THREE COUNTEREXAMPLES TO DISPEL THE MYTH OF THE UNIVERSAL COMPUTER

2006 ◽  
Vol 16 (03) ◽  
pp. 381-403 ◽  
Author(s):  
SELIM G. AKL
Keyword(s):  
Turing Machine ◽  
Computable Function ◽  
General Purpose ◽  
Fixed Number ◽  
Quantum Computers ◽  
Addition Formula ◽  
Models Of Computation ◽  
Infinite Memory

It is shown that the concept of a Universal Computer cannot be realized. Specifically, instances of a computable function [Formula: see text] are exhibited that cannot be computed on any machine [Formula: see text] that is capable of only a finite and fixed number of operations per step. This remains true even if the machine [Formula: see text] is endowed with an infinite memory and the ability to communicate with the outside world while it is attempting to compute [Formula: see text]. It also remains true if, in addition, [Formula: see text] is given an indefinite amount of time to compute [Formula: see text]. This result applies not only to idealized models of computation, such as the Turing 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) cannot be universal.

Computational speed-up by effective operators

1972 ◽  
Vol 37 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Albert R. Meyer ◽  
Patrick C. Fischer
Keyword(s):  
Turing Machine ◽  
Computable Function ◽  
Turing Machines ◽  
Complexity Measures ◽  
Computational Speed ◽  
Partial Recursive Functions ◽  
Speed Up ◽  
Recursive Equations ◽  
Computable Functions ◽  
Effective Operators

The complexity of a computable function can be measured by considering the time or space required to compute its values. Particular notions of time and space arising from variants of Turing machines have been investigated by R. W. Ritchie [14], Hartmanis and Stearns [8], and Arbib and Blum [1], among others. General properties of such complexity measures have been characterized axiomatically by Rabin [12], Blum [2], Young [16], [17], and McCreight and Meyer [10].In this paper the speed-up and super-speed-up theorems of Blum [2] are generalized to speed-up by arbitrary total effective operators. The significance of such theorems is that one cannot equate the complexity of a computable function with the running time of its fastest program, for the simple reason that there are computable functions which in a very strong sense have no fastest programs.Let φi be the ith partial recursive function of one variable in a standard Gödel numbering of partial recursive functions. A family Φ0, Φ1, … of functions of one variable is called a Blum measure on computation providing(1) domain (φi) = domain (Φi), and(2) the predicate [Φi(x) = m] is recursive in i, x and m.Typical interpretations of Φi(x) are the number of steps required by the ith Turing machine (in a standard enumeration of Turing machines) to converge on input x, the space or number of tape squares required by the ith Turing machine to converge on input x (with the convention that Φi(x) is undefined even if the machine fails to halt in a finite loop), and the length of the shortest derivation of the value of φi(x) from the ith set of recursive equations.

scholarly journals Computability Theory and Differential Geometry

2004 ◽  
Vol 10 (4) ◽  
pp. 457-486 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Turing Machine ◽  
Riemannian Metrics ◽  
Computable Function ◽  
Computability Theory ◽  
Settling Time ◽  
Point Of View ◽  
Connected Components ◽  
Local Minima ◽  
Smooth Compact Manifold ◽  
Space Of Riemannian Metrics

Abstract. LetMbe a smooth, compact manifold of dimensionn≥ 5 and sectional curvature ∣K∣ ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics onMmodulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machineTe,eϵ ω, there is a sequence (uniformly effective ine) of homologyn-sphereswhich are also hypersurfaces, such thatis diffeomorphic to the standardn-sphereSn(denoted)iffTehalts on inputk, and in this case the connected sum, so, andis associated with a local minimum of the diameter function on Met(M) whose depth is roughly equal to the settling time ae σe(k)ofTeon inputsy<k.At their request Soare constructed a particular infinite sequence {Ai}ϵωof c.e. sets so that for allithe settling time of the associated Turing machine forAidominates that forAi+1, even when the latter is composed with an arbitrary computable function. From this, Nabutovsky and Weinberger showed that the basins exhibit a “fractal” like behavior with extremely big basins, and very much smaller basins coming off them, and so on. This reveals what Nabutovsky and Weinberger describe in their paper on fractals as “the astonishing richness of the space of Riemannian metrics on a smooth manifold, up to reparametrization.” From the point of view of logic and computability, the Nabutovsky-Weinberger results are especially interesting because: (1) they use c.e. sets to prove structuralcomplexityof the geometry and topology, not merelyundecidabilityresults as in the word problem for groups, Hilbert's Tenth Problem, or most other applications; (2) they usenontrivialinformation about c.e. sets, the Soare sequence {Ai}iϵωabove, not merely Gödel's c.e. noncomputable set K of the 1930's; and (3)withoutusing computability theory there is no known proof that local minima exist even for simple manifolds like the torusT5(see §9.5).

scholarly journals A (Basis for a) Philosophy of a Theory of Fuzzy Computation

2018 ◽  
Vol 20 (1) ◽  
pp. 181-201 ◽  
Author(s):  
Apostolos Syropoulos
Keyword(s):  
Mathematical Model ◽  
Turing Machine ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Computing Device ◽  
Models Of Computation ◽  
Model Based ◽  
Physical Objects ◽  
Linguistic Phenomenon

Abstract Vagueness is a linguistic phenomenon as well as a property of physical objects. Fuzzy set theory is a mathematical model of vagueness that has been used to define vague models of computation. The prominent model of vague computation is the fuzzy Turing machine. This conceptual computing device gives an idea of what computing under vagueness means, nevertheless, it is not the most natural model. Based on the properties of this and other models of vague computing, an attempt is made to formulate a basis for a philosophy of a theory of fuzzy computation.

scholarly journals Introduction to Quantum Computing

2020 ◽  
Author(s):  
Surya Teja Marella ◽  
Hemanth Sai Kumar Parisa
Keyword(s):  
Quantum Computing ◽  
Cyber Security ◽  
Quantum Computer ◽  
General Purpose ◽  
Small Scale ◽  
Quantum Computers ◽  
Ongoing Research ◽  
Advantages And Disadvantages ◽  
Traffic Optimization ◽  
Insight Into

Quantum computing is a modern way of computing that is based on the science of quantum mechanics and its unbelievable phenomena. It is a beautiful combination of physics, mathematics, computer science and information theory. It provides high computational power, less energy consumption and exponential speed over classical computers by controlling the behavior of small physical objects i.e. microscopic particles like atoms, electrons, photons, etc. Here, we present an introduction to the fundamental concepts and some ideas of quantum computing. This paper starts with the origin of traditional computing and discusses all the improvements and transformations that have been done due to their limitations until now. Then it moves on to the basic working of quantum computing and the quantum properties it follows like superposition, entanglement and interference. To understand the full potentials and challenges of a practical quantum computer that can be launched commercially, the paper covers the architecture, hardware, software, design, types and algorithms that are specifically required by the quantum computers. It uncovers the capability of quantum computers that can impact our lives in various viewpoints like cyber security, traffic optimization, medicines, artificial intelligence and many more. At last, we concluded all the importance, advantages and disadvantages of quantum computers. Small-scale quantum computers are being developed recently. This development is heading towards a great future due to their high potential capabilities and advancements in ongoing research. Before focusing on the significances of a general-purpose quantum computer and exploring the power of the new arising technology, it is better to review the origin, potentials, and limitations of the existing traditional computing. This information helps us in understanding the possible challenges in developing exotic and competitive technology. It will also give us an insight into the ongoing progress in this field.

Engineering design is a computable function

1994 ◽  
Vol 8 (1) ◽  
pp. 35-44 ◽  
Author(s):  
Patrick A. Fitzhorn
Keyword(s):  
Engineering Design ◽  
Turing Machine ◽  
Computable Function ◽  
Abstract Model ◽  
Design Specifications ◽  
Design Paradigm ◽  
State Changes ◽  
Computational Methodology

AbstractComputational abstraction of engineering design leads to an elegant theory defining (1) the process of design as an abstract model of computability, the Turing machine; (2) the artifacts of design as enumerated strings from a (possibly multidimensional) grammar; and (3) design specifications or constraints as formal state changes that govern string enumeration. Using this theory, it is shown that engineering design is a computable function. A computational methodology based on the theory is then developed that can be described as a form follows function design paradigm.

scholarly journals An Extensible Computing Platform for Heavy Quantum Simulation Workloads

2021 ◽  
Author(s):  
Andrés García ◽  
José Ranilla ◽  
Raul Alonso Alvarez ◽  
Luis Meijueiro
Keyword(s):  
Quantum Computing ◽  
High Performance ◽  
Quantum Computer ◽  
General Purpose ◽  
Quantum Circuits ◽  
Quantum Computers ◽  
Computing Platform ◽  
Current State ◽  
Classical Computing ◽  
Computing Simulation

The shortage of quantum computers, and their current state of development, constraints research in many fields that could benefit from quantum computing. Although the work of a quantum computer can be simulated with classical computing, personal computers take so long to run quantum experiments that they are not very useful for the progress of research. This manuscript presents an open quantum computing simulation platform that enables quantum computing researchers to have access to high performance simulations. This platform, called QUTE, relies on a supercomputer powerful enough to simulate general purpose quantum circuits of up to 38 qubits, and even more under particular simulations. This manuscript describes in-depth the characteristics of the QUTE platform and the results achieved in certain classical experiments in this field, which would give readers an accurate idea of the system capabilities.

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