SIGACT News Complexity Theory Column 14
Lane A. H e m a s p a a n d r a
Dept. of C o m p u t e r Science, University of Rochester
Rochester, NY 14627, USA [email protected]
I n t r o d u c t i o n to C o m p l e x i t y T h e o r y C o l u m n 14
As you probably already know, there is an active discussion going o n - - i n forums ranging from
lunch-table conversations to workshops on "strategic directions" to formal r e p o r t s l - - r e g a r d i n g the
future of theoretical c o m p u t e r science. Since your complexity columnist does not know T h e Answer,
I've asked a n u m b e r of people to contribute their comments on the narrower issue of the future of
complexity theory. The only ground rule was a loose 1-page limit; each c o n t r i b u t o r could choose
w h a t aspect(s) of the future to address, and the way in which to address th em. T h e first installment
of contributions appears in this issue, and one or two more installments will a p p e a r a m o n g t h e next
few issues.
Also coming during the next few issues: the search for the perfect theory journal, a n d (for
the sharp-eyed) Lance Fortnow dons a clown suit. Finally, let me men tio n t h a t work of Russell
Impagliazzo resolves one of the open questions from Complexity T h e o r y C o l u m n 11.2
G u e s t Column:
T h e F u t u r e of C o m p u t a t i o n a l C o m p l e x i t y T h e o r y :
Part I
C o m m e n t s by C. P a p a d i m i t r i o u , O. G o l d r e i c h / A . W i g d e r s o n ,
A. R a z b o r o v , and M. Sipser
1
On Extroverted Complexity Theory, by Christos H. Papadimitriou 3
Complexity theory has come a long way in the thirty years since its inception. It has identified
some of the most f u n d a m e n t a l and deep problems related to c o m p u t a t i o n , it has developed
a powerful me t h odol ogy for attacking them, an d it is broadly considered as one of the most
challenging m a t h e m a t i c a l frontiers. Considered as a pure m a t h e m a t i c a l discipline in th e p u rs u i t
of m a t h e m a t i c a l insight and depth, complexity is successful and well-established. In this no,e,
however, I would like to concentrate on complexity as an applied m a t h e m a t i c a l discipline whose
function is to gain insights into the problems of the natural, social, and applied an d engineering
1For
a
quick
and
bracingly
v~ied
path
into
the
discussion,
I ' d point to two d o c u m e n t s :
and h t t p : / / t h e o r y . l c s . m i t . e d u / - o d e d / t o c - s p . h t m l .
2In more detail: Impagliazzo [Imp96] resolves Problem 3 of the "Worlds to Die For" Complexity T h e o r y
C o l u m n [HRZ95]. In fact, his result is even stronger t h a n what we conjectured, as he shows t h a t with probability
1 -- 2 -n(~) over the set of oracles there is a relativized pseudora~dom generator t h a t is secure against oracle circuits of
size 2 c~ for some constant c ~ 0. He first shows t h a t a r a n d o m function behaves for almost all oracles like a one-way~
function and, thus, the general m e t h o d of H~stad, Impagliazzo, Levin, and L u b y [HILL91] for converting a one-way
function into a p s e u d o r a u d o m generator can be used to obtain the above result. As we noted in the column, this
m e t h o d provides a provably secure way of privatizing r a n d o m bits (for more details, see [Zim96]).--L. H e m a ~ p a a u d r a
and M. Z i m a n d
SComputer Science Division, University of California Berkeley; christosacs.berkeley.edu. A d a p t e d from the
introduction of [Pap96].
ftp://ftp.cs.washington.edu/tr/1996/O3/UW-CSE-96-O3-O3.PS.Z
6
sciences. This aspect has been somewhat peripheral to what we usually m e a n by "complexity
theory," but I believe it is important for its future.
Understanding the position of complexity theory within the realm of scientific inquiry is an
important project which is, of course, well beyond the scope of this note; here I shall only refer
anecdotally to six distinct ways in which complexity theory has reached out and touch other fields:
Comple~ty as NP-completeness. O n e of the major achievements of complexity theory is the living
connection it has forged between application problems and modes of resource-bounded computation.
This is typically done through the key notion of completeness, of which NP-colnpleteness is of course
the most popular kind. Completeness comes so natural to a complexity theorist, that it is easy to
forget what an important and influential concept it has been. In fact, outside theoretical computer
science, "complexity theory" is often understood--unjustly, to be sure---as synonymous to "NPcompleteness." 4
Comple~ty as mathematical poverty. One of the fundamental theses that seems to be almost
universally accepted and practiced in computer science is that a l g o r i t h m s - - a n d efficient algorithms
in particular--are the natural outflow of mathematical structure discovered in applications. 5 If we
accept this implication, then we must also espouse the contrapositive one, namely, that complexity is
the manifestation of mathematical nastiness. Complexity has been often and brilliantly used within
computer science and mathematics in this allegorical way; a computational problem is formulated
and proved hard for the sole purpose of pointing out the mathematical difficulties involved in an
area or approach (see [HLYS0] for an early example from database theory).
Complezit# as metaphor. Often the implication discussed in the previous paragraph is composed
with the metaphors of an application domain or other scientific discipline with sometimes
exquisite results. "Complexity" may mean "chaos" in the domain of dynamical systems [BPT91],
'~unbounded rationality" in gaxne theory [PY94]--and perhaps "genetic indeterminism" in genetics,
"cognitive implausibility" in artificial intelligence, and so on.
CompIezity as blessing in disguise. Cryptography is of course the best-known example here, but
not the only one. For example, [BTT92] point out that complexity can be desirable in political
science, as evidence that an electoral protocol is resistant to manipulation.
Complezity as herculean sword. The mythical monster Hydra grows three heads for each one cut
off by Hercules. s W h e n complexity theorists point out obstacles to an approach, several novel
alternative approaches typically develop: "It's NP-complete? Then we'll concentrate on planar
graphs, or on random graphs, or else we'll approximate." A n d so on.
Complexity as bea~tzl contest judge. W h e n m a n y alternative approaches have been proposed
for a problem (computational or conceptual), rigorous criteria for evaluating them are needed.
Complexity can come in handy in such a situation. For example, in [DP94] we proposed complexity
as a criterion for comparing the feasibility and desirability of "solution concepts" (approaches to
w h e n a proposed protocol for splitting goods is fair) in mathematical economics, and in [GKPS95]
complexity was used in sorting out alternative proposals in knowledge representation--an important
subfield of artificial intelligence.
4 A m o n g treatments of complexity theory by scientistsoutside our fieldthis one is not the most unfair or ignorant;
compare with [CPM94], for example.
5The only challenge of this principle within computer science comes, I think, ~ o m neural networks and other
metaphor-based algorithmic paradigms.
SIncidentally, this is not complexity's firstbrush with Hercules: K n u t h [Knu74] had proposed i~herculean" as one
of the possible terms for the concept that is now, thankfully, k n o w n as "NP-complete." According to Knuth, K e n
Steiglitz counterproposed ~'augean," a term which Greek mythology buffs will find both hilarious and appropriate.
7
2
On The Usefulness of Hard Problems, by Oded Goldreich 7 and Avi
Wigderson s
We w e r e a s k e d t o w r i t e o n t h e f u t u r e of C o m p l e x i t y T h e o r y . G i v e n t h e p a s t ( a n d p r e s e n t ) of o u r
field, w h i c h c e l e b r a t e d so m a n y a c h i e v e m e n t s , m a n y in s u r p r i s i n g , u n a n t i c i p a t e d d i r e c t i o n s , we feel
it u n w i s e to p r e d i c t t h e p r e v a i l i n g d i r e c t i o n s in say 10 or 20 years. N e v e r t h e l e s s , we axe c o n f i d e n t
t h a t if o u r field c o n t i n u e s to a t t r a c t t h e s a m e c a l i b r e of c r e a t i v e m i n d s , a n d is given t h e f r e e d o m
( a n d m i n u t e r e s o u r c e s ) to p u r s u e its i n t e r n a l a g e n d a , it will c o n t i n u e to t h r i v e . M o r e o v e r , t h e
i m p o r t a n c e of its f i n d i n g s to o t h e r areas of c o m p u t e r science a n d e n g i n e e r i n g , as well as to o t h e r
sciences a n d (yes!) h u m a n i t i e s will c o n t i n u e to grow.
To j u s t i f y t h i s s t r o n g p r e d i c t i o n , we c o n s i d e r w h a t c o m p l e x i t y t h e o r y h a s d o n e w i t h t h e classical
p r o b l e m o f i n t e g e r f a c t o r i z a t i o n : given an integer N, find its prime/actors. M a t h e m a t i c i a n s h a v e
s t u d i e d t h i s p r o b l e m for c e n t u r i e s , s e a r c h i n g for a n efficient f a c t o r i n g a l g o r i t h m e v e n b e f o r e t h e
n o t i o n of a n efficient algorit.hm was defined. T h i s t a s k h a s failed so far, w h i c h m a y v e r y well m e a n
t h a t f a c t o r i n g is infeasible. S o WHAT? C o m p l e x i t y T h e o r y h a s m a n a g e d to u s e t h i s i n f e a s i b i l i t y
as a p i v o t for a v a r i e t y of f u n d a m e n t a l discoveries a n d t h e o r i e s of v e r y g e n e r a l n a t u r e . T h i s is t r u e
to s u c h a n e x t e n t t h a t a list of t o p i c s as b e l o w c a n b e u s e d for a g r a d u a t e c o u r s e w h i c h will b e
offered n o t o n l y to all c o m p u t e r science s t u d e n t s , b u t also to s t u d e n t s of o t h e r d i s c i p l i n e s . I n fact,
it is h i g h t i m e t h a t o u r c o m m u n i t y s t a r t s to d i s s e m i n a t e t h e i n t e l l e c t u a l c o n t e n t s o f t h e t h e o r y o f
c o m p u t a t i o n , a n d c o u r s e s of t h e a b o v e f o r m m a y b e a g o o d s t a r t .
T h e i t e m s b e l o w all follow f r o m t h e a s s u m e d in/easibility o/factoring. T h i s , as well as t h e
c o n s e q u e n c e s , are s t a t e d o n l y i n f o r m a l l y for o b v i o u s r e a s o n s , w i t h t h e u n d e r s t a n d i n g t h a t t h e y all
have precise statements which make them mathematical theorems.
D a t a Representation is Important. This point, which is the main focus of our courses on
Data Structures and Efficient Algorithms, is driven h o m e forcefully here. If Y~j aj2 j = N =
H i Piei , then each side of this equation represents the integer N, and the two representations
(binary expansion and prime factorization) axe equivalent from the information theoretic
viewpoint. However, they are drastically different computationally: the L H S can be easily
computed from the RI-IS, but the reverse direction in general is infeasible. Moreover, some
computational problems, like solving certain polynomial equations modulo N , axe feasible
given the RI-IS but infeasible given the LHS.
P s e u d o r a n d o m n e s s Exists. There are efficient deterministic procedures that take a few
r a n d o m bits and stretch them to a m u c h longer (pseudorandom) string, which looks r a n d o m
to every efficient algorithm. Thus deterministic procedures can greatly expand computational
entropy. This is in stark contrast to the information theoretic analog (deterministic procedures
can never increase entropy), and physical intuition (cf.,the preservation of mass and energy).
Randomness can be Eliminated when Computing Functions.
A straightforward
c o n s e q u e n c e o f t h e p r e v i o u s i t e m is t h a t a n y p r o b a b i l i s t i c a l g o r i t h m (for c o m p u t i n g a f u n c t i o n )
c a n b e r e p l a c e d by a d e t e r m i n i s t i c o n e w h i c h is a l m o s t as efficient. T h e l a t t e r will s i m p l y
e n u m e r a t e all t h e p o s s i b l e p s e u d o r a n d o m s t r i n g s , a r i s i n g f r o m all p o s s i b l e s h o r t seeds, a n d
d e c i d e by a m a j o r i t y vote.
7 D e p a r t m e n t of C o m p u t e r Science a n d Applied M a t h e m a t i c s , W e i z m a n n I n s t i t u t e of Science, R e h o v o t , ISRAEL.
E-mail: oded~,~isdom, vei~m~nn, ac. i l SInstituLe for C o m p u t e r Science, H e b r e w University, Givat B~m, Jerusalem, ISaAEL. E - m a i h a v i © c s . h u j £. ac. i1_
8
Cryptography
is P o s s i b l e . Secure public-key encryption, unforgeable digital signatures,
d i s t r i b u t e d coin-flipping, electronic voting schemes a n d electronic cash transfer can all be
p e r f o r m e d digitally, by c o m m u n i c a t i n g computers. In fact, e v e r y d i s t r i b u t e d protocol w i t h
a r b i t r a r y privacy constraints 9 can be i m p l e m e n t e d in a way t h a t is resilient to a r b i t r a r y faults
by any n u m b e r of the players. In other words, players may e m u l a t e the existence of a t r u s t e d
p a r t y in a setting in which no such t r u s t e d p a r t y exists (and f u r t h e r m o r e in which m a n y
parties c a n n o t be t r u s t e d at all). T h e effect of these techniques on t h e real world is already
i m m e n s e a n d is further growing rapidly.
Z e r o - K n o w l e d g e P r o o f s . A p a r t y having a proof of an a r b i t r a r y m a t h e m a t i c a l t h e o r e m
T can convince anyone that T is true, w i t h o u t giving away a n y t h i n g besides the validity of
T. I n particular, after getting such a proof, one will be convinced t h a t T is t r u e a n d still
be u n a b l e to prove this to others. This paradoxical notion contradicts p o p u l a r beliefs t h a t
o b t a i n i n g a proof of an u n k n o w n t r u t h necessitates learning s o m e t h i n g new.
Is L e a r n i n g F e a s i b l e ? W i t h i n several s t a n d a r d models of learning it is infeasible to learn
even simple concepts described by short formulae. This sheds new light on the f u n d a m e n t a l
scientific task of u n d e r s t a n d i n g the learning process.
P ¢ N P . In particular, a variety of i m p o r t a n t c o m p u t a t i o n a l problems a d m i t no feasible
(exact a n d even approximate) solutions. T h o u s a n d s of such problems are known, a n d
h u n d r e d s are discovered each year, in a variety of scientific and engineering disciplines. T h e s e
problems include famous examples like Satisfiability of Boolean Formulas, T h e Traveling
S a l e s m a n P r o b l e m , and Integer Programming_
n o " N a t u r a l P r o o f s " o f P ~ / V P . This recent research (partly) explains our
failure so far to prove t h a t the problems above are indeed infeasible. First, it abstracts all proof
techniques which were used in d e m o n s t r a t i n g the k n o w n lower bounds, b o t h s t r u c t u r a l l y (as
N a t u r a l Proofs) and logically (as a certain fragment of P e a n o Arithmetic). T h e n it shows t h a t
any proof falling into either of these categories cannot be used to establish a s u p e r p o l y n o m i a l
lower b o u n d for general circuits.
There
are
W h e n considering the contrapositive of the above implications, one is a m a z e d to discover t h a t
any efficient learning a l g o r i t h m for short formulae, any t h e o r e m w i t h no zero-knowledge proof,
any function w h i c h requires r a n d o m n e s s for efficient c o m p u t a t i o n , and any nontrivial circuit lower
b o u n d using k n o w n m e t h o d s c a n a l l be converted into an efficient factoring algorithm!
A n d w h a t if factoring has an efficient algorithm? Well, first of all, due to the nearly-universal
use of the RSA C r y p t o s y s t e m , the consequences on information privacy a n d world e c o n o m y can be
devastating. B u t scientifically speaking, this alone would leave all items above intact, since o t h e r
"one-way" functions would serve j u s t as well as factoring. Moreover, while some of these items are
essentially equivalent to the existence of one-way functions, others hold even u n d e r seemingly m u c h
weaker conditions.
Meanwhile, the search for efficient factoring algorithms continues. Active c o l l a b o r a t i o n of
M a t h e m a t i c i a n s and C o m p u t e r Scientists has led to impressive progress in t h e last decade, using
k n o w n results in N u m b e r T h e o r y as well as establishing new ones. On an o r t h o g o n a l direction, the
recent Q u a n t u m p o l y n o m i a l - t i m e a l g o r i t h m for factoring has greatly e n h a n c e d efforts of Physicists
towards the s t u d y of the feasibility of the Q u a n t u m C o m p u t e r model. W h a t will be next?
gFor example, suppose some parties wish to play a standard
g a m e of
Poker over the telephone.
3
Proofs, Computations
Razborov l°
and Practical Applications,
by Alexander A.
T h e g e n e r a l i d e a t h a t c o m p u t a t i o n s a n d proofs have m u c h m o r e in c o m m o n t h a n it m a y a p p e a r f r o m
t h e first sight is b e i n g d e v e l o p e d in at least t h r e e sister c o m m u n i t i e s d e s c e n d i n g f r o m t h e classical
m a t h e m a t i c a l logic. C o m p l e x i t y T h e o r y c o n t r i b u t e d to t h a t p u r p o s e n o t i o n s like interactive proofs,
probabilistically checkable proofs and transparent proofs c h a l l e n g i n g such f u n d a m e n t a l issues as
r i g o r o u s n e s s a n d verifiability. Feasible P r o o f T h e o r y s u g g e s t e d t h a t even classical proofs have
their own intrinsic complexity, a n d it is e x t r e m e l y closely, r e l a t e d to t h e o r d i n a r y c o m p l e x i t y of
a l g o r i t h m s . Finally, t h e "LICS c o m m u n i t y " analyses proofs in f o r m a l s y s t e m s w h o s e semantics is
i n t e n d e d to c a p t u r e t h e process of c o m p u t a t i o n h a p p e n i n g in the real software.
I n several last years we s t a r t e d to see some r e n e w e d c o m m u n i c a t i o n on this t o p i c b e t w e e n t h e
fields once u n i t e d in t h e f r a m e w o r k of m a t h e m a t i c a l logic. M y p r e d i c t i o n is t h a t in u p c o m i n g years
t h e s e r e l a t i o n s b e t w e e n c o m p u t a t i o n s a n d proofs will receive even closer a t t e n t i o n o n t h e side of
C o m p l e x i t y T h e o r y ( o t h e r p a r t i e s involved a l r e a d y do it on t h e professional basis). O u r c o m m u n i t y
c e r t a i n l y has s o m e t h i n g h e r e b o t h to say (e.g., how to prove a n y t h i n g a b o u t t h e c o m p l e x i t y of
proofs? O r h o w to prove efficiently t h e correctness of p r o g r a m s , r i g o r o u s l y defining first w h a t it
m e a n s ? ) a n d to ask (e.g., w h a t makes o u r o w n f u n d a m e n t a l p r o b l e m s so difficult to solve?)
C o m i n g to a n o t h e r p a r t of t h e question, " W h e r e should C o m p l e x i t y T h e o r y go?", t h e r e ' s a lot
of effort n o w to t r y to increase t h e i m p a c t of our field on key a p p l i c a t i o n areas. B u t in m y o p i n i o n it
is n o t q u i t e clear w h e t h e r C o m p l e x i t y T h e o r y s h o u l d go a n y w h e r e at all or it w o u l d b e m o r e useful
s t a y i n g w h e r e it is. T h e m i s s i o n of this discipline is to p r o v i d e a b r i d g e for t h e traffic of ideas a n d
c o n c e p t s ( w i t h a h a n d f u l of exceptions, not the results themselves!) b e t w e e n p u r e m a t h e m a t i c s a n d
C o m p u t e r Science. T r y to p u l l it to one side (say, for t h e p u r p o s e of m a k i n g a j u n c t i o n i n s t e a d ) , a n d
firstly you will no longer have a n y t h i n g to cross t h e river u p o n , a n d s e c o n d l y y o u m a y discover t h a t
t h e b r i d g e ' s r e m n a n t s axe less useful on t h e l a n d t h a n e x p e c t e d as t h e c o n s t r u c t i o n was d e s i g n e d
for different p u r p o s e s . However, in m y o p i n i o n t h a t p a r t of s e m i - a p p l i e d r e s e a r c h in C o m p l e x i t y
T h e o r y w h i c h develops according to the i n t e r n a l logic of our field has to b e s t r o n g l y e n c o u r a g e d
( c o n t i n u i n g t h e a n a l o g y w i t h t h e bridge, it is s i m p l y o u r d u t y to p r o v i d e as c o n v e n i e n t access to
t h e traffic across it as we c a n m a n a g e ) . T h e w o r k on efficient p r o g r a m verification m e n t i o n e d a b o v e
in q u i t e a different c o n t e x t is one g o o d e x a m p l e of this.
4
Is t h e H a n d w r i t i n g
on the Wall ?
Reflections on the future of theoretical computer science
by Michael Sipser al
D u r i n g d i n n e r one t i m e w i t h several M I T c o m p u t e r science t h e o r i s t s at Joyce C h e n ' s C h i n e s e
r e s t a u r a n t in C a m b r i d g e , discussion t u r n e d to t h e f u t u r e of theory. O n e of t h e g r o u p p a i n t e d a
r a t h e r bleak p i c t u r e , b e c a u s e of a tight j o b m a r k e t a n d a c o n c e r n t h a t few t h e o r e t i c a l results b e a r
u p o n p r a c t i c e . He said, " t h e h a n d w r i t i n g is on t h e wall" for theory. I recall it q u i t e d i s t i n c t l y
b e c a u s e j o b i n t e r v i e w s are m e m o r a b l e events, a n d t h a t d i n n e r o c c u r r e d w h e n I was i n t e r v i e w e d for
a p o s i t i o n at M I T . I was a fresh P h . D . t h e n fxom Berkeley. It was 1979.
l°Steklov Mathematical Institute, Vavilova 42, 117966, GSP-1, Moscow, RUSSIA. Supported by grant ~96-0101222 of the Russian Foundation for Fundamental Research.
11Mathematics Department, MIT. Supported by NSF Grant 9503322 CCR.
10
The future, viewed from 1979, turned out to be much brighter. Indeed, our field has been
rich with beautiful, big ideas. We can take credit for major applications of theory, invented by
theorists, that soon will affect everyone's daily life. Our currency has become high among scientists
and mathematicians in other fields. Talented students have been electing to study theory and
nearly all have found positions as theorists in academia or industry. Theory lives!
Today, I sometimes hear pessimistic views similar to those expressed at that 1979 dinner. Will
our field remain intellectually alive? I believe the future of theory remains as bright now as it was
then. The pace of discovery shows no sign of slowing. Students still find our questions challenging
and wish to join us in solving them. The shortage of jobs and grants is an important problem. But
any field that is interesting enough to continue to attract new people must eventually cope with
such shortages, painful though they remain.
As to where theoretical computer science, or complexity theory, should go, I would be happy to
see them continue on their current paths. Good research aims at depth, elegance, and practicality.
By its nature, theory focuses on the first two, but even in complexity theory, many papers do
not ignore the third. Some even manage to attain all three simultaneously, though such results
are rare in any field. Various people argue that we ought to tilt our field toward practice, but I
disagree. Competing with industry on its tuff becomes increasingly harder as it grows richer_ We
must continue to reserve some of our academic effort for highly speculative and pure research that
industrial companies will never pursue.
As to where theory will go, I'll make only one prediction. The next few years, say five to
be conservative, will see another theoretical breakthrough as amazing and unexpected as were
those previous. And to those still writing by hand on walls, I say an upgrade in technology--and
~.hinking--is overdue.
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Probability and
Information
An Integrated Approach
David Applebaum
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Noisy Information and
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Leszek Plaskota
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1996
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Computability.
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S.B. Cooper, T.A. Slaman, and
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London Mathematical Sorlety Lecm~ Note
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Camhridgs Internatimml ~
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