Algorithmic complexity and randomness in elastic solids

Algorithmic complexity and randomness in elastic solids J. Ratsaby† and J. Chaskalovic‡ August 9, 2021 †Department of Electrical and Electronics Engineering, ‡Department of Mathematics and Computer Science, Ariel University Center, Ariel 40700, ISRAEL and IJLRDA, University Pierre and Marie Curie - Paris VI, FRANCE †ratsaby@ariel.ac.il, ‡jch@ariel.ac.il Abstract A system comprised of an elastic solid and its response to an exter- nal random force sequence is shown to behave based on the principles of the theory of algorithmic complexity and randomness. The solid dis- torts the randomness of an input force sequence in a way proportional to its algorithmic complexity. We demonstrate this by numerical anal- ysis of a one-dimensional vibrating elastic solid (the system) on which we apply a maximally random input force. The level of complexity of the system is controlled via external parameters. The output re- sponse is the eld of displacements observed at several positions on the body. The algorithmic complexity and stochasticity of the result- ing output displacement sequence is measured and compared against the complexity of the system. The results show that the higher the sys- tem complexity the more random-decient the output sequence. This agrees with the theory introduced in [16] which states that physical systems such as this behave as algorithmic selection-rules which act on random actions in their surroundings. 1 Introduction Consider an elastic beam having a length L, (for instance, a bridge). It has some nite descriptive complexity consisting of all the information contained in the engineering design documents. These documents can be put into a single computer le that can be represented by a nite binary string z. This 1 arXiv:0811.0623v1 [cs.CC] 4 Nov 2008 binary sequence has an algorithmic complexity which is dened as the length of the shortest computer program that can generate the sequence. This is dened as the Kolmogorov complexity K(z) of the string z (see [9]). Now consider a random input force sequence applied at one of the two ends of the bridge, for instance, suppose there is a person jumping up and down sporadically on the bridge at its entrance (position 0). Denote by x the binary sequence representing this up/down symbols over some xed time- interval [0, T]. Intuitively, being that x is random makes its complexity K(x) maximal and hence close to its actual length ℓ(x) since there is no redundancy in the patterns of x that can be used to compress it signicantly below its length. Now consider an observer which measures the displacements on the beam at its other end (position L). He records this over the time interval [0, T] and compares it to a xed threshold thereby producing a binary output sequence y consisting of up/down symbols that represent the movement of the beam at position L. This sequence has a nite algorithmic complexity K(y). In this paper we show that for such a physical system, the system complexity K(z), the output complexity K(y) and its level of randomness are all related and there exist statistically signicant correlations between them. Ratsaby [16] introduced a quantitative denition of the information con- tent of a static structure (a solid) and explained its relationship to the sta- bility and symmetry of the solid. His model is based on concepts of the theory of algorithmic information and randomness. He modeled a solid as a selection rule of a nite algorithmic complexity which acts on an incoming random sequence of particles in the surroundings. This selection mechanism is intrinsically connected to the solid's complex non-linear structure (partly a consequence of its internal atomic vibrations) and its intricate time-response to external stimulus. As postulated in [16], a simple solid is one whose infor- mation content is small. Its selection behavior is of low complexity since it can be described by a more concise time-response model (shorter computer program). The solid's stability over time is explained in [16] by using the stochastic property of the frequency stability of a random sequence. Accord- ingly, the physical stability of the system (solid) is intrinsically and inversely proportional to the ability of the solid to deform (or distort) the input se- quence and make it less random, i.e., more random-decient. The current paper presents rst evidence that validate the model of [16]. We choose to simulate a solid structure which consists of a one-dimensional vibrating solid-beam to which we apply a random input force sequence and observe the displacement of the beam at its other end for a nite interval of time. We determine empirically the relationship between the algorithmic 2 complexity of the structure to the stochasticity of the output response. The relationship conrms the theory of [16]. The paper is organized as follows: in section 2 we give a brief introduction to the main concepts of the area of algorithmic complexity and randomness.

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