Efficient Coding Theorem via Probabilistic Representations
Explore the Efficient Coding Theorem through Probabilistic Representations and its applications in algorithms and complexity. Discover the pillars of Kolmogorov Complexity and the time-bounded version. Learn about efficiently samplable objects and their short probabilistic representations. Applications include instance-wise search, probabilistic equivalence, and time hierarchy for sampling distributions.
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An Efficient Coding Theorem via Probabilistic Representations and Its Applications Zhenjian Lu Igor Oliveira University of Warwick ICALP 2021 Supported by a Royal Society University Research Fellowship
Which objects can be compressed? string ? short encoding of ? 01001101 0111010111 010001110 Kolmogorov Complexity: K ? =minimum length of a program that prints ?
In Algorithms and Complexity: We would like to recover objects from their representations in bounded time. Which objects admit short and effective representations?
Time-bounded Kolmogorov complexity [Levin 84] [O 19] A randomized analogue: ?? ? = min . P + log Time P ??? ? = min . P + log Time P Program ? that prints ? ?????????? Program ?, Pr ? prints ? .99 short effective probabilistic representation of ? Many applications, but basic questions remain open: [OS 17, LOS 21]n-bit Primes of rKt complexity n Primes of small Kt complexity? Can we compute Kt(?) in poly time? [O 19] Computing rKt(?) is not in BPP.
Pillars of Kolmogorov Complexity Time-bounded version? Hardness Assumption Language Compression Theorem ? ? ? ?,??+ ?(????) Hardness Assumption Symmetry of Information ? ?,? ? ? + ? ?|? ? ? + ? ?|? ? Coding Theorem Samplable objects have short representations :
This Work: An Efficient Coding Theorem for rKt EfficientlySamplable objects admit short and effectiveprobabilistic representations. Theorem Efficient generation of representation: Given ?,?,? and ?, w.h.p, we can compute in time poly(|?|) an rKt representation of ? with complexity ? = ?(log 1/? + log? + log?). Program P that runs in time ? and prints ? w.h.p such that P + log? ?. Extremely useful in applications! Magic : Running time does not depend on ?.
Applications 1. Polynomial time instance-wisesearch-to-decision reduction 2. Equivalence between probabilistic representations and samplability 3. Strong time hierarchy for sampling distributions (conditional) Thank you! Thank you!
