Computation: The Mathematical Story Christos H ... · Pascal™s Calculator 1650 Babbage & Ada,...
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Computation: The Mathematical Story Christos H. Papadimitriou UC Berkeley christos
Transcript of Computation: The Mathematical Story Christos H ... · Pascal™s Calculator 1650 Babbage & Ada,...
Com
puta
tion:
The
Mat
hem
atic
al S
tory
Chr
isto
s H. P
apad
imitr
iou
UC
Ber
kele
y�c
hris
tos�
HoC
, 12/
6/07
Out
line
�Th
e Fo
unda
tiona
l Cris
is in
Mat
h (1
900
�31
)�
How
it L
ed to
the
Com
pute
r (19
31 �
46)
�A
nd to
P v
sNP
(19
46 �
72)
HoC
, 12/
6/07
The
preh
isto
ry
of c
ompu
tatio
n
Pasc
al�s
Cal
cula
tor
1650
Bab
bage
& A
da, 1
850
the
anal
ytic
al e
ngin
e
Jacq
uard
�s lo
oms
1805
HoC
, 12/
6/07
Trou
ble
in M
ath
Non
-euc
lidea
nge
omet
ries
Can
tor,
1880
: se
ts a
nd in
finity∞
HoC
, 12/
6/07
The
ques
t for
foun
datio
ns
Hilb
ert,
1900
:�W
e m
ust k
now
,w
e ca
n kn
oww
e sh
all k
now
!�
HoC
, 12/
6/07
The
two
ques
ts
An
axio
mat
icsy
stem
that
com
pris
esal
l of M
athe
mat
ics
A m
achi
neth
at fi
nds
a pr
oof f
orev
ery
theo
rem
HoC
, 12/
6/07
The
disa
ster
Göd
el 1
931
The
Inco
mpl
eten
ess T
heor
em�s
omet
imes
, we
cann
ot k
now
�Th
eore
ms t
hat h
ave
no p
roof
HoC
, 12/
6/07
Rec
all t
he tw
o qu
ests
Find
an
axio
mat
icsy
stem
that
com
pris
esal
l of M
athe
mat
ics
?
Find
a m
achi
neth
at fi
nds a
proo
f for
ever
y th
eore
m
HoC
, 12/
6/07
Als
o im
poss
ible
?
but w
hat i
s a m
achi
ne?
HoC
, 12/
6/07
The
mat
hem
atic
al m
achi
nes (
1934
�37
)
Post
Kle
eneC
hurc
h
Turin
g
HoC
, 12/
6/07
Uni
vers
al T
urin
g m
achi
ne
Pow
erfu
l and
cru
cial
idea
whi
ch a
ntic
ipat
es so
ftwar
e
�an
d ra
dica
l too
:de
dica
ted
mac
hine
sw
ere
favo
red
at th
e tim
e
HoC
, 12/
6/07
�If i
t sho
uld
turn
out
that
the
basi
c lo
gics
of a
mac
hine
des
igne
d fo
r the
num
eric
also
lutio
n of
diff
eren
tial e
quat
ions
coi
ncid
ew
ith th
e lo
gics
of a
mac
hine
inte
nded
tom
ake
bills
for a
dep
artm
ent s
tore
, I w
ould
rega
rd th
is a
s the
mos
t am
azin
g co
inci
denc
eth
at I
have
eve
r enc
ount
ered
�
How
ard
Aik
en, 1
939
HoC
, 12/
6/07
In a
wor
ld w
ithou
t Tur
ing�
SPE
CIA
L T
OD
AY
: All
num
ber
crun
cher
s 40%
off
!
Bas
emen
t:G
ame
engi
nes,
Vid
eo a
nd M
usic
com
pute
rs
Thi
rd F
loor
:A
ccou
ntin
g co
mpu
ters
, Bus
ines
s mac
hine
s
Seco
nd F
loor
:D
atab
ase
engi
nes,
Wor
d pr
oces
sors
Firs
t Flo
or:
Web
bro
wse
rs, e
-mai
lers
WE
LC
OM
E T
O T
HE
CO
MPU
TE
R S
TO
RE
!
HoC
, 12/
6/07
And
fina
lly�
von
Neu
man
n 19
46ED
VA
C a
nd re
port
HoC
, 12/
6/07
John
ny c
ome
late
ly
�vo
n N
eum
ann
and
the
Inco
mpl
eten
ess T
heor
em�
�Tur
ing
has d
one
good
wor
k on
the
theo
ries
of a
lmos
t pe
riodi
c fu
nctio
ns a
nd o
f con
tinuo
us g
roup
s� (1
939)
�Zu
se(1
936
�44
) , T
urin
g (1
941
�52
), A
tana
soff
/Ber
ry
(193
7 �
42),
Aik
en (1
939
�45
), et
c.�
The
mee
ting
at th
e A
berd
een,
MD
trai
n st
atio
n�
The
�log
icia
ns�
vsth
e �e
ngin
eers
� at
UPe
nn�
Ecke
rt, M
auch
ly, G
olds
tine,
and
the
Fir
st D
raft
Mad
ness
in th
eir m
etho
d?th
e pa
infu
l hum
an st
ory
G. C
anto
rD
. Hilb
ert
K. G
ödel
E. P
ost
A. M
. Tur
ing
J. V
on N
eum
ann
HoC
, 12/
6/07
Theo
ry o
f Com
puta
tion
sinc
e Tu
ring:
Effic
ient
alg
orith
ms
�So
me
prob
lem
s can
be
solv
ed in
pol
ynom
ial
time
(n, n
log
n, n
2 , n3 ,
etc.
)�
Oth
ers,
like
the
trave
ling
sale
sman
pro
blem
an
d B
oole
an sa
tisfia
bilit
y, a
ppar
ently
can
not
(bec
ause
they
invo
lve
expo
nent
ial s
earc
h)�
Impo
rtant
dic
hoto
my
(von
Neu
man
n 19
52,
Edm
onds
196
5, C
obha
m19
65, o
ther
s)
HoC
, 12/
6/07
Poly
nom
ial a
lgor
ithm
s del
iver
M
oore
�s L
aw to
the
wor
ld�
A 2
nal
gorit
hm fo
r SA
T, ru
n fo
r 1 h
our:
n=
53n
= 45
n=
38n
= 31
n=
23n
= 15
2006
1996
1986
1976
1966
1956
×2
×5
×10
0 ev
ery
deca
de
n7n3
An
nor
nlo
g n
algo
rithm
HoC
, 12/
6/07
NP-
com
plet
enes
sC
ook,
Kar
p, L
evin
(197
1 �
73)
�Ef
ficie
ntly
solv
able
pro
blem
s: P
�Ex
pone
ntia
l sea
rch:
NP
�M
any
com
mon
pro
blem
s cap
ture
the
full
pow
er
of e
xpon
entia
l sea
rch:
NP-
com
plet
e�
Arg
uabl
y th
e m
ost i
nflu
entia
l con
cept
to c
ome
out o
f Com
pute
r Sci
ence
�Is
P =
NP?
Fun
dam
enta
l que
stio
n an
d m
athe
mat
ical
pro
blem
HoC
, 12/
6/07
Inte
llect
ual d
ebt t
o G
ödel
/Tur
ing?
�N
egat
ive
resu
lts a
re a
n im
porta
nt
inte
llect
ual t
radi
tion
in C
ompu
ter S
cien
ce
and
Logi
c�
The
Inco
mpl
eten
ess T
heor
em a
nd T
urin
g�s
halti
ng p
robl
em a
re th
e ar
chet
ypic
al
nega
tive
resu
lts�
The
Göd
el le
tter (
disc
over
ed 1
992)
HoC
, 12/
6/07
HoC
, 12/
6/07
HoC
, 12/
6/07
Rec
all:
Hilb
ert�s
Que
st
axio
ms
+co
njec
ture
alw
ays a
nsw
ers
�yes
/no�
Turi
ng�s
hal
ting
prob
lem
HoC
, 12/
6/07
Göd
el�s
revi
sion
axio
ms
+co
njec
ture
ifth
ere
is a
pro
of
of le
ngth
nit
finds
it
in ti
me
k n
(this
is tr
ivia
l,ju
st tr
y al
l pro
ofs)
HoC
, 12/
6/07
Hilb
ert�s
last
stan
d
�G
ödel
ask
ed v
on N
eum
ann
in th
e 19
56
lette
r:�C
an th
is b
e do
ne in
tim
e n
?
n 2 ?
n c ?�
�Th
is w
ould
still
mec
hani
ze M
athe
mat
ics�
HoC
, 12/
6/07
Surp
rise!
�G
ödel
�s q
uest
ion
is e
quiv
alen
t to
�P =
NP�
�H
e se
ems t
o be
opt
imis
tic a
bout
it�
HoC
, 12/
6/07
So�
�H
ilber
t�s fo
unda
tions
que
st a
nd th
e In
com
plet
enes
s The
orem
hav
e st
arte
d an
in
telle
ctua
l Rub
e G
oldb
erg
that
eve
ntua
lly
led
to th
e co
mpu
ter
�So
me
of th
e m
ost i
mpo
rtant
con
cept
s in
toda
y�s C
ompu
ter S
cien
ce, i
nclu
ding
P vs
NP,
ow
e a
debt
to th
at tr
aditi
on
HoC
, 12/
6/07
And
this
is th
e st
ory
we
tell
in�
LOG
ICO
MIX
A g
raph
ic n
ovel
of r
easo
n, m
adne
ssan
d th
e bi
rth o
f the
com
pute
r
by�
HoC
, 12/
6/07
LOG
ICO
MIX
: A g
raph
ic n
ovel
of r
easo
n,
mad
ness
and
the
birt
h of
the
com
pute
rB
y A
post
olos
Dox
iadi
sand
Chr
isto
s Pap
adim
itrio
uA
rt: A
leco
sPap
adat
osan
d A
nnie
Di D
onna
Blo
omsb
ury,
200
7
HoC
, 12/
6/07
Than
k yo
u!
