On the power of equivalence queries

Abstract

In 1990, Angluin showed that no class exhibiting a combinatorial property called approximate fingerprints can be identified exactly using polynomially many Equivalence queries. Here we show that this is a necessary condition: every class without approximate fingerprints has an identification strategy that makes a polynomial number of Equivalence queries. Furthermore, if the class is reasonable in a technical sense, the computational power required by the strategy is within the polynomial-time hierarchy, so proving non-learnability is at least as hard as showing P different from NP.