On Computability and Learnability of the Pumping Lemma Function

Abstract

On the basis of the well known pumping lemma for regular languages we define such a partial function f: ${\mbox{I\!N}} \rightarrow{\mbox{I\!N}}$ that for every e it yields the least pumping constant for the language W _e . We ask whether f is computable. Not surprisingly f turns out to be non-computable. Then we check whether f is algorithmically learnable. This is also proved not to be the case. Further we investigate how powerful oracle is necessary to actually learn f. We prove that f is learnable in 0′. We also prove some facts relating f to arithmetical hierarchy.