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generally speaking, a proof is going to have
seven characteristics that you want to keep in mind. Good proofs are correct--
that's obviously important-- complete-- you've got to
have all the details there. All of the key steps
have to be there. They should be clear so we can
understand what's going on. Brief is good. You don't want to crush
somebody with all the details. You want to get to the
key points, and be crisp. It's really nice
if they're elegant. Now, that means clever. It's the mathematician's
notion of beauty. Like you go to the art museum,
and the artists will say, wow, that's a
beautiful painting. In mathematics, you say,
wow, that's an elegant proof. It's crisp, clever,
short, to the point. And it's really the
highest compliment you can get from a
mathematician, anyway. And there's a lot of
judgment that goes into that. Just like in art--
there's judgment over what is great art. The proof should
be well organized. For example, use
lemmas the same way you would use
subroutines in writing code-- helps to make it clear. And the proof
should be in order. Sometimes you'll see
proofs where things are done in a haphazard fashion. All the pieces are there, but
they're in the wrong order. Sometimes-- and they teach
this in some high schools-- they teach you to
do proofs backwards. And the classic thing
there is, say you're trying to prove a equals b. Well, the proof sometimes
will start with what you're trying to prove. And then they'll do
a bunch of steps. And then you'll end
up with 1 equals 1, and you'll write a
check, because you went from a equals b to 1 equals 1. And that's, of course, true. Now, that's not a
good thing to do. It can be correct if the
implications go this way. Because really, you're starting
with a fact-- 1 equals 1-- and deriving a equals b. So if, in fact, the
implications work this way, then your proof is right. But most people, especially
as you get farther along, think about it going
from top to bottom. So don't use this technique,
because it'll confuse people. You're liable to make a mistake. Just start with 1 equals 1
and work your way from there-- top down for being in order. Now, good proofs are
very much like good code. In fact, one of the reasons we
care so much about teaching you how to write a good
proof in computer science is so that later on,
you'll be able to prove that your programs are doing
what you expect-- what they're supposed to do. Now, there are many famous
examples where programs did not do what they were
supposed to do, with disastrous consequences. The Airbus A300 was one of the
first commercial jets totally operated by software. It could take off, fly, and
land totally by software. It was a major advance
in the airline industry. The only problem
was, that on one of the first
flights of the A300, the software accidentally
opened the rear door just before landing. And the plane
crashed as a result. It was the first plane crash
in commercial history because of a software bug. There's a famous radiation
device for cancer patients called the Therac-25. It's famous because it
got into a race condition occasionally, which
caused the device to just slam the patient
with radiation-- so much so, it killed the patient. And they had multiple examples
of this, and of course, a lot of lawsuits afterwards. How many of you all
remember the 2000 election? A few of you do. OK, this will probably be the
last class that remembers that. But that's where Al Gore was
going up against George Bush-- very close election-- all
came down to recounting the votes in Florida. But in fact, Al Gore got
negative 16,000 votes in one county because
of a software bug in the electronic
voting booths, which I think they got rid of a
lot of them in the election after that. Because the software was buggy. So poor Al had enough
problem which chads and funny business in Florida. But getting negative
16,000 votes certainly didn't help his chances. Several years ago, a
single faulty command in a computer system used by
United and American Airlines grounded the entire fleets
to both airlines for close to a day. Because they
couldn't do anything. They're all run by computer, and
the whole thing was screwed up. So there's lots more examples. We run into this issue
all the time at Akamai. Akamai is a company started
by MIT folks-- by myself, and Danny Lewin, and a dozen
undergrads in the late 1990s. And we deliver a
lot of the content you get on the web-- Facebook,
all the search engines. A lot of the stuff you go
to comes from our servers. So we've got to be
very careful that we don't have software bugs. Now, in fact we do. And we catch them
every once in awhile. But if we got a bad
one, it would bring down all the sites you go to you. You wouldn't be able to
go to those sites anymore. And everybody would notice. It would be sort
of embarrassing. Now, this really does matter. And this is going to sound a
little scary, but someday-- probably 30, 20 years
from now, somewhere in there-- it's possible
that all of us, our lives may depend on the software
that some of you write. In fact, to bring this
home how scary it is, look at the person
sitting next to you. And imagine that in 25
years, your life depends on whether their code
does what it's supposed to do-- little scary. That's why we are
very motivated to help you learn how to make
rock-solid arguments, so you don't have code that
fries one of your classmates someday, or puts him in
a bad plane situation. Now unfortunately,
writing rock-solid proofs is a very hard thing to do. Even the world's
best mathematicians mess them up on a regular basis. In fact, it's estimated that one
third of all published proofs have bugs, have flaws, that
render the proof incorrect. The trouble often arises
because we get lazy. We don't write down all the
details or all the steps. Because, wow, it's clear. Let's just move on. Now, this can be OK. Not such a good practice,
but it dramatically increases the chances
of making a mistake. And there's some
very famous examples in the math literature
from the world's most famous mathematicians. Gauss-- and we'll
talk about Gauss later during the term--
he's one of the most famous mathematicians ever. He wrote his PhD thesis in 1799. And it's usually referred to
as the first rigorous proof of the fundamental
theorem of algebra. And that says that
every polynomial has a 0 over the complex
numbers-- something probably a lot of you
learned in high school. You get a polynomial. You can find roots
of the polynomial over the complex numbers. But his thesis contains
the following quote, "If a branch of an algebraic
curve enters a bounded region, it must necessarily leave again. Nobody, to my knowledge,
has ever doubted this fact." Warning signs-- buzzers-- should
be going off in your brain. "But if anybody desires it,
then on another occasion, I intend to give a demonstration
which will leave no doubt." So he's using something
that he believes to be true. He sort of thinks it's clear,
that everybody knows it. But he writes this down. And when you're
writing that down, you know there's a problem. And in fact, there's
another Fields medalist-- we talked about Fields medals
last week-- Stephen Smale writes that this was an immense
gap in the proof that was not filled until 1920-- more
than 100 years later. So Gauss never could
give the proof, and nobody did for
over a century more. Remember the Poincare
conjecture from last week? We had Colbert
talking about that. In 1900, Poincare claimed
it was a simple fact. Four years later, he
decided it wasn't so simple, and he demoted his claim to
the status of a conjecture. And of course, this became the
famous Poincare conjecture, which took another
century to solve. So when you think you
see yourself doing this, famous mathematicians do it
too, and it catches them. So you want to not
try to do that. Now, in fact, just to
really bring home the point, we've made a top 10
list of proof techniques you should not use in 6.042. So everybody go to
the handout here. In fact, there's lot of
them on the back, too, but on the front page,
there's the top 10. So we're going to
go through these. These are the ones
we've most observed in 6.042 over the years. Number 10-- proof by
throwing in the kitchen sink. The author writes down every
theorem known to mankind, and then adds a few
more for good measure. Now, this is good, because
when you're questioned later-- you're trying to get a
better score on your exam-- you say, look, the proof
contains all the key facts. They're all here. And so that does help. You get extra credit if
all the facts are there. We are literally seen students
copy over their crib sheet, if they got enough time,
for a problem they're not knowing how to solve, just to
get all of the facts in there. Number nine-- proof by example--
the author gives the case n equals 2 and
suggests it contains most of the ideas of
the general proof. In fact, a student
before a class was telling me on one of
the homework problems, he talked about it to
his non-math friends, and they gave examples,
not the proof. Number eight-- proof by
vigorous hand-waving-- one of my favorites-- I get
up here and wave my hands. And it must be true. Number seven-- proof by
cumbersome notation-- here the reader gets
hopelessly confused, gives up, and says OK. In fact, I once
had a grad student, and we called him the
encryptor, because he could take the simplest proof
and so encrypt it in notation and God knows what, you could
never understand the proof and figure out if it was
really right or wrong. That goes well with number
six-- proof by exhaustion-- so does throwing in
the kitchen sink. Number five--
proof by omission-- you will see that a lot,
also by faculty-- "The reader may easily supply the details. The other cases are
analogous-- trivial." Experts use that all the time. You can find a lot
of my papers online. "The proof is
trivial"-- very bad. Every once in a while, it
turned out not to be so trivial. Number four-- proof by
picture-- we've seen those, saw one on Tuesday. Number three-- proof by
vehement assertion-- sort of like the hand-waving. The more forceful you
are in your argument, the higher you raise your
voice, the more intimidating you become, doesn't
make it more true. Number two-- proof by appeal
to intuition-- you say, "any moron knows that." Well, you're sort of
reluctant to now question it. And then number one-- proof by
reference to eminent authority. I saw Fermat on the elevator
and he said he had a proof. I don't think so. Fermat's been dead
for over 300 years. And he was not so reliable
in the first place. In fact, he has one of
the most famous assertions that turned into a conjecture,
called Fermat's last theorem. And it was not really a
theorem by the time he died. How many people have heard
of Fermat's last theorem? Raise your hand. Yeah, pretty famous. It's the one theorem he proved
that actually, he didn't prove. It says, for all
n bigger than 2, there does not exist an x, y,
and z in the natural numbers plus the positive natural
numbers such that x to the n plus y to the n
equals z to the n. Now of course, for
n equals 2, that's just the Pythagorean theorem. You can find three,
four, and five, when you square them satisfy that. But it says this does not
work for any higher value, any higher power, than 2. Now in 1637, Fermat wrote
in the margin of a block that he had discovered
a proof of this result, but the proof was too
long to fit in the margin. And so he wasn't going to
supply the details there. And he never did
supply the details. And it took 350 years and
100s and 100s of pages. A fellow named Andrew
Wiles-- took him personally over 10 years. He more or less locked
himself in his room for 10 years, produced
100s of pages, and he finally did prove
Fermat's Last Theorem. So Fermat was right
about one thing-- it would not fit in the margin. At least, there's
no proof that we know of that would fit
in the margin of a book. Any questions on
proof technique? We're going to pound a lot on
this over the next few weeks. All right, next
we're going to look at a class of puzzles that was
very popular in the late 1800s. Now, in these puzzles, you have
a grid of letters or numbers. And you've got to slide the
letters and numbers around to put them in order. And so as an example, here's
the problem on a three by three grid, or
an eight puzzle. You want to find a
sequence of moves to go from this configuration. I'll do a three by three case. I have a here, b
here, c, d, e, f. And I've put g and
h out of order. And that square is blank, and
I can move tiles or letters into it. I'm going to start here. And I eventually want to
get to the configuration where the letters are all
in alphabetical order. So g and h are in order. Now, a legal move means
you slide a letter into an adjacent blank square. And you can go in
a row or a column. OK, everybody
understand the game, what you're supposed to do? You slide these things. That could go down. That could go across. How many people played
a game like this before? Good, a lot of you. OK, so I need some
volunteers who are good at solving
puzzles that can do this, have played games like this,
that are pretty quick at it. So I've got it here. Who would like to come down? There are prizes if
you can solve this. What we're going to
do is solve this. And see if you can solve
it in three minutes. But I need three students
to come on down and do this. Who'd like to volunteer? Who thinks they can handle
puzzles under pressure? So you have the g
and h out of order. So it's not so hard. You move the f down there. I don't know, play
around like that. So I need some volunteers. Who'd like to come
down and do this? You can win candy. I've got candy here. There's even a prize if you
don't make it in three minutes. Who wants some candy? Nobody? Is there a prize if
we don't make it? What's that? Is there a prize if
we don't make it? There is. Come on down. All right, now, who's
going to help him out? I need a couple more people
here just to mess him up, because by himself--
there we go-- good. Anybody else like
to come on down? Because if we get
three, it really gets hard, because you fight
over what moves to make. Now, what I'd like you
all to do is sort of scream out
possibilities for him. Like he figures it out
there, you scream out, that'll really screw him up. OK, now, you can't pick them up. You've just got to
slide them around. And you can go, and
we'll go to 2:56, and see if you can solve it. If you can solve it,
you guys get candy. And if you can figure it out
out there, let them know. Slide to the gray area. No, you can't use the gray area. All right, you've got
about two minutes. They have the g and h in order. Everything else is pretty
well screwed up, though. What do you got? A, b, c, d, e, f, h g-- no, no,
no, that's where you started, guys. That's not it. It was close, just one
was out of order is all. Just one letter at a time. There you go. I think they need help. We got about a minute ago. Cannot see anything. You can't see anything. Well, I don't know
if it would help. You want to show
them where you are? This is not looking good. We're pretty close to
the end here, guys. A, b, c, f, e, d,
g, h-- g, h is good. F is not so good. All right, you got
about 30 seconds. Whoa, what are you guys doing? One at a time here. One at the time. In fact, I think
time is up here. All right, 2:56. Oh, no, no. That we don't do. No diagonal moves--
no, no, no, no. All right, well, you
guys were good sports, so you do get a prize. It's not the candy, but I have
these wonderful plastic Nerd Pride pocket protectors for you. There you go, very good sports. Thank you, well done. All right. [APPLAUSE] Now, I was a little
cruel to these guys, for a lot of reasons. It's impossible? You think? [INDISTINCT CHATTER] Yeah it is impossible. This is impossible to do. Let's see if we can
prove it's impossible. Because it doesn't seem
so hard, necessarily. But let's take
that as a theorem. There is no sequence
of legal moves-- diagonal moves makes this a
lot easier-- to invert g and h and also return all the other
letters to their original order or position. Now, to prove this,
we're going to use what's called an invariant. It's a very powerful
and commonly used concept in computer science,
very closely tied to induction, as we'll see. In order to show that
your system can never reach a particular
special state, it is sufficient to show
there's some property called the invariant that holds
at the initial state, and that is preserved
by every legal move, and is not present-- does not
hold-- in that special state. The idea is, if you get
this magic property called the invariant and it
holds at the start, and it holds across
every step, then the only states
you can reach have to have to have the
property, be invariant. And if the special state
doesn't have that property, you can't reach it. All right, so there
is going to be some property we're going
to look for that held at the beginning here-- and
the beginning had this set up-- that holds, is
preserved, by every move, every legal move, but does
not hold in this state. And therefore, you can never
reach this state legally. So they were doomed. So we've got to figure
out what that property is. And that's always the
trick in analyzing systems or
algorithms-- is what's that key property, the
invariant property? So to figure out the
invariant, we sort of have to look at what
happens during a move in this system-- a transition. Well, there's two
kinds of moves. There's a row move. And an example of a
row move is you might have a, b, c, d, g, e, f, h. And I'm going to move the g
its row to the blank square. So it would become this state. So g moves rightward
into the blank, and the rest stays the same. Now, when I make a row
move, did the relative order of the items change? No. Of course, to be
precise, I better define what relative order
means, or the natural order. So by the natural order, I mean
this ordering-- 1, 2 3, 4, 5, 6, 7, 8, 9. So if I look at the order
here, g moved from 5 to 6, but didn't change
order with respect to any of the other guys. Let's state that as a lemma. Because that'll be very useful. Lemma 1-- a row move
does not change the order of the items or the letters. Proof-- well, it's obvious. No, you can't let me get
away with that, right? Because I'm not to let you get
away with that on homework. That's not so good. So we've got to be a little
more careful in the proof here. Otherwise, we just head
down the path to trouble. So let's sort of specify,
what does a row move mean-- just get a little bit
of math and specification around it. Well, in a row move,
we move an item-- a letter-- from some
cell i, whatever i-- it could be anything 1
to 9-- into an adjacent cell. What are the possibilities
for the adjacent cell? Plus or minus 1, yeah, into
cell i minus 1 or i plus 1. Nothing else moves. Hence, the order of
the items is preserved. Because if you're going
with i to i plus 1, everything else is in i minus
1 or less, i plus 2 or more. Relative order does not
change in this case. And I'll claim that's enough. You know that I've thought
about it a little bit. I've really quantified that i
goes to i minus 1 or i plus 1. Nothing else moves. Therefore, we're done. Now, I could've added
some more sentences here, like I talked about. But this is probably enough
in this case, for this guy. So in fact, if we
only had row moves, there's not much that happens. The order never
changes with row moves. So we never get there. But column moves are
more interesting. So as an example of a column
move, a, b, c, d, f, h, e, g-- say I move the g up. Then I would get this one--
a, b, c, d, f, g-- because g moved up-- h, e. So g went to there. Did the ordering
change for this move? Yeah. How so? What things changed order? Which pairs of
letters changed order? AUDIENCE: G changed [INAUDIBLE]. PROFESSOR: G changed by three--
moved three in the order. And what letters did it
change relative order with? AUDIENCE: [INAUDIBLE]. AUDIENCE: The two preceding it. PROFESSOR: Yeah, the
two preceding it. g used to be after h and e. Now it's before h and e. All right, so it changed
relative order with two items. It changed its
position by three. It moved up three
in the ordering. And so it changed order
with the two in between. Let's look at another one. Because we're really
doing these examples to hunt for an
invariant, is what we're trying to do
here-- something that will let us prove we can
never get to the desired state. Let's look at this one. a, b, c, d, g, h,
e, f goes to-- I'm going to move b down this
time-- a, c, d, b, g, h, e, f. So b moves down. The relative order changes. Which pairs got changed
in the relative order? AUDIENCE: b. PROFESSOR: b changed with-- AUDIENCE: c and [INAUDIBLE]. PROFESSOR: c and d. b used to be before c and
d. when it moves down here, it goes after c and d. Any guess about what
Lemma Two is going to be? What happens when we
make a column move? AUDIENCE: [INAUDIBLE]. PROFESSOR: Change the order of-- AUDIENCE: [INAUDIBLE]. PROFESSOR: Good,
so a column move changes the order of a
guy with the previous two or the next two, which means
the relative order of two pairs change. So let's take that as Lemma Two. When we say don't
do proof by example, we don't mean
don't try examples. Because by trying
examples, you find out what you're trying to prove. So that's good to do. A column move changes the
relative order of precisely two pairs of items. So let's prove that. In a column move, we move
an item in cell i for some i to a blank spot in cell what? AUDIENCE: [INAUDIBLE]. PROFESSOR: i minus
3 or i plus 3? AUDIENCE: [INAUDIBLE]. PROFESSOR: OK, and just to
see that, let's draw out the natural order. I've got 1 2, 3,
4, 5, 6, 7, 8, 9. Column moves could be here--
1 to 4, and 4 to 1, 2 and 5, 3 and 6, 4 and 7,
5 and 8, 6 and 9. All 12 possible column
moves, this works for. They're always separated by 3. And when an item
moves three positions, it changes relative order
with two other items. And you can even write
down what they are. It's i minus 1, i minus
2, or i plus 1, i plus 2. Those are the guys it
changes order with. All right, we've
got two lemmas here. That one's done now. In a row move, order
does not change. In a column move, two pairs
of letters get flipped. That's all you can do
with this puzzle now. So can anybody
think of something we can work with to get an
invariant here-- something that won't change? What we should be focusing on? Yeah. AUDIENCE: You can only switch
two pairs of items at a time. So if you don't have an
even number [INAUDIBLE] switch [INAUDIBLE]. PROFESSOR: Yeah, that's good. So really, to focus
in on there is, if you have an even number
of things out of order to start, you're going to
have an even number forever, because you can only
change two at a time. That's a great idea. Let's really specify that
now, and define that. A pair of letters or items,
call them L1 and L2-- they form an inversion, also
known as an inverted pair, if L1 precedes L2
in the alphabet, but L1 appears after
L2 in the puzzle. All right, so that
is an inversion. So for example, let's
see how many inversions do we have in this case? We've got a, b,
c, f, d, g, e, h. How many inversions are there
in that state of the puzzle? Yeah? AUDIENCE: Three. PROFESSOR: Three, good. And what are they? AUDIENCE: d is after f. PROFESSOR: Yeah, d,
f is an inversion. e, f is an inversion. And what's the last one? AUDIENCE: e, g. PROFESSOR: e, g is an inversion. Yes, so the answer is
there's three inversions in this puzzle. And if I keep doing
row and column moves to this, what do
I know about the parity of the number of inversions? AUDIENCE: [INAUDIBLE]. PROFESSOR: Will always be odd--
that's sort of interesting. OK, now, how many inverted pairs
are there in the start state way back over here? How many inverted pairs here? One inversion. How many inversions here? Zero. All right, and can you see where
we're going to head here now? If I started with
one, that's odd. And every time I do a
row move or column move, it's going to stay odd. And I can never get here. Is that clear what
we're trying to do? So let's keep doing it. Let's go over here. All right, so now we'll
use Lemma One and Lemma Two create this notion that we only
change the number of inversions by an even number. And Lemma Three-- during a
move, the number of inversions can only increase by
two, decrease by two, or stay the same. And this proof is pretty easy. There's a couple of cases. In a row move, what happens? How does the number
of inversions change in a row move? AUDIENCE: Stays the same. PROFESSOR: Stays the
same-- no changes-- and that's by Lemma One. Now, in a column move,
there's three cases. But we know two pairs
change order-- that we know. That's Lemma Two. All right, by Lemma
Two, we know that. But the three cases are-- let
me start on the next board. So Case A is that both
of the inverted pairs were in order originally,
before the column move. What happens to the number
of inversions in this case, as a result of the column move? AUDIENCE: [INAUDIBLE]. PROFESSOR: Goes up by two. Case B is that both
pairs were inverted before I made the column move. What happens to the number
of inversions in that case? AUDIENCE: Decreases by two. PROFESSOR: Decreases by
two-- they were in order, so they were inverted. Now when I flop them,
they become in order. So inversions drops by two. And Case C is
there's one of each. One of the pairs was inverted. The other wasn't. What happens to the number
of inversions in that case? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yes, stays the same. All right, and we're done. That's what the lemma says-- I
can go up by two, down by two, or stay the same. All right, now a simple
corollary of this is that during a move, the
parity-- i.e. even or odd of the number of
inversions stays the same. It can't change at all. During a move, the parity--
and the parity means even, odd. Sometimes it's called
0, 1, but the evenness and the oddness of the number
of inversions does not change. AUDIENCE: Instead of two pairs
you mean one pair, right? PROFESSOR: Where? AUDIENCE: [INAUDIBLE] pairs. PROFESSOR: Oops, I mean
one pair's inverted, one pair's not. AUDIENCE: [INAUDIBLE]. PROFESSOR: No, I mean two pairs
change order in a column move. Let's go back and look at that. So I'm doing a column move. For example, g goes up. g, h changes order. g, e changes order. g changes order
with these two guys, so there's two pairs
that got flipped. So in a column move, two
pairs reverse their order. If they were in order,
they become inverted. If they were inverted,
they become in order. Any other questions? Yeah. AUDIENCE: So we're saying
that letters can be numbers in more than one pair? PROFESSOR: Yes, a letter
can be in-- a letter is in seven pairs, because
there's seven other letters. And in fact, we're looking
at a space of 8 times 7 divided by 2-- 28
pairs of letters. So there could be 28 inversions
if they're all out of order. Any other questions? OK, well, we're getting there. We're almost done. I've got to prove this
corollary that during a move, the evenness or oddness of
the number of inversions does not change. And that's because adding or
subtracting 2 from a number does not change its parity. It stays odd or stays even. So adding or subtracting 2
does not change the parity. All right, now we're ready
to state the invariant. Anybody tell me what the
invariant is going to be? What's the invariant going
to be in this system? AUDIENCE: [INAUDIBLE]. PROFESSOR: Yeah, the parity
of the number of inversions is odd, because it
starts that way. It's preserved. And it won't hold in
the desired end state that we're looking at. So we can state that as follows. In every state or
configuration reachable from the start
state, which is a, b, c, d, e, f, h, g--
which are out of order-- the parity of the number
of inversions is odd. And the proof will
be by induction. And invariant proofs
are always by induction. And the inductive
hypothesis-- and this is very typical in
a proof by using invariants-- is, so P of n is
after any sequence of n moves from the start state-- in
fact, just the rest of this is what it is. All right, so our inductive
hypothesis is P of n, after any sequence of n moves--
so n is the number of moves you took to get there--
from the start state, the parity of the number
of inversions is odd. That is the
inductive hypothesis. This is the invariant. And they become
one and the same. And the parameter n is the
number of moves you've taken. And that's how you set up
a proof using an invariant. It always looks just like this. OK, so we're doing a
proof by induction. We've got the
inductive hypothesis. What's the next step? AUDIENCE: Base case. PROFESSOR: Base case. And in these circumstances,
where the proof by invariant, the base case is always 0. We haven't made any moves yet. Now in this case, if
we haven't made moves, the number of inversions in
the start state now is 1. That means the parity is
odd, and the hypothesis is satisfied. P of 0 is true, because in
any sequence of zero moves, after this-- i.e. in this state-- you've
got one inversion. That's odd. The last step is
the inductive step. And here we need to show for
any n bigger or equal to 0, we need to show P of n implies
p of n plus 1 is true-- the standard thing. So let's look at P n plus 1. That's talking
about where we are after a sequence of n plus
1 moves in the puzzle, from the start state. So let's consider any
sequence of n plus 1 moves. And call these
moves-- label them by M1 all the way to M n plus 1. Those are the moves we made. Now, by the inductive
hypothesis, because of P n, we know that after the first n
moves, the parity is still odd. That's what P n says. That's P n. After any sequence
of n moves from here, the parity of the number
of inversions is odd. So we could say by the
inductive hypothesis, or we could say by P n, we know
that the parity after moves M1 out to M n is odd. Now, by Corollary One-- do
we still have Corollary One? [INAUDIBLE] AUDIENCE: [INAUDIBLE]. PROFESSOR: [INAUDIBLE], yeah,
good, we've got Corollary One. During any move, the
parity doesn't change. That says from after the nth
move to after the n plus 1st move-- one extra move--
doesn't change the parity. So it's still odd. So next, by Corollary
One, we know that the parity of the
number of inversions does not change
during M n plus 1. Therefore, this implies that
the parity after all n plus 1 moves-- these guys-- is odd. It started odd. It stayed odd after the first n. It stayed odd during
the n plus 1st. So it's odd. And that's exactly the
statement of P n plus 1. So we've completed
the induction. We've shown that P n
implies P n plus 1. All right, so now, any
questions so far on this? Yeah. AUDIENCE: What's the
difference between a lemma and a corollary? PROFESSOR: Really, there's
not a lot of difference. Corollary usually
is something that's a simple consequence
of something else. And the corollary had
a pretty short proof. Usually when you see corollary,
that's not a 10-page proof there. A lemma is something
that you're going to use as a tool for
a bigger thing, often a theorem or another lemma. In fact, now we're going to
get to the theorem, which is sort of the final thing--
the thing we're really after. And that's sort of the big deal. And now the proof
will be simple, because we've done all these
lemmas and corollaries. Any other questions? All right. In fact, the theorem
is up here, right? There is no sequence of legal
moves to invert g and h. In other words, the
students were screwed. They couldn't do it fairly. So let's prove the theorem now. The parity of the
number of inversions in the desired state-- i.e. in order-- the target
state-- is even. Nothing's out of order. There's zero inversions. By Lemma Four, the desired
state cannot be reached from the start state, because
its parity is odd, using legal moves. So we're done. So the proof of the
theorem-- by now, it's short, because we did four
lemmas, a corollary, and lots of argument. Could you have solved the puzzle
if I allow the blank to be somewhere else at the end? Is the puzzle solvable then? No. Why not? Doesn't impact the parity at
all, so you can't do that. Again, the idea
here-- and you'll get some practice
in homework-- is you're looking for a property
that holds at the beginning, is preserved by every
step, but is not present in the target state. Now actually, this puzzle
was enormously popular in the late 1880s. In fact there was $1,000
prize offered for anybody who could solve
the larger version. Back then, it was the 15
puzzle, which you can still buy today-- same problem
on a 4 by 4 grid-- exactly the same problem. A pair is out of order. You've got to get them in order. You can't do it. Now, the proof
you can't do it is a little harder-- same
idea-- one extra idea in it-- one extra lemma. And that's for homework. So you'll go through and
see exactly the structure with one extra little trick it. But back then, I guess most
people didn't-- of course, most people didn't
realize it wasn't doable. So they could offer
this price safely. $1,000 then is probably worth a
quarter million dollars today. Any questions on
the eight puzzle? Yeah. AUDIENCE: The
inductive steps-- when you said, "by inductive
hypothesis," [INAUDIBLE] parity after, shouldn't the
first term be M0? PROFESSOR: Let's see. No, the first move is M1. The second is M2. Now, when I talk
about zero moves, that means no moves
are taking place. I'm still in the start state. So a move is a transition
between states. So you could have state 0. That's the start state. State one would be the
move after move one. Have I got that right here? So the inductive step, I need
to show P0 zero implies P1. P1 implies P2, and so forth. And P n is the parity of
inversions after n moves. So you could have state s0. We didn't label the states here,
but you could have state s0. Yeah. AUDIENCE: So in this problem,
we assume that everything should be defined as rows or columns. How do we know when the
assumptions are valid? PROFESSOR: That's
a good question. OK, so really,
there's two cases. To be really precise,
we should have argued there's only two types of
moves, a row move and a column move. And in fact, if you look at what
can move into a blank square, there's four guys that
can do it-- the guy above, the guy to the right, the guy
to the left, and the guy below. Two of those are row
moves, two are columns. So that's right-- to really
pin down the details, we should have checked those
are the only two possible moves. And if there was a third kind of
move that I hadn't considered, this proof would be bogus. That's a good question. Any other questions? OK. Now, I think
probably, you see why the invariants are important. But say that someday,
you're building software to run a nuclear reactor. Now, there's a certain
state you'd really like to avoid-- meltdown. And you'd like to
be able to prove that, in any sequence of moves
that happen in your reactor program, you never reach
the meltdown state. Well you'd use
invariants to do that. Or if you're building
software for an airplane, you'd like to never
reach the crash state. Or if you're building
a radiation device, you never want to get in the
state that fries the patient. So this is an important notion. So for the rest
of the day, we're going to talk about a
different kind of induction called strong induction. It's very similar to
ordinary induction, but it's a little easier to use
when solving certain problems. Now like regular
induction, strong induction can be expressed with an axiom. So let me show you the
strong induction axiom. You have a predicate,
P n, like before. If P0 is true-- your base
case-- and for all n, it's not P n implies P
n plus 1, it's P0 and P1 and dot, dot, dot, and P n
are all true, then P n plus 1. If this is true for all n,
then P n is true for all n. Now, the only difference
between strong induction and ordinary induction that
we did last time is this part. In ordinary induction,
you're showing that P n implies P n plus 1. In strong induction, you're
showing that all these facts put together imply P n plus 1. Now, to show
implication, remember, you get to assume that
all these things are true. So you know how in
ordinary induction, we say, "assume P n is true and you're
going to prove P n plus 1?" In strong induction, you get
to say, "assume P n is true, P n minus 1 is true, dot,
dot, dot, P1 is true, and P0 is true. You get to assume a lot more. So it's a stronger
proof technique, because it allows you to do
more-- or so you would think. Now in fact, any proof you
can do with strong induction, you can do with
ordinary induction. It just might be harder. So you can't prove
any more with it, but it makes your
proofs much easier. And we're going to do
an example in a minute where the proof is
much easier by getting to assume all of these
things are true in order to prove P n plus 1. Are there any questions
about the difference here with strong induction-- that
now you get to assume all these are true as part of
the inductive step? All right. Our first example of
a strong induction is going to be a simple game. We've got a stack of
eight blocks here. Now, in this game,
what you're going to do is divide the stack
into two sub-stacks. So I just took eight, and
split it into three and five. For that move, you get
15 points-- 3 times 5. Let's write that down. OK, so this is called
the Unstacking Game. And in the first move, we
went 8 split to 5 and 3. And that's worth 15 points. Now in the next move, I'm going
to split the stack of five into four and one. And for that, you get
4 points-- 4 times 1. And I keep on going until I
have a stacks of height one. And the last move, I'll be
splitting a two into two ones. That'll give me one point. Then I add up all the points,
and that's your score. And the goal is to get
the most number of points when you add up
the entire score. Now, just so this game is
clear, we're going to play it. And I'm going to
raise the stakes here. What we're going to
do is we're going to have a competition
between three members of the class and the TAs. Now, if the class team
wins-- they get more points-- we're going to give
everybody in the class candy. And I've got lots of candy here. If the TAs beat you,
they get the candy. AUDIENCE: What if it's a tie? PROFESSOR: We'll
figure that out. We'll come to that later. But you want to win this thing. So I need some
volunteers from the class who think they can do well. Yeah, come on down. You two guys come on down. All right, TAs come on up. Now, I want you
to make some noise if you think your three class
reps here can beat the TAs. [CHEERS AND APPLAUSE] All right, how many
people-- make some noise you think the TAs are going to win. AUDIENCE: Ooh. PROFESSOR: Ooh, that's nasty. It must've been a rough
recitation last week on Wednesday. All right, so let's see. I think we're going to
let the class go first. This is your stack. Now think about what move
you want to make first, because that matters. And you guys can give them
some advice if you want, as to the move they
should make first. And I'm going to
keep score over here. Oh, OK, they went
for the maximum move. That's good. They realized 4 times
4 is 16-- very good. So let's write that up here. You're Off to a good start. In fact, that's the
maximum number of points for the first move,
I think-- good job. All right, so here's
the class here. And here are the TAs. And the class went 8 into
4 by 4, and they got 16. Now, the TAs are working
hard here, as you can see. And they've got a move. That's a pretty pathetic
first move for the TAs. Looking good for the
candy for you guys. They got 7 points for the TAs. Class, what would you like
to do for your next move? AUDIENCE: [INAUDIBLE]. PROFESSOR: You can't start
over now-- too late for that. [SIDE CONVERSATION] A lot of people
counting on you guys. [SIDE CONVERSATION] Oh, no, no, one at a time. No, just one. One stack gets
split-- either one. AUDIENCE: [INAUDIBLE]. PROFESSOR: There you go. Oh, well, what do you do now? [SIDE CONVERSATION] AUDIENCE: Their strategy's
obviously better than ours. AUDIENCE: Yeah, but we've
figured it out already. AUDIENCE: We're going to lose. PROFESSOR: The TAs are
convinced they're going to lose. Three and one, OK,
good, interesting. So we have three and one. That gives you 3 points. All right, TAs, we need a move. All right, so they've gone
six and one-- 6 points. Class, you think
you're dead already? So you've gotten three
and one now, again, for 3. So you're up to 22. You're way ahead-- 22 to 13. All right, TAs. They look a little
disorganized to me. What do you think? AUDIENCE: [INAUDIBLE]. PROFESSOR: Another one--
you can tell where they're headed here-- five and one. They have 5 points. They're up to 18. They're 4 behind you. All right, class. All right, you got 2 points. TAs-- you really think they
know how to do this best? I don't know. AUDIENCE: [INAUDIBLE]. PROFESSOR: Oh, big
change in strategy there. All that thinking, and
that's what we got. OK, 4 points. Class. All right, they
pick up 2 more here. The three just got split into
two and one for 2 points. How are you doing? You've got 19, 22, 24, up to 26. 13, 18, 22-- you're
still doing good. TAs are behind. Let's see what they've
got up their sleeves here. All right, they definitely
have a strategy going here. That's 3 points. Ooh, there's not much
option left for you guys. Yeah. [SIDE CONVERSATION] AUDIENCE: Choose the right. AUDIENCE: Air blocks. PROFESSOR: No, no,
no, that'll be a 0. All right, that gave
you a point-- two went to one and one-- 1 point. Not a lot of choice for
you guys either, I guess. There's only one way
to split that stack. All right, 2 points,
and your final move for the class and all the candy. You get 1 more point. Let's see what your total is. You've got 22, 24, 26, 27, 28. And the TAs are going to get
one last move for the TAs. All right, and you
got 2, 1, 1, 2. That's interesting. 1, 2, 3, 4, 5, 6, 7-- 7 times 8
over 2-- no, that's not right. Did I do that right? Oh, 28-- we got a tie. Oh, geez. I tell you what--
I'll give you guys one more chance to start over. You've seen that. You've got to beat them. You tied them. You've got to beat them. One more chance here. [SIDE CONVERSATION] You got to beat them, though. You got 28. You already tied them. They came from
behind and tied you. You've got to beat them. You got off to a good start,
but then they caught up. [SIDE CONVERSATION] Now, your class reps had an
interesting thought here. They ask, is it possible
that any strategy gives 28? Hmm. I wouldn't do that
to you, would I? [LAUGHTER] Yeah, probably. Any ideas? Can you beat 28? They're already
eating the candy. In fact, you can't beat 28. And you can't do
worse than 28, either. Because every strategy gives 28. So I guess we're going to
have to throw it to the class to vote who should
get the candy. So if you think your reps
really should win this and you should get the
candy, make some noise. [CHEERS AND APPLAUSE] If you think your TAs should
get the candy make some noise. [CHEERS AND APPLAUSE] All right, so you get the candy. Here, help pass
out the candy here. We've got plenty of
bags and baskets. Take it all up on both
sides, and pass it out. [SIDE CONVERSATION] OK, is everybody getting
some candy up there? We should have
enough for everybody, unless somebody got really
hungry down here in front. OK, so while you're
doing that, let's try to prove this theorem. So the theorem is that all
strategies for the n-block game produce the same score. It wasn't a coincidence for 8. And the score, we'll call it
S of n for the n-block game. So for example, we're trying
to show that S of 8 was 28. So what strategy
do you think we're going to use for the proof? AUDIENCE: Strong induction. PROFESSOR: Strong
induction-- so you want to write that down--
proof by strong induction. What's the next step in a
proof by any kind of induction? AUDIENCE: Base case. PROFESSOR: Base case-- not yet. AUDIENCE: The predicate. PROFESSOR: The predicate--
what's the predicate here? All right, so for the
inductive hypothesis, the predicate-- any guesses as
to what P of n is going to be? AUDIENCE: [INAUDIBLE]. PROFESSOR: What's that? AUDIENCE: n should
be 2n minus 1 by 2. PROFESSOR: Yeah,
that's part of it, but we're going to use this
as a predicate to start with. You've gone a step ahead there. So I'm going to start with
this as being a predicate, because I'm trying to
show they're all the same. And I haven't figured out
what the scores going to be. But I think you're already
on a good track here for what the score is going to be. Now we do the base case. So we take n equals
1-- the one-block game. S of 1 is 0. We never took a move. We never got a score. There was only one block. We ended before we started. So that's OK. And now we have
the inductive step. And here we get to assume P1,
P2, all the way up to P n, to prove P n plus 1. And so now we look
at n plus 1 blocks, because that's we're starting. So we have n plus 1,
and we split it somehow. And it could be any split. We might have K on one
side, then n plus 1 minus K on the other side for
any K between 1 and n. And let's figure out our
score for that scenario. Well, we get the
product of these numbers for the first step. We get K times n plus 1
minus K for the first step. And then as we recurse,
and we split up that block of K blocks,
how many points do we get for splitting
that all the way down? AUDIENCE: P of K. PROFESSOR: P of K, because
that's the induction. We assume that as part of
the induction hypothesis. And how many points do
we get for splitting this stack all the way down? Yeah, P of n plus 1 minus K. And you see why
strong induction's coming in handy here? For the TA's strategy,
K would be 1. We know P of 1 is 0. And you would just need
ordinary induction. This would be P of n. But for a general strategy,
it could be any split. So we need to have all
these assumed here. It makes it much easier
when using strong induction. All right, so this
is our total score for the game for
n plus 1 blocks. And we're trying to show
it just depends on n, and that it doesn't depend
on K. Does that depend on K? Looks like it to me, right? I got K in every single term. So I'm not there. I'm stuck. You see why I'm stuck? Because I'd like to be able
to say that the score I get for the n plus one
block game-- this is S n plus 1-- is the same
for any sequence of moves. It doesn't even depend on
K. But I can't do that. What do you do when you're
stuck with an induction proof? What's one of the tricks? AUDIENCE: Make it stronger. PROFESSOR: Make
it stronger, which means I get a stronger
induction hypothesis. Now, somebody gave me
one up there before. How could I strengthen my
induction hypothesis here? AUDIENCE: [INAUDIBLE]. PROFESSOR: Give the
formula for S n-- that would make it stronger. I tell you what it is. What's a good guess
for the formula here? AUDIENCE: Factorial. PROFESSOR: Factorial-- Well,
is S of 8 equal 8 factorial? No, not so good. [SIDE CONVERSATION] AUDIENCE: n equals
n minus 1 over 2. PROFESSOR: Yeah,
that's a better guess-- equals n times n minus 1 over 2. And in fact, what's
8 times 7 over 2? It's 28. It works here. Let's check another one. What's S of 2? AUDIENCE: 1. PROFESSOR: 1-- well, 2
times 1 over 2-- it works. What's S of 3? 2 the first move, then 1 more. 3-- 3 times 2 over
2 is 3-- looks good. So now I've got a stronger
induction hypothesis. Not only is it
always the same, it is that number is your score. Let's see if that works. Let's plug that in now. So base case--
let's check it out-- is 1 times 1 minus
1 over 2 equal to 0. Yes. Let's plug it in here now. P of K would be now
K, K minus 1 over 2. This would be n plus 1 minus K,
n minus K over 2 plus this one here. All right, I've got to add
all these things up, and see if it equals that
expression for n plus 1. Let's do that. All right, I get 2 K n
plus 2 K minus 2 K squared. I'm putting all this over 2--
one giant mess over 2 here-- plus K squared minus
K plus n squared. Well, let's write it just as
n plus 1 times n, minus K n, minus K, minus K n, all over 2. And now let's cancel. Two K n's here cancel K n, K n. 2K cancels K and K. 2K
squared minus 2 K squared cancels K squared. And I forgot, plus K
squared from here and here. So I'm left with n
plus 1 times n over 2. That is S n plus 1. And the K disappears. We've established, now, the
stronger induction hypothesis. So not only did we prove
that every set of unstackings gives you the same score. We proved the score is, for
n blocks, n times n minus 1 over 2. That's was pretty good
for strong induction. It lets you do some
pretty powerful things. OK, very good,
that's it for today.
