On optimal language compression for sets in PSPACE/poly

TL;DR

This paper investigates the limits of optimal compression for sets in PSPACE/poly, showing conditions under which near-optimal compression is achievable and highlighting complexity barriers beyond PSPACE/poly.

Contribution

It establishes a conditional method for near-optimal compression of strings in PSPACE/poly and demonstrates that such optimal compression cannot be extended to more complex sets.

Findings

Achieves near-information-theoretic optimal compression for PSPACE/poly sets.

Shows that optimal compression is impossible for sets beyond PSPACE/poly.

Depends on a complexity-theoretic assumption relating DTIME and DSPACE classes.

Abstract

We show that if DTIME[2^O(n)] is not included in DSPACE[2^o(n)], then, for every set B in PSPACE/poly, all strings x in B of length n can be represented by a string compressed(x) of length at most log(|B^{=n}|)+O(log n), such that a polynomial-time algorithm, given compressed(x), can distinguish x from all the other strings in B^{=n}. Modulo the O(log n) additive term, this achieves the information-theoretic optimum for string compression. We also observe that optimal compression is not possible for sets more complex than PSPACE/poly because for any time-constructible superpolynomial function t, there is a set A computable in space t(n) such that at least one string x of length n requires compressed(x) to be of length 2 log(|A^=n|).