Abstract.
The language compression problem asks for succinct descriptions of the strings in a language A such that the strings can be efficiently recovered from their description when given a membership oracle for A. We study randomized and nondeterministic decompression schemes and investigate how close we can get to the information theoretic lower bound of \(\log {\left\| {A^{{ = n}} } \right\|}\) for the description length of strings of length n.
Using nondeterminism alone, we can achieve the information theoretic lower bound up to an additive term of \(O{\left( {{\left( {{\sqrt {\log {\left\| {A^{{ = n}} } \right\|}} } + \log n} \right)}\log n} \right)};\) using both nondeterminism and randomness, we can make do with an excess term of \(O{\left( {\log ^{3} n} \right)}.\) With randomness alone, we show a lower bound of \(n - \log {\left\| {A^{{ = n}} } \right\|} - O{\left( {\log n} \right)}\) on the description length of strings in A of length n, and a lower bound of \(2 \cdot \log {\left\| {A^{{ = n}} } \right\|} - O(1)\) on the length of any program that distinguishes a given string of length n in A from any other string. The latter lower bound is tight up to an additive term of \(O{\left( {\log n} \right)}.\)
The key ingredient for our upper bounds is the relativizable hardness versus randomness tradeoffs based on the Nisan–Wigderson pseudorandom generator construction.
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Manuscript received 12 August 2004
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Buhrman, H., Lee, T. & Melkebeek, D.v. Language compression and pseudorandom generators. comput. complex. 14, 228–255 (2005). https://doi.org/10.1007/s00037-005-0199-5
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DOI: https://doi.org/10.1007/s00037-005-0199-5