In this paper we further explore the relationship between the PAC and $k$-RFA models, with several interesting results. First, we develop a characterization of $k$-RFA learnability upon which we build a general tool for proving hardness results. We then apply this and other new techniques for studying RFA learning to two particularly expressive function classes, $k$-decision-lists ($k$-DL) and $k$-TOP, the class of thresholds of parity functions in which each parity function takes at most $k$ inputs. Among other results, we prove a hardness result for $k$-RFA learnability of $k$-DL, $k \leq n-2$. In sharp contrast, an $(n-1)$-RFA algorithm for learning $(n-1)$-decision-lists is presented. Similarly, we prove that 1-DL is learnable if and only if at least half of the inputs are visible in each instance. In addition, we show that there is a uniform-distribution $k$-RFA learning algorithm for the class of $k$-decision-lists in which the output value alternates at most $j$ times. The latter algorithm is efficient for $j$ and $k$ constant. For $k$-TOP we show weak learnability by a $k$-RFA algorithm (efficient for constant $k$) and strong uniform-distribution $k$-RFA learnability of $k$-TOP with polynomial sample complexity. Finally, by combining some of our $k$-DL and $k$-TOP results, we show that, unlike the PAC model, weak learning does {\em not} imply strong learning in the $k$-RFA model.