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We investigate the properties of a Block Decomposition Method (BDM), whichextends the power of a Coding Theorem Method (CTM) that approximates localestimations of algorithmic complexity based upon Solomonoff-Levin's theory ofalgorithmic probability providing a closer connection to algorithmic complexitythan previous attempts based on statistical regularities e.g. as spotted bysome popular lossless compression schemes. The strategy behind BDM is to findsmall computer programs that produce the components of a larger, decomposedobject. The set of short computer programs can then be artfully arranged insequence so as to produce the original object and to estimate an upper bound onthe length of the shortest computer program that produces said original object.We show that the method provides efficient estimations of algorithmiccomplexity but that it performs like Shannon entropy when it loses accuracy. Weestimate errors and study the behaviour of BDM for different boundaryconditions, all of which are compared and assessed in detail. The measure maybe adapted for use with more multi-dimensional objects than strings, objectssuch as arrays and tensors. To test the measure we demonstrate the power of CTMon low algorithmic-randomness objects that are assigned maximal entropy (e.g.$\pi$) but whose numerical approximations are closer to the theoretical lowalgorithmic-randomness expectation. We also test the measure on larger objectsincluding dual, isomorphic and cospectral graphs for which we know thatalgorithmic randomness is low. We also release implementations of the methodsin most major programming languages---Wolfram Language (Mathematica), Matlab,R, Perl, Python, Pascal, C++, and Haskell---and a free online algorithmiccomplexity calculator.

A Decomposition Method for Global Evaluation of Shannon Entropy and Local Estimations of Algorithmic Complexity