Positive Characteristic Sets for Relational Pattern Languages
S. Mahmoud Mousawi, Sandra Zilles
TL;DR
The paper investigates learning relational pattern languages from positive examples by introducing positive characteristic sets and linking them to telltales. It establishes that positive characteristic sets exist exactly when telltales exist, and studies two core relations—reversal and equal-length—within erasing and non-erasing settings. For reversal, the authors show a sharp contrast: non-erasing reversal languages admit positive characteristic sets, while erasing reversal languages over a binary alphabet do not, highlighting fundamental learnability barriers. For equal-length, they prove linear-size positive characteristic sets exist when the alphabet is at least three, and identify a binary-subclass P_{2,3} where similar linear-size sets exist under careful structural constraints. Overall, the work advances understanding of efficient learning from positive data in relational pattern languages and maps out key open questions for broader alphabet regimes and relation types.
Abstract
In the context of learning formal languages, data about an unknown target language L is given in terms of a set of (word,label) pairs, where a binary label indicates whether or not the given word belongs to L. A (polynomial-size) characteristic set for L, with respect to a reference class L of languages, is a set of such pairs that satisfies certain conditions allowing a learning algorithm to (efficiently) identify L within L. In this paper, we introduce the notion of positive characteristic set, referring to characteristic sets of only positive examples. These are of importance in the context of learning from positive examples only. We study this notion for classes of relational pattern languages, which are of relevance to various applications in string processing.