My Favorite Ten Complexity Theorems
of the Past Decade
Lance Fortnow
Department of Computer Science
East th Street
Chicago Illinois
Abstract We review the past ten years in computational complexity
theory by fo cusing on ten theorems that the author enjoyed the most
We use each of the theorems as a springb oard to discuss work done in
various areas of complexity theory
Intro duction
Just ab out ten years ago in the spring of I enrolled in a graduate compu
tational complexity course taught by Juris Hartmanis at Cornell That course
marked the b eginning of study and research in computational complexity that
has b een a ma jor part of my life ever since
As a decade has passed I nd it useful to review where complexity theory
has come during those years I have found that complexity theory has ourished
during that time No twenty page pap er could p ossibly do justice to the past ten
years in complexity theory
Instead I have taken a di erent approach I have isolated ten theorems that I
have enjoyed the most during the past decade following a few self imp osed guide
lines See Section I have then used these theorems as a basis to describ e
many other results in complexity theory using each theorem as a springb oard
into a subarea of computational complexity
Please note that each area could have a twenty page survey of its own I
cannot p ossibly mention all the imp ortant results in any given area Nor am I
able to bring in all the di erent subareas in complexity theory
I have found the past ten years in complexity theory quite exciting Cir
cuit complexity has come of age during the past ten years and has provided
us with a rich source of combinatorial problems Interactive pro of systems have
surprised us all with their ability to use randomness to simulate alternation and
their imp ortant connections to program testing and hardness of approximation
algorithms Algebraic techniques now seem to p ervade all areas of complexity
esp ecially circuit complexity interactive pro of systems and counting complex
ity We have also seen a lot of interest in probabilistic computation b oth in its
?
Email fortnow cs uchicago edu Partially supp orted by NSF grant CCR
p ower in interactive pro of systems and the many successful attempts to reduce
and eliminate randomness in various computation mo dels
Computational complexity theory has had steady growth since its inception
in the s I would guess that the numb er of active researchers in computational
complexity theory has doubled over the past decade Complexity theory now has
a ma jor conference The IEEE Structure in Complexity Theory Conference as
well as many smaller conferences and workshops
However complexity theory has had its disapp ointments We seem no closer
to solving the famous P NP problem than we were ten years ago I have
found that really only one theorem Theorem gives us such fundamental re
sults ab out complexity classes that would make it into an undergraduate course
Despite these shortcomings this pap er shows that complexity theory has pro
vided us with great theorems throughout the past ten years
Guidelines
In cho osing the ten theorems I used the following guidelines
Theorems chosen from the broad area of computational complex
ity theory There are several go o d theorems in other areas of theory and
computer science However I have my exp ertise in computational complexity
theory I would not make a go o d judge in other areas
Theorems chosen from the past ten years It would not b e fair to judge
theorems b efore I was actively involved in complexity theory
Theorems chosen for imp ortance of result originality and di culty
of technique and relationships with other results in complexity
theory
No theorems proven at The University of Chicago I do not want to
play favorites
Theorems chosen for diversity I wanted to get a representation of many
di erent areas of complexity theory
Theorems chosen as results instead of for p eople I did not make any
attempt to cho ose p eople instead I chose results that I enjoyed
The Theorems
We present the ten favorite theorems in rough chronological order
Branching Programs
In the last decade we have seen algebra play a much larger role in complexity
theory than ever b efore Usually in these results algebra plays no role in the
statement of the theorem but simple algebraic prop erties lead to very b eautiful
results
No theorem captures this algebraic b eauty more than our rst theorem a
surprising characterization of b ounded width branching programs due to Bar rington
Favorite Theorem Bar Bounded width polynomial size branching pro
grams recognize exactly those languages in NC
A branching program is a ro oted directed acyclic graph where each internal
vertex is lab elled by a variable name and for each input symb ol there is a single
edge leaving that vertex lab elled by that symb ol The leaf vertices are lab elled
by Accept or Reject The machine pro ceeds along a ro ot to leaf path by
examining the input of the current vertex s lab el and then follows the edge
corresp onding to the value of that input The machine accepts or rejects when
it comes to the corresp onding leaf vertex
A language L has b ounded width branching programs if for some k for all
O
n there is a width k branching program using n inputs and at most n no des
that accepts exactly those strings of length n in L
Recall that NC consists of the languages accepted by constant fan in and
logarithmic depth circuits
One can easily show that every language that accepts b ounded width p oly
size branching programs must lie in NC by divide and conquer However it did
not seem p ossible to count in a b ounded width program so many b elieved that
ma jority an NC function did not have b ounded width branching programs In
fact Yao Yao showed a sup er p olynomial lower b ound for width two branch
ing programs to compute ma jority
Barrington used the fact that S the set of p ermutations on ve elements
formed a nonsolvable group More precisely there are two ve cycles
is a ve cycle Bar and in S such that
rington uses this simple fact to capture the computation of a circuit by just
rememb ering one of these cycles
Barrington s result has some implications for the complexity class PSPACE
By the Chandra Kozen and Sto ckmeyer CKS result relating PSPACE with
alternating p olynomial time one can view PSPACE computation as a uniform
NC circuit of p olynomial depth
Cai and Furst CF showed that every PSPACE language L can b e ac
cepted by a Turing machine that has a p olynomial size clo ck makes logarith
mic space computation then erases all but a few bits of its tap e b etween clo ck
ticks They prove their result by simulating an exp onentially long but easily de
scribable b ounded width branching program for the NC circuit that simulates
the PSPACE machine
Bovet Crescenzi and Silvestri BCS develop ed the concept of leaf lan
guages Think of a nondeterministic p olynomial time Turing machine M on in
put x as generating an exp onentially long string M x read o of the end of the
computation paths leaves of M on x Given a machine M and a set of strings
A we de ne the leaf language L as the set of x such that M x is in A
Hertrampf Lautemann Schwentick Vollmer and Wagner HLS show
that any language in PSPACE has a leaf language de ned via a regular lan
guage Their pro of uses Barrington s result by simulating the branching program with a nite automaton
Bounded Depth Circuits
In the mid s theorems ab out circuits provided great excitement among
complexity theorists The techniques used in complexity theory b efore then did
not seem to help prove the big theorems such as P NP Many theorists
b elieved that the new combinatorial and algebraic to ols used in circuits could
prove some nontrivial separations in complexity classes
Although this promise has remained unful lled we have seen several sepa
ration results in low level circuit classes as well as many new combinatorial and
algebraic techniques Because of the vast amount of e ort devoted to circuit com
plexity in the last decade we devote Sections and to circuit complexity
Circuit complexity also comes up in connection with many of the other theorems
mentioned in this pap er
The early work in circuit complexity came out of an attempt to create an
oracle to separate the p olynomial time hierarchy from PSPACE In order to
create such an oracle an interesting combinatorial problem arose One would
need to show that a simple problem like parity did not have small constant
depth circuits
Furst Saxe and Sipser FSS and Ajtai Ajt indep endently showed that
parity do es not have constant depth p olynomial size circuits Yao Yao showed
a strong enough b ound to get the desired oracle separation H astad greatly
simpli ed Yao s pro of and achieved near tight b ounds
Favorite Theorem H as Parity does not have depth d circuits with
1 d
n
less than gates
H astad s pro of used random restrictions where some of the variables in a p o
tential circuit for parity are randomly set to zero or one with some probability
H astad s main switching lemma showed that a random restriction of an AND
gate of small OR gates resulted in an OR gate of small AND gates H astad
then used an inductive argument by converting any depth d circuit claiming to
solve parity into a depth d circuit for parity This switching lemma has also
found several other applications subsequently
Razb orov Raz and Smolensky Smo show that for primes p and q
p q the MOD function requires an exp onential numb er of gates for b ounded
q
depth circuits with AND OR and MOD gates Their pro of uses a new idea
p
of approximating b ounded depth circuits by low degree p olynomials over a nite
eld
Linial Mansour and Nisan LMN show using Fourier Transform meth
o ds how to approximate b ounded depth circuits by low degree p olynomials over
the rationals and reals Their pro of uses the H astad switching lemma Their re
sult has had some applications in circuit complexity and learning theory Tarui
Tar shows how to approximate b ounded depth circuits by low degree p oly
nomials over the integers His pro of uses the Valiant Vazirani VV lemma
describ ed in Section and can b e used to give an alternative pro of of part of Theorem
Monotone Circuits
H astad s Theorem Theorem led hop e that circuit complexity could now
lead to p erhaps a combinatorial separation of machine based complexity classes
Since it is well known that every language in P has p olynomial size circuits we
could separate P from NP by exhibiting some NP problem that do es not have
p olynomial size circuits We b elieve that such problems exist b ecause Karp and
Lipton KL showed that if every language in NP has p olynomial size circuits
then the p olynomial hierarchy collapses
However proving sup erp olynomial or even sup erlinear lower b ounds for
such an NP problem seems extremely di cult Razb orov lo oked at answer
ing this question in a restricted mo del In particular he lo oked at monotone
circuits i e circuits with no negations or negated variables Of course a non
monotone problem cannot have such circuits so Razb orov lo oked at a monotone
NP complete problem Clique
Favorite Theorem Raz b The general clique function requires expo
nential ly large monotone circuits
Instead of the restriction metho d used by H astad Theorem Razb orov uses
an approximation metho d Razb orov shows how to approximate each AND and
OR with approximate AND and approximate OR Each approximation cannot
cause to o many errors in the inputs However he shows the nal approximated
circuit must error in many inputs Razb orov then concludes that the circuits
must have had lots of gates Alon and Boppana AB strengthen Razb orov s
b ounds
Initially many thought that p erhaps we could extend these techniques into
the general case Now it seems that Razb orov s theorem says much more ab out
the weakness of monotone mo dels than ab out the hardness of NP problems
Razb orov Raz a showed that matching also do es not have p olynomial size
monotone circuits However we know that matching do es have a p olynomial
time algorithm Edm and thus p olynomial size nonmonotone circuits Tardos
Tar exhibited a monotone problem that has an exp onential gap b etween its
monotone and nonmonotone circuit complexity
Other results on monotone complexity come out of communication complex
ity Karchmer and Wigderson KW show a direct connection b etween a com
munication complexity game and circuit depth They use this characterization to
show that graph connectivity requires log n depth in p olynomial size mono
tone circuits Razb orov Raz uses more general matrix metho ds to prove sim
ilar lower b ounds for other problems Raz and Wigderson RW show that
monotone circuits for matching require linear depth
We also have seen several arguments that the techniques used in Theorems
and will not help us settle the big questions like P NP Razb orov Raz
himself showed that approximation techniques like those used in the pro of of
Theorem will not work in the general case Razb orov Raz later shows a
fragment of arithmetic strong enough to prove Theorems and cannot show
that NP do es not have p olynomial size circuits Razb orov and Rudich RR
show that under a strong cryptographic assumption combinatorial pro ofs ful ll
ing certain largeness and constructivity prop erties cannot show that NP do es
not have p olynomial size circuits
This last decade started with the promise of great separation results using
combinatorial techniques but ends nding us no closer to solving the big op en
questions in complexity theory
Nondeterministic Space
We usually use many factors to determine the quality of a result If we lo ok
solely at the imp ortance of the statement of the theorem one result stands out
Immerman and Szelep cs enyi indep endently proved the following fundamental
theorem ab out nondeterministic space complexity
Favorite Theorem Imm Sze Nondeterministic Space is closed un
der complementation
From the very b eginnings of complexity theory theorists have thought hard
ab out the relationship among the various deterministic and nondeterministic
time and space classes The fundamental theorems in complexity theory relate
these classes such as the basic time and space hierarchy theorems due to Hart
manis and Stearns HS in
While the relationships b etween deterministic and nondeterministic time
classes remain the single most imp ortant op en area in complexity we know much
more ab out the related space complexity questions In Savitch Sav
showed that one can simulate a S n space nondeterministic Turing machine
by a S n space deterministic machine Thus for example NPSPACE
PSPACE
In the two and a half decades since no one has improved up on this quadratic
increase from nondeterministic to deterministic space From Immerman and
Szelep cs enyi we do get that only a linear blow up going from nondeterministic
existential space to conondeterministic universal space
We can think of the computation of a s n space b ounded Turing machine M
O s jxj
on input x as a directed graph G on no des where each no de represents
a con guration of M x and each edge represents a transition Note that M x
accepts if and only if there is a directed path from the initial con guration to
an accepting con guration
The pro of of Theorem uses inductive counting to show that there is no
path from s to t in an n no de graph in nondeterministic O log n space For a
directed graph G and a no de s of G de ne the value c as the numb er of no des
i
reachable from s in G by a path of length at most i They show how using a
nondeterministic log space machine they can verify the value of c inductively as
i
i go es from to n They can then use c to determine if a no de t is not reachable
n
from no de s
Though Immerman and Szelep cs enyi have similar pro ofs they each went
ab out this theorem from di erent angles Immerman has built up a theory giv
ing logical characterizations of complexity classes Immerman lo oked at a char
acterization of nondeterministic logarithmic space NL as sets of languages
describable by rst order expressions with transitive closure Immerman showed
that having rst order with the complement of transitive closure led to the same
class He then translated this pro of to show that NL coNL and then that all
nondeterministic space classes are closed under complementation
Szelep cs enyi lo oked at the question of whether context sensitive languages
are closed under complement This was one of the few op en formal languages
questions in Hop croft and Ullman HU Landweb er Lan showed that con
text sensitive languages accepted exactly those sets in nondeterministic linear
space Thus by proving that nondeterministic space is closed under complement
Immerman and Szelep cs enyi also showed that context sensitive languages are
closed under complementation
Boro din Co ok Dymond Ruzzo and Tompa BCD b extended the in
ductive counting arguments used in the pro of of Theorem to show two other
results
LOGCFL the class of languages log space reducible to context free lan
guages is closed under complementation
Undirected s t connectivity has an errorless log space exp ected p olynomial
time probabilistic algorithm
Cryptographic Assumptions
In another trend during the past decade researchers have lo oked at complexity
issues arising from cryptography Cryptographers used many di erent hardness
assumptions in order to prove the security of their various proto cols Some com
plexity theorists lo oked at the relative hardness of these assumptions
Cryptographers designed their proto cols with either hardness assumptions
ab out particular languages like factoring or discrete logarithm or with some
general assumption Usually a general assumption to ok one of the following
three forms
One way functions exist
Pseudorandom generators exist
Trap do or functions exist
We will not de ne these notions formally in this pap er but keep in mind that
cryptographers usually require that hardness o ccurs for most inputs It was well
known that the second two assumptions imply the rst but ten years ago the
other directions remained op en
Impagliazzo Levin and Luby settled one relationship
Favorite Theorem ILL Pseudorandom generators can be constructed
from any one way function
This problem has a very interesting history Blum and Micali BM show how
to create pseudorandom generators based on the hardness of discrete logarithm
Yao Yao generalizes their algorithm to create a generator based on any one
way p ermutation Levin Lev shows a technical one way prop erty of functions
that he can use to create secure pseudorandom generators Goldreich Krawczyk
and Luby GKL show how to convert a regular one way function to a pseu
dorandom generator Impagliazzo Levin and Luby then showed Theorem Fi
nally H astad H as extended the techniques of Impagliazzo Levin and Luby
to show how to construct pseudo random functions from any uniformly one way
function
The pro of of Theorem uses a new idea of computational entropy In other
words they use a one way function to create a distribution that may not have
large real entropy or randomness but do es have large entropy in a computational
sense They then use this pseudo entropy distribution to create the pseudoran
dom generator Their techniques make imp ortant use of a result of Goldreich
and Levin GL that shows how to get a hard core bit out of any one way
function
The relationship b etween one way functions and trap do or functions app ear
much more di cult to resolve Impagliazzo and Rudich IR give some rela
tivization results that shed light on this di culty
Isomorphism Conjecture
Back in the late s Berman and Hartmanis BH lo oked at the structure
of the known NP complete sets via many one reductions They develop ed some
conditions under which one could show two sets were isomorphic i e there
existed a p olynomial time computable p olynomial time invertible bijection re
ducing one to the other They then showed that all the NP complete problems
known at that time were isomorphic They conjectured that all NP complete
problems are isomorphic Since their conjecture implies P NP or one would
have nite sets isomorphic to in nite sets they thought that mayb e that di cult
problem could b e attacked by lo oking at the structure of complete sets
The isomorphism conjecture would imply that every NP complete set A has
(1)
n n
exp onential density i e for all n jA j Mahaney Mah gave
some evidence to this direction by showing that there are no sparse p olynomial
dense NP complete sets unless P NP
What ab out complete sets via other notions of reductions Karp and Lipton
KL show that if there exist sparse sets NP hard via Turing reductions then
the p olynomial time hierarchy collapses to the second level However what kind
of reductions to sparse sets can we rule out by only assuming P NP
Many p eople tried extending the techniques of Mahaney Ogiwara and Watan
ab e made large progress with a brand new trick to solve the problem for b ounded
truth table reductions i e nonadaptive reductions that make a constant numb er of queries to the set
Favorite Theorem OW There are no sparse sets hard for NP via
polynomial time bounded truth table queries unless P NP
The pro of uses a new technique known as left sets For a satis able formula
Ogiwara and Watanab e create a step function based on the lexicographically
least witness They then use the reduction to nd this step and thus a witness
Homer and Longpr e HL give a very clear pro of of Theorem with stronger
b ounds
Complexity theorists have proven many other interesting results relating to
the isomorphism conjecture in the past decade Joseph and Young JY con
jecture that one could use certain one way functions to create NP complete sets
nonisomorphic to SAT Ko Long and Du KLD show that one way func
tions exist if and only if there exist two sets reducible to each other via injective
length increasing reductions but not isomorphic to each other Kurtz Mahaney
and Royer KMR show that relative to a random oracle those two sets could
b e NP complete On the other hand Hartmanis and Hemachandra HH show
that there exists a relativized world where no one way functions exist but the
isomorphism conjecture fails
Kurtz Mahaney and Royer KMR show there exists some set such that all
sets many one equivalent to it are isomorphic to it Fenner Fortnow and Kurtz
FFK give a relativized world where all the NP complete sets are isomorphic
i e the Berman Hartmanis conjecture holds
Simulating Randomness
Probabilistic computation has played a ma jor role in complexity theory during
the last decade In Section we will see how interactive pro of system mo dels
use randomness as a p owerful veri cation to ol In this section we will lo ok at an
opp osite approach to lo ok at how we can reduce or eliminate randomness from
some computational mo dels
We have seen many imp ortant results in this area over the past decade Noam
Nisan has played an imp ortant role in many of these results so in this section I
would like to highlight one of his more general results
Favorite Theorem Nis a For any r n and s n there exists a pseu
dorandom generator that converts a random seed of length O s n log r n to
r n bits that looks random to any algorithm using s n space
Nisan s generator builds on universal hashing a technique to generate pairwise
indep endence or close to it without using many random bits Nisan then builds
a random string by recursively applying a sp eci c universal hash function
Theorem has imp ortant applications to the problem of universal traversal
sequences Aleliunas Karp Lipton Lov asz and Racko AKL show that for
every n there exist p olynomial in n length universal traversal sequences that tra
verse every connected undirected graph on n no des Constructing such sequences
has remained an imp ortant op en question Istrail Ist requires a di cult pro of
O log n
just to show a constructible sequence for cycles Theorem implies that n
length universal traversal sequences can b e constructed in O log n space
Nisan Nis b extends the techniques of Theorem to show that every lan
guage accepted in randomized logarithmic space like undirected graph connec
tivity can b e accepted by a deterministic Turing machine running in p olynomial
time and O log n space Also building on the generator from Theorem Nisan
Szemer edi and Wigderson NSW show that undirected connectivity can b e
computed in O log n space
Nisan and Zuckerman NZ show how to simulate any randomized s n
space b ounded Turing machine that uses pol y s n random bits in determin
istic space s n Their pro of uses a pro cedure that extracts randomness from a
defective random source using a small additional numb er of truly random bits
Supp ose we had a probabilistic algorithm A that gave the correct answer
with probability using r random coins If we wanted to get this probability
k
up to we can use the standard trick of running AO k times and taking
ma jority vote However this metho d will take O nk random coins
Can we achieve the same result using fewer random coins Impagliazzo and
Zuckerman IZ give a tight answer to this question They show how to achieve
k
the error using only O n k random coins Their pro of uses a new idea
of taking a random walk on an expander graph
Counting Complexity
In Valiant Val a lo oked at the question of computing the p ermanent of
a matrix Unlike the determinant where we knew p olynomial time algorithms
the p ermanent seemed a much more di cult problem to handle
In order to capture the p ower of the p ermanent Valiant develop ed a new
function class P A function is in P if there exists a nondeterministic
p olynomial time Turing machine M such that f x equals the numb er of accept
ing computations of M x Valiant showed that the p ermanent is P complete
In future work Valiant Val b showed that several other natural counting
questions are P complete
Clearly P functions are hard for NP But we knew little ab out their
relationship to other complexity classes In p erhaps the b est complexity result of
the last decade To da showed a surprising relationship b etween the p olynomial
time hierarchy and P functions
Favorite Theorem To d Every language in the polynomial time hier
archy can be reduced in polynomial time to a single query of a P function
In other words one can use counting to simulate a constant numb er of alterna
tions
To da s pro of uses several new and exciting ideas making this theorem one
of the prettiest structural complexity results in the last decade To da actually
proves two separate theorems First he shows how to randomly reduce every
language in the p olynomial time hierarchy to a P question where L P if
there exists a P function f where x L if and only if f x is o dd Then To da
shows how to reduce languages probabilistically reducible to P to a single P
question
Valiant and Vazirani proved the following extremely useful lemma
Lemma VV There exists a random reduction mapping CNF formulas
to CNF formulas such that for al l formulas of n variables
If is not satis able then is never satis able
has exactly one If is satis able then with probability at least
n
satisfying assignment
P
Papadimitriou and Zachos PZ show that P P i e a P machine
that asks arbitrary P questions to an oracle
To da uses Lemma in a novel way combined with the Papadimitriou and
Zachos result to show how to probabilistically reduce every NP language to a P
language To da then uses a nice induction again with some new tricks to show
that every language in PH the p olynomial time hierarchy probabilistically
reduces to a P set
To da then shows for any p olynomial q how to mo dify a P function f to
q jxj
a new function g such that f x mo d g x mo d for all x Using
this fact To da can combine the randomness and the P question into a single
P question thus completing his pro of
To da s theorem sparked renewed interest in counting complexity Much of
the work has centered on classes de ned by counting functions We will discuss
this work in Section
To da s result also has implications in circuit complexity Allender All
shows how to use To da s pro of to show that every constant depth circuit has an
equivalent depth three quasip olynomial size threshold circuit The class ACC
consists of those circuits accepted by b ounded depth circuits with AND OR
and MOD gates where q is a xed integer not necessarily prime Yao Yao
q
and Beigel and Tarui BT extend the ideas of Theorem to show that every
language in ACC is recognized by depth two circuits with a symmetric inde
p endent of input order gate at the ro ot and quasip olynomial AND gates of
p olylog fan in at the leaves
Fenner Fortnow and Kurtz FFK develop ed the notion of GapP The
function class GapP consists of functions f x where there exists a nondeter
ministic p olynomial time Turing machine M where f x is equal to the numb er
of accepting paths of M x minus the numb er of rejecting paths of M x Equiv
alently GapP consists of the closure of P functions under subtraction Fenner
Fortnow and Kurtz show that many of the P closure prop erties also hold for
GapP and that lo oking at GapP functions simpli ed many counting complexity
arguments
To da and Ogiwara TO extend part of To da s work to show that every
PH
function in GapP probabilistically reduces to a GapP function Their re
sult shows that in addition to P PH probabilistically reduces to many other
counting classes such as PP see Section
To da s result leads to one of the more intriguing op en questions to arise in
P
the last decade Do es P PSPACE In other words can counting simulate
p olynomial alternations
Counting Classes
As describ ed in Section counting complexity played a ma jor role in com
plexity theory in the past decade Instead of asking whether an NP question
has a solution we now ask how many In Section we lo oked at the complexity
of functions de ned this way In this section we will lo ok at complexity classes
based on counting functions
Gill Gil in his seminal pap er on probabilistic complexity classes de ned
the basic probabilistic classes that we lo ok at to day including the class PP
Probabilistic Polynomial time The class PP consists of those languages ac
cepted by probabilistic p olynomial time Turing machines that accept with prob
ability greater than a half
This class also plays an imp ortant role in counting complexity A language
L is in PP if there is some nondeterministic p olynomial time Turing machine
M such that x is in L if and only if the accepting paths of M x outnumb er the
rejecting paths or equivalently when some GapP function f x is greater than
zero
PP P
Such a class contains NP By binary search one can show that P P
PP
Thus by Theorem we have that PH P and thus if PP PH then the
p olynomial time hierarchy collapses
Clearly PP is closed under complementation but surprisingly there is no
easy way to show that PP is closed under intersection That question remained
op en until the s when Beigel Reingold and Spielman showed us that PP do es
indeed have this closure prop erty
Favorite Theorem BRS PP is closed under intersection
The pro of of Theorem really makes use of the connection b etween GapP
functions and low degree p olynomials Beigel Reingold and Spielman make use
of the fact that there exist rational functions that very closely approximate
the absolute value function and use it to create a rational function of GapP
functions that take on a p ositive value if and only if b oth GapP functions are
p ositive They then show how to test for this condition with a PP predicate
Fortnow and Reingold FR extend the techniques of Beigel Reingold and
Spielman to show that PP is closed under truth table reductions Beigel Bei
NP
created a very useful oracle relative to which P PP and thus PP is not
closed under Turing reductions
We have seen many other counting classes arise in complexity theory recently
We unfortunately only have ro om to review a few of them
The class Mo d P consists of those languages L where there exists a P
k
function f such that x L if and only if f x mo d k Papadimitriou and
P
Zachos PZ lo ok at the sp ecial case of P Mo d P and show that P
P Beigel and Gill BG and Hertrampf Her extend this result for Mo d P
k
where k is prime We have also already seen the imp ortance that P plays in
the pro of of Theorem
Fenner Fortnow and Kurtz FFK formalize a notion of complexity classes
de ned via GapP functions see Section They de ne a class SPP where
a language L lies in SPP if there exists a GapP function f such that f x
x They show that SPP is contained in every reasonable Gap de nable
L
class
Interactive Pro of Systems
No review of complexity theory in the past decade could b e complete without
mentioning interactive pro of systems In order to keep this section manageable
we will concentrate only on the complexity theoretical asp ects of interactive pro of
systems I recognize the imp ortance of many interesting zero knowledge program
testing and approximation results that arise from the theory of interactive pro of
systems but will leave most of the discussion of them to other surveys
We highlight the last of the truly great results in interactive pro of systems
a pap er by Arora Lund Motwani Sudan and Szegedy that characterizes NP
by a simple veri cation pro cedure
Favorite Theorem ALM Every language in NP has a probabilis
tical ly checkable proof system where the veri er uses only O log n random coins
and asks only a constant number of queries to the proof
In order to appreciate Theorem we rst give a history of the complexity of
interactive pro of systems
Goldwasser Micali and Racko GMR and Babai Bab BM inde
p endently develop ed interactive proof systems The mo del has an arbitrarily
p owerful prover that tries to convince an untrusting p olynomial time probabilis
tic veri er ab out whether some string is in a language Goldwasser and Sipser
GS show that the p ower of the mo del do es not dep end on whether or not
the prover can see the veri er s coins
A bounded round interactive pro of system only allows a xed numb er of
p olynomial length messages b etween the prover and the veri er Babai Bab
shows that every b ounded round pro of system has an equivalent system consist
ing of a single message from the veri er followed by a single message from the
prover Boppana H astad and Zachos BHZ show that if every coNP language
has b ounded round interactive pro of systems then the p olynomial time hierar
p
chy collapses to Goldreich Micali and Wigderson GMW show that graph
nonisomorphism has a b ounded round interactive pro of system As an immedi
ate corollary we get that graph isomorphism is not NP complete unless the
p olynomial time hierarchy collapses
Feldman Fel showed that in interactive pro of systems we can assume the
prover runs in p olynomial space and thus every language accepted by interactive
pro of systems lies in PSPACE Lund Fortnow Karlo and Nisan LFKN
show that every language in PH has an unb ounded round interactive pro of
system Shamir Sha extends the techniques of Lund Fortnow Karlo and
Nisan to show that every language in PSPACE has an interactive pro of system
The pro of uses the low degree structure of some P and PSPACE complete
problems These results are some of the very few known that do not relativize
Fortnow and Sipser FS exhibit an oracle relative to which some coNP lan
guage do es not have an interactive pro of system
Ben Or Goldwasser Kilian and Wigderson BGKW develop ed a multiple
prover interactive pro of system where many separated provers try to convince
the veri er that a string lies in a language Fortnow Romp el and Sipser FRS
showed that every language accepted by such pro of systems lies in NEXP
Babai Fortnow and Lund BFL showed that every language in NEXP has
such a pro of system Babai Fortnow and Lund s pro of uses a relativizable form
of the Lund Fortnow Karlo and Nisan proto col combined with a multilinearity
test Feige and Lov asz FL show that very language in NEXP in fact has a
two prover one round pro of system
Arora and Safra AS de ne a probabilistical ly checkable proof system where
the prover must write down a p erhaps exp onentially long pro of system that
the p olynomial time probabilistic veri er can check using random access to the
pro of Fortnow Romp el and Sipser FRS show the equivalence b etween the
multiple prover interactive pro of system mo del and an arbitrary probabilistically
checkable pro of Arora Lund Motwani Sudan and Szegedy building on ideas
of Arora and Safra AS proved Theorem
Arora Lund Motwani Sudan and Szegedy s result also has implications
for approximation problems Papaditimitriou and Yannakakis PY de ne a
class MAXSNP of NP optimization problems such as nding an assignment
that maximizes the numb er of clauses made true in a formula A corollary of
the result of Arora Lund Motwani Sudan and Szegedy shows that for every
MAXSNP hard language L there is some constant such that one could
not approximate problems in L within a factor unless P NP
Conclusions
One can draw several interesting conclusions ab out the theorems on the list
No theorem stands alone Every theorem chosen either has a long line of
results leading up to it and or has several results that use the theorem or
technique in an imp ortant way
No one p erson has dominated complexity theory Though a few strong the
orists do stand out from the last decade no single p erson has proven more
than one of the theorems I have chosen
No single country dominates the list The twenty authors of these ten results
represent no fewer than nine separate countries
Here s to hoping that computational complexity theory can achieve as many great results in the next decade as it has in this past one
Acknowledgments
I would like to thank Satyanarayana V Lokam Dieter Van Melkeb eek and Sophie
Laplante for their comments and help on this pap er
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