This dissertation develops a cognitive model describing when children learn to group distinct sound segments (allophones) into abstract equivalence classes (phonemes). The allophones an individual acquires are arbitrary and determined by their particular input, yet are intricately involved in language cognition once learned. The proposed acquisition model characterizes the role of surface segment alternations in children's input by using the Tolerance Principle (Yang 2016) to evaluate the cognitive cost of possible phoneme inventory structures iteratively as a child's vocabulary grows. This Alternation-sensitive Phoneme Learning model therefore traces the emergence of abstract representations from concrete speech stimuli, starting from a default representation where underlying contrasts simply mirror surface-segment contrasts (Invariant Transparency Hypothesis, Ringe & Eska 2013). A longitudinal corpus study of four children's alveolar stop and flap productions establishes that English medial flap allophony follows a U-shaped acquisition course, which is characteristic of learning linguistic rules or generalizations. The Alternation-sensitive Phoneme Learning cognitive model is then validated by accurately predicting the timing of changes in each child's productions, which signal allophone acquisition. A second case study models the historical process of secondary split in Menominee mid and high back vowels. Here, the acquisition model serves as an independently motivated quantitative test for the occurrence of phonemic split, providing an alternative to traditional reliance on linguists' case-specific subjective judgements about when it might occur. A third case study examines the phonemic status of the velar nasal in German, showing how this acquisition model can discriminate between tolerable grammars and the subset of tolerable grammars that are learnable, with implications for the relationship between formal language description and psychological representation. This dissertation's approach synthesizes insights from computational modelling, naturalistic corpus data, historical linguistics, and experimental research on child language acquisition. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]