Brian Hayes

GRAPH THEORY IN PRACTICE: PART II

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A Lucid Interval

ive a digital computer a problem in arithmetic , and it will grind away methodically , tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. The cause of this sorry situation is not that software is full of bugs—although that is very likely true as well— nor is it that hardware is unreliable. The problem is simply that computers are discrete and finite machines, and they cannot cope with some of the continuous and infinite aspects of mathematics. Even an innocent-looking number like 1 ⁄10 can cause no end of trouble: In most cases, the computer cannot even read it in or print it out exactly, much less perform exact calculations with it. Errors caused by these limitations of digital machines are usually small and inconsequential, but sometimes every bit counts. On February 25, 1991, a Patriot missile battery assigned to protect a military installation at Dahrahn, Saudi Arabia, failed to intercept a Scud missile, and the malfunction was blamed on an error in computer arithmetic. The Patriots control system kept track of time by counting tenths of a second; to convert the count into full seconds, the computer multiplied by 1 ⁄ 10. Mathematically, the procedure is unassailable, but computationally it was disastrous. Because the decimal fraction 1 ⁄ 10 has no exact finite representation in binary notation, the computer had to approximate. Apparently, the conversion constant stored in the program was the 24-bit binary fraction 0.00011001100110011001100, which is too small by a factor of about one ten-millionth. The discrepancy sounds tiny, but over four days it built up to about a third of a second. In combination with other peculiarities of the control software , the inaccuracy caused a miscalculation of almost 700 meters in the predicted position of the incoming missile. Twenty-eight soldiers died. Of course it is not to be taken for granted that better arithmetic would have saved those 28 lives. (Many other Patriots failed for unrelated reasons; some analysts doubt whether any Scuds were stopped by Patriots.) And surely the underlying problem was not the slight drift in the clock but a design vulnerable to such minor timing glitches. Nevertheless, the error in computer multiplication is mildly disconcerting. We would like to believe that the mathematical machines that control so much of our lives could at least do elementary arithmetic correctly. One approach to dealing with such numerical errors is a technique …

THE EASIEST HARD PROBLEM

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The Spectrum of Riemannium

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Randomness as a Resource

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HOW TO AVOID YOURSELF

PAFAS (Process Algebra for Faster Asynchronous Systems) is a process algebra where actions are assumed to occur within a given time bound, which for simplicity is taken to be 1. A testing-based faster-than relation is presented that compares asynchronous systems according to their worst-case eÆciency [4,3]. Main results are: the resulting pre-order based on the more realistic real-valued time is the same as the pre-order based on the simpler integer-valued time { so we use the latter; the faster-than relation can be characterized with some sort of refusal traces. A larger example studying implementations of a bu er can be found in [1]. While the testing de nition is qualitative, we point out that it can also be seen as considering a quantitative performance measure. Then we adapt the PAFASapproach to a setting, where user behaviour is known to belong to a very speci c, but often occurring class of request-response behaviours, and show how to determine an asymptotic performance measure for nite-state processes [2].

First Links in the Markov Chain

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The Best Bits

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Computation and the Human Predicament

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The World in a Spin

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