Ashby, Townsend, and colleagues have developed an important model for the psychophysical identification task, termed General Recognition Theory (also called Decision Bound Theory ) based on multidimensional signal detection theory, that has been extended to categorization, Garner filtering, similarity judgments, and preference successfully. The virtue of this theory (and its extensions) is the ability to assess the nature of the perceptual representations of stimuli (e.g., their dimensional interactions), within a single framework and across paradigms. This paper further extends this theory to the problem of same different judgments. Three tasks are described for the example of two-dimensional stimuli. In two of these tasks only one dimension is relevant (component same-different). In the third, both dimensions are deemed relevant (standard same-different). Several theorems concerning both response accuracy and response time have been proved relating a property termed perceptual separability and certain response time invariance that can be observed in the component same-different tasks. This property, generally speaking, is the constancy of perception of one dimension when another irrelevant dimension changes in the stimulus. Also, under the assumption of perceptual independence, response probabilities and response times of the two component tasks can be meaningfully related to those measures in the standard same-dfifferent paradigm.