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Towards applying computational complexity to foundations of physics
 Notes of Mathematical Seminars of St. Petersburg Department of Steklov Institute of Mathematics
, 2004
"... In one of his early papers, D. Grigoriev analyzed the decidability and computational complexity of different physical theories. This analysis was motivated by the hope that this analysis would help physicists. In this paper, we survey several similar ideas that may be of help to physicists. We hope ..."
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In one of his early papers, D. Grigoriev analyzed the decidability and computational complexity of different physical theories. This analysis was motivated by the hope that this analysis would help physicists. In this paper, we survey several similar ideas that may be of help to physicists. We hope that further research may lead to useful physical applications. 1
Fast Quantum Algorithms for Handling Probabilistic and Interval Uncertainty
, 2003
"... this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether (and how) "hypothetic" physical devices can compute NPhard problems faster than in exponential time ..."
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Cited by 9 (8 self)
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this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether (and how) "hypothetic" physical devices can compute NPhard problems faster than in exponential time
Designing, Understanding, and Analyzing Unconventional Computation: The Important Role of Logic and Constructive Mathematics
"... 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. ..."
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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.
mlq header will be provided by the publisher Fast Quantum Algorithms for Handling Probabilistic and Interval Uncertainty
, 2003
"... In many reallife situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easiertomeasure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn). Measurements are never 100 % ac ..."
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In many reallife situations, we are interested in the value of a physical quantity y that is difficult or impossible to measure directly. To estimate y, we find some easiertomeasure quantities x1,..., xn which are related to y by a known relation y = f(x1,..., xn). Measurements are never 100 % accurate; hence, the measured values ˜xi are different from xi, and the resulting estimate ˜y = f(˜x1,..., ˜xn) is different from the desired value y = f(x1,..., xn). How different can it be? Traditional engineering approach to error estimation in data processing assumes that we know the probabildef ities of different measurement errors ∆xi = ˜xi − xi. In many practical situations, we only know the upper bound ∆i for this error; hence, after the measurement, the only information that we have about xi is that it def belongs to the interval xi = [˜xi − ∆i, ˜xi + ∆i]. In this case, it is important to find the range y of all possible values of y = f(x1,..., xn) when xi ∈ xi. We start the paper with a brief overview of the computational complexity of the corresponding interval computation problems. Most of the related problems turn out to be, in general, at least NPhard. In this paper, we show how the use of quantum computing can speed up some computations related to interval and probabilistic uncertainty. We end the paper with speculations on whether (and how) “hypothetic ” physical devices can compute NPhard problems faster than in exponential time. Most of the paper’s results were first presented at NAFIPS’2003 [30]. Copyright line will be provided by the publisher 1 Introduction: Data Processing
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"... Communicated by (xxxxxxxxxx) In theoretical computer science, researchers usually distinguish between feasible problems (that can be solved in polynomial time) and problems that require more computation time. A natural question is: can we use new physical processes, processes that have not been used ..."
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Communicated by (xxxxxxxxxx) In theoretical computer science, researchers usually distinguish between feasible problems (that can be solved in polynomial time) and problems that require more computation time. A natural question is: can we use new physical processes, processes that have not been used in modern computers, to make computations drastically faster – e.g., to make intractable problems feasible? Such a possibility would occur if a physical process provides a superpolynomial ( = faster than polynomial) speedup. In this direction, the most active research is undertaken in quantum computing. It is well known that quantum processes can speed up computations; however, the only proven quantum speedups are polynomial. Parallelization is another potential source of speedup. In Euclidean space, parallelization only leads to a polynomial speedup. We show that in quantum spacetime, parallelization can potentially leads to superpolynomial speedup of computations.