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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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Cited by 17 (16 self)
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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
To Properly Reflect Physicists’ Reasoning about Randomness, We Also Need a Maxitive (Possibility
 Measure”, Proceedings of the 2005 IEEE International Conference on Fuzzy Systems FUZZIEEE’2005
"... According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, so this cu ..."
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Cited by 4 (2 self)
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According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, so this cup will start lifting up. In contrast, physicists believe that events with extremely small probability cannot occur. In this paper, we show that to get a consistent formalization of this belief, we need, in addition to the original probability measure, to also consider a maxitive (possibility) measure. We also show that the resulting advanced and somewhat difficulttodescribed definition can be actually viewed as a particular case of something very natural: the general notion of boundedness. 1
Application of Kolmogorov Complexity to Advanced Problems in Mechanics
, 2004
"... correct solution to a system of differential equations may be not physically possible: Traditional mathematical analysis tacitly assumes that all numbers, no matter how large or how small, are physically possible. From the engineering viewpoint, however, a number like 10 is not possible, becau ..."
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Cited by 2 (2 self)
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correct solution to a system of differential equations may be not physically possible: Traditional mathematical analysis tacitly assumes that all numbers, no matter how large or how small, are physically possible. From the engineering viewpoint, however, a number like 10 is not possible, because it exceeds the number of particles in the Universe. In this paper, we extend Kolmogorov's ideas from discrete objects to continuous objects known with given accuracy ", and show how this extension can clarify the analysis of dynamical systems.
I. PHYSICISTS ASSUME THAT INITIAL CONDITIONS AND
"... Abstract — According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, ..."
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Abstract — According to the traditional probability theory, events with a positive but very small probability can occur (although very rarely). For example, from the purely mathematical viewpoint, it is possible that the thermal motion of all the molecules in a coffee cup goes in the same direction, so this cup will start lifting up. In contrast, physicists believe that events with extremely small probability cannot occur. In this paper, we show that to get a consistent formalization of this belief, we need, in addition to the original probability measure, to also consider a maxitive (possibility) measure.
Negative Results of Computable Analysis Disappear If We Restrict Ourselves to Random (Or, More Generally, Typical) Inputs
"... such as weight, speed, etc., are characterized by real numbers. To get information about the corresponding value x, we perform measurements. Measurements are never absolute accurate. As a result of each measurement, we get a measurement result ˜x; for each measurement, we usually also know the upper ..."
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such as weight, speed, etc., are characterized by real numbers. To get information about the corresponding value x, we perform measurements. Measurements are never absolute accurate. As a result of each measurement, we get a measurement result ˜x; for each measurement, we usually also know the upper bound ∆ on the (absolute value of) the measurement error ∆x def = ˜x − x: x − ˜x  ≤ ∆. To fully characterize a value x, we must measure it with a higher and higher accuracy. As a result, when we perform measurements with accuracy 2 −n with n = 0, 1,..., we get a sequence of rational numbers rn for which x − rn  ≤ 2 −n. From the algorithmic viewpoint, we can view this sequence as an oracle that, given an integer n, returns a rational number rn. Such sequences represent real numbers in computable analysis; see, e.g., [9, 10]. First negative result. In computable analysis, several negative results are known. For example, it is known that no algorithm is possible that, given two numbers x and y, would check whether these numbers are equal or not.
Towards Formalizing . . . to Kolmogorov Complexity
"... To formalize some types of nonmonotonic reasoning in physics, researchers have proposed an approach based on Kolmogorov complexity. Inspired by Vladimir Lifschitz’s belief that many features of reasoning can be described on a purely logical level, we show that an equivalent formalization can be des ..."
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To formalize some types of nonmonotonic reasoning in physics, researchers have proposed an approach based on Kolmogorov complexity. Inspired by Vladimir Lifschitz’s belief that many features of reasoning can be described on a purely logical level, we show that an equivalent formalization can be described in purely logical terms: namely, in terms of physical induction. One of the consequences of this formalization is that the set of notabnormal states is (pre)compact. We can therefore use Lifschitz’s result that when there is only one state that satisfies a given equation (or system of equations), then we can algorithmically find this state. In this paper, we show that this result can be extended to the case of approximate uniqueness.
Towards Formalizing NonMonotonic Reasoning . . . to Kolmogorov Complexity
"... To formalize some types of nonmonotonic reasoning in physics, researchers have proposed an approach based on Kolmogorov complexity. Inspired by Vladimir Lifschitz’s belief that many features of reasoning can be described on a purely logical level, we show that an equivalent formalization can be de ..."
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To formalize some types of nonmonotonic reasoning in physics, researchers have proposed an approach based on Kolmogorov complexity. Inspired by Vladimir Lifschitz’s belief that many features of reasoning can be described on a purely logical level, we show that an equivalent formalization can be described in purely logical terms: namely, in terms of physical induction. One of the consequences of this formalization is that the set of notabnormal states is (pre)compact. We can therefore use Lifschitz’s result that when there is only one state that satisfies a given equation (or system of equations), then we can algorithmically find this state. In this paper, we show that this result can be extended to the case of approximate uniqueness.
Expanding Algorithmic Randomness to the Algebraic Approach to Quantum Physics: Kolmogorov Complexity and Quantum Logics
 JOURNAL OF UNCERTAIN SYSTEMS VOL.5, NO.X, PP.XX, 2011
, 2011
"... Physicists usually assume that events with a very small probability cannot occur. Kolmogorov complexity formalizes this idea for nonquantum events. We show how this formalization can be extended to quantum events as well. ..."
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Physicists usually assume that events with a very small probability cannot occur. Kolmogorov complexity formalizes this idea for nonquantum events. We show how this formalization can be extended to quantum events as well.