Circuit Expressions of Low Kolmogorov Complexity

We study circuit expressions of logarithmic and poly-logarithmic polynomial-time Kolmogorov complexity, focusing on their complexity-theoretic characterizations and learnability properties. They provide a nontrivial circuit-like characterization for a natural nonuniform complexity class that lacked it up to now. We show that circuit expressions of this kind can be learned with membership queries in polynomial time if and only if every NE-predicate is E-solvable. Thus they are learnable given that the learner is allowed the extra use of an oracle in NP. The precise way of accessing the oracle is shown to be optimal under relativization. We present a precise characterization of the subclass de ned by Kolmogorov-easy circuit expressions that can be constructed from membership queries in polynomial time, with some consequences for the structure of reduction and equivalence classes of tally sets of very low density. Preliminary, sometimes weaker versions of the results in this paper were announced at EuroColt'95. Partially supported by the E.U. through the ESPRIT Long Term Research Project 20244 (ALCOM-IT) and through the HCM Network CHRX-CT93-0415 (COLORET); by the Spanish DGICYT through project PB95-0787 (KOALA), and by Acciones Integradas Hispano-Alemanas HA-119-B and AL-201-B.