## Designing, Understanding, and Analyzing Unconventional Computation: The Important Role of Logic and Constructive Mathematics

### BibTeX

@MISC{Kreinovich_designing,understanding,,

author = {Vladik Kreinovich},

title = {Designing, Understanding, and Analyzing Unconventional Computation: The Important Role of Logic and Constructive Mathematics},

year = {}

}

### OpenURL

### Abstract

In this paper, we explain why, in our opinion, logic and constructive mathematics are playing – and should play – an important role in the design, understanding, and analysis of unconventional computation.

### Citations

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- 1988
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Citation Context ...a solution to the problem. For example, several logic programming languages (widely used in AI applications) make it possible to automatically transform logical specifications into a code; see, e.g., =-=[25, 26, 33]-=-. Comment. It is worth mentioning that a related work was done at Microsoft Research on Spec Explorer and Abstract State Machine. Logic has been efficiently used in program verification. Not only the ... |

1795 | An Introduction to Kolmogorov Complexity and its Applications, 2 nd edition
- Li, Vitanyi
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Citation Context ... a Nobelist M. Gell-Mann suggested that physical equations should include terms explicitly depending on complexity [9]. A natural formalization of this complexity is Kolmogorov complexity (see, e.g., =-=[24]-=-: the shortest length of a program that generates a given sequence of symbols: K(x) def = min{len(p) : p generates x}. Under this assumption, by observing physical and biological processes, we can mea... |

442 | Foundations of Constructive Analysis
- Bishop
- 1967
(Show Context)
Citation Context ...es already – actually, starting with the 1950s when the first computers appeared. There is a special branch of mathematics called constructive mathematics that deals with such definitions; see, e.g., =-=[1, 2, 3, 4, 6, 18, 23]-=-. At present, most research in constructive mathematics is devoted to specific problems in which specific algorithms are needed. These results are scattered around, are motivated mostly by specific pr... |

230 |
The Quark and the Jaguar
- Gell-Mann
- 1994
(Show Context)
Citation Context ...cult to describe on the level of fundamental physics. To facilitate this description, a Nobelist M. Gell-Mann suggested that physical equations should include terms explicitly depending on complexity =-=[9]-=-. A natural formalization of this complexity is Kolmogorov complexity (see, e.g., [24]: the shortest length of a program that generates a given sequence of symbols: K(x) def = min{len(p) : p generates... |

171 |
Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics
- Jaulin, Kieffer, et al.
- 2001
(Show Context)
Citation Context ..., the accuracy is usually fixed. In this case, it makes sense to develop simplified algorithms that work only for specific accuracy values. This is, in essence, the main idea of interval computations =-=[10, 11, 12, 13, 21, 29]-=- – what Yu. Matiyasevich has called applied constructive mathematics. The name comes from the fact that for a single quantity, when we know the measurement result ˜x with a known accuracy ∆, then all ... |

108 |
Measurement Errors and Uncertainty. Theory and Practice
- Rabinovich
- 2005
(Show Context)
Citation Context ...ons with measurement inaccuracy is to assume that we know the probability distribution for the measurement error ∆x def = ˜x − x. Usually, it is assumed that this distribution is Gaussian; see, e.g., =-=[38]-=-. However, there are practical situations when we do not know this distribution. Indeed, the distribution for ∆x usually comes from the calibration of the corresponding measuring instrument (MI). To p... |

95 | Introduction to Interval Analysis
- Moore, Kearfott, et al.
(Show Context)
Citation Context ..., the accuracy is usually fixed. In this case, it makes sense to develop simplified algorithms that work only for specific accuracy values. This is, in essence, the main idea of interval computations =-=[10, 11, 12, 13, 21, 29]-=- – what Yu. Matiyasevich has called applied constructive mathematics. The name comes from the fact that for a single quantity, when we know the measurement result ˜x with a known accuracy ∆, then all ... |

72 |
Foundations of constructive mathematics. Metamathematical studies., volume 6 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3
- Beeson
- 1985
(Show Context)
Citation Context ...es already – actually, starting with the 1950s when the first computers appeared. There is a special branch of mathematics called constructive mathematics that deals with such definitions; see, e.g., =-=[1, 2, 3, 4, 6, 18, 23]-=-. At present, most research in constructive mathematics is devoted to specific problems in which specific algorithms are needed. These results are scattered around, are motivated mostly by specific pr... |

56 |
Applied Proof Theory: Proof Interpretations and their Use in Mathematics
- Kohlenbach
- 2008
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Citation Context ...s xi can serve as a 2ε-approximation to x0. This possibility of “mining” a non-constructive proof for possible algorithms has been actively used in many areas of computational mathematics; see, e.g., =-=[14]-=-. However, this area of research is only now developing its potential; more applications are potentially possible, more work is needed. This “proof mining” makes it possible to go beyond the situation... |

55 |
The Wave Equation with Computable Initial Data such that its Unique Solution is not Computable
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- 1981
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Citation Context ...cal equations such as the wave equation. It turned out that even for the wave equation, there exist computable initial conditions u(x, 0) for which the solution u(x, T ) is not computable; see, e.g., =-=[34, 35, 36, 37]-=-. At present, the related research is mainly aimed at analyzing how physical processes can, in principle, “compute” functions which are not computable in the usual sense. From the viewpoint of our mai... |

45 |
Black Holes and Time Warps: Einstein’s Outrageous Legacy
- Thorne
- 1994
(Show Context)
Citation Context ...olve NP-hard problems in polynomial time. Acausal processes. The simplest example of such a scheme is to use acausal processes, i.e., processes that go back in time and influence the past; see, e.g., =-=[39]-=-. The idea is to spend as much time as needed on computations, and then send the result of the computation back in time, to the moment when the user formulated the problem. Thus, the user will receive... |

44 |
Kearfott and
- B
- 1996
(Show Context)
Citation Context ..., the accuracy is usually fixed. In this case, it makes sense to develop simplified algorithms that work only for specific accuracy values. This is, in essence, the main idea of interval computations =-=[10, 11, 12, 13, 21, 29]-=- – what Yu. Matiyasevich has called applied constructive mathematics. The name comes from the fact that for a single quantity, when we know the measurement result ˜x with a known accuracy ∆, then all ... |

31 |
A computable ordinary differential equation which possesses no computable solution
- Richards
- 1979
(Show Context)
Citation Context ...cal equations such as the wave equation. It turned out that even for the wave equation, there exist computable initial conditions u(x, 0) for which the solution u(x, T ) is not computable; see, e.g., =-=[34, 35, 36, 37]-=-. At present, the related research is mainly aimed at analyzing how physical processes can, in principle, “compute” functions which are not computable in the usual sense. From the viewpoint of our mai... |

25 |
Theory of Deductive Systems and Its Applications
- Maslov
- 1987
(Show Context)
Citation Context ...rive at a conclusion that time travel can trigger events with very small probability p0 ≪ 1. Let us show how this conclusion can be used to solve NP-hard problems in polynomial time; for details, see =-=[15, 16, 20, 27, 30]-=-. As an example of an NP-hard problem, we can take the propositional satisfiability problem SAT: given a propositional formula F (x1, . . . , xn), find the values of the propositional variables that m... |

23 |
A Notion of Mechanistic Theory
- Kreisel
- 1974
(Show Context)
Citation Context ...ann’s scheme can indeed potentially speed up computations.Unconventional Computation: The Role of Logic 13 Other schemes using new physical phenomena are based on: • quantum field theory (G. Kreisel =-=[22]-=-), • natural idea that every theory is approximate [16, 17], etc. Unconventional computations and constructive mathematics. All above schemes use or propose a radically new physical process. It is wor... |

20 |
Formalizing Common Sense
- McCarthy
- 1990
(Show Context)
Citation Context ...a solution to the problem. For example, several logic programming languages (widely used in AI applications) make it possible to automatically transform logical specifications into a code; see, e.g., =-=[25, 26, 33]-=-. Comment. It is worth mentioning that a related work was done at Microsoft Research on Spec Explorer and Abstract State Machine. Logic has been efficiently used in program verification. Not only the ... |

18 |
Modal intervals
- Gardenes, Sainz, et al.
(Show Context)
Citation Context ..., this means that the control leading to the desired state is possible. Because of this connection, modal logic has been efficiently used in designing algorithms for interval computations; see, e.g., =-=[5, 8, 18]-=-. There is even a special term modal interval analysis for such applications. From direct to indirect methods of algorithm design: proof mining. Historically, the first existence proofs were direct in... |

17 | Which algorithms are feasible and which are not depends on the geometry of space-time
- Morgenstein, Kreinovich
- 1995
(Show Context)
Citation Context ...eral computer work in parallel to perform the same task. Parallelization does lead to a drastic speedup, but, alas, in Euclidean space, parallelization only leads to a polynomial speed-up; see, e.g., =-=[21, 31]-=-. Indeed, the speed of all the physical processes is bounded by the speed of light c. Thus, in time T , we can only reach computational units at a distance ≤ R = c · T . The volume V (R) of this area ... |

13 |
The wave equation with computable initial data whose unique solution is nowhere computable
- Zhong
- 1997
(Show Context)
Citation Context ...cal equations such as the wave equation. It turned out that even for the wave equation, there exist computable initial conditions u(x, 0) for which the solution u(x, T ) is not computable; see, e.g., =-=[34, 35, 36, 37]-=-. At present, the related research is mainly aimed at analyzing how physical processes can, in principle, “compute” functions which are not computable in the usual sense. From the viewpoint of our mai... |

7 |
Precise numerical methods in C
- Aberth
- 1998
(Show Context)
Citation Context ...es already – actually, starting with the 1950s when the first computers appeared. There is a special branch of mathematics called constructive mathematics that deals with such definitions; see, e.g., =-=[1, 2, 3, 4, 6, 18, 23]-=-. At present, most research in constructive mathematics is devoted to specific problems in which specific algorithms are needed. These results are scattered around, are motivated mostly by specific pr... |

7 |
Some relations between classical and constructive mathematics
- Beeson
- 1987
(Show Context)
Citation Context |

7 |
What can physics give to constructive mathematics
- Kosheleva, Kreinovich
- 1981
(Show Context)
Citation Context ...rive at a conclusion that time travel can trigger events with very small probability p0 ≪ 1. Let us show how this conclusion can be used to solve NP-hard problems in polynomial time; for details, see =-=[15, 16, 20, 27, 30]-=-. As an example of an NP-hard problem, we can take the propositional satisfiability problem SAT: given a propositional formula F (x1, . . . , xn), find the values of the propositional variables that m... |

7 | Fast quantum algorithms for handling probabilistic and interval uncertainty
- Kreinovich, Longpré
- 2004
(Show Context)
Citation Context ...rive at a conclusion that time travel can trigger events with very small probability p0 ≪ 1. Let us show how this conclusion can be used to solve NP-hard problems in polynomial time; for details, see =-=[15, 16, 20, 27, 30]-=-. As an example of an NP-hard problem, we can take the propositional satisfiability problem SAT: given a propositional formula F (x1, . . . , xn), find the values of the propositional variables that m... |

7 |
Time travel and computing
- Moravec
(Show Context)
Citation Context |

6 | In some curved spaces, one can solve NP-hard problems in polynomial time
- Kreinovich, Margenstern
- 2009
(Show Context)
Citation Context |

5 |
From interval computations to modal mathematics: applications and computational complexity
- Bouchon-Meunier, Kreinovich
- 1998
(Show Context)
Citation Context ..., this means that the control leading to the desired state is possible. Because of this connection, modal logic has been efficiently used in designing algorithms for interval computations; see, e.g., =-=[5, 8, 18]-=-. There is even a special term modal interval analysis for such applications. From direct to indirect methods of algorithm design: proof mining. Historically, the first existence proofs were direct in... |

5 |
Maximum entropy and acausal processes: astrophysical applications and challenges
- Koshelev
- 1998
(Show Context)
Citation Context |

5 | Why Kolmogorov complexity in physical equations
- Kreinovich, Longpré
- 1998
(Show Context)
Citation Context ...h of a program that generates a given sequence of symbols: K(x) def = min{len(p) : p generates x}. Under this assumption, by observing physical and biological processes, we can measure the value K(x) =-=[19]-=-. However, it is well known that K(x) is not algorithmically computable [24], and it is also known that the ability to get non-computable values can speed up computations. Thus, Gell-Mann’s scheme can... |

4 |
A Short Introduction to Modal Logic, Center for the Study
- Mints
- 1992
(Show Context)
Citation Context ...omputations. Many problems of interval computations can be naturally reformulated in terms of modal logic – specifically, it terms of the original modal logic of necessity and possibility; see, e.g., =-=[7, 28]-=-. Specifically, in situations like robust control, we want to make sure that the control is stable for all possible values of the parameters from the given intervals, i.e., in terms of modal logic, th... |

2 |
On the logic of using observable events in decision making
- Kosheleva, Soloviev
- 1981
(Show Context)
Citation Context ...ns.Unconventional Computation: The Role of Logic 13 Other schemes using new physical phenomena are based on: • quantum field theory (G. Kreisel [22]), • natural idea that every theory is approximate =-=[16, 17]-=-, etc. Unconventional computations and constructive mathematics. All above schemes use or propose a radically new physical process. It is worth noticing that some of the unconventional computation sch... |