This paper explores the language acquisition mechanism within a recent theoretical nativist framework that assumes some computational principles. We will review previous accounts of the logical problem of language acquisition, arguing that language acquisition is part of general cognitive mechanism or at least associated with maturation of cognitive skills. For a theoretical framework, we will adopt the minimalist program and its principles. To support our theoretical argument, we will introduce empirical evidence from ESL (English as a Second Language) and SLI (Specific Language Impairment) data. The two types of data will illustrate that there might be some relationship between the development of language skills and that of the cognitive skills.