1,529 research outputs found
The Synthesis and Analysis of Stochastic Switching Circuits
Stochastic switching circuits are relay circuits that consist of stochastic
switches called pswitches. The study of stochastic switching circuits has
widespread applications in many fields of computer science, neuroscience, and
biochemistry. In this paper, we discuss several properties of stochastic
switching circuits, including robustness, expressibility, and probability
approximation.
First, we study the robustness, namely, the effect caused by introducing an
error of size \epsilon to each pswitch in a stochastic circuit. We analyze two
constructions and prove that simple series-parallel circuits are robust to
small error perturbations, while general series-parallel circuits are not.
Specifically, the total error introduced by perturbations of size less than
\epsilon is bounded by a constant multiple of \epsilon in a simple
series-parallel circuit, independent of the size of the circuit.
Next, we study the expressibility of stochastic switching circuits: Given an
integer q and a pswitch set S=\{\frac{1}{q},\frac{2}{q},...,\frac{q-1}{q}\},
can we synthesize any rational probability with denominator q^n (for arbitrary
n) with a simple series-parallel stochastic switching circuit? We generalize
previous results and prove that when q is a multiple of 2 or 3, the answer is
yes. We also show that when q is a prime number larger than 3, the answer is
no.
Probability approximation is studied for a general case of an arbitrary
pswitch set S=\{s_1,s_2,...,s_{|S|}\}. In this case, we propose an algorithm
based on local optimization to approximate any desired probability. The
analysis reveals that the approximation error of a switching circuit decreases
exponentially with an increasing circuit size.Comment: 2 columns, 15 page
The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies
Boolean satisfiability problems are an important benchmark for questions
about complexity, algorithms, heuristics and threshold phenomena. Recent work
on heuristics, and the satisfiability threshold has centered around the
structure and connectivity of the solution space. Motivated by this work, we
study structural and connectivity-related properties of the space of solutions
of Boolean satisfiability problems and establish various dichotomies in
Schaefer's framework.
On the structural side, we obtain dichotomies for the kinds of subgraphs of
the hypercube that can be induced by the solutions of Boolean formulas, as well
as for the diameter of the connected components of the solution space. On the
computational side, we establish dichotomy theorems for the complexity of the
connectivity and st-connectivity questions for the graph of solutions of
Boolean formulas. Our results assert that the intractable side of the
computational dichotomies is PSPACE-complete, while the tractable side - which
includes but is not limited to all problems with polynomial time algorithms for
satisfiability - is in P for the st-connectivity question, and in coNP for the
connectivity question. The diameter of components can be exponential for the
PSPACE-complete cases, whereas in all other cases it is linear; thus, small
diameter and tractability of the connectivity problems are remarkably aligned.
The crux of our results is an expressibility theorem showing that in the
tractable cases, the subgraphs induced by the solution space possess certain
good structural properties, whereas in the intractable cases, the subgraphs can
be arbitrary
Modality and expressibility
When embedding data are used to argue against semantic theory A and in favor of semantic theory B, it is important to ask whether A could make sense of those data. It is possible to ask that question on a case-by-case basis. But suppose we could show that A can make sense of all the embedding data which B can possibly make sense of. This would, on the one hand, undermine arguments in favor of B over A on the basis of embedding data. And, provided that the converse does not hold—that is, that A can make sense of strictly more embedding data than B can—it would also show that there is a precise sense in which B is more constrained than A, yielding a pro tanto simplicity-based consideration in favor of B. In this paper I develop tools which allow us to make comparisons of this kind, which I call comparisons of potential expressive power. I motivate the development of these tools by way of exploration of the recent debate about epistemic modals. Prominent theories which have been developed in response to embedding data turn out to be strictly less expressive than the standard relational theory, a fact which necessitates a reorientation in how to think about the choice between these theories
Unsupervised Generative Modeling Using Matrix Product States
Generative modeling, which learns joint probability distribution from data
and generates samples according to it, is an important task in machine learning
and artificial intelligence. Inspired by probabilistic interpretation of
quantum physics, we propose a generative model using matrix product states,
which is a tensor network originally proposed for describing (particularly
one-dimensional) entangled quantum states. Our model enjoys efficient learning
analogous to the density matrix renormalization group method, which allows
dynamically adjusting dimensions of the tensors and offers an efficient direct
sampling approach for generative tasks. We apply our method to generative
modeling of several standard datasets including the Bars and Stripes, random
binary patterns and the MNIST handwritten digits to illustrate the abilities,
features and drawbacks of our model over popular generative models such as
Hopfield model, Boltzmann machines and generative adversarial networks. Our
work sheds light on many interesting directions of future exploration on the
development of quantum-inspired algorithms for unsupervised machine learning,
which are promisingly possible to be realized on quantum devices.Comment: 11 pages, 12 figures (not including the TNs) GitHub Page:
https://congzlwag.github.io/UnsupGenModbyMPS
Approximating Holant problems by winding
We give an FPRAS for Holant problems with parity constraints and
not-all-equal constraints, a generalisation of the problem of counting
sink-free-orientations. The approach combines a sampler for near-assignments of
"windable" functions -- using the cycle-unwinding canonical paths technique of
Jerrum and Sinclair -- with a bound on the weight of near-assignments. The
proof generalises to a larger class of Holant problems; we characterise this
class and show that it cannot be extended by expressibility reductions.
We then ask whether windability is equivalent to expressibility by matchings
circuits (an analogue of matchgates), and give a positive answer for functions
of arity three
An Exploratory Study of Forces and Frictions affecting Large-Scale Model-Driven Development
In this paper, we investigate model-driven engineering, reporting on an
exploratory case-study conducted at a large automotive company. The study
consisted of interviews with 20 engineers and managers working in different
roles. We found that, in the context of a large organization, contextual forces
dominate the cognitive issues of using model-driven technology. The four forces
we identified that are likely independent of the particular abstractions chosen
as the basis of software development are the need for diffing in software
product lines, the needs for problem-specific languages and types, the need for
live modeling in exploratory activities, and the need for point-to-point
traceability between artifacts. We also identified triggers of accidental
complexity, which we refer to as points of friction introduced by languages and
tools. Examples of the friction points identified are insufficient support for
model diffing, point-to-point traceability, and model changes at runtime.Comment: To appear in proceedings of MODELS 2012, LNCS Springe
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