27 research outputs found
Compact DSOP and partial DSOP Forms
Given a Boolean function f on n variables, a Disjoint Sum-of-Products (DSOP)
of f is a set of products (ANDs) of subsets of literals whose sum (OR) equals
f, such that no two products cover the same minterm of f. DSOP forms are a
special instance of partial DSOPs, i.e. the general case where a subset of
minterms must be covered exactly once and the other minterms (typically
corresponding to don't care conditions of ) can be covered any number of
times. We discuss finding DSOPs and partial DSOP with a minimal number of
products, a problem theoretically connected with various properties of Boolean
functions and practically relevant in the synthesis of digital circuits.
Finding an absolute minimum is hard, in fact we prove that the problem of
absolute minimization of partial DSOPs is NP-hard. Therefore it is crucial to
devise a polynomial time heuristic that compares favorably with the known
minimization tools. To this end we develop a further piece of theory starting
from the definition of the weight of a product p as a functions of the number
of fragments induced on other cubes by the selection of p, and show how product
weights can be exploited for building a class of minimization heuristics for
DSOP and partial DSOP synthesis. A set of experiments conducted on major
benchmark functions show that our method, with a family of variants, always
generates better results than the ones of previous heuristics, including the
method based on a BDD representation of f
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Capturing patterns of spatial and temporal autocorrelation in ordered response data : a case study of land use and air quality changes in Austin, Texas
textMany databases involve ordered discrete responses in a temporal and spatial context, including, for example, land development intensity levels, vehicle ownership, and pavement conditions. An appreciation of such behaviors requires rigorous statistical methods, recognizing spatial effects and dynamic processes. This dissertation develops a dynamic spatial ordered probit (DSOP) model in order to capture patterns of spatial and temporal autocorrelation in ordered categorical response data. This model is estimated in a Bayesian framework using Gibbs sampling and data augmentation, in order to generate all autocorrelated latent variables. The specifications, methodologies, and applications undertaken here advance the field of spatial econometrics while enhancing our understanding of land use and air quality changes. The proposed DSOP model incorporates spatial effects in an ordered probit model by allowing for inter-regional spatial interactions and heteroskedasticity, along with random effects across regions (where "region" describes any cluster of observational units). The model assumes an autoregressive, AR(1), process across latent response values, thereby recognizing time-series dynamics in panel data sets. The model code and estimation approach is first tested on simulated data sets, in order to reproduce known parameter values and provide insights into estimation performance. Root mean squared errors (RMSE) are used to evaluate the accuracy of estimates, and the deviance information criterion (DIC) is used for model comparisons. It is found that the DSOP model yields much more accurate estimates than standard, non-spatial techniques. As for model selection, even considering the penalty for using more parameters, the DSOP model is clearly preferred to standard OP, dynamic OP and spatial OP models. The model and methods are then used to analyze both land use and air quality (ozone) dynamics in Austin, Texas. In analyzing Austin's land use intensity patterns over a 4-point panel, the observational units are 300 m Γ 300 m grid cells derived from satellite images (at 30 m resolution). The sample contains 2,771 such grid cells, spread among 57 clusters (zip code regions), covering about 10% of the overall study area. In this analysis, temporal and spatial autocorrelation effects are found to be significantly positive. In addition, increases in travel times to the region's central business district (CBD) are estimated to substantially reduce land development intensity. The observational units for the ozone variation analysis are 4 km Γ 4 km grid cells, and all 132 observations falling in the study area are used. While variations in ozone concentration levels are found to exhibit strong patterns of temporal autocorrelation, they appear strikingly random in a spatial context (after controlling for local land cover, transportation, and temperature conditions). While transportation and land cover conditions appear to influence ozone levels, their effects are not as instantaneous, nor as practically significant as the impact of temperature. The proposed and tested DSOP model is felt to be a significant contribution to the field of spatial econometrics, where binary applications (for discrete response data) have been seen as the cutting edge. The Bayesian framework and Gibbs sampling techniques used here permit such complexity, in world of two-dimensional autocorrelation.Civil, Architectural, and Environmental Engineerin
Differentiable Quantum Programming with Unbounded Loops
The emergence of variational quantum applications has led to the development
of automatic differentiation techniques in quantum computing. Recently, Zhu et
al. (PLDI 2020) have formulated differentiable quantum programming with bounded
loops, providing a framework for scalable gradient calculation by quantum means
for training quantum variational applications. However, promising parameterized
quantum applications, e.g., quantum walk and unitary implementation, cannot be
trained in the existing framework due to the natural involvement of unbounded
loops. To fill in the gap, we provide the first differentiable quantum
programming framework with unbounded loops, including a newly designed
differentiation rule, code transformation, and their correctness proof.
Technically, we introduce a randomized estimator for derivatives to deal with
the infinite sum in the differentiation of unbounded loops, whose applicability
in classical and probabilistic programming is also discussed. We implement our
framework with Python and Q#, and demonstrate a reasonable sample efficiency.
Through extensive case studies, we showcase an exciting application of our
framework in automatically identifying close-to-optimal parameters for several
parameterized quantum applications.Comment: Codes are available at https://github.com/njuwfang/DifferentiableQP
Synthesis of Linear Reversible Circuits and EXOR-AND-based Circuits for Incompletely Specified Multi-Output Functions
At this time the synthesis of reversible circuits for quantum computing is an active area of research. In the most restrictive quantum computing models there are no ancilla lines and the quantum cost, or latency, of performing a reversible form of the AND gate, or Toffoli gate, increases exponentially with the number of input variables. In contrast, the quantum cost of performing any combination of reversible EXOR gates, or CNOT gates, on n input variables requires at most O(n2/log2n) gates. It was under these conditions that EXOR-AND-EXOR, or EPOE, synthesis was developed.
In this work, the GF(2) logic theory used in EPOE is expanded and the concept of an EXOR-AND product transform is introduced. Because of the generality of this logic theory, it is adapted to EXOR-AND-OR, or SPOE, synthesis. Three heuristic spectral logic synthesis algorithms are introduced, implemented in a program called XAX, and compared with previous work in classical logic circuits of up to 26 inputs. Three linear reversible circuit methods are also introduced and compared with previous work in linear reversible logic circuits of up to 100 inputs
Group gradings on classical lie superalgebras
Assuming the base field is algebraically closed, we classify, up to isomorphism, gradings
by arbitrary groups on non-exceptional classical simple Lie superalgebras, excluding
those of type A(1, 1), and on finite dimensional superinvolution-simple associative
superalgebras. We assume the characteristic to be 0 in the Lie case, and different from 2
in the associative case. Our approach is based on a version of Wedderburn Theorem for
graded-simple associative superalgebras satisfying a descending chain condition, which
allows us to classify superinvolutions using nondegenerate supersymmetric sesquilinear
forms on graded modules over a graded-division superalgebra. To transfer the results
from the associative case to the Lie case, we use the duality between G-gradings and
b G-actions for finite dimensional universal algebras
Short periodic variations in the first-order semianalytical satellite theory.
Thesis. 1979. M.S.--Massachusetts Institute of Technology. Dept. of Aeronautics and Astronautics.MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERONAUTICS.Includes bibliographical references.M.S