Wednesday, October 24, 2007

Catchy: "Put the Turing in Manufac-turing"

From an op-ed piece by Ken Goldberg and Vijay Kumar:

But U.S. manufacturing today is where database technology was in the early 1960's, a patchwork of ad hoc solutions that lacked the rigorous methodology that leads to scientific innovation. That all changed in 1970 when Ted Codd, an IBM mathematician, invented relational algebra, an elegant mathematical database model that galvanized federally funded research leading to today's $14 billion database industry.

Manufacturing needs the same treatment. Just as the method to add two numbers together doesn't depend on what kind of pencil you use, manufacturing abstractions can be wholly independent of the product one is making or the assembly line systems used to assemble it. Another precedent is the Turing Machine, an elegant abstract model invented by Alan Turing in the 1930s, which established the mathematical and scientific foundations for our now-successful high-tech industries. Without Turing's theoretical work, the system that typeset this line wouldn't exist.

What's needed today is an analogy to the Turing Machine for design, automation and manufacturing. Recent developments in computing and information science have now made it possible to model and reason about physical manufacturing processes, setting the stage for us to "put the Turing into Manufacturing". The result, as was the case with databases and computers, would be higher quality, more reliable products, reduced costs, and faster delivery

As they say, RTWT.

Untangling a geometric graph

Planarity is a rather addictive game developed by John Tantalo. The idea is that you're given a planar graph, drawn with lots of crossings, and your goal is to move vertices around so as to recover a planar embedding. The game can get surprisingly challenging, and has even spawned a photoshop tribute called, 'Planarity, My Anti-drug'.

If we put on our theory hats, the first question that pops out is: how many vertices might you need to move in order to move from an embedding of a planar graph to a planar embedding ? This process, of moving vertices around to remove crossings, is called untangling. Technically, H is an untangling of G if H is crossing free and is isomorphic as G (in other words, untangling might allow more general moves than the sliding allowed by Planarity). A vertex of G was said to be fixed if it was mapped (by the isomorphism) to a vertex of H with the same coordinates.

Back in 1998, it was first asked whether a graph could be made planar while keeping a constant fraction of vertices fixed. The answer, shown by Pach and Tardos, was no, via a graph in which at most roughly n^{2/3} of the vertices could stay fixed (which is one way of explaining the difficulty of Planarity). Of course, the next question was whether it was the case that all planar graphs could be untangled by fixing at least some n^epsilon vertices.

A recent paper by Bose, Dujmovic, Hurtado, Langerman, Morin and Wood answers this question in the affirmative, showing that any geometric planar graph can be untangled while keeping at least n^{1/4} vertices fixed. The proofs involved are intricate, but are graph-theoretic and "elementary" in nature.

Wolfram (2,3) machine exists

Back in May, Bill and David discussed the new Wolfram challenge: Show that a particular 2-state 3-color TM is universal. Apparently, this question has now been resolved affirmatively, by Alex Smith, an undergraduate at the University of Birmingham.

Read more about here, and here.

(via BB)

Monday, October 15, 2007

I foresee mass murders of cell phone users...

From the NYT:
It’s been a long time, but Led Zeppelin, one of the last superstar acts to refrain from selling its music online, is finally offering its catalog to digital-music fans. The shift by Led Zeppelin, whose reunion concert in London next month has already incited a frenzy for tickets, highlights the clout of digital sales in the music market as mass merchants reduce the shelf space devoted to compact discs. Under a series of new agreements expected to be announced today, the band will make its songs available first as ringtones and similar mobile features starting this week in an exclusive deal with Verizon Wireless.
And if you don't believe me, watch Wayne's World.

Thursday, October 11, 2007

Popularizing computing

Brian Hayes writes for the American Scientist (and on his blog, bit-player), and in my mind is one of the best writers of popular material about computing. Every time I read something he's written, I learn something new, even if it's about a topic I'm familiar with. What's even more impressive is that he often focuses on the mathematical aspects of computing and finds interesting ways to explain them to his audience.

Two recent articles of his are exemplars:

1. Sorting out the genome, from September-October 2007. Here's a key extract:
Given two arrangements of a set of genes—say a b c d e f g and f e b a g c d—how do you determine what sequence of reversals produced the transformation? This example has a three-step solution, which you can probably find in a few minutes with pencil and paper. For larger genomes and longer chains of reversals, however, trial-and-error methods soon falter. Is there an efficient algorithm for identifying a sequence of reversals that converts one permutation into another?

The genetic reversal problem lies at the intersection of biology, mathematics and computer science. For some time, the prospects for finding a simple and efficient solution seemed dim, even with the most powerful tools of all three disciplines. But the story has a happy ending. A little more than a decade ago, computing gene reversals was still a subtle research problem; now it can be done with such ease that it's a matter of routine technology. If you need to know the "reversal distance" between two genomes, you can go to a Web site and get the answer in seconds.
The article proceeds to lay out the entire history of sorting with reversals, pancake sorting, and related problems. It's a great case study for the study of algorithms in an application domain: how to model a problem, come up with algoritms, discover that the model doesn't quite capture what you want, rethink the model, come up with more algorithms, and so on. It touches on dynamic programming, greedy algorithms, NP-hardness, and approximations. I liked it so much I'll be devoting a lecture in my algorithms class to it, to illustrate how algorithms research works "in the real world".

2. Conquering divide, bit-player, October 9th, 2007.
The "hook" for this article is is Eric Allender's review article on the true complexity of division (short summary: Division is complete for DLOGTIME-uniform TC^0), which is a must-read in its own right. What Hayes discusses in his article is the reason to even ask this question, starting with the general lack of symmetry between grade-school methods for multiplication and division. I found this particularly insightful:

Why is division so hard?

At times I feel this is a dumb question, and that the universe doesn’t owe us an answer to such questions. Some things are just harder than others; that’s the way the world works. We can count ourselves lucky that multiplication has a polynomial-time algorithm; we have no right to expect that our luck will hold when we turn to division.

But then I get to thinking about physical systems that multiply and divide, where the two operations seem perfectly symmetrical. An example is an electrical circuit governed by Ohm’s law.

If we attach the terminals of a voltmeter to opposite ends of a resistor, we observe Ohm’s law in the form E = IR: Voltage is equal to the product of current and resistance. If we install an ammeter in series with the resistor, we see the effects of the law I = E/R: Current is equal to voltage divided by resistance. If you prefer pipes to wires, you can do the hydraulic version of the experiment, and there are also lots of mechanical schemes, where the same principle is illustrated with levers, gears, pulleys and such. In all of these systems, nature seems to divide just as easily as it multiplies. Why can’t we do the same with pencil and paper, or transistors, or neurons?

Saturday, October 06, 2007

CoM #18

The latest installment of the Carnival of Mathematics is out, hosted by jd2718. It's an amazing effort, with 52 posts from 49 different authors. The first few installments of the CoM were quite sparse, and for a while I wondered whether there was enough chatter in the blathosphere to sustain it. The answer, looking at the variety and quality of posts that Jonathan has put together, is clearly yes !

Well done !

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