An Unsolvable Problem of Elementary Number Theory

Church, Alonzo

doi:10.2307/2371045

Cited by 1,231 publications

(534 citation statements)

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“…A general approach (Schmidhuber, 1997) for both SL and RL seeks to compactly encode weights of large NNs (Schmidhuber, 1997) through programs written in a universal programming language (Gödel, 1931;Church, 1936;Turing, 1936;Post, 1936). Often it is much more efficient to systematically search the space of such programs with a bias towards short and fast programs (Levin, 1973b;Schmidhuber, 1997Schmidhuber, , 2004, instead of directly searching the huge space of possible NN weight matrices.…”

Section: Deep Hierarchical Rl (Hrl) and Subgoal Learning With Fnns Anmentioning

confidence: 99%

“…The most general type of RL is constrained only by the fundamental limitations of computability identified by the founders of theoretical computer science (Gödel, 1931;Church, 1936;Turing, 1936;Post, 1936). Remarkably, there exist blueprints of universal problem solvers or universal RL machines for unlimited problem depth that are time-optimal in various theoretical senses (Hutter, 2005(Hutter, , 2002Schmidhuber, 2002Schmidhuber, , 2006b).…”

Section: Universal Rlmentioning

confidence: 99%

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Deep learning in neural networks: An overview

2015

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In recent years, deep artificial neural networks (including recurrent ones) have won numerous contests in pattern recognition and machine learning. This historical survey compactly summarises relevant work, much of it from the previous millennium. Shallow and deep learners are distinguished by the depth of their credit assignment paths, which are chains of possibly learnable, causal links between actions and effects. I review deep supervised learning (also recapitulating the history of backpropagation), unsupervised learning, reinforcement learning & evolutionary computation, and indirect search for short programs encoding deep and large networks.LATEX source: http://www.idsia.ch/˜juergen/DeepLearning8Oct2014.tex Complete BIBTEX file (888 kB): http://www.idsia.ch/˜juergen/deep.bib Preface This is the preprint of an invited Deep Learning (DL) overview. One of its goals is to assign credit to those who contributed to the present state of the art. I acknowledge the limitations of attempting to achieve this goal. The DL research community itself may be viewed as a continually evolving, deep network of scientists who have influenced each other in complex ways. Starting from recent DL results, I tried to trace back the origins of relevant ideas through the past half century and beyond, sometimes using "local search" to follow citations of citations backwards in time. Since not all DL publications properly acknowledge earlier relevant work, additional global search strategies were employed, aided by consulting numerous neural network experts. As a result, the present preprint mostly consists of references. Nevertheless, through an expert selection bias I may have missed important work. A related bias was surely introduced by my special familiarity with the work of my own DL research group in the past quarter-century. For these reasons, this work should be viewed as merely a snapshot of an ongoing credit assignment process. To help improve it, please do not hesitate to send corrections and suggestions to juergen@idsia.ch.

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Section: Deep Hierarchical Rl (Hrl) and Subgoal Learning With Fnns Anmentioning

confidence: 99%

Section: Universal Rlmentioning

confidence: 99%

Deep learning in neural networks: An overview

2015

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“…Recursion has been argued to be a key human ability (Hauser, Chomsky, & Fitch, 2002) and is a core component of many computational systems (e.g. Church, 1936). L is the name of the function the learner is trying to infer-the one that maps sets to number words.…”

Section: Recursion (L S)mentioning

confidence: 99%

“…The representational system we use is lambda calculus, a formal language for compositional semantics (e.g. Heim & Kratzer, 1998;Steedman, 2001), computation more generally (Church, 1936), and other natural-language learning tasks (Zettlemoyer & Collins, 2007Piantadosi, Goodman, B.A., & J.B., 2008). The core inductive part of the model uses Bayesian statistics to formalize what inferences learners should make from data.…”

Section: Introductionmentioning

confidence: 99%

Bootstrapping in a language of thought: A formal model of numerical concept learning

Piantadosi¹,

Tenenbaum²,

2012

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In acquiring number words, children exhibit a qualitative leap in which they transition from understanding a few number words, to possessing a recursive system of interrelated numerical concepts. We present a computational framework for understanding this inductive leap as the consequence of statistical inference over a sufficiently powerful representational system. We provide an implemented model that is powerful enough to learn number word meanings and other related conceptual systems from naturalistic data. The model shows that bootstrapping can be made computationally and philosophically well-founded as a theory of number learning. Our approach demonstrates how learners may combine core cognitive operations to build sophisticated representations during the course of development, and how this process explains observed developmental patterns in number word learning.

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“…Accuracy conditions are also discussed. The complexity of even a simple measurement, as a task for a quantum robot, and the relation to a possible Church Turing hypothesis [15,16] applicable to the carrying out of physical experiments [8] is noted in the last section.…”

Section: Introductionmentioning

confidence: 99%

Quantum Robots Plus Environments

Benioff

Quantum Communication, Computing, and Measurement 2

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A quantum robot is a mobile quantum system, including an on board quantum computer and needed ancillary systems, that interacts with an environment of quantum systems. Quantum robots carry out tasks whose goals include making specified changes in the state of the environment or carrying out measurements on the environment. The environments considered so far, oracles, data bases, and quantum registers, are seen to be special cases of environments considered here. It is also seen that a quantum robot should include a quantum computer and cannot be simply a multistate head.A model of quantum robots and their interactions is discussed in which each task, as a sequence of alternating computation and action phases,is described by a unitary single time step operator T = Ta + Tc (discrete space and time are assumed). The overall system dynamics is described as a sum over paths of completed computation (Tc) and action (Ta) phases. A simple example of a task, measuring the distance between the quantum robot and a particle on a 1D lattice with quantum phase path dispersion present, is analyzed. A decision diagram for the task is presented and analyzed.

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An Unsolvable Problem of Elementary Number Theory

Cited by 1,231 publications

References 0 publications

Deep learning in neural networks: An overview

Deep learning in neural networks: An overview

Bootstrapping in a language of thought: A formal model of numerical concept learning

Quantum Robots Plus Environments

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