We study expression learning problems with syntactic restrictions
and introduce the class of \emph{finite-aspect checkable languages} to
characterize symbolic languages that admit decidable learning. The
semantics of such languages can be defined using a bounded amount of
auxiliary information that is independent of expression size but
depends on a fixed structure over which evaluation occurs. We
introduce a generic programming language for writing programs that
evaluate expression syntax trees, and we give a meta-theorem that
connects such programs for finite-aspect checkable languages to finite
tree automata, which allows us to derive new decidable learning
results and decision procedures for several expression learning
problems by writing programs in the programming language.