the gods that evening, a bulletin on the radio announced the crash of an airliner in Florida, USA. The pilot was another Captain Clark and the flight was Flight 23."
"Burroughs began collecting odd 23s after this gruesome synchronicity," Wilson wrote in the May 2007 issue of Fortean Times.
No scientist would consider Wilson, with his peculiar numerological theories, as a credible source, nor the Fortean Times as a reputable forum.
I have no interest in promoting any form of numerology, but I would say that such a scenario might reflect Burroughs' own mind. We don't know what he'd been thinking about or doing before this symmetrical coincidence set, but in our model reality signals at later times reflect interference of signals at an earlier time. The "meaningful coincidences" are similar subsignals triggered by a previous interference.
Traveling in Tennessee
One morning several years ago my son and I were driving from Knoxville in order to do some hiking in Tennessee mountain country. We had decided to take a scenic route and were barely aware that that route intersected a major interstate highway that linked to Knoxville. As we go to the intersection -- having slowed down after a wrong turn -- a man came down the embankment from the interstate. My son recognized him and we stopped to talk. The man, who my son had met with the day before, had had some difficulties and was hitch-hiking back to Ohio. My son was able to wish him well, say goodbye and give him some money for his journey.
I don't recall what the two of us had been discussing in the car, but I do recall thinking that that intersection of events was no random coincidence.
Safe conduct
A friend I have known for years recently confided this experience: He had gone with a group to go whitewater rafting in Pennsylvania. But there was no room for him on the raft and he rented a one-person kayak so he could accompany his friends.
When hitting some rapids, he capsized and spilled out of his vessel; his friends were nowhere in sight.
My friend is neurologically impaired and so was in quite a predicament, even though the water was not over his head.
At that point, a man appeared and from my friend's vantage point, "it looked like he was walking on water." The man helped my friend out of the stream and then took his leave. My friend was worried about the kayak, because ordinarily such an untended floatable shoots downstream. But, he turned and noticed it floating idly nearby.
My friend was clearly much taken with this experience.
My collage days
A few years ago I regularly indulged my artistic bent my artistic bent by composing collages in ways that I considered interesting and original.
On a number of occasions the "echoes" were quite strong, and occasionally gruesome. In fact, one collage was followed by an extraordinarily gruesome event that reflected the collage quite obviously (to me), and I have decided not to give details out of concern for the feelings of the survivors.
My thinking is that the juxtaposition of images that individually or collectively contain high E-values can have a profound impact on one's virtual world.
In fact, I became reluctant to make more of these and threw out almost all the ones I had made.
The collage technique underscores the interference effect. That is, the images form a superposed set, a collage, that influences the reality formulator, usually within 24 hours.
A cold day in hell
This could fit into the 'experiments' appendix, except that I was too irritable to consider what I was doing a detached experiment.
On the evening of January 23, 2009, I was in the San Diego Public Library glancing through a book on superconductivity, which occurs at extremes of cold. I noticed the word "transpire" used as a synonym for "occur" and was irritated at this unusual and unprofessional usage. In fact, I was so annoyed that I did "numerology" on the word and several anagrams of the word. But once the anger passed, I threw the paper with these scribblings in the trash bin.
The next morning, I encountered an internet story from the Independent about two young fishermen who survived shipwreck inside a floating icebox used for storing fish. They were enormously thirsty when rescued from the shark-infested South Pacific waters and had survived by drinking rainwater deposited by a typhoon that had failed to capsize their icebox.
Bad company
A year or two after the attacks of Sept. 11, 2001, I wrote a web page about the divisibility of numbers by a 9 or an 11.
The following summer afternoon I was enjoying a picnic put on by a spiritually inclined group. In the adjacent picnic grove was a gathering of members of the Pagans motorcycle gang, including women and children.
The Pagans are considered an outlaw gang and are known for their Nazi and occult symbolism.
They weren't raucous but rather subdued. They gave off "strange vibes."
My estimate is that the association that I and others have with the numbers 9 and 11 went into a reality projection of people who many, including myself, think of as engaging in evil.
I have noticed that when doing math, the projections, or "echoes," can be quite vivid.
Two Pied Pipers
When working on this page, I switched over to take a look at the news and noticed that Michael Jackson's heart had stopped and a short time later he was reported dead. I'm wondering if this sudden evil report is an echo of the "collage days" anecdote, which talks about the artist and gruesome happenings; in light of the singer's troubled public image, the Pagans anecdote might also have been a contributor.
After learning about Jackson's death, I stepped out of the library for a bite to eat and upon returning noticed a sign on the door advertising a rock performer who would sing for children that evening. The performer is a friend who is a member of the spiritually inclined group I was with at the picnic (though I don't recall that he was at the picnic). In fact, I later overheard some of his performance but did not attend.
The two singers who entertain children represent two spiritual paths offered in life, on the one hand represented by the hedonistic Pagans and on the other by the spiritually inclined group I was with (we like pleasure but strive against self-defeating pleasure).
Would all this mean that I think I caused a man's death? This type of thinking is akin to the questions that can and cannot properly be asked in quantum mechanics. The best that can be said is that the interference effects are followed by a possibly synchronistic report. The report very often is about an event that precedes the interference, just as would be expected in the discussion in my main article.
What do you know?
I had been studying differential equations the morning of April 23, 2009, and later jotted down a thought that impressed me taken from a book by Werner Heisenberg (Physics and beyond). He had written, according to my notes, that there were no determinants that could give the position of an emitted radium electron, that the quantum information gave all the knowledge there was.
Later, I visited a church where the speaker had the unusual first name, Knowledge.
I have many, many more personal anecdotes. But I don't have the notes of them and they are only randomly stored in my memory. So some of the more entertaining and perhaps compelling ones aren't here. If I happen to get around to it, I may add some more.
First published Tuesday, June 26, 2007
On Hilbert's sixth problem
There is no consensus on whether Hilbert's sixth problem: Can physics be axiomatized? has been answered.
From Wikipedia, we have this statement attributed to Hilbert:
6. Mathematical treatment of the axioms of physics. The investigations of the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics.
Hilbert proposed his problems near the dawn of the Planck revolution, while the debate was raging about statistical methods and entropy, and the atomic hypothesis. It would be another five years before Einstein conclusively proved the existence of atoms.
It would be another year before Russell discovered the set of all sets paradox, which is similar to Cantor's power set paradox. Though Cantor uncovered this paradox, or perhaps theorem, in the late 1890s, I am uncertain how cognizant of it Hilbert was.
Interestingly, by the 1920s, Zermelo, Fraenkel and Skolem had axiomatized set theory, specifically forbidding that a set could be an element of itself and hence getting rid of the annoying self-referencing issues that so challenged Russell and Whitehead. But, in the early 1930s, along came Goedel and proved that ZF set theory was either inconsistent or incomplete. His proof actually used Russell's Principia Mathematica as a basis, but generalizes to apply to all but very limited mathematical systems of deduction. Since mathematical systems can be defined in terms of ZF, it follows that ZF must contain some theorems that cannot be tracked back to axioms. So the attempt to axiomatize ZF didn't completely succeed.
In turn, it would seem that Goedel, who began his career as a physicist, had knocked the wind out of Problem 6. Of course, many physicists have not accepted this point, arguing that Goedel's incompleteness theorem applies to only one essentially trivial matter.
In a previous essay, I have discussed the impossibility of modeling the universe as a Turing machine. If that argument is correct, then it would seem that Hilbert's sixth problem has been answered. But I propose here to skip the Turing machine concept and use another idea.
Conceptually, if a number is computable, a Turing machine can compute it. Then again Church's lamda calculus, a recursive method, also allegedly could compute any computable. So are the general Turing machine and the lamda calculus equivalent? Church's thesis conjectures that they are, implying that it is unknown whether either misses some computables (rationals or rational approximations to irrationals).
But Boolean algebra is the real-world venue used by computer scientists. If an output can't be computed with a Boolean system, no one will bother with it. So it seems appropriate to define an algorithm as anything that can be modeled by an mxn truth table and its corresponding Boolean statement.
The truth table has a Boolean statement where each element is above the relevant column. So a sequence of truth tables can be redrawn as a single truth table under a statement combined from the sub-statements. If a sequence of truth tables branches into parallel sequences, the parallel sequences can be placed consecutively and recombined with an appropriate connective.
One may ask about more than one simultaneous output value. We regard this as a single output set with n output elements.
So then, if something is computable, we expect that there is some finite mxn truth table and corresponding Boolean statement. Now we already know that Goedel has proved that, for any sufficiently rich system, there is a Boolean statement that is true, but NOT provably so. That is, the statement is constructible using lawful combinations of Boolean symbols, but the statement cannot be derived from axioms without extension of the axioms, which in turn implies another statement that cannot be derived from the extended axioms, ad infinitum.
Hence, not every truth table, and not every algorithm, can be reduced to axioms. That is, there must always be an algorithm or truth table that shows that a "scientific" system of deduction is always either inconsistent or incomplete.
Now suppose we ignore that point and assume that human minds are able to model the universe as an algorithm, perhaps as some mathematico-logical theory; i.e., a group of "cause-effect" logic gates, or specifically, as some mxn truth table. Obviously, we have to account for quantum uncertainty. Yet, suppose we can do that and also suppose that the truth table need only work with rational numbers, perhaps on grounds that continuous phenomena are a convenient fiction and that the universe operates in quantum spurts.
Yet there is another proof of incompleteness. The algorithm, or its associated truth table, is an output value of the universe -- though some might argue that the algorithm is a Platonic idea that one discovers rather than constructs. Still, once scientists arrive at this table, we must agree that the laws of mechanics supposedly were at work so that the thoughts and actions of these scientists were part of a massively complex system of logic gate equivalents.
So then the n-character, grammatically correct Boolean statement for the universe must have itself as an output value. Now, we can regard this statement as a unique number by simply assigning integer values to each element of the set of Boolean symbols. The integers then follow a specific order, yielding a corresponding integer.
(The number of symbols n may be regarded as corresponding to some finite time interval.)
Now then, supposing the cosmos is a machine governed by the cosmic program, the cosmic number should be computable by this machine (again the scientists involved acting as relays, logic gates and so forth). However, the machine needs to be instructed to compute this number. So the machine must compute the basis of the "choice." So it must have a program to compute the program that selects which Boolean statement to use, which in turn implies another such program, ad infinitum.
In fact, there are two related issues here: the Boolean algebra used to represent the cosmic physical system requires a set of axioms, such as Hutchinson's postulates, in order to be of service. But how does the program decide which axioms it needs for itself? Similarly, the specific Boolean statement requires its own set of axioms. Again, how does the program decide on the proper axioms?
So then, the cosmos cannot be fully modeled according to normal scientific logic -- though one can use such logic to find intersections of sets of "events." Then one is left to wonder whether a different system of representation might also be valid, though the systems might not be fully equivalent.
At any rate, the verdict is clear: what is normally regarded as the discipline of physics cannot be axiomatized without resort to infinite regression.
*************
So, we now face the possibility that two scientific systems of representation may each be correct and yet not equivalent.
To illustrate this idea, consider the base 10 and base 2 number systems. There are some non-integer rationals in base 10 that cannot be expressed in base 2, although approximation can be made as close as we like. These two systems of representation of rationals are not equivalent.
(Cantor's algorithm to compute all the rationals uses the base 10 system. However, he did not show that all base n rationals appear in the base 10 system.)
First published Monday, June 25, 2007
The world of null-H
Some thoughts on data compression, implicate order and entropy (H):Consider a set of automated machines. We represent each machine as a sequence of logic gates. Each gate is assigned an integer. In that case each machine can be represented as a unique number, to wit its sequence of logic gate numbers. If subroutines are done in parallel, we can still find some way to express the circuit as a single unique number, of course.
In fact, while we're at it, let us define an algorithm as any procedure that can be modeled by a truth table. Each table is a constructable mxn matrix which clearly is unique and hence can be written as a sequence of 0's and 1's read off row by row with a precursor number indicating dimensions. In that case, the table has a bit value of more than nxm. On the other hand, each machine's table is unique and can be assigned an index number, which may have a considerably lower bit value than nxm.
Now suppose, for convenience, we choose 10 machines with unique truth table numbers that happen to be n bits in length. We then number these machines from 1 to 10.
Now, when we send a message about one of the machines to someone who has the list of machines and reference numbers stored, the message can be compressed as a number between 1 and 10 (speaking in base-10 for convenience). So the information value of, say 7, is far lower than that for n=25 base-2 digits. Suppose that it is equally probable that any machine description will be sent. In that case the probability, in base 2, is 2-25, and the information value is 25 bits.
However, if one transmits the long-form description, it is no better than transmitting the three-bit representation of 7 (111). Clearly the information value of 3 bits for 7 is far lower than the 25 bits for the long description.
Of course Shannon's memoryless channel is a convenient fiction, allowing a partial desription which is often useful (he did some pioneering work on channels with memory also, but I haven't seen it). These days, few channels are memoryless, since almost every computer system comes with look-up functions.
So what we have is data compression. The 25 bits have been compressed into some low number of bits. But, we don't have information compression, or do we we?
If one transmits the long form message to the receiver, that information is no more useful to the receiver than the information in the abbreviated form. Iimplicit will do. Iexplicit has no additional surprisal value to the receiver.
So Iexplicit - Iimplicit might be considered the difference between the stored information value and the transmitted information value. But, once the system is up and running, this stored information is useless. It is dead information -- unless someone needs to examine the inner workings of the machine and needs to look it up. Otherwise the persons on each end can talk in compressed form about Machine X and Machine Y all the time without ever referring to the machine details. So Ie - Ii = -Ii under those conditions.
So stored information is dead information for much of the time. It has zero value unless resurrected by someone with an incomplete memory of the long integer string.
Now we have not bothered with the issue of the average information (entropy) of the characters, which here is a minor point. But clearly the entropy of the messaging system increases with compression. Information is "lost" to the storage unit (memory).
However, if someone consults the memory unit, stored information is recovered and the entropy declines.
The point is that entropy in this sense seems to require an observer. Information doesn't really have a single objective value.
Yes, but perhaps this doesn't apply to thermodynamics, you say. The entropy of the universe always declines. Turned around, that statement really means that the most probable events will usually occur and the least probable usually won't. So many scientists seek to redefine sets of "events" in order to discover more intersections. They seek to reduce the number of necessary sets to a minimum.
Note that each set might be treated as a machine with its set language elements denoted by numbers. In that case sets with long integer numbers can be represented by short numbers. In that case, again, entropy seems to be observer-dependent.
Of course one can still argue that the probability of the memory unit remaining intact decreases with time. Now we enter into the self-referencing arena -- as in Russell's set of all sets -- in that we can point out that the design of the memory unit may well require another look-up system, again implying that information and entropy are observer-dependent, not sometimes but always.
Consider a quantum experiment such as a double-slit experiment one photon at a time.
The emitted photon will land in a region of the interference pattern with a specific probability derived from the square of the photon's wave amplitude.
If we regard the signal as corresponding to the coordinates of the detected photon, then the received signal carries an information value equal to -log(p), where p is the probability for those coordinates. (I am ignoring the unneeded term "qubit" here.)
In that case, the entropy of the system is found by -p1log(p1) +...+ -pnlog(pn).
So the entropy corresponds to the wave function and the information corresponds to the collapse of the wave function -- and we see that information is observer-dependent. The observer has increased the information and decreased the general entropy.
On the other hand, information is a fleeting thing in both the classical and quantum arenas. "Shortly" after the information is received, it loses its surprisal value and dies -- until a new observer obtains it ("new observer" could mean the same observer who forgets the information).
Star Wars, AI and quantum computing
Basics of interceptor design by the Federation of American Scientists
Coyle report on National Missile Defense
Boost-phase issues cited by FAS
Garwin's case for 'boost-phase' vs. N. Korea
MIT's Postol shoots down midcourse interception
NMD technical challenges discussed by Center for Defense Information
Australian parliament report on technical and political issues
Quantum factoring with Shor's algorithm
Starwars feud behind Jason's shootdown?
American Physical Society report skeptical of boost-phase interception (search APS site)
The National Missile Defense proposal looks a bit iffy. Given missile-to-missile problems, are laser weapons really an inferior idea? Will advances in artificial intelligence spare NMD from failure? Can quantum computation play a role in NMD? Essayist Paul R. Conant first published ca. 2000 explores these questions.
Laser beams or missile-to-missile collision?
As proposed, the National Missile Defense has, at best, little bang for the buck. Every phase of interception, using direct collision as the kill mechanism, has serious drawbacks.However, a system combining boost-phase interceptors with high-powered midcourse laser beams is not implausible, though the expense of an effective system may turn out to be unacceptable. Such a system has been found unpromising in a July 2003 report from a 12-member panel of the American Physical Society.
NMD is not designed to be effective against Russia, which can fire enough ICBMs to simply overwhelm the battery of interceptors. Though some have argued that the system might work against Chinese missiles, this seems unlikely, since the Chinese are well able to deploy effective decoys and other 'penetration aids.' The assumption that more primitive states will be unable to deploy such countermeasures is highly unconvincing.
A test intercept, rated successful by the Pentagon, was widely criticized for using a decoy heat source that would not be used in a real-world situation.
NMD is not capable of preventing delivery of biowar toxins, such as smallpox and anthrax, because the warhead, when entering space, can divide into numerous small bomblets invisible to interceptor detectors.
NMD's apparent purpose is to prevent fanatical elements of 'rogue' states -- Iraq, Iran and North Korea -- from successful nuclear warhead launches. North Korea has a missile that can reach Alaska and Hawaii and is presumed to be developing one that might reach the lower 48. It is known to have diverted nuclear materials for bomb research. Neither Iraq nor Iran has the capability to hit American cities, though an unstated purpose of development of NMD might be, as a corollary, to develop weaponry for defense of Israel, as part of a theater missile defense program. Despite George Bush Senior's enthusiastic statements about the success of the Patriots against Iraq's scuds, the kill rate was apparently very low.
NMD might also be justified as a means of countering terrorists who gain control of a missile or two. But, as the events of September 11 show, such a catastrophe needn't happen only in a distrusted nation; it might happen anywhere, raising very hard questions as to where to position interceptors.
The Air Force is reported to have overcome optical turbulence problems with its long-range (250-mile) laser heat ray, which, well clear of a foe's air defenses, would incinerate missiles, either during boost-phase or during free fall. However, the experts with the American Physical Society have noted continuing technical difficulties with an airborne laser.
There has been much debate as to whether spaceborne or airborne lasers are practical because land-based weapons are easier to defend. On the other hand, nothing beats the speed of light and missiles sent to kill a laser can be zapped by laser beam. The question then becomes, how many lasers are required?
An important advantage of space-based lasers is low maintenance. Another is that they can fire during a foe's boost phase as well as during midcourse free fall and even during the end course.
Also, the heat ray needn't incinerate every missile instantly. Rather, the missile would be heated above its air friction threshhold and would burn up like a meteorite. During boost phase, it might be advantageous to hit the missile in the lower atmosphere, though the APS panel said that a laser beam would be ineffective during boost phase against solid-fuel rockets, which are more heat resistant than liquid-fuel rockets. If the intruder is past boost phase, a laser intercept might be triggered sometime after atmosphere re-entry.
Multiple beams would hit decoys, which will likely be balloons, that will either explode harmlessly or overheat and burn up on re-entry.
A penetration aid that could render rocket kill vehicles useless would be a device to jam the KV's end-game detector, which is operated by an onboard computer. That is, the warhead carries a device that sends out a high-amplitude pulse of radio waves to create a power surge in the KV's onboard computer. The jammer might be set to go off some specific time after a surveillance signal (x-band radar) is detected.
The laser's computer will tend to be much farther away from the foe's warhead, and, of course, jamming ability diminishes by the inverse square of the distance.
Space-based lasers would be vulnerable to similar lasers fired from the surface or above it, though lower atmosphere optical turbulence can impede strength and accuracy of pulses.
Still, laser satellites can be cloaked with stealth technology; that is, a satellite can be shaped to promote the bending of surveillance waves around it and coated with a skin designed to absorb radar and perhaps other surveillance waves. In addition, small rocket-motor firings can be programmed to occur pseudorandomly in order to make minor orbit changes, thus increasing difficulty of detection.
Light and fast
The exo-atmospheric kill vehicle is designed to hit a warhead near the top of its trajectory in space. The advantage of an EKV (or of an atmospheric kill vehicle) is its light payload, which consists principally of a compact infrared (heat) sensor, coupled with an onboard computer. If an EKV's initial velocity equals that of its target, it will move much faster.The EKV is guided to the vicinity of the target by ground-based x-band radar. The EKV's high-sensitivity heat sensor and onboard computer then orchestrate the 'end-game' intercept.
In space, the target is following a simple, largely unperturbed trajectory, so that course projection problems are less than they would be for a conventional aircraft or for a cruise missile taking evasive actions.
A salvo of EKV's would be launched at slightly different times in order to reduce likelihood of correlated errors (a miss by one KV won't imply a miss by all).
Initially, the Pentagon plans to deploy 20 interceptors, to be followed by 100 ground-based EKV's in Alaska. However, if North Korea's reach improves, it is reported that North Dakota is better positioned for an interceptor battery.
Infrared-sensing satellites would be used to detect launches. Though geosynchronous orbits are envisioned, the 22,000-mile altitude means a 2.5-second data delay. That couple of seconds adds to some minimum time between launch A and launch B.
At lower altitudes, more satellites are required for keeping inhabited areas under surveillance.
Boost-phase concerns
Boost-phase missile-to-missile interception -- knocking out the adversary missile while its booster is still firing -- is viewed by some experts as the next step in missile defense. The booster's rocket flame is easy for a KV to detect and the missile's course is highly predictable, improving likelihood of a hit. Boost-phase interception of ICBM's also avoids the decoy problem associated with midcourse interception.The American Physical Society analysts argued that boost-phase interceptors are unlikely to be fast enough to catch missiles using solid fuel. If they are, they would be impractically large, the scientists said. They also said that, even against liquid-fuel missiles, boost-phase KV's would be unable to reach missiles fired from Iran, which is too far from submarine-launch or air-launch areas.
A major concern of boost-phase is protection of the weapon platforms, which tend to be vulnerable to adversary attacks. The KV's must be close enough for interception, meaning ships or aircraft must patrol within a specified region continuously.
Spaceborne lasers avoid this problem, though attack by satellite killer missiles, perhaps armed with jamming devices, is a concern. A foe with ground-based lasers might destroy any airborne or spaceborne weapons, but it is assumed that such systems are beyond the capability of 'rogue' nations. It would seem that if ground-based lasers were effective, the Pentagon would not be seeking to build a battery of EKV's.
It has been estimated that a ship-borne Aegis KV, loitering in the North Pacific, could knock out a North Korean missile headed for America in one shot, though weapons expert Richard L. Garwin favors stationing ships in the Japan basin or operating a joint U.S.-Russian missile base on Russian soil near the North Korean border.
Ships in the Persian Gulf would be positioned to fire interceptors at missiles boosting from Iraq or Iran, though distances to Iran's hinterlands could pose problems. Though such ships would be well-defended by batteries of short-range missiles, attack is still plausible. If Saddam rained enough missiles on the ships, they would run out of defensive rockets, thus leaving him free to fire at other targets.
Even if the United States placed boost-phase interceptors on submarines -- a formidable task that would require a new class of subs -- an adversary might 'smoke out' the subs by firing missiles in various directions, tracking the origins of KV launches, raining missiles down on the subs, and then firing a new volley at other targets. These scenarios are very unlikely because of the costs involved, but cannot be totally discounted.
An important use of boost-phase interceptors might be to deter an Indo-Pak nuclear war. India and Pakistan have missiles in the 1,500-mile range capable of carrying nuclear warheads. The political, ethnic and religious passions in the region could well be sufficient to spark such a cataclysm. As we know, a Pakistan nuclear scientist assisted the Taliban and al Queda in their nuclear weapon queries.
Because such a conflagration would imperil the security of every nation and pose radioactive fallout perils for the entire globe, it seems that America keeping a boost-phase anti-missile system primed might be worthwhile.
Midcourse hassles
The biggest hurdle to interception of an ICBM near the top of its trajectory in space is the likelihood of effective decoys.A simple countermeasure is to have the warhead deploy reflective balloons in space and set them to rotating ('tumbling'). The sunless side of a balloon is far cooler than the warhead, which has been heated by air friction. But if the balloon rotates and its reflectivity is proportionately brighter than the warhead's, it will give off the same heat signature as the warhead, which is also rotating. In space, where there is no air resistance, the balloons will travel at the same velocity as the warhead.
Hence, the KV can't obtain sufficient data to distinguish among the objects.
As MIT's Ted Postol says, faster computers and better detectors are of no use here. 'Getting better information that is irrelevant doesn't help you.'
However, General Ronald Kadish, head of the Pentagon's missile defense program, thinks improvements in complexity theory might answer this obstacle. That issue is discussed below under 'artificial intelligence.'
As for atmospheric midcourse interception, it would appear to be a useful backup option in the event of boost-phase misses. It is necessary however that the KV not also be near the top of its potential arc, other wise relative velocity might be too low, resulting in a light tap, rather than a destructive hit. Apparently, Patriot intercepts in the Persian Gulf war suffered from low impact collisions, though some scuds may have been gently steered away from their original target. Patriot warheads, however, explosively fragmented near interception in order to increase probability of a hit. Hence, greater overall relative velocity was needed.
A not-foolproof countermeasure for short and medium range missiles is use of thrusts higher than anticipated at correspondingly changed launch angles. That is, say, if the Israelis are geared to aim near the top (height H) of scud trajectory A, Saddam might redesign his scuds to go to H+x, where x varies.
Because the warhead falls from a higher altitude, the increased acceleration and associated buffeting may increase course projection error, though this may not be a serious issue. Arms expert Andrew Sessler is persuasive in arguing that buffeting-related error is liable to be insignificant for endcourse ICBM's.
The Israelis, assuming they have been aiming for what they believe is the top of the scud trajectory, would then have to be prepared for a range of maximum altitudes. Their options:
*Store enough fuel in each interceptor for any plausible altitude, possibly complicating design and rendering a lot of costly equipment obsolete.
*Add interceptors, each of which varies in thrust, to the defensive system.
*Aim low, intercepting any comer at a bit below the lowest possible maximum altitude, and tolerate disadvantages, if any.
Three-phase interception not in cards
If a missile is missed in boost phase, try again in midcourse. If missed there, try again at endcourse, on the downward arc. That seems reasonable, but it's not in the cards.The military is unenthusiastic about the endcourse option, perhaps because of costs associated with missile-to-missile systems. However, a laser system should be able to fire at enemy missiles during any phase of flight with little difference in cost.
Artificial intelligence and course-plotting
Gen. Kadish has argued that real-world tests, along with advances in complexity theory, would likely overcome the the midcourse decoy scenarios offered by Postol and others.As this reporter understands it, complexity theory focuses on special cases of negative entropy: conditions that yield perceived order out of seemingly random inputs. Perhaps the general has been reading reports from Los Alamos National Laboratory, which has close collegial ties with Murray Gell-Mann's Santa Fe Instititute, a private think tank devoted to the study of complexity theory. Yet it seems unlikely that Gell-Mann and his associates have come up with revolutionary (and presumably classified) mathematical theorems that would allow the government such capability. Revolutionary theorems are hard to come by -- and they surface independently of government research grants.
It may be that Kadish was referring to advances in artificial intelligence, a research area no doubt entwined with government secrecy and military funding.
AI -- or quasi-AI (which is a less-controversial notion) -- is related to complexity theory because a top issue of complexity theory is the discovery of how life forms, including how the perceptive apparatus called the human mind, might evolve from self-replicating parts. So, if you can devise a program of self-contained automata to interact and build larger systems, you might be able to obtain a program that thinks, or quasi-thinks.
Development in World War II of radar-guided anti-aircraft fire contributed importantly to the development of computers (and inspired Norbert Weiner in his philosophy of cybernetics). Aircraft course-prediction systems use various weighting methods (more weight to more recent data, for example), filtration of signal noise and smoothing algorithms (methods of approximating an output curve closely through occasional input values). These calculations are judgmental in nature, the computer using various criteria to guess the next move (and perhaps countermove). We might think of the Deep Blue program, which defeated world chess champion Gary Kasparov, as a variant of an advanced radar detection program.
Quasi-AI, like human intelligence, is useful for anticipating a position based on imperfect data. If we consider 'random' to imply 'no datum on which to base a judgment,' then neither a human mind nor a computer program can make a useful prediction. Of course, we usually mean 'random within constraints,' in which case useful prediction is limited by the constraints.
We may view 'chaotic' or 'pseudorandom' to mean outputs that cannot be ascertained without knowing previous output values, as with a recursion function of the X-next type.
So if the next move is random within constraints or pseudorandom, we may face an exponential rise in computing work to either curtail the constraints or implement a recursion function.
In the case of the midcourse decoy scenarios, pattern recognition would have to go beyond course and brightness clues.
Pattern recognition can be, like the traveling salesman problem, a computational quagmire, though not necessarily so. If a pattern is composed of independent elements, computational work rises, essentially, by n. But if a pattern is a set of interdependent elements, then computational work can rise exponentially by number of elements. That is, a detector identifying two co-dependent elements has 2! (=2) units of work to do; a detector coping with 10 interdependent elements might have in the vicinity of 3.6 million (10!) units of work to do.
Pattern recognition of the decoy balloons tumbling through space looks to be an iffy task. Detection equipment would need to identify a set of small differences and analyze them, but possible clues are so negligible that it seems recognition of interdependent subsets would be required. (The total number of possible subsets of a set is 2^n, an exponential quantity.)
And suppose an 'advance in complexity theory' makes decoy recognition feasible? What countermeasure might yield another computational quagmire?
Bottom-up artificial neural networks have proved effective at 'learning' to discriminate among patterns. For example, in 'The Engine of Reason, the Seat of the Soul' (MIT Press, 1995), Paul M. Churchland cites programs in which a vector quantity is assigned to each element of a limited set of faces. The program then uses a process of error reduction until it is able to match a face with a numerical code (name) most or all the time. Such a system requires repeated trials (run by a serial computer).
The technology's appeal to weapons designers lies in the fact that the program is effective at identifying the correct pattern (target) even with degraded (noisy) data.
This technology, while fascinating, is unlikely to help much in the discrimination of Postol-type decoys from warheads. Neither human nor classical machine will do well at such a task because the differences in input data are so miniscule.
As of June 12, 2002, the Pentagon, taking advantage of the post-9/11 secrecy craze, had classified data on future tests of decoys, leading some to charge the Rumsfeld contingent of trying to shield NMD from legitimate technical criticism. Nevertheless, the Pentagon has never directly refuted the point made by Postol and others, but rather relied on the notion that technical breakthroughs will save the day. The possibility that NMD can be made to work only against inferior decoys but never against the best decoys designable is not addressed.
Does quantum computing lurk behind NMD
In December 2001, the New York Times reported that an IBM research team was about to announce the factoring of the number 15 using quantum computation, an experiment with defense repercussions. No confirmation of this initial report can be found in the usual places, such as science magazines or even at the relevant IBM web site. However, an IBM scientist referred this writer to a Nature article by the IBM researchers (Jan. 4, 2002), which discusses the potential, using lasers and beam splitters, for quantum computing using photon quantum effects. Nothing about the actual factorization of the number 15 is noted, though the IBM researcher does not deny the accuracy of the Times report.The team reputedly used a quantum device to test and validate Shor's algorithm, which shows a way to use the quantum phenomenon of superposition to do simultaneous factoring. The team factored the number 15 into its primes of 3 and 5. Of course, this is a far cry from factoring hundred digit numbers but the validation of such a method sent shockwaves around the world.
If a classical supercomputer can crack a code by factoring numbers with 10^x digits, code-makers simply use primes that, when multiplied together, result in numbers with (10^x) + 1 digits. On a classical computer, the work of cracking such a code rises exponentially by digit place. It seems likely that decoys may present similar computational issues, particularly if the course-plotting program is already souped-up with nonlinear methods.
Is there a way out?
There remains the intriguing concept of quantum computation, which still appears an elusive quarry, despite its apparent validation in principle.
Essentially, the idea behind such a device is that though a quantum particle, once observed, appears to have taken a specific path, we can NEVER predict which path it will take with absolute certainty. For example, if a photon goes through a symmetrical interferometer, it is said to take one of, say, two paths each with a probability of perhaps 1/2, but, if left undetected enroute, the photon emerges as if it takes one path with probability 1. Some sort of interference causes the photon to have a nonprobabilistic final path.
Can this quantum weirdness be harnessed? In principle, yes, as the experimenters proved.
According to the 'many worlds' theory proposed by Hugh Everett (who, incidentally, spent his career as a Pentagon scientist), for each path the photon MIGHT take in our world, it actually DOES take in 'another world' (or 'parallel universe'). Our universe differentiates into two universes at the time a quantum particle, such as a photon, seems to make a 'random' choice. It is possible for differentiated universes to merge back into a single wave under the right conditions. If somehow these split universes could be merged back into 'our' universe, you might be able to compute classically exponentially hard problems in the blink of an eye.
Supposing the 'many worlds' idea holds, just imagine that you could somehow get nearly identical computers in a large set of universes to parallel-process a tough problem -- one universe/computer per route in the traveling salesman problem, for example.
A more usual view is that the photon, before detection, has various possible positions superposed. On detection, the wavelike nature collapses, and the photon's position can be known.
In the quantum computation experiment, factors are associated with superposed quantum states called spins.
At any rate, though quantum computation might someday prove a boon to code-crackers and AI program designers, it seems at this point unlikely that the code-busting National Security Agency would be very happy at the prospect of the Pentagon squandering such an asset on an easy-to-spy-on weapons system.
For deep insights into the worlds of quantum theory and AI, see The Emperor's New Clothes and Shadows of the Mind by Roger Penrose and The Fabric of Reality by David Deutsch. Also see Deutsch's Frontiers article.
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