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Let G be a context-free grammar (CFG) in Chomsky normal form. We take the number of rules in G to be the size of G. We also assume all CFGs are in Chomsky normal form. We consider the question of, given a string w of length n, what is the smallest CFG such that L(G)={w}? We show the following: 1) For all w, |w|=n, there is a CFG of size with O(n/log n) rules, such that L(G)={w}. 2) There exists a string w, |w|=n, such that every CFG G with L(G)={w} is of size Omega(n/log n). We give two proofs of: one nonconstructive, the other constructive., Comment: Most results already known

Herein we explore a dual tree algorithm for matrix multiplication of $A\in \mathbb{R}^{M\times D}$ and $B\in\mathbb{R}^{D\times N}$, very narrowly effective if the normalized rows of $A$ and columns of $B$, treated as vectors in $\mathbb{R}^{D}$, fall into clusters of order proportionate to $\Omega(D^{\tau})$ with radii less than $\arcsin(\epsilon/\sqrt{2})$ on the surface of the unit $D$-ball. The algorithm leverages a pruning rule necessary to guarantee $\epsilon$ precision proportionate to vector magnitude products in the resultant matrix. \textit{ Unfortunately, if the rows and columns are uniformly distributed on the surface of the unit $D$-ball, then the expected points per required cluster approaches zero exponentially fast in $D$; thus, the approach requires a great deal of work to pass muster.}

Hybrid circuit and packet switching for data center networking (DCN) has received considerable research attention recently. A hybrid-switched DCN employs a much faster circuit switch that is reconfigurable with a nontrivial cost, and a much slower packet switch that is reconfigurable with no cost, to interconnect its racks of servers. The research problem is, given a traffic demand matrix (between the racks), how to compute a good circuit switch configuration schedule so that the vast majority of the traffic demand is removed by the circuit switch, leaving a remaining demand matrix that contains only small elements for the packet switch to handle. In this paper, we propose two new hybrid switch scheduling algorithms under two different scheduling constraints. Our first algorithm, called 2-hop Eclipse, strikes a much better tradeoff between the resulting performance (of the hybrid switch) and the computational complexity (of the algorithm) than the state of the art solution Eclipse/Eclipse++. Our second algorithm, called BFF (best first fit), is the first hybrid switching solution that exploits the potential partial reconfiguration capability of the circuit switch for performance gains.

The Kolmogorov complexity of x, denoted C(x), is the length of the shortest program that generates x. For such a simple definition, Kolmogorov complexity has a rich and deep theory, as well as applications to a wide variety of topics including learning theory, complexity lower bounds and SAT algorithms. Kolmogorov complexity typically focuses on decompression, going from the compressed program to the original string. This paper develops a dual notion of compression, the mapping from a string to its compressed version. Typical lossless compression algorithms such as Lempel-Ziv or Huffman Encoding always produce a string that will decompress to the original. We define a general compression concept based on this observation. For every m, we exhibit a single compression algorithm q of length about m which for n and strings x of length n >= m, the output of q will have length within n-m+O(1) bits of C(x). We also show this bound is tight in a strong way, for every n >= m there is an x of length n with C(x) about m such that no compression program of size slightly less than m can compress x at all. We also consider a polynomial time-bounded version of compression complexity and show that similar results for this version would rule out cryptographic one-way functions.

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 show that no market-scoring rule, under reasonable conditions, always moves the price directly towards beliefs of the agents. We present a modified version of a market scoring rule for budget-limited traders, and show that it does have the property that, from any starting position, optimal trade by a budget-limited trader will result in the market being moved towards the trader's true belief. This mechanism also retains several attractive strategic properties of the market scoring rule.

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, for any e in [0, 1], the game has a e-Nash equilibrium if and only if both players have (1 - e)n random coins. When players are bound to run in polynomial time, Nash equilibria can exist if and only if one-way functions exist. It is possible to trade-off randomness for running time. In particular, under reasonable assumptions, if we give one player only O(log n) random coins but allow him to run in arbitrary polynomial time and we restrict his opponent to run in time n^k, for some fixed k, then we can sustain an Nash equilibrium. When the game is played for an infinite amount of rounds with time discounted utilities, under reasonable assumptions, we can reduce the amount of randomness required to achieve a e-Nash equilibrium to n, where n is the number of random coins necessary to achieve an approximate Nash equilibrium in the general case., Comment: To appear in Proceedings of the 12th ACM Conference on Electronic Commerce. ACM, New York, 2011

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 with L on arbitrarily large polynomial stretches of input lengths. There is a significant separation of L from C if there is no robust simulation of L in C. The new notion of simulation is a cleaner and more natural notion of simulation than the infinitely-often notion. We show that various implications in complexity theory such as the collapse of PH if NP = P and the Karp-Lipton theorem have analogues for robust simulations. We then use these results to prove that most known separations in complexity theory, such as hierarchy theorems, fixed polynomial circuit lower bounds, time-space tradeoffs, and the theorems of Allender and Williams, can be strengthened to significant separations, though in each case, an almost everywhere separation is unknown. Proving our results requires several new ideas, including a completely different proof of the hierarchy theorem for non-deterministic polynomial time than the ones previously known.

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, $O(n^{2/3}\epsilon^{-8/3}\log n)$, independent samples from each distribution, runs in time linear in the sample size, makes no assumptions about the structure of the distributions, and distinguishes the cases when the distance between the distributions is small (less than $\max\{\epsilon^{4/3}n^{-1/3}/32, \epsilon n^{-1/2}/4\}$) or large (more than $\epsilon$) in $\ell_1$ distance. This result can be compared to the lower bound of $\Omega(n^{2/3}\epsilon^{-2/3})$ for this problem given by Valiant. Our algorithm has applications to the problem of testing whether a given Markov process is rapidly mixing. We present sublinear for several variants of this problem as well., Comment: 26 pages, A preliminary version of this paper appeared in the 41st Symposium on Foundations of Computer Science, 2000, Redondo Beach, CA, A comment from W.D. Smith has been added on the title page

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-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.

Consider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the hardness of factoring. We define a new notion of bounded rationality, where the payoffs of players are discounted by the computation time they take to produce their actions. We use this notion to give a direct correspondence between the existence of equilibria where Alice has a winning strategy and the hardness of factoring. Namely, under a natural assumption on the discount rates, there is an equilibriumwhere Alice has a winning strategy iff there is a linear-time samplable distribution with respect to which Factoring is hard on average. We also give general results for discounted games over countable action spaces, including showing that any game with bounded and computable payoffs has an equilibrium in our model, even if each player is allowed a countable number of actions. It follows, for example, that the Largest Integer game has an equilibrium in our model though it has no Nash equilibria or epsilon-Nash equilibria., Comment: To appear in Proceedings of The First Symposium on Innovations in Computer Science

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, and the equality of permutation groups given by generators. To determine if two graphs are cospectral (have the same eigenvalues), however, we compute their characteristic polynomials and see if they are the same; the characteristic polynomial is a complete invariant for the equivalence relation of cospectrality. This is weaker than a canonical form, and it is not known whether a polynomial-time canonical form for cospectrality exists. Note that it is a priori possible for an equivalence relation to be decidable in polynomial time without either a complete invariant or canonical form. Blass and Gurevich (SIAM J. Comput., 1984) ask whether these conditions on equivalence relations -- having an FP canonical form, having an FP complete invariant, and simply being in P -- are in fact different. They showed that this question requires non-relativizing techniques to resolve. Here we extend their results, and give new connections to probabilistic and quantum computation., Comment: 21 pages

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 strong hypotheses concerning randomness and genericity of disjoint pairs.

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 outcomes are permutations of objects, and Boolean combinatorics, where outcomes are combinations of binary events. We look at three restrictive languages that limit what traders can bet on. Even with severely limited languages, we find that LMSR pricing is $\SP$-hard, even when the same language admits polynomial-time matching without the market maker. We then propose an approximation technique for pricing permutation markets based on a recent algorithm for online permutation learning. The connections we draw between LMSR pricing and the vast literature on online learning with expert advice may be of independent interest.

A workshop sponsored by NSF C-CR and organized by the Center for Discrete Math and Theoretical Computer Science (DIMACS) was held at Elmsford, New York, January 17-18, 2002 where we had several discussions that gave structure to this report. The workshop immediately followed the Fifth Annual Quantum Information Processing Workshop that was held January 14-17 at IBM Yorktown. This is the report from that workshop. The report recommends that the NSF Division of Computer-Communications Research (C-CR) develop a new initiative in "Theory of Quantum Computing and Communication" and describes several research directions that this initiative could support., Comment: 9 pages. A report from the NSF sponsored workshop held January 17-18, 2002 in Elmsford, New York. Workshop date corrected from previous version

The complexity of quantum computation remains poorly understood. While physicists attempt to find ways to create quantum computers, we still do not have much evidence one way or the other as to how useful these machines will be. The tools of computational complexity theory should come to bear on these important questions. Quantum computing often scares away many potential researchers from computer science because of the apparent background need in quantum mechanics and the alien looking notation used in papers on the topic. This paper will give an overview of quantum computation from the point of view of a complexity theorist. We will see that one can think of BQP as yet another complexity class and study its power without focusing on the physical aspects behind it., Comment: 13 pages. Based on a talk presented at the Second Workshop on Algorithms in Quantum Information Processing at DePaul University, Chicago, January 21, 1999

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. 1. BQP is low for PP, i.e., PP^BQP=PP. 2. There exists a relativized world where P=BQP and the polynomial-time hierarchy is infinite. 3. There exists a relativized world where BQP does not have complete sets. 4. There exists a relativized world where P=BQP but P is not equal to UP intersect coUP and one-way functions exist. This gives a relativized answer to an open question of Simon., Comment: 13 pages, no figures; presented at the 13th annual Conference on Computational Complexity (1998); submitted to the Journal of Computer and System Sciences

A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from polynomial space by showing that all Turing-complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing-complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's Program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing-complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic and nonuniform reductions. We show how settling some of these autoreducibility questions will also read to new complexity class separations.