Kolmogorov complexity and Shannon entropy are conceptually different measures. However, for any recursive probability distribution, the expected value of Kolmogorov complexity equals its Shannon entropy, up to a constant. We study if a similar relationship holds for Rényi and Tsallis entropies of order α, showing that it only holds for α = 1. Regarding a time-bounded analogue relationship, we show that, for some distributions we have a similar result. We prove that, for universal time-bounded distribution m t (x), Tsallis and Rényi entropies converge if and only if α is greater than 1. We also establish the uniform continuity of these entropies.
Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program in absence of auxiliary information. Variants known as logical depth and computational depth are expressed in Kolmogorov complexity theory.We derive quantitative relation between logical depth and computational depth and unify the different depth notions by relating them to A. Kolmogorov and L. Levin's fruitful notion of randomness deficiency. Subsequently, we revisit the computational depth of infinite strings, introducing the notion of super deep sequences and relate it with other approaches.
We prove several results relating injective one-way functions, timebounded conditional Kolmogorov complexity, and time-bounded conditional entropy. First we establish a connection between injective, strong and weak one-way functions and the expected value of the polynomial time-bounded Kolmogorov complexity, denoted here by E(K t (x|f (x))). These results are in both directions. More precisely, conditions on E(K t (x|f (x))) that imply that f is a weak one-way function, and properties of E(K t (x|f (x))) that are implied by the fact that f is a strong oneway function. In particular, we prove a separation result: based on the concept of time-bounded Kolmogorov complexity, we find an interval in which every function f is a necessarily weak but not a strong one-way function. Then we propose an individual approach to injective one-way functions based on Kolmogorov complexity, defining Kolmogorov one-way functions and prove some relationships between the new proposal and the classical definition of one-way functions, showing that a Kolmogorov one-way function is also a deterministic one-way function. A relationship between Kolmogorov one-way functions and the conjecture of polynomial time symmetry of information is also proved. Finally, we relate E(K t (x|f (x))) and two forms of time-bounded entropy, the unpredictable entropy H unp , in which "one-wayness" of a function can be easily expressed, and the Yao + entropy, a measure based on compression/decompression schema in which only the decompressor is restricted to be time-bounded.
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