Computing the incomputable has always been a challenge. For example, in finding the busiest Turing machines (Rado) given a number of symbols and states (whimsically called busy beavers). This means either finding Turing machines that, starting from an empty input, produce more non-blank symbols in their output tapes before halting than any other Turing machine of the same size, or Turing machines that, also starting from an empty input, have the greatest runtime before halting than any other Turing machine of the same size. Both problems are ultimately undecidable because of the Turing-complete capabilities of Turing machines, as proven by Alan Turing himself (that is, the capability of some Turing machines to simulate any other Turing machine).
In this new paper we describe how we have managed to calculate an approximation of a so-called Universal Distribution (aka Levin’s semi-measure) which connects the frequency of production of a string to its Kolmogorov complexity (K). The chief advantage of calculating an approximation of the Universal Distribution is that it is an incremental process over an average of a large number of Turing machines. One doesn’t get rid of the constant from the invariance theorem in the theory of algorithmic information theory (for example when Kolmogorov complexity is measured using 2 different universal Turing machines), yet one seems to have to make fewer arbitrary decisions.
One of the main advantages is that one can better deal with strings of very short lengths. Think about it! If one wished to approximate K for a single bit by using compression algorithms, the lossless compression algorithm would not be able to compress the single bit any further. And this not only happens for a single bit but for all strings up to a certain minimal length for which lossless compression algorithms are simply unsuitable (recall that a compression algorithm includes the decompression instructions together with the data in the new compressed object in order to make it self-decompressible).
The usefulness of lossless compression algorithms as a method for approximating K derives from the fact that compression is a sufficient test of non-randomness. This is because K is, more precisely than an uncomputable function, upper semi-computable, meaning that one can estimate upper bounds. The lossless compressed length of an object s (e.g. a string) is therefore an upper bound on K(s). The usefulness of the Coding Theorem Method (the theorem presented in this paper) will ultimately come down to whether it is useful in applications, which is the main motivation behind its conception, given the failure of compression algorithms to provide information about the possible K(s) values for s that are too short (shorter, say, than the typical size of the length of the instructions that a lossless compression algorithm has to add to the compressed object).
The state of the art of the Coding Theorem Method can be gleaned from this paper, recently made public by my colleagues and I, and announcing the release of the calculation of a universal distribution based on (5,2), that is, all Turing Machines with 5 states and 2 symbols: Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines
It represents a major improvement over this previous calculation of mine, that required new and more powerful techniques to deal with a semi-computable distribution. It improves our previous work in terms both of accuracy and coverage of number of short strings and validates previous calculations of universal distributions, showing the incremental nature of the method to be fully compatible with the other calculated universal distributions with smaller samples of small Turing machines (and for which the known Busy Beaver values could be used).
In this other paper we explain why our approximations of K are real-number values, showing that strict integer-value program size follows our Coding Theorem Method, and thus that ours constitutes a more fine-grained measure. It is also shown that logical depth departs from both strict program-size and the Coding Theorem Method evaluations, being in agreement with the theory for all these 3 measures. The paper is available online at: Correspondence and Independence of Numerical Evaluations of Algorithmic Information Measures
In the next post I will be announcing and also briefly explaining the results from another paper showing that not only can our Coding Theorem deal with short strings, but that we have found a way to validate the method by lossless compressibility. Moreover, we have found that in the transition period, where the Coding Theorem Method starts to be too expensive to be of practical use whereas the compression method starts to provide some results, the 2 methods are in great agreement with each other. Like an expensive microscope of great power (e.g. the LHC at CERN), our Coding Theorem Method requires an incredible amount of calculation. The trick is to know when to use a microscope–to know when a microscope is required rather than a telescope. We believe we are providing the tools to deal with the information content of the smallest entities in the computational universe.