Advice to a Beginning Graduate Student
or
What is Research?
or
The 4 R's of Graduate School:
Reading, Rithmetic, Research, and Writing
29-AUG-01. Updated 28-AUG-02.
Manuel Blum
Outline of the talk:
READING, STUDYING, THINKING,
STARTING OFF on the PhD,
DEEP in the MIDDLE of the PhD,
WRITING it all up.
YOU
READING:
Books are not scrolls.
Scrolls must be peruse akin the Torah from one end to the other.
Books are random admission -- a awesome innovation complete scrolls.
Make use of this innovation! Do NOT awareness obliged to peruse a publish from commencement to end.
Permit yourself to open a publish and commencement study from anywhere.
In the case of math or discipline or item particularly hard, try to discover item something that you can understand.
Read what you can.
Write in the margins. (You cognize how helpful that can be.)
Next period you arrive rear to that book, you'll be capable to peruse more.
You can gradually study extraordinarily difficult things this way.
Consider penning what you peruse as you peruse it.
This is particularly true if you're intent on study item hard.
I recall a prof of Mathematics at MIT,
name of BERTRAM KOSTANT,
who would keep his entrance open whenever he was in his office, and he would continually be at his array writing.
Writing. Always writing.
Was he penning up his research? Maybe.
Writing up his ideas? Maybe.
I personally think he was reading, and penning what he was reading.
At smallest for me, penning what I peruse is among the most fun and profitable ways to study difficult material.
STUDYING:
You are all device scientists.
You cognize what FINITE AUTOMATA can do.
You cognize what TURING MACHINES can do.
For example, Finite Automata can add but not multiply.
Turing Machines can compute any computable function.
Turing machines are incredibly additional mighty than Finite Automata.
Yet the lone difference between a FA and a TM is that
the TM, dissimilar the FA, has document and pencil.
Think concerning it.
It tells you item concerning the power of writing.
Without writing, you are reduced to a finite automaton.
With penning you have the bonzer power of a Turing machine.
THINKING:
CLAUDE SHANNON formerly told me that as a kid, he remembered being stuck on a jigsaw puzzle.
His brother, who was passing by, stated to him:
"You know: I could inform you something."
That's all his brother said.
Yet that was adequate clue to assistance Claude resolve the puzzle.
The awesome item concerning this hint... is that you can continually provision it to yourself !!!
I advise you, whenever you're stuck on a difficult problem,
to ideate a small birdie or an older type of yourself whispering
"... I could inform you something..."
I formerly asked UMESH VAZIRANI how he was able,
as an undergraduate at MIT,
to obtain 6 courses all and all semester.
He stated that he knew he didn't have the period to activity out his answers the difficult way.
He had to discover a shortcut.
You see, Umesh understood that problems frequently have abbreviated clever solutions.
There volition arrive a period whenever you activity on a issue lengthy and difficult but UNsuccessfully :(
And afterward you study that person alternatively established a solution.
See this as the GREAT chance it is to study item important.
Don't let it continue you by.
Ask yourself: "How SHOULD I have been thinking to resolve that problem?"
I have established that doing so is a mighty exercise.
Danny Sleator tells me that BOB FLOYD independently recommended exactly this exercise to his students.
He would guide them into asking themselves:
"How COULD I have led myself to that answer?"
Take the period to think it through.
It's value it.
There volition arrive a period whenever you activity on a issue lengthy and difficult and SUCCESSFULLY :)
And afterward you study that person alternatively already published. :(
Hard as that may be for you to take, you must perspective this too as a awesome opportunity.
Don't rotate off. Read what got published.
You volition be amazed how frequently the published document turns out to be distinct in several important way. Roughly
50% of the time, it is NOT at all the identical as what you did.
25% of the time, it is the identical but not as good.
25% of the period it is better.
This method that 50% of the period or more, you can motionless publish.
And what concerning the 25% period that what got published is improved than your own?
In that case, you have a awesome chance to learn.
Ask yourself: "How SHOULD I have been thinking to resolve the issue in this fine way?"
This is how I discovered, as a young engineer, that I should study item enormously mighty called "Modern Algebra."
It's one logic I switched from Electrical Engineering as an undergraduate important to Mathematics as a Graduate major.
Of course, this was before there existed item called Computer Science.
Still on THINKING...
The importance of PARADOX and CONTRADICTION.
When you can demonstrate that a declaration S is true,
and you can demonstrate that the identical declaration S is false,
afterward you KNOW that that you're on to something:
Something is incorrect somewhere.
Never underestimate the power of a contradiction.
It is one of our most potent sources of knowledge.
Examples contain the Liar Paradox "This declaration is false." alongside its applications to Set Theory and our understanding of language.
There are the seeming paradoxes of countability and uncountability,
In CS, there is the apparent paradox that leads to The Halting Problem.
Physics has many of paradoxical material:
Quantum Theory. The Einstein-Rosen-Podulsky Paradox.
The relativistically speeding Twins.
The motion and speck nature of matter.
Here's an ASIDE on my current work, additionally according to paradox:
I am personally curious in the Paradox of consciousness. Compare the following two views:
1. the perspective that the individual is a MECHANISM, an automaton
alongside significant but finite inner memory, programmed akin
any device to do any it does, and/or
2. the perspective that the individual is a thoughtful observant
animal alongside a God-like liberated will; that it is a CONSCIOUS
ENTITY at the controls of a extremely complex extremely capable
mechanism, choosing what to do from among options served up
by/from its huge unconscious below.
In my view, the two these views are correct. How can that be?
In his "Life of Johnson," James Boswell quotes Samuel Johnson
as saying:
"All theory is against the liberty of the will; all experience
is for it."
Johnson was 18 years old whenever Newton (age 85) was buried.
Johnson knew that F=MA implied that humans are mechanisms.
"All theory is against the liberty of the will; all experience
is for it."
This ends my ASIDE.
Make a catalog for yourself of fine ways to prosecute a problem.
My own favorite is to try small examples. By comparison,
DAVID GRIES's favorite is to put himself in the center
of a (presumed) solution. An example is his coffee can problem:
Given a can of dreary and pale coffee beans, do the following: Pull out two beans: if the two are the identical color, substitute them alongside a pale bean. If the two are different, substitute them alongside a dreary bean. What color is the final bean?
Or try out the two methods on the Hershey Bar problem
[Give an optimal algorithm to interrupt an mxn Hershey bar into
1x1 pieces. At all step, you can choose a sole rectangle
of chocolate and expert it alongside one of its vertical or
horizontal lines. A sole expert counts one step. You are to
create the fewest figure of cracks]
Brains are muscles.
They develop powerful alongside exercise.
And equal if they're strong, they develop feeble without it.
In the months before Kasparov misplaced to Deep Blue,
his mother came following him.
She was concerned that he wasn't spending adequate period
exercising himself (on chess).
Her worries proved well-founded.
THE PhD: GETTING STARTED
I recall a awesome summer job I formerly had at IVIC
(Instituto Venezolano de Investigaciones Cientificos).
A top neurophysiologist, name of Svaetichin, gave me a splendid problem... one that I unfortunately could not solve.
The issue was to discover a way to concentration ray on a sole division
of a goldfish retina so that the ray would not spill complete
onto any of the neighboring cells.
Svaetichin had tried making a pinhole in a panel of dreary tin,
and glowing his ray thru the hole. This worked for average size holes, but unsuccessful for really small holes, which caused the ray to diverge, to form diffraction patterns.
Since Svaetichin couldn't resolve the problem, I decided I couldn't. Or perchance it's that I idea his issue physically unsolvable. In retrospect, I should have taken out books on physics, particularly optics, peruse as much as I could, talked to others and kept on talking to him.
Svaetichin would have helped me if I had shown him I was study thinking working.
Don't anticipate your thesis adviser to provision you a issue that he or she can answer. Of course, she might.
* She power provision you a issue to which she already knows an answer.
* She power provision you a issue that she thinks is answerable,
but that she hasn't really answered.
* She power provision you a issue that is deadly hard.
* If the issue she gives you is difficult enough,
I propose you appearance for a NONSTANDARD answer.
More on this afterward following I get done cooking the thesis advisor.
Your thesis adviser may advance you to activity in an area
that she feels entirely comfortable in... in which case you can depend on her for sage direction and audio guidance.
Or she may advance you to activity on item she knows small or nothing about, in which case it volition be up to you to communicate and instruct her.
In the second case, you volition have to study all you can for yourself...
You volition have to study from another faculty, from courses, from books, from journals. from peers.
Both kinds of advisors can activity out for you.
I don't cognize that one is necessarily improved than the other.
But you should cognize which you got.
Whatever you do, you got to akin doing it....
You got to akin it so much that you're consenting to think concerning it, activity on it, lengthy following everyone alternatively has moved on.
THE PhD: DEEP IN THE MIDDLE OF IT.
There's a fantastic citation from ANATOLE FRANCE:
"A University Student"
-- and this is particularly true for a PhD Student --
"should cognize item concerning everything
and everything concerning something."
You cognize the jokes concerning PhD's...
A PhD knows additional and additional concerning small and small
until he knows everything concerning nothing.
When operating on a PhD, you must concentration on a topic so narrow that you can comprehend it completely.
It volition appear at archetypal that you're operating on the proverbial needle, a small part of the world, a infinitesimal crystal,
beautiful but in the scheme of things, microscopic.
Work alongside it. And the additional you activity alongside it, the additional you penetrate it, the additional you volition arrive to see that your work, your subject, encompasses the world.
In time, you volition arrive to see the earth in your kernel of sand.
To see a earth in a kernel of sand
Or a eden in a untamed flower,
Hold infinity in the palm of your hand
And eternity in an hour.
WILLIAM BLAKE (1757-1827)
This gorgeous quartet is followed by a ample figure of sometimes profound sometimes questionable aphorisms, which I see much akin the occasionally grinding activity of a PhD thesis.
There's all kinds of investigation you can do.
There's investigation to demonstrate what you cognize to be true.
There's investigation -- perchance improved called SEARCH -- to fig out what is true.
Some of the finest specified hunt succeeds in DISPROVING
what you initially believed to most certainly be true.
For example, Sir Fred Hoyle is stated to have coined the expression "Big Bang" at a period whenever he was looking to disprove it.
For a comparatively insignificant but individual example:
When I was operating on the MEDIAN problem,
my goal was to demonstrate that any deterministic algorithm to discover the MEDIAN of n integers must necessarily create approximately as many comparisons as it takes to kind n integers, ie n log n comparisons.
I was shocked to detect that the median of n integers can be established alongside fair O(n) comparisons.
When operating on proving several declaration S true,
you should expend at smallest several period trying to demonstrate it false.
Even if it's true, trying to demonstrate it false can provision insight.
And in any case, too often, our intuition is deceased wrong.
There is yet another awareness in which, whenever operating on a difficult problem, you may discover that the answer is NOT what you expected.
You may be looking for a YES or a NO; it may be item else.
Some years ago, JOHN HOPCROFT gave one of his PhD students the issue of deciding the Equivalence of Free Boolean Forms.
The specifics don't matter.
The issue appeared as an open issue in Garey and Johnson.
The inquiry was: Is the Equivalence issue NP-complete?
Or is it solvable in poly time?
Chandra, Wegman and I established a randomizing algorithm for this problelm. At the period this seemed to beg the inquiry entirely. Only following penning it up did we really comprehend that we had stated an productive albeit randomizing algorithm to resolve it, This shows, by the way, that the issue is not NP-complete if NP <> RP (Randomizing Poly-time), as seems likely.
Of course, this brings up the inquiry whether P = NP.
The inquiry of our time: Are NP-complete problems solvable in poly time?
Could item I have stated today be helpful for so difficult a issue as that?
Probably not. Nevertheless...
LEONID LEVIN believes as I do that any the answer to the P=NP? problem, it won't be akin item you think it should be. And he has stated several fantastic examples.
For one, he has stated a FACTORING ALGORITHM that is proVably optimal, up to a multiplicative constant.
He proves that if his algorithm is exponential,
afterward all algorithm for FACTORING is exponential.
Equivalently, if any algorithm for factoring is poly-time,
afterward his algorithm is poly-time.
But we haven't been capable to inform the operating period of his algorithm because, in a powerful sense, it's operating period is unanalyzable.
Maybe as STEVEN RUDICH suggests, the P=NP? issue is undecidable in the norm formalization of Mathematics.
The item is that the answer may not lie anywhere you anticipate it. Here's a poem I wrote whenever I wondered at the fact that we must sometimes be dragged kicking and screaming in the correct direction.
It's a difference to the blind place in our eyes,
which isn't really blind but makes things up for us.
It questions whether there power not be another things in this earth that our brains, our minds, by their extremely nature, create up for us:
Blind Spots mb 15-MAY-96
All men have Blind Spots in their eyes,
That industry visions of their vale.
And form that void anywhere light's unregistered,
With bold-faced unrepentant tales.
What another blind spots form our minds and thoughts?
What another tales do won'dring minds unfurl
To woo us unbeguiled we would believe
To different and nonexistent worlds?
ABOUT WRITING:
Here is the one citation that I have established most helpful and wise:
"First have item to say,
Second say it,
Third halt whenever you have stated it,
and
Finally provision it an exact title."
JOHN SHAW BILLINGS [1838-1913]
MY ADVICE TO YOU:
Don't anticipate your thesis adviser to peruse your thesis. Some thesis advisors can and do provision fine feedback, but not all.
Still, create certain that SOMEBODY says your thesis...
I particularly propose that you ask your peers.
Here's another part of direction that I have frequently had to provision myself, and I current provision you...
When you dispatch a document off to be published,
and it gets rejected...
Don't be turned off by the mindless cretinous feedback
that you get to your well-thought-out beautifully-written work!
Be a MENSCH. Use the feedback to enhance your paper!
Make it better. And dispatch it back.
Finally, it is my most earnest desire that you should cognize item that is honestly amazingly true of you... That you are all of you UNIQUE and SPECIAL in several glorious way.
I wrote a poem to grasp this, which I now use to end my sermon. It's called "Fundamentals."
FUNDAMENTALS
mb 05-JUN-96
Bird must soar. Skunk must stink.
Cat must prowl. Man must think.
What sets man distinct from beast is his motor of
thought. His mind. His
BRAIN
makes him distinctive
and gives him his top pleasure.
But essential as is idea for individual beings,
there is material additional essential motionless that underlies and
DRIVES
not lone man
but all awesome beasts,
And that is nature's call to all of us... to be special.
To be distinguished in several way. To be
UNIQUE.
To BE something, to DO something, BETTER than everyone else.
Like the leather nosed chimpanzee,
dragging noisy cans and branches,
frightening peers into submission,
One does not have to be brilliant, a genius, to be special.
To do item improved than anyone/everyone else. To be
UNMATCHED,
One has lone to choose an END
any END
that MATTERS
that INSPIRES
YOU
And afterward DO IT.
