Language compression and pseudorandom generators

Abstract

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 A =n 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(( log A =n + log n) log n); using both nondeterminism and randomness, we can make do with an excess term of O(log 3 n). With randomness alone, we show a lower bound of n − log A =n − O(log n) on the description length of strings in A of length n, and a lower bound of 2 · log A =n − 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(log n). The key ingredient for our upper bounds is the relativizable hardness versus randomness tradeoffs based on the Nisan-Wigderson pseudorandom generator construction.

On Optimal Language Compression for Sets in PSPACE/poly

“…This leads to the general idea of considering time-bounded versions of Kolmogorov complexity. An interesting line of research [Sip83,BFL01,BLvM05,LR05] in time-bounded Kolmogorov complexity, which we also pursue in this paper, focuses on a notion called time-bounded distinguishing Kolmogorov complexity, CD t (·), introduced by Sipser [Sip83]. We say that a description (called a program in the Kolmogorov complexity literature) p distinguishes a string x if p accepts x and only x. CD t,A (x) is the size of the smallest program that distinguishes x and that runs in time t(|x|) with access to the oracle A. Sipser showed that, for every set B and every length n, there is a string w of length poly(n) such that, for every x ∈ B =n , CD poly,B =n (x | w) ≤ log(|B =n |)+log log(|B =n |)+O(1).…”

“…The precision term poly log(n · 1/ε) has been improved to O(log(n · 1/ε)) in [BMVZ13]. In addition, Buhrman, Lee, and van Melkebeek [BLvM05] showed that for all B and x ∈ B =n , CND poly,B =n (x) ≤ log(|B =n |) + O(( log(|B =n |) + log n) log n), where CND is similar to CD except that the distinguisher program is nondeterministic.…”

On the Optimal Compression of Sets in PSPACE

“…This leads to the idea of considering a time-bounded version of Kolmogorov complexity. An interesting line of research [Sip83,BFL01,BLvM05,LR05], which we also pursue in this paper, focuses on the time-bounded distinguishing Kolmogorov complexity, CD t (·). We say that a program p distinguishes x if p accepts x and only x. CD t,A (x) is the size of the smallest program that distinguishes x and that runs in time t(|x|) with access to the oracle A. Buhrman, Fortnow, and Laplante [BFL01] show that for some polynomial p, for every set B, and every string x ∈ B =n , CD p,B =n (x) ≤ 2 log(|B =n |) + O(log n).…”

“…Buhrman, Fortnow, and Laplante [BFL01] show that log(|B =n |) can be achieved if we allow a few exceptions: For any B, any ǫ, for all except a fraction of ǫ strings x ∈ B =n , CD poly,B =n (x) ≤ log(|B =n |) + poly log(n · 1/ǫ). Buhrman, Lee, and van Melkebeek [BLvM05] show that for all B and x ∈ B =n , CND poly,B =n (x) ≤ log(|B =n |) + O(( log(|B =n |) + log n) log n), wher CND is similar to CD except that the distinguisher program is nondeterministic.…”

Compression of Samplable Sources

“…The study of language compression (see e.g. Buhrman et al 2004 for recent results and references to earlier work) focuses on the worst-case compressibility (in the above sense) for sources that are flat over an efficiently decidable support (i.e. sources with membership oracles, just as we study).…”