From: gagan@cs.UMD.EDU (Gagan Agrawal)
Newsgroups: cs.grad
Subject:  Theory results are berkeley
Date: 8 Apr 92 16:52:02 GMT

type: NYT (Copyright 1992 The New York Times)
priority: Regular
date: 04-06-92 1806EDT
category: Domestic
subject: BC MATH PROOF SCI ART (2takes)
title: NEW SHORT CUT FOUND FOR LONG MATH PROOFS
text:

ART ADV: Diagram of proofs has been sent to NYT graphic clients.
Non-subscribers can make individual purchase by calling
212-556-4204 or 1927.
kd
By GINA KOLATA
c.1992 N.Y. Times News Service
       In a discovery that overturns centuries of mathematical
tradition, a group of graduate students and young researchers has
discovered a way to check even the longest and most complicated
proof by scrutinizing it in just a few spots.
       The finding, which some mathematicians say seems almost magical,
depends upon transforming the set of logical statements that
constitute a proof into a special mathematical form in which any
error is so amplified as to be easily detectable.
       Using this new result, the researchers have already made a
landmark discovery in computer science.
       They showed that it is impossible to compute even approximate
solutions for a large group of practical problems that have long
foiled researchers.
       Even that negative finding is very significant, experts say,
because in mathematics, a negative result, showing something is
impossible, can be just as important and open just as many new
areas of research as a positive one.
       The discovery was made by Sanjeev Arora and Madhu Sudan,
graduate students at the University of California at Berkeley, Dr.
Rajeev Motwani, an assistant professor at Stanford University, and
Dr. Carsten Lund and Dr. Mario Szegedy, young computer scientists
at AT&T Bell Laboratories.
       Motwani, who is the senior member of the group, just turned 30
on March 26.
       ``With the conventional notion of a proof, you had to check it
line by line,'' said Dr. Michael Sipser, a theoretical computer
scientist at the Massachusetts Institute of Technology. ``An error
might be buried on page 475, line 6. A `less than or equal to'
should have been a `less than.' That would totally trash the whole
proof. But you'd have to dig through the whole thing to find it,''
Sipser said.
       Now, he added, ``the new idea is that there is a way to
transform any proof so that if there is an error, it appears almost
everywhere. I'd say, `You have a proof? Show me a page.' If there
is an error, it will be there.''
       The finding, which is built on two and a one half years work by
leading researchers, is expected to have a profound impact. But
because it is so new and unexpected, mathematicians and computer
scientists cannot yet predict its scope of application.
       ``This is philosophically important,'' said Dr. Mihalis
Yannakakis, a theoretical computer scientist at AT&T Bell
Laboratories.
       Dr. Manuel Blum, a mathematician at the University of California
at Berkeley, agreed that the result was ``really exciting,'' with
ramifications that were hard to predict.
       The new verification method can show up errors in calculations
as well as in proofs, experts said, and Blum suggested it could
have a role in checking long computer computations.
       Dr. Umesh Vazirani, a theoretical computer scientist at the
University of California at Berkeley, said that the discovery was
``one of the most outstanding in the past decade,'' because it
allowed investigators to decide, almost at a glance, whether it was
worthwhile to try to find an approximate solution to a problem they
were unable to solve exactly.
       ``As soon as you formulate a problem, you can know if it is
intractable,'' Vazirani said.
       The finding is so recent that it has not yet been published.
``We just wrote it in the last month or so,'' Motwani said.

(STORY CAN END HERE. OPTIONAL MATERIAL FOLLOWS.)

        As is the custom in mathematics and theoretical computer
science, the result gains credibility not by having been sent to a
journal and reviewed by other scientists but by having been vouched
for by leaders in the field. In this case, at least a dozen experts
say they are convinced by the result, and amazed by it.
       ``It is absolutely stunning,'' said Dr. Laszlo Babai, a
theoretical computer scientist at the University of Chicago whose
work helped lay the groundwork for the new discovery.
       Dr. Richard Karp, a theoretical computer scientist at the
University of California at Berkeley agreed. ``Personally, to me
it's very surprising,'' he said.
       Karp added that the method's application to show that certain
problems have no approximate solutions is the most important
discovery in his field of theoretical computer science in more than
two decades.
       The insight into the nature of proofs shows how a connection can
be created between each and every logical statement of a proof, no
matter how long or complicated the proof may be.
       It applies to the 15,000-page proof that is the longest ever
published as well as to the single sentence proof that can be
grasped in an instant. And it is no harder to check a long proof
than a short one.
       ``Even if your proof is the size of the universe, you still
would not have an increased number of places to check,'' Babai
said.
       MORE (Optional 2ndtake follows.)




            *** End of Article ***
type: NYT (Copyright 1992 The New York Times)
priority: Regular
date: 04-06-92 1806EDT
category: Domestic
subject: BC MATH PROOF SCI
title: UNDATED: Babai said.
text:

       The investigators used a variety of mathematical tricks to
transform a proof so that its errors, if any, will show up almost
everywhere. One method they used was a way of comparing lists of
data.
       Data comparisons are an integral part of long proofs. For
example, Babai pointed out, a typical mathematical proof might have
a line that says, ``We know by theorem 31 that a equals b and by
theorem 72 that b equals c. Therefore, we conclude that a equals
c.''
       To see if this statement is correct, Babai said, ``We have to go
back to theorem 31 and see that it does indeed say b equals c, and
similarly we have to look up theorem 72 and see that it has been
copied correctly.''
       One way to compare two lists of numbers is to arrange them in a
grid and then extend each row and column in an arithmetic
progression.
       To compare the data list 1,3,-1,4 with 1,5,-1,4, in which a
single digit has been miscopied, the lists would be formed into
arrays with 1,3 and 1,5 in the top rows.
       The rows would then be extended in arithmetic progression,
forming 1,3,5,7,9,11 and 1,5,7,9,11,13. Already the one-digit
difference has been amplified to five differences.
       By increasing the number of items listed in the arrays, the
number of differences in an extended array can be made so great
that sampling just a few squares will be bound to detect a
difference and thus an error in one of the original data sets.
       Proofs are not data but they can be manipulated in analogous
ways by representing each step of a proof as a term in a
mathematical equation.
       For example, Arora said, a logical statement in a typical proof
might say ``either statement x and statement y are true, or
statement z is not true.'' This can be converted to the equation,
xy pluas (1-z).
       Using these equations, the mathematicians formulated
relationships that would have to hold if the proof contained no
error.
       Blum said that the new finding does have in it an intrinsic
uncertainty. An error will show up in almost everywhere in a
suitably transformed proof, but not absolutely everywhere. But, he
said, it is easy to probe the proof in enough places to make the
chance of missing the error negligible.
       ``Realistically, you can make the probability of missing an
error as small as 1 divided by the number of atoms in the
universe,'' Blum said. That means you can make the chance of
mistaking an incorrect proof for a correct one, ``virtually zero,''
he said.
       The investigators also noticed that their proof was
mathematically equivalent to a profound statement of how difficult
it would be to solve certain problems that plague mathematicians.
       Their proof about proofs showed that it was just as hard to get
an approximate solution to a class of very hard problems as it was
to get an exact solution.
       Since computer scientists have pretty much given up on exact
solutions to these problems, the new result means they must abandon
even their remaining shreds of hope of obtaining rough solutions to
a wide group of practical problems.
       This group of thousands of problems involves finding the best
ways to do things like design telephone networks or airline routes
or to set up sequences of jobs in factories.
       Although no one has ever proved that an exact solution is
impossible, the consensus is that they are. Babai said
mathematicians had tried for centuries to find an exact solution,
to no avail.
       ``Literally thousands of people have considered these problems,
including the greatest minds in mathematics, including Gauss, and
have gotten nowhere,'' he said, referring to Karl F. Gauss, an
18th-century German mathematician and astronomer.
       In 1971, investigators showed that a wide range of these very
hard problems were mathematically equivalent: a solution to one
would solve all. Without hope of an exact solution that would solve
them all the time, they began a search for an approximate solution
that would always come very close, at least, to the best answer.
       Babai explained: ``There was a feeling that, `All right, we are
not interested in the best possible solution. If we can get within
1 percent of an exact solution, that's good enough.' ''
       Investigators found some good approximate solutions that worked
for some problems some of the time, but they could not find a good
approximation that would always work.
       Now they know there is a reason for that. The solutions they
were seeking were as hard to find as exact solutions.
       Karp said this finding alone was a monumental achievement,
equivalent in its impact to the finding in 1971 that the problems
were equivalent. Until now, he said, ``One could always cling to
the hope that one could get what one wanted with a fast algorithm
that gives a good approximation. Now we know that even that is
unachievable.''





            *** End of Article ***
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