Abstract Approaches based on the idea generically called distributional learning have been making great success in the algorithmic learning of several rich subclasses of context-free languages and their extensions. Those language classes are defined by properties concerning string-context relation. In this paper, we present a distributional learning algorithm for conjunctive grammars with the $k$-finite context property ( $k$-fcp ) for each natural number $k$. We also compare our result with the closely related work by Clark et al. (JMLR 2010) [5] on learning some context-free grammars using contextual binary feature grammars ( cbfgs). We prove that the context-free grammars targeted by their algorithm have the $k$-fcp . Moreover, we show that every exact cbfg has the $k$-fcp , too, while not all of them are learnable by their algorithm. Clark et al. conjectured a learning algorithm for exact cbfgs should exist. This paper answers their conjecture in a positive way.