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

Abstract

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' 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.

Computing with New Resources

“…Some related results are also proved in [JK06] and some proofs in [JK07b] are based on the proofs in [JK06]. The results of this section are from the above papers.…”

“…In [JK07b], several variants of BNCEx model are explored that allow only uniformly bounded number of bounded negative counterexamples. Specifically, -in the model BNC n Ex, a learner makes a subset query for every new conjecture until n negative counterexamples have been received (still the learner can change its mind if new positive data have been received) -in the model BGNC n Ex, a learner is allowed to make subset queries (not necessarily for every conjecture) until n counterexamples have been received.…”

“…The paper [JK07b] is devoted to a study of relationships between all these models and how they are related to the similar models when counterexamples are of arbitrary size (specifically, the variants NC n Ex and GNC n Ex corresponding to the above BNC n Ex and BGNC n Ex are considered). Some related results are also proved in [JK06] and some proofs in [JK07b] are based on the proofs in [JK06].…”

“…The first problem studied in [JK07b] is the following: under which circumstances, more limited learners of the above types can simulate more capable learners. The major result is that BNCEx-type and BNCBc-type learners getting up to n least bounded counterexamples can be simulated by the learners of the same type and getting up to 2n − 1 just answers "yes" if a bounded counterexample exists and "no", otherwise.…”

“…Another issue addressed in [JK07b] is how short counterexamples fair against arbitrary or least counterexamples (this includes also the cases when just answers 'no' are returned instead of counterexamples). As it was established in [JK06], one answer 'no' used by an NCEx-learner can sometimes do more than unbounded number of least (short) counterexamples used by Bc * -learners.…”