Title: Bridging the Gap between Semi-supervised and Supervised Continual Learning via Data Programming. Abstract: Semi-supervised continual learning (SSCL) has shown its utility in learning cumulative knowledge with partially labeled data per task. However, the state-of-the-art has yet to explicitly address how to reduce the performance gap between using partially labeled data and fully labeled. In response, we propose a general-purpose SSCL framework, namely DP-SSCL, that uses data programming (DP) to pseudo-label the unlabeled data per task, and then cascades both ground-truth-labeled and pseudo-labeled data to update a downstream supervised continual learning model. The framework includes a feedback loop that brings mutual benefits: On one hand, DP-SSCL inherits guaranteed pseudo-labeling quality from DP techniques to improve continual learning, approaching the performance of using fully supervised data. On the other hand, knowledge transfer from previous tasks facilitates training of the DP pseudo-labeler, taking advantage of cumulative information via self-teaching. Experiments show that (1) DP-SSCL bridges the performance gap, approaching the final accuracy and catastrophic forgetting as using fully labeled data, (2) DP-SSCL outperforms existing SSCL approaches at low cost, by up to $25\%$ higher final accuracy and lower catastrophic forgetting on standard benchmarks, while reducing memory overhead from $100$ MB level to $1$ MB level at the same time complexity, and (3) DP-SSCL is flexible, maintaining steady performance supporting plug-and-play extensions for a variety of supervised continual learning models.