Improving the Space-Bounded Version of Muchnik’s Conditional Complexity Theorem via “Naive” Derandomization

Abstract

Many theorems about Kolmogorov complexity rely on existence of combinatorial objects with specific properties. Usually the probabilistic method gives such objects with better parameters than explicit constructions do. But the probabilistic method does not give "effective" variants of such theorems, i.e. variants for resource-bounded Kolmogorov complexity. We show that a "naive derandomization" approach of replacing these objects by the output of Nisan-Wigderson pseudo-random generator may give polynomial-space variants of such theorems. Specifically, we improve the preceding polynomial-space analogue of Muchnik's conditional complexity theorem. I.e., for all a and b there exists a program p of least possible length that transforms a to b and is simple conditional on b. Here all programs work in polynomial space and all complexities are measured with logarithmic accuracy instead of polylogarithmic one in the previous work.

Kolmogorov Complexity and Algorithmic Randomness

“…Эта идея была предложена (в другой ситуации и ещё до Мучника) в статье [14], и применена к доказательству теоремы Мучника в [125]. Преимущество такого подхода в том, что можно использовать известные явные конструкции экстракторов и другую тех-нику теории сложности (например, псевдослучайные генераторы, как предложил А. Ромащенко) для доказательства варианта теоремы Мучника, где используется сложность с ограничением на используемую память [123,124]. (Вообще сложность с ограничением на ресурсы | важная отдельная тема, которую мы в нашей книге не обсуждаем.)…”

Short lists for shortest descriptions in short time

“…They show that the description p in (iii) can be computed efficiently from x given O(log |x|) bits of advice, and we will improve their result further in Corollary 9 by showing that the number of advice bits in (i) can be reduced from O(log |x|) to O(1). In a similar vein, Musatov and Romashchenko [7,8] investigated Muchnik's Theorem in the context of space complexity.…”

On Algorithmic Statistics for Space-Bounded Algorithms

“…To reduce it we will use the Nisan-Wigderson generator. The same idea was used in [7]. • Function G m is computable in space poly(f );…”