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

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 difficult-to-described definition can be actually viewed as a particular case of something very natural: the general notion of boundedness. c ○ 2007 World Academic Press, UK. All rights reserved. 1

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.

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.

Physicists usually assume that events with a very small probability cannot occur. Kolmogorov complexity formalizes this idea for non-quantum events. We show how this formalization can be extended to quantum events as well.

Abstract. To formalize some types of non-monotonic 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.

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 Kolmogorov-Martin-Löf 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”.

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.