## Transcending the Limits of Turing Computability (1998)

Citations: | 18 - 7 self |

### BibTeX

@MISC{Adamyan98transcendingthe,

author = {Vadim A. Adamyan and Cristian S. Calude and Boris S. Pavlov},

title = {Transcending the Limits of Turing Computability},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

Hypercomputation or super-Turing computation is a “computation ” that transcends the limit imposed by Turing’s model of computability. The field still faces some basic questions, technical (can we mathematically and/or physically build a hypercomputer?), cognitive (can hypercomputers realize the AI dream?), philosophical (is thinking more than computing?). The aim of this paper is to address the question: can we mathematically build a hypercomputer? We will discuss the solutions of the Infinite Merchant Problem, a decision problem equivalent to the Halting Problem, based on results obtained in [9, 2]. The accent will be on the new computational technique and results rather than formal proofs. 1

### Citations

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Citation Context ... essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Landauer [22, 23] (information is inevitably physical) and Deutsch =-=[12, 13]-=- (the reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic ...is that the laws of physics “happen” to permit the existence of physi... |

170 |
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Citation Context ... essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Landauer [22, 23] (information is inevitably physical) and Deutsch =-=[12, 13]-=- (the reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic ...is that the laws of physics “happen” to permit the existence of physi... |

161 |
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Citation Context ...s like {OR, NOT} or {NAND}. A set of quantum gates S is called universal if any unitary operation can be approximated with an arbitrary accuracy by a quantum circuit involving gates in S; see more in =-=[16, 17, 19, 8]-=-. 5spectral point p2 = λ > 0 is played by the scattered waves −→ Ψν, ←− Ψν, labeled with vectors ν ∈ E. The components of the scattered waves Ψν(p) in the outer space L2(R) are presented as linear co... |

125 |
Lectures on Computation
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Citation Context ...s like {OR, NOT} or {NAND}. A set of quantum gates S is called universal if any unitary operation can be approximated with an arbitrary accuracy by a quantum circuit involving gates in S; see more in =-=[16, 17, 19, 8]-=-. 5spectral point p2 = λ > 0 is played by the scattered waves −→ Ψν, ←− Ψν, labeled with vectors ν ∈ E. The components of the scattered waves Ψν(p) in the outer space L2(R) are presented as linear co... |

111 |
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Citation Context ...ops”): the equidistant sequence tl = ∑l m=1 δm, for the first space, and the perturbed sequence ˜tl = ∑l m=1 ∆m, for the second space. Both spaces are equipped with proper Wiener measures W, ˜ W (see =-=[29]-=-). The measure W is defined on the algebra of all finite-dimensional cylindrical sets Ct1,t2,...,tN of trajectories with fixed initial point ∆ 1 ,∆ 2 ,...,∆ N x0 = 0 and “gates” ∆l, l = 1, . . .,N (wh... |

89 |
Information is Physical
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(Show Context)
Citation Context ...pecific class of computing procedures, [14, 9, 21, 2] make essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Landauer =-=[22, 23]-=- (information is inevitably physical) and Deutsch [12, 13] (the reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic ...is that the... |

70 | Non-Turing computations via Malament–Hogarth spacetimes
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Citation Context ... are various proposals to break Turing’ Barrier by showing that certain classes of computing procedures have super-Turing power (see [10, 31, 6, 7, 30, 11]). A specific class of computing procedures, =-=[14, 9, 21, 2]-=- make essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Landauer [22, 23] (information is inevitably physical) and Deu... |

59 | Quantum algorithm for hilbert’s tenth problem
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Citation Context ... are various proposals to break Turing’ Barrier by showing that certain classes of computing procedures have super-Turing power (see [10, 31, 6, 7, 30, 11]). A specific class of computing procedures, =-=[14, 9, 21, 2]-=- make essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Landauer [22, 23] (information is inevitably physical) and Deu... |

55 |
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Citation Context ... that the time of the computation grows indefinitely or by the fact that the space of computation grows indefinitely or both. The classical theories of computability and complexity (see, for example, =-=[5]-=-) do not give any indication in this respect. In the following section we will show that time can be made finite provided we use a specific probabilistic strategy. 24 A Probabilistic Solution In this... |

36 |
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Citation Context ...ue of the journal Minds and Machines (12, 4, 2002). Currently there are various proposals to break Turing’ Barrier by showing that certain classes of computing procedures have super-Turing power (see =-=[10, 31, 6, 7, 30, 11]-=-). A specific class of computing procedures, [14, 9, 21, 2] make essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Lan... |

19 |
Computing with Cells and Atoms
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(Show Context)
Citation Context ...s like {OR, NOT} or {NAND}. A set of quantum gates S is called universal if any unitary operation can be approximated with an arbitrary accuracy by a quantum circuit involving gates in S; see more in =-=[16, 17, 19, 8]-=-. 5spectral point p2 = λ > 0 is played by the scattered waves −→ Ψν, ←− Ψν, labeled with vectors ν ∈ E. The components of the scattered waves Ψν(p) in the outer space L2(R) are presented as linear co... |

14 |
Scattering matrices for microschemes, Oper.Theory
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(Show Context)
Citation Context ...did not click after running the experiment for the time T, is P non-click (N) > 1 − 1 − P(N) P(N) · ) . √ ε √ √∏ (1 + γ) T ∞ − 1 − ε m=1 ∆m . 5 A Brownian Solution Based on Resonance Amplification In =-=[1]-=- the idea to consider a single act of quantum computation as a scattering process was suggested. We will first illustrate the method by describing a simple quantum scattering system realizing the quan... |

13 |
quantum measurements
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(Show Context)
Citation Context ...estion: can we mathematically build a hypercomputer? We will discuss the solutions of the Infinite Merchant Problem, a decision problem equivalent to the Halting Problem, based on results obtained in =-=[9, 2]-=-. The accent will be on the new computational technique and results rather than formal proofs. 1 Introduction Hypercomputation or super-Turing computation is a “computation” that transcends the limit ... |

13 | Transcending Turing Computability - MacLennan - 2002 |

13 |
Hypercomputation: hype or computation
- TEUSCHER, SIPPER
(Show Context)
Citation Context ...ue of the journal Minds and Machines (12, 4, 2002). Currently there are various proposals to break Turing’ Barrier by showing that certain classes of computing procedures have super-Turing power (see =-=[10, 31, 6, 7, 30, 11]-=-). A specific class of computing procedures, [14, 9, 21, 2] make essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Lan... |

11 |
Quantum mechanical computers, Optics News 11
- Feynman
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(Show Context)
Citation Context |

11 |
Two-body scattering on a graph and application to simple nanoelectronic devices
- Melnikov, Pavlov
- 1995
(Show Context)
Citation Context ...nerated by the total Hamiltonian He + Hd + Hint with a proper interaction term. This would lead to a quite sophisticated problem of quantum mechanics, similar to three-body problems, see for instance =-=[26]-=-. We assume now that the state of the dot is selected independently and thus reduce the above problem to the corresponding one-body problem for an electron scattered in the quantum wire depending on t... |

9 |
The theory of extensions and explicitly-solvable models (In Russian) Uspekhi Mat
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(Show Context)
Citation Context ...(the jump [u ′ ](0) and the value u(0) at the origin) of the component of the wave-function in the outer space and the symplectic coordinates ⃗ ξ± ∈ E of the inner component of the wave-function (see =-=[28]-=-): ( [u ′ ](0) − ⃗ ξ− ) = ( 0 β β + 0 ) ( u(0) ⃗ξ+ ) , (9) we obtain the scattering matrix in the form ( −→T −→ (p) R(p) Sβ(p) = ←− ←− R(p) T (p) ) , with equal transmission and reflection coefficient... |

8 | Computing the uncomputable - Casti - 1997 |

8 | Bound states and scattering in quantum waveguides coupled laterally through a boundary window
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(Show Context)
Citation Context ...iting into a specific quantum wire depends on the length of the window. This “switching phenomenon” was discovered by del Alamo and Eugster [3] and intensely discussed in literature, see for instance =-=[15, 4]-=-. We can arrange the setup in such a way that, under normal conditions, the electron exits from the same wire it enters, but switches to the other wire when a classical extra potential is applied, a r... |

5 |
Information is Inevitably Physical
- Landauer
- 2002
(Show Context)
Citation Context ...pecific class of computing procedures, [14, 9, 21, 2] make essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Landauer =-=[22, 23]-=- (information is inevitably physical) and Deutsch [12, 13] (the reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic ...is that the... |

4 | Quantum electronic devices based on metal-dielectric transition in low-dimensional quantum structures
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(Show Context)
Citation Context ...iting into a specific quantum wire depends on the length of the window. This “switching phenomenon” was discovered by del Alamo and Eugster [3] and intensely discussed in literature, see for instance =-=[15, 4]-=-. We can arrange the setup in such a way that, under normal conditions, the electron exits from the same wire it enters, but switches to the other wire when an extra potential is applied, a realizatio... |

4 |
Model of a zero-range potential with internal structure Sov
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(Show Context)
Citation Context ...es. To avoid the discussion of the general situation we use here a scattering matrix for a solvable model of the quantum dot which is based on zero-range potential with inner structure, introduced in =-=[27]-=- and already used in [4] for the description of typical features of nano-devices, see also [2]. 3 The transmission coefficients appear on the main diagonal of the matrix to fit the physical meaning of... |

4 | Quantum fieldeffect directional coupler - Alamo, Eugster - 1990 |

3 |
Hypercomputation. Minds and machines
- Copeland
- 2002
(Show Context)
Citation Context ...ue of the journal Minds and Machines (12, 4, 2002). Currently there are various proposals to break Turing’ Barrier by showing that certain classes of computing procedures have super-Turing power (see =-=[10, 31, 6, 7, 30, 11]-=-). A specific class of computing procedures, [14, 9, 21, 2] make essential use of some physical theory, relativity theory in [14], quantum theory [9, 21]; they all reflect an attitude advocated by Lan... |

3 |
Introduction to solid-state theory. Translated from German by
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- 1978
(Show Context)
Citation Context ...lectrons have the same constant spin on the quantum circuit R− ∪ R+, with the quantum dot attached. 4 We may assume that the circuit lies on the surface of a semiconductor with Fermi-level α 2 1 (see =-=[25]-=-). We assume that the state of the dot with the level α 2 1 vacant corresponds to Iin = Iout = 0 and the state of the dot with the level α 2 1 occupied corresponds to Iin = Iout = 1. We identify these... |

1 | A Quantum Scattering Approach to Undecidable Problems
- Adamyan, Calude, et al.
(Show Context)
Citation Context ...estion: can we mathematically build a hypercomputer? We will discuss the solutions of the Infinite Merchant Problem, a decision problem equivalent to the Halting Problem, based on results obtained in =-=[9, 2]-=-. The accent will be on the new computational technique and results rather than formal proofs. 1 Introduction Hypercomputation or super-Turing computation is a “computation” that transcends the limit ... |

1 |
Personal communication to
- Ignatov
- 1970
(Show Context)
Citation Context ...tack, then the weight of the combination will be 55 + N 100 g; otherwise the weight is just 55 g. The Merchant Problem quoted above was widely spread in allies armies during the Second World War, cf. =-=[20]-=-. Probably the elegant solution described above was the very first solution of a computational problem bearing typical features of quantum computing, see an extended discussion in [9]. 3 The Infinite ... |