False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al. [ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time", and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely

Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients

Discovering the underlying mathematical expressions describing a dataset is a core challenge for artificial intelligence. This is the problem of $\textit{symbolic regression}$. Despite recent advances in training neural networks to solve complex tasks, deep learning approaches to symbolic regression are underexplored. We propose a framework that leverages deep learning for symbolic regression via a simple idea: use a large model to search the space of small models. Specifically, we use a recurrent neural network to emit a distribution over tractable mathematical expressions and employ a novel risk-seeking policy gradient to train the network to generate better-fitting expressions. Our algorithm outperforms several baseline methods (including Eureqa, the gold standard for symbolic regression) in its ability to exactly recover symbolic expressions on a series of benchmark problems, both with and without added noise. More broadly, our contributions include a framework that can be applied to optimize hierarchical, variable-length objects under a black-box performance metric, with the ability to incorporate constraints in situ, and a risk-seeking policy gradient formulation that optimizes for best-case performance instead of expected performance.Comment: Published at International Conference on Learning Representations, 202

Improving exploration in policy gradient search: Application to symbolic optimization

Many machine learning strategies designed to automate mathematical tasks leverage neural networks to search large combinatorial spaces of mathematical symbols. In contrast to traditional evolutionary approaches, using a neural network at the core of the search allows learning higher-level symbolic patterns, providing an informed direction to guide the search. When no labeled data is available, such networks can still be trained using reinforcement learning. However, we demonstrate that this approach can suffer from an early commitment phenomenon and from initialization bias, both of which limit exploration. We present two exploration methods to tackle these issues, building upon ideas of entropy regularization and distribution initialization. We show that these techniques can improve the performance, increase sample efficiency, and lower the complexity of solutions for the task of symbolic regression.Comment: Published in 1st Mathematical Reasoning in General Artificial Intelligence Workshop, ICLR 202

Model Checking CTL is Almost Always Inherently Sequential

The model checking problem for CTL is known to be P-complete (Clarke, Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of CTL obtained by restricting the use of temporal modalities or the use of negations---restrictions already studied for LTL by Sistla and Clarke (1985) and Markey (2004). For all these fragments, except for the trivial case without any temporal operator, we systematically prove model checking to be either inherently sequential (P-complete) or very efficiently parallelizable (LOGCFL-complete). For most fragments, however, model checking for CTL is already P-complete. Hence our results indicate that, in cases where the combined complexity is of relevance, approaching CTL model checking by parallelism cannot be expected to result in any significant speedup. We also completely determine the complexity of the model checking problem for all fragments of the extensions ECTL, CTL+, and ECTL+

Model Cortical Association Fields Account for the Time Course and Dependence on Target Complexity of Human Contour Perception

Can lateral connectivity in the primary visual cortex account for the time dependence and intrinsic task difficulty of human contour detection? To answer this question, we created a synthetic image set that prevents sole reliance on either low-level visual features or high-level context for the detection of target objects. Rendered images consist of smoothly varying, globally aligned contour fragments (amoebas) distributed among groups of randomly rotated fragments (clutter). The time course and accuracy of amoeba detection by humans was measured using a two-alternative forced choice protocol with self-reported confidence and variable image presentation time (20-200 ms), followed by an image mask optimized so as to interrupt visual processing. Measured psychometric functions were well fit by sigmoidal functions with exponential time constants of 30-91 ms, depending on amoeba complexity. Key aspects of the psychophysical experiments were accounted for by a computational network model, in which simulated responses across retinotopic arrays of orientation-selective elements were modulated by cortical association fields, represented as multiplicative kernels computed from the differences in pairwise edge statistics between target and distractor images. Comparing the experimental and the computational results suggests that each iteration of the lateral interactions takes at least ms of cortical processing time. Our results provide evidence that cortical association fields between orientation selective elements in early visual areas can account for important temporal and task-dependent aspects of the psychometric curves characterizing human contour perception, with the remaining discrepancies postulated to arise from the influence of higher cortical areas

Evenness mediates the global relationship between forest productivity and richness

1. Biodiversity is an important component of natural ecosystems, with higher species richness often correlating with an increase in ecosystem productivity. Yet, this relationship varies substantially across environments, typically becoming less pronounced at high levels of species richness. However, species richness alone cannot reflect all important properties of a community, including community evenness, which may mediate the relationship between biodiversity and productivity. If the evenness of a community correlates negatively with richness across forests globally, then a greater number of species may not always increase overall diversity and productivity of the system. Theoretical work and local empirical studies have shown that the effect of evenness on ecosystem functioning may be especially strong at high richness levels, yet the consistency of this remains untested at a global scale. 2. Here, we used a dataset of forests from across the globe, which includes composition, biomass accumulation and net primary productivity, to explore whether productivity correlates with community evenness and richness in a way that evenness appears to buffer the effect of richness. Specifically, we evaluated whether low levels of evenness in speciose communities correlate with the attenuation of the richness–productivity relationship. 3. We found that tree species richness and evenness are negatively correlated across forests globally, with highly speciose forests typically comprising a few dominant and many rare species. Furthermore, we found that the correlation between diversity and productivity changes with evenness: at low richness, uneven communities are more productive, while at high richness, even communities are more productive. 4. Synthesis. Collectively, these results demonstrate that evenness is an integral component of the relationship between biodiversity and productivity, and that the attenuating effect of richness on forest productivity might be partly explained by low evenness in speciose communities. Productivity generally increases with species richness, until reduced evenness limits the overall increases in community diversity. Our research suggests that evenness is a fundamental component of biodiversity–ecosystem function relationships, and is of critical importance for guiding conservation and sustainable ecosystem management decisions

Cortical Surround Interactions and Perceptual Salience via Natural Scene Statistics

Spatial context in images induces perceptual phenomena associated with salience and modulates the responses of neurons in primary visual cortex (V1). However, the computational and ecological principles underlying contextual effects are incompletely understood. We introduce a model of natural images that includes grouping and segmentation of neighboring features based on their joint statistics, and we interpret the firing rates of V1 neurons as performing optimal recognition in this model. We show that this leads to a substantial generalization of divisive normalization, a computation that is ubiquitous in many neural areas and systems. A main novelty in our model is that the influence of the context on a target stimulus is determined by their degree of statistical dependence. We optimized the parameters of the model on natural image patches, and then simulated neural and perceptual responses on stimuli used in classical experiments. The model reproduces some rich and complex response patterns observed in V1, such as the contrast dependence, orientation tuning and spatial asymmetry of surround suppression, while also allowing for surround facilitation under conditions of weak stimulation. It also mimics the perceptual salience produced by simple displays, and leads to readily testable predictions. Our results provide a principled account of orientation-based contextual modulation in early vision and its sensitivity to the homogeneity and spatial arrangement of inputs, and lends statistical support to the theory that V1 computes visual salience

Climatic controls of decomposition drive the global biogeography of forest-tree symbioses

The identity of the dominant root-associated microbial symbionts in a forest determines the ability of trees to access limiting nutrients from atmospheric or soil pools1,2, sequester carbon3,4 and withstand the effects of climate change5,6. Characterizing the global distribution of these symbioses and identifying the factors that control this distribution are thus integral to understanding the present and future functioning of forest ecosystems. Here we generate a spatially explicit global map of the symbiotic status of forests, using a database of over 1.1 million forest inventory plots that collectively contain over 28,000 tree species. Our analyses indicate that climate variables—in particular, climatically controlled variation in the rate of decomposition—are the primary drivers of the global distribution of major symbioses. We estimate that ectomycorrhizal trees, which represent only 2% of all plant species7, constitute approximately 60% of tree stems on Earth. Ectomycorrhizal symbiosis dominates forests in which seasonally cold and dry climates inhibit decomposition, and is the predominant form of symbiosis at high latitudes and elevation. By contrast, arbuscular mycorrhizal trees dominate in aseasonal, warm tropical forests, and occur with ectomycorrhizal trees in temperate biomes in which seasonally warm-and-wet climates enhance decomposition. Continental transitions between forests dominated by ectomycorrhizal or arbuscular mycorrhizal trees occur relatively abruptly along climate-driven decomposition gradients; these transitions are probably caused by positive feedback effects between plants and microorganisms. Symbiotic nitrogen fixers—which are insensitive to climatic controls on decomposition (compared with mycorrhizal fungi)—are most abundant in arid biomes with alkaline soils and high maximum temperatures. The climatically driven global symbiosis gradient that we document provides a spatially explicit quantitative understanding of microbial symbioses at the global scale, and demonstrates the critical role of microbial mutualisms in shaping the distribution of plant species

Evenness mediates the global relationship between forest productivity and richness

1. Biodiversity is an important component of natural ecosystems, with higher species richness often correlating with an increase in ecosystem productivity. Yet, this relationship varies substantially across environments, typically becoming less pronounced at high levels of species richness. However, species richness alone cannot reflect all important properties of a community, including community evenness, which may mediate the relationship between biodiversity and productivity. If the evenness of a community correlates negatively with richness across forests globally, then a greater number of species may not always increase overall diversity and productivity of the system. Theoretical work and local empirical studies have shown that the effect of evenness on ecosystem functioning may be especially strong at high richness levels, yet the consistency of this remains untested at a global scale. 2. Here, we used a dataset of forests from across the globe, which includes composition, biomass accumulation and net primary productivity, to explore whether productivity correlates with community evenness and richness in a way that evenness appears to buffer the effect of richness. Specifically, we evaluated whether low levels of evenness in speciose communities correlate with the attenuation of the richness–productivity relationship. 3. We found that tree species richness and evenness are negatively correlated across forests globally, with highly speciose forests typically comprising a few dominant and many rare species. Furthermore, we found that the correlation between diversity and productivity changes with evenness: at low richness, uneven communities are more productive, while at high richness, even communities are more productive. 4. Synthesis. Collectively, these results demonstrate that evenness is an integral component of the relationship between biodiversity and productivity, and that the attenuating effect of richness on forest productivity might be partly explained by low evenness in speciose communities. Productivity generally increases with species richness, until reduced evenness limits the overall increases in community diversity. Our research suggests that evenness is a fundamental component of biodiversity–ecosystem function relationships, and is of critical importance for guiding conservation and sustainable ecosystem management decisions