1,721,141 research outputs found
Critical Behavior in Physics and Probabilistic Formal Languages
No full textWe show that the mutual information between two symbols, as a function of the number of symbols between the two, decays exponentially in any probabilistic regular grammar, but can decay like a power law for a context-free grammar. This result about formal languages is closely related to a well-known result in classical statistical mechanics that there are no phase transitions in dimensions fewer than two. It is also related to the emergence of power law correlations in turbulence and cosmological inflation through recursive generative processes. We elucidate these physics connections and comment on potential applications of our results to machine learning tasks like training artificial recurrent neural networks. Along the way, we introduce a useful quantity, which we dub the rational mutual information, and discuss generalizations of our claims involving more complicated Bayesian networks. Keywords: formal languages; statistical mechanics; criticalit
Consciousness as a state of matter
No full textWe examine the hypothesis that consciousness can be understood as a state of matter, “perceptronium”, with distinctive information processing abilities. We explore four basic principles that may distinguish conscious matter from other physical systems such as solids, liquids and gases: the information, integration, independence and dynamics principles. If such principles can identify conscious entities, then they can help solve the quantum factorization problem: why do conscious observers like us perceive the particular Hilbert space factorization corresponding to classical space (rather than Fourier space, say), and more generally, why do we perceive the world around us as a dynamic hierarchy of objects that are strongly integrated and relatively independent? Tensor factorization of matrices is found to play a central role, and our technical results include a theorem about Hamiltonian separability (defined using Hilbert–Schmidt superoperators) being maximized in the energy eigenbasis. Our approach generalizes Giulio Tononi’s integrated information framework for neural-network-based consciousness to arbitrary quantum systems, and we find interesting links to error-correcting codes, condensed matter criticality, and the Quantum Darwinism program, as well as an interesting connection between the emergence of consciousness and the emergence of time.National Science Foundation (U.S.) (AST-090884)National Science Foundation (U.S.) (AST-1105835
How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem
No full textWe analyze cosmology assuming unitary quantum mechanics, using a tripartite partition into system,
observer, and environment degrees of freedom. This generalizes the second law of thermodynamics to
‘‘The system’s entropy cannot decrease unless it interacts with the observer, and it cannot increase unless
it interacts with the environment.’’ The former follows from the quantum Bayes theorem we derive. We
show that because of the long-range entanglement created by cosmological inflation, the cosmic entropy
decreases exponentially rather than linearly with the number of bits of information observed, so that a
given observer can reduce entropy by much more than the amount of information her brain can store.
Indeed, we argue that as long as inflation has occurred in a non-negligible fraction of the volume, almost
all sentient observers will find themselves in a post-inflationary low-entropy Hubble volume, and we
humans have no reason to be surprised that we do so as well, which solves the so-called inflationary
entropy problem. An arguably worse problem for unitary cosmology involves gamma-ray-burst constraints
on the ‘‘big snap,’’ a fourth cosmic doomsday scenario alongside the ‘‘big crunch,’’ ‘‘big chill,’’
and ‘‘big rip,’’ where an increasingly granular nature of expanding space modifies our life-supporting laws
of physics. Our tripartite framework also clarifies when the popular quantum gravity approximation
G
8 GhT
i is valid, and how problems with recent attempts to explain dark energy as gravitational
backreaction from superhorizon scale fluctuations can be understood as a failure of this approximation.National Science Foundation (U.S.) (grant nos. AST-0708534, AST-090884 and AST-1105835
Improved Measures of Integrated Information
Get PDFAlthough there is growing interest in measuring integrated information in computational and cognitive systems, current methods for doing so in practice are computationally unfeasible. Existing and novel integration measures are investigated and classified by various desirable properties. A simple taxonomy of Φ-measures is presented where they are each characterized by their choice of factorization method (5 options), choice of probability distributions to compare (3 × 4 options) and choice of measure for comparing probability distributions (7 options). When requiring the Φ-measures to satisfy a minimum of attractive properties, these hundreds of options reduce to a mere handful, some of which turn out to be identical. Useful exact and approximate formulas are derived that can be applied to real-world data from laboratory experiments without posing unreasonable computational demands.United States. Army Research Office (Grant W911NF-15-1-0300
Ensemble inhibition and excitation in the human cortex: An Ising-model analysis with uncertainties
Get PDFThe pairwise maximum entropy model, also known as the Ising model, has been widely used to analyze the collective activity of neurons. However, controversy persists in the literature about seemingly inconsistent findings, whose significance is unclear due to lack of reliable error estimates. We therefore develop a method for accurately estimating parameter uncertainty based on random walks in parameter space using adaptive Markov-chain Monte Carlo after the convergence of the main optimization algorithm. We apply our method to the activity patterns of excitatory and inhibitory neurons recorded with multielectrode arrays in the human temporal cortex during the wake-sleep cycle. Our analysis shows that the Ising model captures neuronal collective behavior much better than the independent model during wakefulness, light sleep, and deep sleep when both excitatory (E) and inhibitory (I) neurons are modeled; ignoring the inhibitory effects of I neurons dramatically overestimates synchrony among E neurons. Furthermore, information-theoretic measures reveal that the Ising model explains about 80–95% of the correlations, depending on sleep state and neuron type. Thermodynamic measures show signatures of criticality, although we take this with a grain of salt as it may be merely a reflection of long-range neural correlations
Omniscopes: Large area telescope arrays with only N logN computational cost
Get PDFWe show that the class of antenna layouts for telescope arrays allowing cheap analysis hardware (with correlator cost scaling as Nlog N rather than N[superscript 2] with the number of antennas N) is encouragingly large, including not only previously discussed rectangular grids but also arbitrary hierarchies of such grids, with arbitrary rotations and shears at each level. We show that all correlations for such a 2D array with an n-level hierarchy can be efficiently computed via a fast Fourier transform in not two but 2n dimensions. This can allow major correlator cost reductions for science applications requiring exquisite sensitivity at widely separated angular scales, for example, 21 cm tomography (where short baselines are needed to probe the cosmological signal and long baselines are needed for point source removal), helping enable future 21 cm experiments with thousands or millions of cheap dipolelike antennas. Such hierarchical grids combine the angular resolution advantage of traditional array layouts with the cost advantage of a rectangular fast Fourier transform telescope. We also describe an algorithm for how a subclass of hierarchical arrays can efficiently use rotation synthesis to produce global sky maps with minimal noise and a well-characterized synthesized beam.United States. National Aeronautics and Space Administration (Grant NAG5-11099)United States. National Aeronautics and Space Administration (Grant NNG 05G40G)National Science Foundation (U.S.) (Grant AST-0607597)National Science Foundation (U.S.) (Grant AST-0708534)National Science Foundation (U.S.) (Grant AST-0908848)National Science Foundation (U.S.) (Grant PHY0855425)David & Lucile Packard Foundatio
Sharpening the second law of thermodynamics with the quantum Bayes theorem
Get PDFWe prove a generalization of the classic Groenewold-Lindblad entropy inequality, combining decoherence and the quantum Bayes theorem into a simple unified picture where decoherence increases entropy while observation decreases it. This provides a rigorous quantum-mechanical version of the second law of thermodynamics, governing how the entropy of a system (the entropy of its density matrix, partial-traced over the environment and conditioned on what is known) evolves under general decoherence and observation. The powerful tool of spectral majorization enables both simple alternative proofs of the classic Lindblad and Holevo inequalities without using strong subadditivity, and also novel inequalities for decoherence and observation that hold not only for von Neumann entropy, but also for arbitrary concave entropies.National Science Foundation (U.S.) (Grant AST-1105835)Massachusetts Institute of Technology. Undergraduate Research Opportunities Progra
Fast Fourier transform telescope
No full textWe propose an all-digital telescope for 21 cm tomography, which combines key advantages of both
single dishes and interferometers. The electric field is digitized by antennas on a rectangular grid, after
which a series of fast Fourier transforms recovers simultaneous multifrequency images of up to half the
sky. Thanks to Moore’s law, the bandwidth up to which this is feasible has now reached about 1 GHz, and
will likely continue doubling every couple of years. The main advantages over a single dish telescope are
cost and orders of magnitude larger field-of-view, translating into dramatically better sensitivity for largearea
surveys. The key advantages over traditional interferometers are cost (the correlator computational
cost for an N-element array scales as Nlog[subscript 2]N rather than N[superscript 2]) and a compact synthesized beam. We argue
that 21 cm tomography could be an ideal first application of a very large fast Fourier transform telescope,
which would provide both massive sensitivity improvements per dollar and mitigate the off-beam point
source foreground problem with its clean beam. Another potentially interesting application is cosmic
microwave background polarization.David and Lucile Packard FoundationJohn Templeton foundationNational Science FoundationNAS
Testing multifield inflation: A geometric approach
Get PDFWe develop an approach for linking the power spectra, bispectrum, and trispectrum to the geometric and kinematical features of multifield inflationary Lagrangians. Our geometric approach can also be useful in determining when a complicated multifield model can be well approximated by a model with one, two, or a handful of fields. To arrive at these results, we focus on the mode interactions in the kinematical basis, starting with the case of no sourcing and showing that there is a series of mode conservation laws analogous to the conservation law for the adiabatic mode in single-field inflation. We then treat the special case of a quadratic potential with canonical kinetic terms, showing that it produces a series of mode sourcing relations identical in form to that for the adiabatic mode. We build on this result to show that the mode sourcing relations for general multifield inflation are an extension of this special case but contain higher-order covariant derivatives of the potential and corrections from the field metric. In parallel, we show how the mode interactions depend on the geometry of the inflationary Lagrangian and on the kinematics of the associated field trajectory. Finally, we consider how the mode interactions and effective number of fields active during inflation are reflected in the spectra and introduce a multifield consistency relation, as well as a multifield observable β[subscript 2] that can potentially distinguish two-field scenarios from scenarios involving three or more effective fields.National Science Foundation (U.S.) (Grant AST-0708534)National Science Foundation (U.S.) (Grant AST-0908848)David & Lucile Packard Foundatio
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