Resource-bounded Kolmogorov complexity of hard languages: Extended abstract

Abstract

This paper explores further the relationship between randomness and complexity. The concept of time-/space-bounded Kolmogorov complexity of languages is introduced. Among others we show that there exists a language L in DTIME(22lin) such that the 2poly-time-bounded Kolmogorov complexity of L is exponential almost everywhere. We study time-/space- bounded Kolmogorov complexity of languages that are DTIME(22lin)-(SPACE (2lin)-) hard under polynomial-time Turing reductions. The connection between Kolmogorov-random languages and almost everywhere hard languages is investigated. We also show that Kolmogorov randomness implies Church randomness. Finally, we work out the relationship between time-/space- bounded Kolmogorov complexity and time-/space- bounded descriptional complexity of Boolean circuits and formulas for hard languages. This result provides a classification of exponential-size circuits and formulas in terms of the amount of information contained in them.