On the optimal compression of sets in PSPACE

TL;DR

This paper demonstrates that, under certain complexity assumptions, sets in PSPACE can be compressed to near the information-theoretic limit with polynomial-time decodability, advancing understanding of space-bounded compression.

Contribution

It establishes a near-optimal compression scheme for PSPACE sets assuming a complexity-theoretic separation, linking compression efficiency to complexity class separations.

Findings

Compression length is at most log of the set size plus O(log n)

Decoding can be performed in polynomial time

Achieves near-information-theoretic optimal compression under certain assumptions

Abstract

We show that if DTIME[2^{O(n)}] is not included in DSPACE[2^{o(n)}], then, for every set B in PSPACE, 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 trem, this achieves the information-theoretical optimum for string compression.