L-Printable Sets

Abstract

A language is if there is a logspace algorithm which, on input 1n, prints all members in the language of length n. Following the work of Allender and Rubinstein [SIAM J. Comput., 17 (1988), pp. 1193--1202] on P-printable sets, we present some simple properties of the sets. This definition of L-printable is robust and allows us to give alternate characterizations of the sets in terms of tally sets and Kolmogorov complexity. In addition, we show that a regular or context-free language is if and only if it is sparse, and we investigate the relationship between sets, L-rankable sets (i.e., sets A having a logspace algorithm that, on input x, outputs the number of elements of A that precede x in the standard lexicographic ordering of strings), and the sparse sets in L. We prove that under reasonable complexity-theoretic assumptions, these three classes of sets are all different. We also show that the class of sets of small generalized Kolmogorov space complexity is exactly the class of sets that are L-isomorphic to tally languages.