A Review of Methods for Estimating Algorithmic Complexity and Beyond: Options, and Challenges
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I review previous and novel techniques in the field of applications of algorithmic (Kolmogorov) complexity starting from those dominating the area such as statistical lossless compression algorithms to more recent approaches that advance and complement those based on statistical compression but also bring new challenges and have their own limitations. I present evidence suggesting that different methods complement each other for different regimes. I also make the case that despite the many challenges, some methods can be better motivated by and better grounded in the principles of algorithmic complexity theory. I will explain how different approaches to algorithmic complexity can explore the relaxation of different necessary and sufficient conditions in their exploration of numerical applicability with some of those approaches taking greater risks than others in exchange for application relevancy. I conclude with a discussion of possible directions that may be taken to advance the area and encourage methodological innovation.
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