Signal degradation is ubiquitous and computational restoration of degraded signal has been investigated for many years. Recently, it is reported that the capability of signal restoration is fundamentally limited by the perception-distortion tradeoff, i.e. the distortion and the perceptual difference between the restored signal and the ideal `original' signal cannot be made both minimal simultaneously. Distortion corresponds to signal fidelity and perceptual difference corresponds to perceptual naturalness, both of which are important metrics in practice. Besides, there is another dimension worthy of consideration, namely the semantic quality or the utility for recognition purpose, of the restored signal. In this paper, we extend the previous perception-distortion tradeoff to the case of classification-distortion-perception (CDP) tradeoff, where we introduced the classification error rate of the restored signal in addition to distortion and perceptual difference. Two versions of the CDP tradeoff are considered, one using a predefined classifier and the other dealing with the optimal classifier for the restored signal. For both versions, we can rigorously prove the existence of the CDP tradeoff, i.e. the distortion, perceptual difference, and classification error rate cannot be made all minimal simultaneously. Our findings can be useful especially for computer vision researches where some low-level vision tasks (signal restoration) serve for high-level vision tasks (visual understanding).