On optimal language compression for sets in PSPACE/poly

الأصل · EN

We show that if DTIME[2ᵒ(n)] is not included in DSPACE[2ᵒ(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|).