On Learning Languages from Positive Data and a Limited Number of Short Counterexamples

We consider two variants of a model for learning languages in the limit from positive data and a limited number of short negative counterexamples (counterexamples are considered to be short if they are smaller that the largest element of input seen so far). Negative counterexamples to a conjecture are examples which belong to the conjectured language but do not belong to the input language. Within this framework, we explore how/when learners using n short (arbitrary) negative counterexamples can be simulated (or simulate) using least short counterexamples or just `no\u27 answers from a teacher. We also study how a limited number of short counterexamples fairs against unconstrained counterexamples, and also compare their capabilities with the data that can be obtained from subset, superset, and equivalence queries (possibly with counterexamples). A surprising result is that just one short counterexample can sometimes be more useful than any bounded number of counterexamples of arbitrary size. Most of results exhibit salient examples of languages learnable or not learnable within corresponding variants of our models