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Humans possess the capability to reason at an abstract level and to structure information into abstract categories, but the underlying neural processes have remained unknown. Experimental evidence has recently emerged for the organization of an important aspect of abstract reasoning: for assigning words to semantic roles in a sentence, such as agent (or subject) and patient (or object). Using minimal assumptions, we show how such a binding of words to semantic roles emerges in a generic spiking neural network through Hebbian plasticity. The resulting model is consistent with the experimental data and enables new computational functionalities such as structured information retrieval, copying data, and comparisons. It thus provides a basis for the implementation of more demanding cognitive computations by networks of spiking neurons., Comment: 23 pages, 5 figures

"Net neutrality" often refers to the policy dictating that an Internet service provider (ISP) cannot charge content providers (CPs) for delivering their content to consumers. Many past quantitative models designed to determine whether net neutrality is a good idea have been rather equivocal in their conclusions. Here we propose a very simple two-sided market model, in which the types of the consumers and the CPs are {\em power-law distributed} --- a kind of distribution known to often arise precisely in connection with Internet-related phenomena. We derive mostly analytical, closed-form results for several regimes: (a) Net neutrality, (b) social optimum, (c) maximum revenue by the ISP, or (d) maximum ISP revenue under quality differentiation. One unexpected conclusion is that (a) and (b) will differ significantly, unless average CP productivity is very high.

Grid cells, discovered more than a decade ago [5], are neurons in the brain of mammals that fire when the animal is located near certain specific points in its familiar terrain. Intriguingly, these points form, for a single cell, a two-dimensional triangular grid, not unlike our Figure 3. Grid cells are widely believed to be involved in path integration, that is, the maintenance of a location state through the summation of small displacements. We provide theoretical evidence for this assertion by showing that cells with grid-like tuning curves are indeed well adapted for the path integration task. In particular we prove that, in one dimension under Gaussian noise, the sensitivity of measuring small displacements is maximized by a population of neurons whose tuning curves are near-sinusoids -- that is to say, with peaks forming a one-dimensional grid. We also show that effective computation of the displacement is possible through a second population of cells whose sinusoid tuning curves are in phase difference from the first. In two dimensions, under additional assumptions it can be shown that measurement sensitivity is optimized by the product of two sinusoids, again yielding a grid-like pattern. We discuss the connection of our results to the triangular grid pattern observed in animals.

It was recently shown \cite{STV} that satisfiability is polynomially solvable when the incidence graph is an interval bipartite graph (an interval graph turned into a bipartite graph by omitting all edges within each partite set). Here we relax this condition in several directions: First, we show that it holds for $k$-interval bigraphs, bipartite graphs which can be converted to interval bipartite graphs by adding to each node of one side at most $k$ edges; the same result holds for the counting and the weighted maximization version of satisfiability. Second, given two linear orders, one for the variables and one for the clauses, we show how to find, in polynomial time, the smallest $k$ such that there is a $k$-interval bigraph compatible with these two orders. On the negative side we prove that, barring complexity collapses, no such extensions are possible for CSPs more general than satisfiability. We also show NP-hardness of recognizing 1-interval bigraphs.

We study the Poincare-Bendixson theorem for two-dimensional continuous dynamical systems in compact domains from the point of view of computation, seeking algorithms for finding the limit cycle promised by this classical result. We start by considering a discrete analogue of this theorem and show that both finding a point on a limit cycle, and determining if a given point is on one, are PSPACE-complete. For the continuous version, we show that both problems are uncomputable in the real complexity sense; i.e., their complexity is arbitrarily high. Subsequently, we introduce a notion of an "approximate cycle" and prove an "approximate" Poincar\'e-Bendixson theorem guaranteeing that some orbits come very close to forming a cycle in the absence of approximate fixpoints; surprisingly, it holds for all dimensions. The corresponding computational problem defined in terms of arithmetic circuits is PSPACE-complete.

The Adaptive Seeding problem is an algorithmic challenge motivated by influence maximization in social networks: One seeks to select among certain accessible nodes in a network, and then select, adaptively, among neighbors of those nodes as they become accessible in order to maximize a global objective function. More generally, adaptive seeding is a stochastic optimization framework where the choices in the first stage affect the realizations in the second stage, over which we aim to optimize. Our main result is a $(1-1/e)^2$-approximation for the adaptive seeding problem for any monotone submodular function. While adaptive policies are often approximated via non-adaptive policies, our algorithm is based on a novel method we call \emph{locally-adaptive} policies. These policies combine a non-adaptive global structure, with local adaptive optimizations. This method enables the $(1-1/e)^2$-approximation for general monotone submodular functions and circumvents some of the impossibilities associated with non-adaptive policies. We also introduce a fundamental problem in submodular optimization that may be of independent interest: given a ground set of elements where every element appears with some small probability, find a set of expected size at most $k$ that has the highest expected value over the realization of the elements. We show a surprising result: there are classes of monotone submodular functions (including coverage) that can be approximated almost optimally as the probability vanishes. For general monotone submodular functions we show via a reduction from \textsc{Planted-Clique} that approximations for this problem are not likely to be obtainable. This optimization problem is an important tool for adaptive seeding via non-adaptive policies, and its hardness motivates the introduction of \emph{locally-adaptive} policies we use in the main result.

Machine learning relies on the assumption that unseen test instances of a classification problem follow the same distribution as observed training data. However, this principle can break down when machine learning is used to make important decisions about the welfare (employment, education, health) of strategic individuals. Knowing information about the classifier, such individuals may manipulate their attributes in order to obtain a better classification outcome. As a result of this behavior---often referred to as gaming---the performance of the classifier may deteriorate sharply. Indeed, gaming is a well-known obstacle for using machine learning methods in practice; in financial policy-making, the problem is widely known as Goodhart's law. In this paper, we formalize the problem, and pursue algorithms for learning classifiers that are robust to gaming. We model classification as a sequential game between a player named "Jury" and a player named "Contestant." Jury designs a classifier, and Contestant receives an input to the classifier, which he may change at some cost. Jury's goal is to achieve high classification accuracy with respect to Contestant's original input and some underlying target classification function. Contestant's goal is to achieve a favorable classification outcome while taking into account the cost of achieving it. For a natural class of cost functions, we obtain computationally efficient learning algorithms which are near-optimal. Surprisingly, our algorithms are efficient even on concept classes that are computationally hard to learn. For general cost functions, designing an approximately optimal strategy-proof classifier, for inverse-polynomial approximation, is NP-hard.

We conjecture that PPAD has a PCP-like complete problem, seeking a near equilibrium in which all but very few players have very little incentive to deviate. We show that, if one assumes that this problem requires exponential time, several open problems in this area are settled. The most important implication, proved via a "birthday repetition" reduction, is that the n^O(log n) approximation scheme of [LMM03] for the Nash equilibrium of two-player games is essentially optimum. Two other open problems in the area are resolved once one assumes this conjecture, establishing that certain approximate equilibria are PPAD-complete: Finding a relative approximation of two-player Nash equilibria (without the well-supported restriction of [Das13]), and an approximate competitive equilibrium with equal incomes [Bud11] with small clearing error and near-optimal Gini coefficient., Comment: Revision 2 derandomizes the main reduction

We propose a primitive called PJOIN, for "predictive join," which combines and extends the operations JOIN and LINK, which Valiant proposed as the basis of a computational theory of cortex. We show that PJOIN can be implemented in Valiant's model. We also show that, using PJOIN, certain reasonably complex learning and pattern matching tasks can be performed, in a way that involves phenomena which have been observed in cognition and the brain, namely memory-based prediction and downward traffic in the cortical hierarchy.

We propose an optimum mechanism for providing monetary incentives to the data sources of a statistical estimator such as linear regression, so that high quality data is provided at low cost, in the sense that the sum of payments and estimation error is minimized. The mechanism applies to a broad range of estimators, including linear and polynomial regression, kernel regression, and, under some additional assumptions, ridge regression. It also generalizes to several objectives, including minimizing estimation error subject to budget constraints. Besides our concrete results for regression problems, we contribute a mechanism design framework through which to design and analyze statistical estimators whose examples are supplied by workers with cost for labeling said examples.

We introduce a dynamic mechanism design problem in which the designer wants to offer for sale an item to an agent, and another item to the same agent at some point in the future. The agent's joint distribution of valuations for the two items is known, and the agent knows the valuation for the current item (but not for the one in the future). The designer seeks to maximize expected revenue, and the auction must be deterministic, truthful, and ex post individually rational. The optimum mechanism involves a protocol whereby the seller elicits the buyer's current valuation, and based on the bid makes two take-it-or-leave-it offers, one for now and one for the future. We show that finding the optimum deterministic mechanism in this situation - arguably the simplest meaningful dynamic mechanism design problem imaginable - is NP-hard. We also prove several positive results, among them a polynomial linear programming-based algorithm for the optimum randomized auction (even for many bidders and periods), and we show strong separations in revenue between non-adaptive, adaptive, and randomized auctions, even when the valuations in the two periods are uncorrelated. Finally, for the same problem in an environment in which contracts cannot be enforced, and thus perfection of equilibrium is necessary, we show that the optimum randomized mechanism requires multiple rounds of cheap talk-like interactions., Comment: Previos versions appeared in SODA'16 and GEB. This version includes the limited commitment model which appeared in the conference proceedings but ommitted from the journal paper

We show that there are simplex pivoting rules for which it is PSPACE-complete to tell if a particular basis will appear on the algorithm's path. Such rules cannot be the basis of a strongly polynomial algorithm, unless P = PSPACE. We conjecture that the same can be shown for most known variants of the simplex method. However, we also point out that Dantzig's shadow vertex algorithm has a polynomial path problem. Finally, we discuss in the same context randomized pivoting rules., Comment: To appear in IPCO 2014

Competitive equilibrium with equal incomes (CEEI) is a well known fair allocation mechanism; however, for indivisible resources a CEEI may not exist. It was shown in [Budish '11] that in the case of indivisible resources there is always an allocation, called A-CEEI, that is approximately fair, approximately truthful, and approximately efficient, for some favorable approximation parameters. This approximation is used in practice to assign students to classes. In this paper we show that finding the A-CEEI allocation guaranteed to exist by Budish's theorem is PPAD-complete. We further show that finding an approximate equilibrium with better approximation guarantees is even harder: NP-complete., Comment: Appeared in EC 2014

We show that, if truth assignments on $n$ variables reproduce through recombination so that satisfaction of a particular Boolean function confers a small evolutionary advantage, then a polynomially large population over polynomially many generations (polynomial in $n$ and the inverse of the initial satisfaction probability) will end up almost certainly consisting exclusively of satisfying truth assignments. We argue that this theorem sheds light on the problem of novelty in Evolution.

For all $n, \epsilon >0$, we show that the set of Poisson Binomial distributions on $n$ variables admits a proper $\epsilon$-cover in total variation distance of size $n^2+n \cdot (1/\epsilon)^{O(\log^2 (1/\epsilon))}$, which can also be computed in polynomial time. We discuss the implications of our construction for approximation algorithms and the computation of approximate Nash equilibria in anonymous games., Comment: PTRF, to appear

To help a user specify and verify quantified queries --- a class of database queries known to be very challenging for all but the most expert users --- one can question the user on whether certain data objects are answers or non-answers to her intended query. In this paper, we analyze the number of questions needed to learn or verify qhorn queries, a special class of Boolean quantified queries whose underlying form is conjunctions of quantified Horn expressions. We provide optimal polynomial-question and polynomial-time learning and verification algorithms for two subclasses of the class qhorn with upper constant limits on a query's causal density., Comment: Extended Version of PODS 2013 paper