On the algorithmic complexity of static structures

Abstract

This paper provides a first indication that this is true for a system comprised of a static structure described by hyperbolic partial differential equations and is subjected to an external random input force. The system deforms the randomness of an input force sequence in proportion to its algorithmic complexity. The authors demonstrate this by numerical analysis of a one-dimensional vibrating elastic solid (the system) on which we apply a maximally-random force sequence (input). The level of complexity of the system is controlled via external parameters. The output response is the field of displacements observed at several positions on the body. The algorithmic complexity and stochasticity of the resulting output displacement sequence is measured and compared against the complexity of the system. The results show that the higher the system complexity, the more random-deficient the output sequence.

Learning, complexity and information density

“…This is explained by the relationship between the complexity of a system and its ability to 'distort' the randomness in its environment. The rst proof of this concept appeared in a recent paper [8] where it was shown that this inverse relationship between system complexity and randomness exists also in a physical system. The particular system investigated consisted of a one-dimensional vibrating solid-beam to which a random sequence of external input forces is applied.…”

“…The current paper is yet another proof of concept of the model of [5]. We proceed along the line of [8] but instead of considering a physical system (the static solid with input force sequence) we consider a decision system and study its inuence on a random binary data sequence on which prediction decisions are made. The decision system is based on the maximum a posteriori probability decision where probabilities are dened by a statistical parametric model which is estimated from data.…”

Random scattering of bits by prediction

“…Following [18] there have been recent theoretical and empirical results that validate his model for specific problem domains. The first empirical proof of his model appeared in [25,26,7] where it was shown that this inverse relationship between system complexity and randomness exists also in a real physical system. The particular system investigated consisted of a one-dimensional vibrating solid-beam to which a random sequence of external input forces is applied.…”

“…where l(π) is the length of the sequence π, φ is a universal partial recursive function which acts as a description method, i.e., when provided with input (π, y) it gives a specification for x (for an introduction see section 2 of [26]). The chaoticity of x (n) is large if its complexity is close to its length n. The classical work of [2,3,14,29] relates chaoticity to stochasticity.…”

“…obtaining the length 0 of the compressed file that contains ξ (n) 0 (henceforth referred to as the estimated algorithmic complexity of ξ (n) 0 since it is an approximation of the Kolmogorov complexity of ξ (n) 0 , see [26]). We measure the KL-divergence ∆ 0 between the probability distribution P (w|p) of binary words w of length 4 and the empirical probability distribution Pm (w) as measured from the mistake subsequence ξ (n) 0 .…”