SAS

Abstract

If we consider a finite set $A$, it is desirable to represent every $x$ in $A$ by another shorter string compressed($x$) such that compressed($x$) describes unambiguously the initial $x$. Regarding the compression rate, ideally, one would like to achieve the information-theoretical bound |compressed($x$)| $\approx \log (|A|)$, for all $x$ in $A$. This optimal rate is achievable for c.e. (and also co-c.e.) sets $A$, because for such a set C($x$) $\leq \log (|A^{=n}|) + O(\log n)$ (C($x$) is the Kolmogorov complexity of string $x$ and $n$ is the length of $x$). In the time-bounded setting, we would like to have a polynomial-time type of unambiguous description. The relevant concept is the time-bounded distinguishing complexity, CD(), introduced by Sipser. The $t$-time bounded distinguishing complexity of $x$, denoted CD$^t(x)$ is the length of the shortest program $p$ that accepts $x$ and only $x$ within time $t(|x|)$. Recently we have shown that for sets in P, NP, PSPACE, optimal compression can be achieved, using some reasonable hardness assumptions. We show that this also holds for sets in P/poly, i.e., for sets computable by polynomial-size circuits. Furthermore, we sketch here a different proof method (even though some key elements are common) suggested by Vinodchandran, which needs a weaker hardness assumption.