We study expression learning problems with syntactic restrictions and introduce the class of 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, which we call semantic aspects, that is independent of expression size but depends on a fixed structure over which evaluation occurs. We introduce a generic programming language for expressing model-checking programs that work over 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 model-checking programs in the programming language.