# Lance FortnowIllinois Institute of Technology | IIT · College of Computing

Lance Fortnow

PhD, MIT Applied Math, 1989

## About

247

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Introduction

Additional affiliations

January 2008 - June 2012

September 1989 - December 2007

## Publications

Publications (247)

We survey recent research on the power of nondeterministic computation and how to use nondeterminism to get new separations of complexity classes. Results include separating \(\mathsf{NEXP}\) from \(\mathsf{NP}\) with limited advice, a new proof of the nondeterministic time hierarchy and a surprising relativized world where \(\mathsf{NP}\) is as po...

We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind [1] who study the consequences of Sat being many-one reducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (Sat
\(\leq_m^p \mathrm{LT}_1\)). They claim that P= N...

I recently completed a general audience book on the P versus NP problem [1]. Writing the book has forced me to step back and take a fresh look at the question from a non-technical point of view. There are really two different P versus NP problems. One is the formal mathematical question, first formulated by Steve Cook in 1971 [2] and listed as one...

The Church-Turing thesis has stood the test of time, capturing computation models Turing could not have conceived of, including
digital computation, probabilistic, parallel and quantum computers and the Internet. The thesis has become accepted doctrine
in computer science and the ACM has named its highest honor after Turing. Many now view computati...

Hanson's market scoring rules allow us to design a prediction market that
still gives useful information even if we have an illiquid market with a
limited number of budget-constrained agents. Each agent can "move" the current
price of a market towards their prediction.
While this movement still occurs in multi-outcome or multidimensional markets
we...

We apply results on extracting randomness from independent sources to “extract” Kolmogorov complexity. For any α,ϵ>0, given a string x with K(x)>α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y|=Ω(|x|), with K(y)>(1-ϵ)|y|. This result holds for both unbounded and space-bounded Kolmogorov complexi...

We consider a repeated Matching Pennies game in which players have limited
access to randomness. Playing the (unique) Nash equilibrium in this n-stage
game requires n random bits. Can there be Nash equilibria that use less than n
random coins?
Our main results are as follows: We give a full characterization of
approximate equilibria, showing that,...

The OR-SAT problem asks, given Boolean formulae ϕ1,…,ϕm each of size at most n, whether at least one of the ϕi's is satisfiable. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n, unless NP⊆coNP/poly, and the Polynomial-Time Hierarchy collapses. This result settles an open pro...

We define and study a new notion of “robust simulations” between complexity classes which is intermediate between the traditional notions of infinitely-often and almost-everywhere, as well as a corresponding notion of “significant separations”. A language L has a robust simulation in a complexity class C if there is a language in C which agrees wit...

In this fifth article in the ACM Ubiquity symposium on What is computation? Paul S. Rosenbloom explains why he believes computing is the fourth great scientific domain, on par with the physical, life, and social sciences. Editor

We study the equilibrium behavior of informed traders interacting with market scoring rule (MSR) market makers. One attractive feature of MSR is that it is myopically incentive compatible: it is optimal for traders to report their true beliefs about the likelihood of an event outcome provided that they ignore the impact of their reports on the prof...

Given samples from two distributions over an n-element set, we wish to test whether these distributions are statistically close. We present an algorithm which uses sublinear in n, specifically, O(n[superscript 2/3]ε[superscript −8/3] log n), independent samples from each distribution, runs in time linear in the sample size, makes no assumptions abo...

The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially
verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in
polynomial-time does this imply the polynomial-time hierarchy collapses? By computing a multivalued func...

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adapti...

Tennenholtz (GEB 2004) developed Program Equilibrium to model play in a finite two-player game where each player can base their strategy on the other player's strategies. Tennenholtz's model allowed each player to produce a "loop-free" computer program that had access to the code for both players. He showed a folk theorem where any mixed-strategy i...

From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 “Algebraic Methods in Computational Complexity “ was held in Schloss Dagstuhl-Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as w...

The methods to handle NP-complete problems and the theory that has developed from those approaches are discussed. The collection of problems that have efficiently verifiable solutions is known as NP (non-deterministic polynomial time). So P=NP means that for every problem that has an efficiently verifiable solution, a solution can be found. An effi...

Computer science has grown to become a mature field where no major university can survive without a strong CS department. It is time for computer science to grow up and publish in a way that represents the major discipline it has become. Computer science should refocus the conference system on its primary purpose of bringing researchers together. I...

To determine if two lists of numbers are the same set, we sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms arise in graph isomorphism algorithms. To determine if two graphs are cospectral (have the same eigenvalues), we compute their characteri...

We show several unconditional lower bounds for exponential time classes against polynomial time classes with advice, including:
1
For any constant c, \({\sf NEXP} \not \subseteq {\rm{\sf P}}^{\sf NP[n^c]}/n^c\)
1
For any constant c, \({\sf MAEXP} \not \subseteq {\rm {\sf MA}}/n^c\)
1
\({\sf BPEXP} \not \subseteq {\sf BPP}/n^{o(1)}\)
It was previous...

in 1982, Kannan showed that ΣP2 does not have nk-sized circuits for any k. Do smaller classes also admit such circuit lower bounds? Despite several improvements of Kannan's result, we still cannot prove that PNP does not have linear size circuits. Work of Aaronson and Wigderson provides strong evidence - the "algebrization" barrier - that current t...

Under a standard hardness assumption we exactly characterize the worst-case running time of languages that are in average polynomial-time over all polynomial-time samplable distributions. More precisely we show that if exponential time is not infinitely often in subexponential space, then the following are equivalent for any algorithm A: • For all...

This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational complexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2^(n^k))-inseparable. We also relate these conditions to...

Any proof of P ≠ NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (e.g., that PP does not have linear-size circuits) that overcome both ...

We describe a new approach for understanding the difficulty of designing efficient learning algorithms. We prove that the existence of an efficient learning algorithm for a circuit class C in Angluin's model of exact learning from membership and equivalence queries or in Valiant's PAC model yields a lower bound against C. More specifically, we prov...

\newcommand{\sharpP}{\mathrm{\# P}}
\newcommand{\parityP}{{\oplus\mathrm{P}}}
\renewcommand{\P}{\mathrm{P}}
\newcommand{\BPP}{\mathrm{BPP}}
$
Toda in his celebrated paper showed that the polynomial-time
hierarchy is contained in $\P^\sharpP$. We give a short and
simple proof of the first half of Toda's Theorem that the
polynomial-time hierarchy is...

We exhibit a new computational-based definition of awareness, informally that our level of unawareness of an object is the amount of time needed to generate that object within a certain environment. We give several examples to show this notion matches our intuition in scenarios where one organizes, accesses and transfers information. We also give a...

Consider a weather forecaster predicting a probability of rain for the next day. We consider tests that, given a finite sequence of forecast predictions and outcomes, will either pass or fail the forecaster. Sandroni showed that any test which passes a forecaster who knows the distribution of nature can also be probabilistically passed by a forecas...

Consider a weather forecaster predicting the probability of rain for the next day. We consider tests that given a finite sequence of forecast predictions and outcomes will either pass or fail the forecaster. It is known that any test which passes a forecaster who knows the distribution of nature can also be probabilistically passed by a forecaster...

Consider a weather forecaster predicting a probability of rain for the next day. We consider tests that given a finite sequence of forecast predictions and outcomes will either pass or fail the forecaster. Sandroni shows that any test which passes a forecaster who knows the distribution of nature can also be probabilistically passed by a forecaster...

We analyze the computational complexity of market maker pricing algorithms for combinatorial prediction markets. We focus on Hanson's popular logarithmic market scoring rule market maker (LMSR). Our goal is to implicitly maintain correct LMSR prices across an exponentially large outcome space. We examine both permutation combinatorics, where outcom...

We exhibit a new computational-based definition of awareness, informally that our level of unawareness of an object is the amount of time needed to generate that object within a certain environment. We give several examples to show this notion matches our intuition in scenarios where one organizes, accesses and transfers information. We also give a...

We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P NP[1] = P NP[2] , then the polynomial-time hierarchy collapses to S p 2 ⊆ Σ p 2 ∩ Π p 2. Even s...

A language L has a property tester if there exists a probabilistic algorithm that given an input x only asks a small number of bits of x and distinguishes the cases as to whether x is in L and x has large Hamming distance from all y in L. We dene a similar notion of quantum property testing and show that there exist languages with quantum property...

Tennenholtz (GEB 2004) developed Program Equilibrium to model play in a finite two-player game where each player can base their strategy on the other player's strategies. Tennenholtz's model allowed each player to produce a "loop-free" computer program that had access to the code for both players. He showed a folk theorem where any mixed-strategy i...

We study the equilibrium behavior of informed traders in- teracting with two types of automated market makers: market scoring rules (MSR) and dynamic parimutuel markets (DPM). Although both MSR and DPM subsidize trade to encourage information aggregation, and MSR is myopically incentive compatible, neither mechanism is in- centive compatible in gen...

We discuss the design of combinatorial betting mechanisms. We characterize
the computational complexity of several variants of the problem and
pose some open research questions.

The class
, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if
is computable in polynomial-time does this imply the polynomial-time hierarchy collapses? (By computing a multivalued function in...

We exhibit a relativized world where NP ∩ SPARSE has no complete sets. This gives the first relativized world where no optimal proof systems exist.
We also examine under what reductions NP ∩ SPARSE can have complete sets. We show a close connection between these issues and reductions from sparse to tally sets. We also consider the question as to wh...

Antunes, Fortnow, van Melkebeek and Vinodchandran captured the notion of non-random information by computational depth, the difference between the polynomial-time- bounded Kolmogorov complexity and traditional Kolmogorov complexity. We show unconditionally how to probabilistically find satisfying assignments for formulas that have at least one assi...

We consider a permutation betting scenario, where people wager on the flnal ordering of n candidates: for example, the outcome of a horse race. We examine the auctioneer problem of risklessly matching up wagers or, equivalently, flnding arbitrage opportunities among the proposed wagers. Requiring bidders to explicitly list the orderings that they'd...

We survey time hierarchies, with an emphasis on recent work on hierarchies for semantic classes.

The seminar brought together almost 50 researchers covering a wide spectrum of complexity theory. The focus on algebraic methods showed once again the great importance of algebraic techniques for theoretical computer science. We had almost 30 talks of length between 15 and 45 minutes. This left enough room for discussions. We had an open problem se...

From 07.10. to 12.10., the Dagstuhl Seminar 07411 ``Algebraic Methods in Computational Complexity'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given...

Unger studied the balanced leaf languages defined via poly-logarithmically sparse leaf pattern sets. Unger shows that NP-complete sets are not polynomial-time many-one reducible to such balanced leaf language unless the polynomial hierarchy collapses to Θ\(^{p}_{\rm 2}\) and that Σ\(^{p}_{\rm 2}\)-complete sets are not polynomial-time bounded-truth...

We apply recent results on extracting randomness from independent sources to “extract” Kolmogorov complexity. For any α, ε> 0, given a string x with K(x) > α|x|, we show how to use a constant number of advice bits to efficiently compute another string y, |y|=Ω(|x|), with K(y) > (1–ε)|y|. This result holds for both classical and space-bounded Kolmog...

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is an k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f . If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which ar...

We show that any 1-round 2-server Private Information Retrieval Protocol where the answers are 1-bit long must ask questions that are at least n 2 bits long, which is nearly equal to the known n 1 upper bound. This improves upon the approximately 0.25n lower bound of Kerenidis and de Wolf while avoiding their use of quantum

We construct an oracle relative to which NP has p-measure 0 but D p has measure 1 in EXP. This gives a strong relativized negative answer to a question posed by Lutz [Lut96]. Secondly, we give strong evidence that BPP is small. We show that BPP has p-measure 0 unless EXP = MA and the polynomial-time hierarchy collapses. This contrasts the work of R...

We introduce Computational Depth, a measure for the amount of “nonrandom” or “useful” information in a string by considering the difference of various Kolmogorov complexity measures. We investigate three instantiations of Computational Depth:•Basic Computational Depth, a clean notion capturing the spirit of Bennett's Logical Depth. We show that a T...

Gap-definability and the gap-closure operator were defined in [FFK91]. Few complexity classes were known at that time to be gap-definable. In this paper, we give simple characterizations of both gap-definability and the gap-closure operator, and we show that many complexity classes are gap-definable, including P#P,
P# P[1] P^{\# P_{[1]} }
, PSPACE...

The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hierarchy collapses? We give a relativized negative...

Kolmogorov Complexity measures the amount of information in a string by the size of the smallest program that generates that string. Antunes, Fortnow, van Melkebeek, and Vinodchandran captured the notion of useful information by computational depth, the difference between the polynomial-time-bounded Kolmogorov complexity and traditional Kolmogorov...

A property tester with high probability accepts inputs satisfying a given property and rejects inputs that are far from satisfying it. A tolerant property tester, as defined by Parnas, Ron and Rubinfeld, must also accept inputs that are close enough to satisfying the property.We construct two properties of binary functions for which there exists a...

We establish the first polynomial time-space lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic random-access Turing machine can solve satisfiability in time nc and space nd, where d approaches 1 when c does. On co...

We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M(i,j) = C(i,j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approximating the value...

We show that for any constant a, ZPP/b(n) strictly contains ZPTIME(na)/b(n) for some b(n) = O(log n log log n). Our techniques are very general and give the same hierarchy for all common semantic time classes including RTIME, NTIME ∩ coNTIME, UTIME, MATIME, AMTIME and BQTIME.We show a stronger hierarchy for RTIME: For every constant c, RP/1 is not...

Shortly after Steve Cook and Richard Karp showed the ex-istence of many natural NP-complete languages, researchers started to realize the great importance of the P versus NP problem and the difficulty of settling it. One graduate student at the Massachusetts Institute of Technology started to look beyond NP, asking what problems have a higher compl...

How much do we have to change a string to increase its Kolmogorov complexity? We show that we can increase the complexity
of any non-random string of length n by flipping O(Ön)O(\sqrt{n}) bits and some strings require W(Ön)\Omega(\sqrt{n}) bit flips. For a given m, we also give bounds for increasing the complexity of a string by flipping m bits.
C...

We show several results about derandomization including: 1. If NP is easy on average then efficient pseudorandom generators exist and P=BPP. 2. If NP is easy on average then given an NP machine M we can easily on average find accepting computations of M(x) when it accepts. 3. For any A in EXP, if NEXPA is in PA/poly then NEXPA=EXPA. 4.If A is pk an...

We show that under a reasonable hardness assumptions, the time-bounded Kolmogorov distribution is a universal samplable distribution. Under the same assumption we exactly characterize the worst-case running time of languages that are in average polynomial-time over all P-samplable distributions.

We show that RL ⊆ L/O(n), i.e., any language computable in randomized logarithmic space can be computed in deterministic logarithmic space with a linear amount of non-uniform advice. To prove our result we use an ultra-low space walk on the Gabber-Galil expander graph due to Gutfreund and Viola.

We consider the problem of computing the weight of a Euclidean minimum spanning tree for a set of points in . We focus on the setting where the input point set is supported by certain basic (and commonly used) geometric data structures that can provide efficient access to the input in a structured way. We present an algorithm that estimates with hi...

We prove a new equivalence between the non-uniform and uniform complexity of exponential time. We show that EXP ⊆ NP/log if and only if EXP = P<sub>||</sub><sup>NP</sup> Our equivalence makes use of a recent result due to Shaltiel and Umans showing EXP in P<sub>||</sub><sup>NP</sup> implies EXP in NP/poly.

We show a hierarchy for probabilistic time with one bit of advice, specifically we show that for all real numbers 1 ≤ α ≤ β, BPTIME(n<sup>α</sup>)/l ⊆ BPTIME(n<sup>β</sup>)/l. This result builds on and improves an earlier hierarchy of Barak using O(log log n) bits of advice. We also show that for any constant d > 0, there is a language L computable...

We show a hierarchy for probabilistic time with one bit of advice, speci cally we show that for all real numbers 1 6 < , BPT IME(n )=1 ( BPT IME(n )=1. This result builds on and improves an earlier hierarchy of Barak using O(log log n) bits of advice.

We review some of quantum algorithms for search problems: Grover's search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks.

We describe the properties of various notions of time-bounded Kolmogorov complexity and other connections between Kolmogorov complexity and computational complexity.

Alice wants to query a database but she does not want the database to learn what she is querying. She can ask for the entire database. Can she get her query answered with less communication? One model of this problem is Private Information Retrieval, henceforth PIR. We survey results obtained about the PIR model including partial # University of Ma...

We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M (i, j) = C(i, j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approximating the va...

We introduce the study of Kolmogorov complexity with error. For a metric d, we define C
a
(x) to be the length of a shortest program p which prints a string y such that d(x,y) ≤ a. We also study a conditional version of this measure C
a, b
(x|y) where the task is, given a string y′ such that d(y,y′) ≤ b, print a string x′ such that d(x,x′) ≤ a. Thi...

We show that for any constant a, ZPP/b(n) strictly contains ZPTIME(n a)/b(n) for some b(n) = O(log n log log n). Our techniques are very general and give the same hierarchy for all the common promise time classes including RTIME, NTIME ∩ coNTIME, UTIME, MATIME, AMTIME and BQTIME. We show a stronger hierarchy for RTIME: For every constant c, RP/1 is...

From 10.10.04 to 15.10.04, the Dagstuhl Seminar 04421 ``Algebraic Methods in Computational Complexity'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations gi...

This is a survey about the title question, written for people who (like the author) see logic as forbidding, esoteric, and remote from their usual concerns. Beginning with a crash course on Zermelo-Fraenkel set theory, it discusses oracle independence; natural proofs; independence results of Razborov, Raz, DeMillo-Lipton, Sazanov, and others; and o...

Given a set X of sequences over a finite alphabet, we investigate the following three quantities.
(i)
The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X.
(ii)
The deterministic feasible predictability of X is the highest success ratio that a polynomial-time det...

We consider the problem of computing the weight of a Euclidean minimum spanning tree for a set of n points in R . We focus on the setting where the input point set is supported by certain basic (and commonly used) geometric data structures that can provide e#cient access to the input in a structured way. We present an algorithm that estimates with...

We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse. We show this hypothesis equivalent to several other complexity conjectures including:•In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept Σ*.•Every total multival...

this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld

We take a fresh look at CD complexity, where CD t (x) is the size of the smallest program that distinguishes x from all other strings in time t(|x|). We also look at CND complexity, a new nondeterministic variant of CD complexity, and time-bounded Kolmogorov complexity, denoted by C complexity. We show several results relating time-bounded C, CD, a...

This survey focuses on the recent developments in the area of derandomization, with the emphasis on the derandomization of time-bounded randomized complexity classes.

The results in this paper show that coNP is contained in NP with 1 bit of advice (denoted NP/1) if and only if the Polynomial Hierarchy (PH) collapses to DP, the second level of the Boolean Hierarchy (BH). Previous work showed that BH ∶DP⇒ coNP ∶ NP/poly. The stronger assumption that PH ∶ DP in the new result allows the length of the advice functio...

Extractors are functions which are able to extract" random bits from arbitrary distributions which contain" sucient randomness. Explicit constructions of extractors have many applications in complexity theory and combinatorics.

We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract de nition of genericity that encompasses a large collection of dierent generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generi...

We consider the question whether there exists a set A such that every set polynomial-time Turing equivalent to A is also many-one equivalent to A. We show that if E = NE then no sparse set has this property. We give the first relativized world where there exists a set with this property, and in this world the set A is sparse.

We give the rst characterization of Turing machines that run in polynomial-time on average. We show that a Turing machine M runs in average polynomial-time if for all inputs x the Turing machine uses time exponential in the computational depth of x, where the computational depth is a measure of the amount of useful" information in x.

We show that there exists a 2-membership comparable set that is not btt-reducible to any pselective set. This is a rare example of an unconditional separation in computational complexity.

We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP coNP All disjoint pairs of NP sets are P-separable.

This article de nes and proves basic properties of the standard quantum circuit model of computation. The model is developed abstractly in close analogy with (classical) deterministic and probabilistic circuits, without recourse to any physical concepts or principles. It is intended as a primer for theoretical computer scientists who do not know|an...