The Coding Theorem of L.A. Levin connects unconditional prefix Kolmogorov complexity with the discrete universal distribution. There are conditional versions referred to in several publications but as yet there exist no written proofs in English. Here we provide those proofs. They use a different definition than the standard one for the conditional version of the discrete universal distribution. Under the classic definition of conditional probability, there is no conditional version of the Coding Theorem.
Diverse applications of Kolmogorov complexity to learning [CIKK16], circuit complexity [OPS19], cryptography [LP20], average-case complexity [Hir21], and proof search [Kra22] have been discovered in recent years. Since the running time of algorithms is a key resource in these fields, it is crucial in the corresponding arguments to consider time-bounded variants of Kolmogorov complexity. While fruitful interactions between time-bounded Kolmogorov complexity and different areas of theoretical computer science have been known for quite a while (e.g., [Sip83, Ko91, ABK+06, AF09], to name a few), t…
Universidad de Castilla-La Mancha（卡斯蒂利亚-拉曼恰大学）；Universitat Politècnica de València（瓦伦西亚理工大学）
Recent research in machine teaching has explored the instruction of any concept expressed in a universal language. In this compositional context, new experimental results have shown that there exist data teaching sets surprisingly shorter than the concept description itself. However, there exists a bound for those remarkable experimental findings through teaching size and concept complexity that we further explore here. As concepts are rarely taught in isolation we investigate the best configuration of concepts to teach a given set of concepts, where those that have been acquired first can be …
