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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 20 (19 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 8 (5 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
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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Cited by 5 (4 self)
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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.
If many physicists are right and no physical theory is perfect, then the use of physical observations can enhance computations
, 2013
"... The questions of what is computable in the physical world are usually analyzed in the context of a physical theory – e.g., what is computable in Newtonian physics, what is computable in quantum physics, etc. Many physicists believe that no physical theory is perfect, i.e., that no matter how many ob ..."
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Cited by 4 (3 self)
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The questions of what is computable in the physical world are usually analyzed in the context of a physical theory – e.g., what is computable in Newtonian physics, what is computable in quantum physics, etc. Many physicists believe that no physical theory is perfect, i.e., that no matter how many observations support a physical theory, inevitably, new observations will come which will require this theory to be updated. We show, somewhat unexpectedly, that if such a noperfecttheory principle is true, then the use of physical data can drastically enhance computations.
Towards a “Generic” Notion of Genericity: From “Typical” and “Random” to Meager, Shy, etc.
 JOURNAL OF UNCERTAIN SYSTEMS VOL.6
, 2012
"... In many application areas, it is important to study “generic” properties, i.e., properties which hold for “typical” examples. For example, if we know the probabilities of different events, we can consider a “random” object – i.e., an object that, crudely speaking, does not belong to any class of “un ..."
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Cited by 4 (4 self)
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In many application areas, it is important to study “generic” properties, i.e., properties which hold for “typical” examples. For example, if we know the probabilities of different events, we can consider a “random” object – i.e., an object that, crudely speaking, does not belong to any class of “unusual” events (i.e., to any class with a small probability). In other cases, “typical” may mean not belonging to an “unusual ” subset which is small in some other sense – e.g., a subset of a smaller dimension. The corresponding notion of “typicalness ” has been formalized for several cases, including the case of random events. In this case, the known KolmogorovMartinLöf definition of randomness captures the idea that properties with probability 0 are impossible. In our previous papers, we modified this definition to take into account that from a practical viewpoint, properties with very small probabilities are often considered impossible as well. In this paper, we extend this definition to a general notion of “generic”.
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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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.
Adding possibilistic knowledge to probabilities makes many problems algorithmically decidable
 Proceedings of the World Congress of the International Fuzzy Systems Association IFSA’2015, Gijon
"... Adding possibilistic knowledge to probabilities makes many problems algorithmically decidable ..."
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Adding possibilistic knowledge to probabilities makes many problems algorithmically decidable
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.
Predictions with Possibilistic Information
"... many problems algorithmically decidable ..."
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1 What Is Computable? What Is Feasibly Computable? Different Aspects of These Questions
"... Abstract In this chapter, we show how the questions of what is computable and what is feasibly computable can be viewed from the viewpoint of physics: what is computable within the current physics? what is computable if we assume – as many physicists do – that no final physical theory is possible? w ..."
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Abstract In this chapter, we show how the questions of what is computable and what is feasibly computable can be viewed from the viewpoint of physics: what is computable within the current physics? what is computable if we assume – as many physicists do – that no final physical theory is possible? what is computable if we consider data processing, i.e., computations based on physical inputs? Our physicsbased analysis of these questions leads to some unexpected answers, both positive and negative. For example, we show that under the nophysicaltheoryisperfect assumption, almost all problems are feasibly solvable – but not all of them.