Rado Serial Number Check
Rene Seiler
List buckets, or, if a bucket is specified with --bucket=,list its objects. Adding --allow-unorderedremoves the ordering requirement, possibly generating results morequickly for buckets with large number of objects.
Maximum concurrent bucket operations. Affects operations thatscan the bucket index, e.g., listing, deletion, and all scan/searchoperations such as finding orphans or checking the bucket index.The default is 32.
A biggest number contest is clearly pointless when the contestants taketurns. But what if the contestants write down their numbers simultaneously,neither aware of the others? To introduce a talk on "Big Numbers," I invite twoaudience volunteers to try exactly this. I tell them the rules:
You have fifteen seconds. Using standard math notation, English words, orboth, name a single whole numbernot an infinityon a blank index card. Beprecise enough for any reasonable modern mathematician to determine exactly whatnumber youve named, by consulting only your card and, if necessary, thepublished literature.
So contestants cant say "the number of sand grains in the Sahara," becausesand drifts in and out of the Sahara regularly. Nor can they say "my opponentsnumber plus one," or "the biggest number anyones ever thought of plusone"again, these are ill-defined, given what our reasonable mathematician hasavailable. Within the rules, the contestant who names the bigger numberwins.
The contests results are never quite what Id hope. Once, a seventh-gradeboy filled his card with a string of successive 9s. Like many other big-numbertyros, he sought to maximize his number by stuffing a 9 into every place value.Had he chosen easy-to-write 1s rather than curvaceous 9s, his number couldhave been millions of times bigger. He still would been decimated, though, bythe girl he was up against, who wrote a string of 9s followed by thesuperscript 999. Aha! An exponential: a number multiplied by itself999 times. Noticing this innovation, I declared the girls victory withoutbothering to count the 9s on the cards.
And yet the girls number could have been much bigger still, had she stackedthe mighty exponential more than once. Take , for example. Thisbehemoth, equal to 9387,420,489, has 369,693,100 digits. Bycomparison, the number of elementary particles in the observable universe has a meager 85digits, give or take. Three 9s, when stacked exponentially, already lift usincomprehensibly beyond all the matter we can observeby a factor of about10369,693,015. And weve said nothing of or.
Place value, exponentials, stacked exponentials: each can express boundlesslybig numbers, and in this sense theyre all equivalent. But the notationalsystems differ dramatically in the numbers they can express concisely.Thats what the fifteen-second time limit illustrates. It takes the same amountof time to write 9999, 9999, and yet the firstnumber is quotidian, the second astronomical, and the third hyper-megaastronomical. The key to the biggest number contest is not swift penmanship, butrather a potent paradigm for concisely capturing the gargantuan.
Such paradigms are historical rarities. We find a flurry in antiquity,another flurry in the twentieth century, and nothing much in between. But when anew way to express big numbers concisely does emerge, its often a byproduct ofa major scientific revolution: systematized mathematics, formal logic, computerscience. Revolutions this momentous, as any Kuhnian could tell you, only happenunder the right social conditions. Thus is the story of big numbers a story ofhuman progress.
This same pattern holds, I think, for big numbers. Curiosity and opennesslead to fascination with big numbers, and to the buoyant view that no quantity,whether of the number of stars in the galaxy or the number of possible bridgehands, is too immense for the mind to enumerate. Conversely, ignorance andirrationality lead to fatalism concerning big numbers. Historian Ilan Vardi cites the ancient Greekterm sand-hundred,colloquially meaning zillion; as well as a passage from PindarsOlympic Ode II asserting that "sand escapes counting."
There are some ... who think that the number of the sand is infinite inmultitude ... again there are some who, without regarding it as infinite, yetthink that no number has been named which is great enough to exceed itsmultitude ... But I will try to show you [numbers that] exceed not only thenumber of the mass of sand equal in magnitude to the earth ... but also that ofa mass equal in magnitude to the universe.
This Archimedes proceeded to do, essentially by using the ancient Greek termmyriad, meaning ten thousand, as a base for exponentials. Adopting aprescient cosmological model of Aristarchus, in which the "sphere of the fixedstars" is vastly greater than the sphere in which the Earth revolves around thesun, Archimedes obtained an upper bound of 1063 on the number of sandgrains needed to fill the universe. (Supposedly 1063 is the biggestnumber with a lexicographically standard American name: vigintillion. Butthe staid vigintillion had better keep vigil lest it be encroached upon by themore whimsically-named googol, or 10100, andgoogolplex, or .) Vast though it was, of course, 1063wasnt to be enshrined as the all-time biggest number. Six centuries later,Diophantus developed a simpler notation for exponentials, allowing him tosurpass . Then, in the Middle Ages, the rise of Arabicnumerals and place value made it easy to stack exponentials higher still. ButArchimedes paradigm for expressing big numbers wasnt fundamentally surpasseduntil the twentieth century. And even today, exponentials dominate populardiscussion of the immense.
Consider, for example, the oft-repeated legend of the Grand Vizier in Persiawho invented chess. The King, so the legend goes, was delighted with the newgame, and invited the Vizier to name his own reward. The Vizier replied that,being a modest man, he desired only one grain of wheat on the first square of achessboard, two grains on the second, four on the third, and so on, with twiceas many grains on each square as on the last. The innumerate King agreed, notrealizing that the total number of grains on all 64 squares would be264-1, or 18.6 quintillionequivalent to the worlds present wheatproduction for 150 years.
Fittingly, this same exponential growth is what makes chess itself sodifficult. There are only about 35 legal choices for each chess move, but thechoices multiply exponentially to yield something like 1050 possible boardpositionstoo many for even a computer to search exhaustively. Thats why ittook until 1997 for a computer, Deep Blue, to defeat the human world chesschampion. And in Go, which has a 19-by-19 board and over 10150possible positions, even an amateur human can still rout the worlds top-rankedcomputer programs. Exponential growth plagues computers in other guises as well.The traveling salesman problem asks for the shortest route connecting a set ofcities, given the distances between each pair of cities. The rub is that thenumber of possible routes grows exponentially with the number of cities. Whenthere are, say, a hundred cities, there are about 10158 possibleroutes, and, although various shortcuts are possible, no known computeralgorithm is fundamentally better than checking each route one by one. Thetraveling salesman problem belongs to a class called NP-complete, which includeshundreds of other problems of practical interest. (NP stands for the technicalterm Nondeterministic Polynomial-Time.) Its known that if theres an efficientalgorithm for any NP-complete problem, then there are efficient algorithms forall of them. Here efficient means using an amount of time proportional to atmost the problem size raised to some fixed powerfor example, the number ofcities cubed. Its conjectured, however, that no efficient algorithm forNP-complete problems exists. Proving this conjecture, called P NP, has been a great unsolved problem of computer sciencefor thirty years.
Although computers will probably never solve NP-complete problemsefficiently, theres more hope for another grail of computer science:replicating human intelligence. The human brain has roughly a hundred billionneurons linked by a hundred trillion synapses. And though the function of anindividual neuron is only partially understood, its thought that each neuronfires electrical impulses according to relatively simple rules up to a thousandtimes each second. So what we have is a highly interconnected computer capableof maybe 1014 operations per second; by comparison, the worldsfastest parallel supercomputer, the 9200-Pentium Pro teraflops machine at SandiaNational Labs, can perform 1012 operations per second. Contrary topopular belief, gray mush is not only hard-wired for intelligence: it surpassessilicon even in raw computational power. But this is unlikely to remain true forlong. The reason is Moores Law, which, in its 1990s formulation, states thatthe amount of information storable on a silicon chip grows exponentially,doubling roughly once every two years. Moores Law will eventually play out, asmicrochip components reach the atomic scale and conventional lithographyfalters. But radical new technologies, such as optical computers, DNA computers,or even quantum computers, could conceivably usurp silicons place. Exponentialgrowth in computing power cant continue forever, but it may continue longenough for computersat least in processing powerto surpass human brains.
To prognosticators of artificial intelligence, Moores Law is a gloriousherald of exponential growth. But exponentials have a drearier side as well. Thehuman population recently passed six billion and is doubling about once everyforty years. At this exponential rate, if an average person weighs seventykilograms, then by the year 3750 the entire Earth will be composed of humanflesh. But before you invest in deodorant, realize that the population will stopincreasing long before thiseither because of famine, epidemic disease, globalwarming, mass species extinctions, unbreathable air, or, entering thespeculative realm, birth control. Its not hard to fathom why physicist AlbertBartlett asserted "the greatest shortcoming of the human race" to be "ourinability to understand the exponential function." Or why Carl Sagan advised usto "never underestimate an exponential." In his book Billions &Billions, Sagan gave some other depressing consequences of exponentialgrowth. At an inflation rate of five percent a year, a dollar is worth onlythirty-seven cents after twenty years. If a uranium nucleus emits two neutrons,both of which collide with other uranium nuclei, causing them to emit twoneutrons, and so forthwell, did I mention nuclear holocaust as a possible endto population growth?
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