Grace et al. (2018) showed that humans could estimate ratios and differences of stimulus magnitudes by feedback and without explicit instruction in a nonsymbolic 'artificial algebra' task, but that responding depended on both operations even though only one was trained. Here we asked whether control by the trained operation would increase over several sessions, that is, if perceptual learning would occur. Observers (n = 16) completed four sessions in which feedback was based on either ratios or differences for stimulus pairs that varied in brightness (Experiment 1) or line length (Experiment 2). Results showed that control by the trained and untrained operations increased and decreased, respectively, over the sessions, indicating perceptual learning. For about two thirds of individual sessions, regressions indicated significant control by both differences and ratios, suggesting that the perceptual system automatically computes two operations. The similarity of results across experiments with both intensive (brightness) and extensive (line length) stimulus dimensions suggests that differences and ratios are computed centrally, perhaps as part of a general system for processing magnitudes (cf. Walsh, 2003).

中文翻译：

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非符号“人工代数”中的差异和比率：扩展训练的影响

格蕾丝等人。(2018) 表明，人类可以通过反馈估计刺激幅度的比率和差异，而无需在非符号“人工代数”任务中得到明确的指导，但即使只有一个经过训练，响应也取决于两种操作。在这里，我们询问训练操作的控制是否会在几个会话中增加，即是否会发生感知学习。观察者 (n = 16) 完成了四个会话，其中反馈基于亮度（实验 1）或线长（实验 2）变化的刺激对的比率或差异。结果表明，受过训练和未受过训练的操作的控制分别在会话中增加和减少，表明感知学习。对于大约三分之二的个人会议，回归表明差异和比率具有显着的控制，这表明感知系统自动计算两个操作。具有密集（亮度）和广泛（线长度）刺激维度的实验结果的相似性表明差异和比率是集中计算的，也许作为处理幅度的一般系统的一部分（参见 Walsh，2003）。