Hypercomputation
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NP-Complete Problems and Physical Reality
NP-complete Problems and Physical Reality Scott Aaronson∗ Abstract Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adia- batic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing.” The section on soap bubbles even includes some “experimental” re- sults. While I do not believe that any of the proposals will let us solve NP-complete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1 Introduction “Let a computer smear—with the right kind of quantum randomness—and you create, in effect, a ‘parallel’ machine with an astronomical number of processors . All you have to do is be sure that when you collapse the system, you choose the version that happened to find the needle in the mathematical haystack.” —From Quarantine [31], a 1992 science-fiction novel by Greg Egan If I had to debate the science writer John Horgan’s claim that basic science is coming to an end [48], my argument would lean heavily on one fact: it has been only a decade since we learned that quantum computers could factor integers in polynomial time. In my (unbiased) opinion, the showdown that quantum computing has forced—between our deepest intuitions about computers on the one hand, and our best-confirmed theory of the physical world on the other—constitutes one of the most exciting scientific dramas of our time. -
Technical Report No. 2005-492 the MYTH of UNIVERSAL COMPUTATION∗
Technical Report No. 2005-492 THE MYTH OF UNIVERSAL COMPUTATION∗ Selim G. Akl School of Computing Queen’s University Kingston, Ontario, Canada K7L 3N6 January 31, 2005 Abstract It is shown that the concept of a Universal Computer cannot be realized. Specif- ically, instances of a computable function are exhibited that cannot be computed on any machine that is capable of only aF finite and fixed number of operations per step. This remainsU true even if the machine is endowed with an infinite memory and the ability to communicate with the outsideU world while it is attempting to compute . It also remains true if, in addition, is given an indefinite amount of time to computeF . This result applies not onlyU to idealized models of computation, such as the TuringF Machine and the like, but also to all known general-purpose computers, including existing conventional computers (both sequential and parallel), as well as contemplated unconventional ones such as biological and quantum computers. Even accelerating machines (that is, machines that increase their speed at every step) are not universal. Keywords: Universal computer, Turing machine, simulation, parallel computation, superlinear speedup, real-time computation, uncertainty, geometric transformation. 1 Introduction Universal computation, which rests on the principle of simulation, is one of the foundational concepts in computer science. Thus, it is one of the main tenets of the field that any computation that can be carried out by one general-purpose computer can also be carried out on any other general-purpose computer. At times, the imitated computation, running on the second computer, may be faster or slower depending on the computers involved. -
Computation with Advice
Computation with Advice Vasco Brattka Arno Pauly∗ Laboratory of Foundational Aspects of Computer Science Computer Laboratory Department of Mathematics & Applied Mathematics University of Cambridge, United Kingdom University of Cape Town, South Africa [email protected] [email protected] Computation with advice is suggested as generalization of both computation with discrete advice and Type-2 Nondeterminism. Several embodiments of the generic concept are discussed, and the close connection to Weihrauch reducibility is pointed out. As a novel concept, computability with random advice is studied; which corresponds to correct solutions being guessable with positive probability. In the framework of computation with advice, it is possible to define computational complexity for certain concepts of hypercomputation. Finally, some examples are given which illuminate the interplay of uniform and non-uniform techniques in order to investigate both computability with advice and the Weihrauch lattice. 1 Introduction An approach to classify the incomputability of some problem is what kind of help we need to overcome it and compute the problem. The general idea is that for some (generally incomputable) function f , we find a computable function g, such that f (x)= g(x,w) holds for suitable w. One can conceive many potential requirements for the suitability of w. Concepts discussed so far in the literature are computability with discrete advice, where basically the set of potential advice values w is restricted ([23]), non-determinism, which amounts to wrong advice being effectively recognizable ([21], [3]), and the uniqueness of suitable advice. Others will be newly introduced in the following. For computability with discrete advice, preservation under Weihrauch reducibility was shown di- rectly ([15]), for the other concepts it was concluded from various arguments, however, as we will see, there is a uniform proof in the generic case. -
Hypercomputation by Definition
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 317 (2004) 191–207 www.elsevier.com/locate/tcs Hypercomputation by deÿnition Benjamin Wells Department of Mathematics, University of San Francisco, 2130 Fulton Street, San Francisco, CA 94117, USA Received 12 August 2003; received in revised form 21 September 2003 Abstract Hypercomputation refers to computation surpassing the Turing model, not just exceeding the von Neumann architecture. Algebraic constructions yield a ÿnitely based pseudorecursive equa- tional theory (Internat. J. Algebra Comput. 6 (1996) 457–510). It is not recursive, although for each given number n, its equations in n variables form a recursive set. Hypercomputation is therefore required for an algorithmic answer to the membership problem of such a theory. Yet Alfred Tarski declared these theories to be decidable. The dilemma of a decidable but not re- cursive set presents an impasse to standard computability theory. One way to break the impasse is to predicate that the theory is computable—in other words, hypercomputation by deÿnition. c 2003 Elsevier B.V. All rights reserved. MSC: primary 03D35; secondary 03C05; 03D15; 03D40; 20M07; 68Q05; 68Q10; 68Q15 Keywords: Hypercomputation; Pseudorecursive theory; Varieties of semigroups; Church–Turing Thesis; Decidable theories; Computability Preface A. I. Mal’tsev’s work inspired the hunt for pseudorecursive varieties, and Alfred Tarski’s encouragement led to publishing the ÿndings. Although the heir of a two-ÿgure fortune, I am still a beginner at transcending duality. 1. Introduction We consider here computability problems on the boundary between logic and algebra. -
Triadic Automata and Machines As Information Transformers
information Article Triadic Automata and Machines as Information Transformers Mark Burgin Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles, CA 90095, USA; [email protected] Received: 12 December 2019; Accepted: 31 January 2020; Published: 13 February 2020 Abstract: Algorithms and abstract automata (abstract machines) are used to describe, model, explore and improve computers, cell phones, computer networks, such as the Internet, and processes in them. Traditional models of information processing systems—abstract automata—are aimed at performing transformations of data. These transformations are performed by their hardware (abstract devices) and controlled by their software (programs)—both of which stay unchanged during the whole computational process. However, in physical computers, their software is also changing by special tools such as interpreters, compilers, optimizers and translators. In addition, people change the hardware of their computers by extending the external memory. Moreover, the hardware of computer networks is incessantly altering—new computers and other devices are added while other computers and other devices are disconnected. To better represent these peculiarities of computers and computer networks, we introduce and study a more complete model of computations, which is called a triadic automaton or machine. In contrast to traditional models of computations, triadic automata (machine) perform computational processes transforming not only data but also hardware and programs, which control data transformation. In addition, we further develop taxonomy of classes of automata and machines as well as of individual automata and machines according to information they produce. Keywords: information; automaton; machine; hardware; software; modification; process; inductive; recursive; superrecursive; equivalence 1. -
Recent Advances in Real Complexity and Computation
604 Recent Advances in Real Complexity and Computation UIMP-RSME Lluís A. Santaló Summer School Recent Advances in Real Complexity and Computation July 16–20, 2012 Universidad Internacional Menéndez Pelayo, Santander, Spain José Luis Montaña Luis M. Pardo Editors American Mathematical Society Real Sociedad Matemática Española American Mathematical Society Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Recent Advances in Real Complexity and Computation UIMP-RSME Lluís A. Santaló Summer School Recent Advances in Real Complexity and Computation July 16–20, 2012 Universidad Internacional Menéndez Pelayo, Santander, Spain José Luis Montaña Luis M. Pardo Editors Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms 604 Recent Advances in Real Complexity and Computation UIMP-RSME Lluís A. Santaló Summer School Recent Advances in Real Complexity and Computation July 16–20, 2012 Universidad Internacional Menéndez Pelayo, Santander, Spain José Luis Montaña Luis M. Pardo Editors American Mathematical Society Real Sociedad Matemática Española American Mathematical Society Providence, Rhode Island Licensed to University Paul Sabatier. Prepared on Mon Dec 14 09:01:17 EST 2015for download from IP 130.120.37.54. -
The Physical Church-Turing Thesis: Modest Or Bold?1
The Physical Church-Turing Thesis: Modest or Bold?1 Gualtiero Piccinini University of Missouri – St. Louis Email: [email protected] This is a preprint of a paper whose final and definitive form will be published in the British Journal for the Philosophy of Science. The British Journal for the Philosophy of Science is available online at http://bjps.oxfordjournals.org/. Abstract: This paper defends a modest version of the Physical Church-Turing thesis (CT). Following an established recent trend, I distinguish between what I call Mathematical CT—the thesis supported by the original arguments for CT— and Physical CT. I then distinguish between bold formulations of Physical CT, according to which any physical process—anything doable by a physical system—is computable by a Turing machine, and modest formulations, according to which any function that is computable by a physical system is computable by a Turing machine. I argue that Bold Physical CT is not relevant to the epistemological concerns that motivate CT and hence not suitable as a physical analogue of Mathematical CT. The correct physical analogue of Mathematical CT is Modest Physical CT. I propose to explicate the notion of physical computability in 1 Thanks to Darren Abramson, Dave Chalmers, Jack Copeland, Martin Davis, John Earman, Corey Maley, John Norton, Toby Ord, Michael Rabin, Oron Shagrir, Wilfried Sieg, and Mark Sprevak, among others, for helpful discussions on this topic. Thanks to Oron Shagrir and the anonymous referees for helpful comments. Material that led to this paper was presented at the 2005 Eastern APA (meeting of the Society for Machines and Mentality) and to the Department of Mathematics and Computer Science at the University of Missouri – St. -
Black Hole Computation: Implementations with Signal Machines
Black hole computation: implementations with signal machines J´erˆome Durand-Lose Laboratoire d’Informatique Fondamentale d’Orl´eans, Universit´ed’Orl´eans, B.P. 6759, F-45067 ORLEANS´ Cedex 2. 1. Introduction No position is taken on the theoretical and practical feasibility of using any potentially existing particular black hole for hyper-computing. The reader is referred to other contributions in this issue as well as Etesi and N´emeti (2002); N´emeti and D´avid (2006); N´emeti and Andr´eka (2006) for clear introductions and surveys on this topic. All the work presented here in done the context of so-called “Malament-Hogarth” space-times and not slowly rotating Kerr black hole. The main differences are that, one, the observer remains out of the black hole and second, nested black holes are considered. A black hole computation works in the following way. At some location ⋆ one observer starts a computing device which he sends on a different world- line such that: at some point, after a finite time on his own world-line the observer has the whole future of the device infinite world-line in its causal past and the device can send some limited information to the observer. The device has an infinite time for computing ahead of it and can send a single piece of information to the observer. Not only can the observer perceive this piece of information, but after a finite duration which he is aware of, the observer knows for certain that if he has not received anything then the machine never sends anything during its whole, possibly infinite, computation.