m-sophistication

The m-sophistication of a finite binary string x is introduced as a generalization of some parameter in the proof that complexity of complexity is rare. A probabilistic near sufficient statistic of x is given which length is upper bounded by the m-sophistication of x within small additive terms. This shows that m-sophistication is lower bounded by coarse sophistication and upper bounded by sophistication within small additive terms. It is also shown that m-sophistication and coarse sophistication can not be approximated by an upper or lower semicomputable function, not even within very large error.Comment: 13 pages, draf

Asymmetry of the Kolmogorov complexity of online predicting odd and even bits

Symmetry of information states that $C(x) + C(y|x) = C(x,y) + O(\log C(x))$. We show that a similar relation for online Kolmogorov complexity does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring $x_1x_2... x_n$ be the length of a shortest program that computes $x_2$ on input $x_1$, computes $x_4$ on input $x_1x_2x_3$, etc; and similar for odd complexity. We show that for all n there exist an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence $x_2x_1x_4x_3\ldots$, decreases the sum of odd and even complexity to $C(x)$.Comment: 20 pages, 7 figure

Influence tests I: ideal composite hypothesis tests, and causal semimeasures

Ratios of universal enumerable semimeasures corresponding to hypotheses are investigated as a solution for statistical composite hypotheses testing if an unbounded amount of computation time can be assumed. Influence testing for discrete time series is defined using generalized structural equations. Several ideal tests are introduced, and it is argued that when Halting information is transmitted, in some cases, instantaneous cause and consequence can be inferred where this is not possible classically. The approach is contrasted with Bayesian definitions of influence, where it is left open whether all Bayesian causal associations of universal semimeasures are equal within a constant. Finally the approach is also contrasted with existing engineering procedures for influence and theoretical definitions of causation.Comment: 29 pages, 3 figures, draf

Relating and contrasting plain and prefix Kolmogorov complexity

In [3] a short proof is given that some strings have maximal plain Kolmogorov complexity but not maximal prefix-free complexity. The proof uses Levin's symmetry of information, Levin's formula relating plain and prefix complexity and Gacs' theorem that complexity of complexity given the string can be high. We argue that the proof technique and results mentioned above are useful to simplify existing proofs and to solve open questions. We present a short proof of Solovay's result [21] relating plain and prefix complexity: $K (x) = C (x) + CC (x) + O(CCC (x))$ and $C (x) = K (x) - KK (x) + O(KKK (x))$, (here $CC(x)$ denotes $C(C(x))$, etc.). We show that there exist $\omega$ such that $\liminf C(\omega_1\dots \omega_n) - C(n)$ is infinite and $\liminf K(\omega_1\dots \omega_n) - K(n)$ is finite, i.e. the infinitely often C-trivial reals are not the same as the infinitely often K-trivial reals (i.e. [1,Question 1]). Solovay showed that for infinitely many $x$ we have $|x| - C (x) \le O(1)$ and $|x| + K (|x|) - K (x) \ge \log^{(2)} |x| - O(\log^{(3)} |x|)$, (here $|x|$ denotes the length of $x$ and $\log^{(2)} = \log\log$, etc.). We show that this result holds for prefixes of some 2-random sequences. Finally, we generalize our proof technique and show that no monotone relation exists between expectation and probability bounded randomness deficiency (i.e. [6, Question 1]).Comment: 20 pages, 1 figur

NMOS-based integrated modular bypass for use in solar systems (NIMBUS): intelligent bypass for reducing partial shading power loss in solar panel applications

NMOS-based Integrated Modular Bypass for Use in Solar systems (NIMBUS) is designed as a replacement for the traditional bypass diode, used in common solar panels. Because of the series connection between the individual solar cells, the power output of a photovoltaic (PV) panel will drop disproportionally under partial shading. Currently, this is solved by dividing the PV panel into substrings, each with a diode bypass placed in parallel. This allows an alternative current path. However, the diodes still have a significant voltage drop (about 350 mV), and due to the fairly large currents in a panel, the diodes are dissipating power that we would rather see at the output of the panel. The NIMBUS chip, being a low-voltage-drop switch, aims to replace these diodes and, thus, reduce that power loss. NIMBUS is a smart bypass: a completely stand-alone system that detects the failing of one or more cells and activates when necessary. It is designed for a 100-mV voltage drop under a 5-A load current. When two or more NIMBUS chips are placed in parallel, an internal synchronization circuit ensures proper operation to provide for larger load currents. This paper will elaborate on the operation, design and implementation of the NIMBUS chip, as well as on the first measurements