Simply turning logic into continuous formulas can break learning in subtle ways J?. Instead, we need hybrid systems that carefully balance learning and reasoning. We focus on three practical questions:

1. When can hybrid systems actually learn?

   Hybrid systems like abductive learning preserve the full capacity of learning and reasoning. We establish a PAC learnability condition for such systems. C?.

2. How do we handle ambiguity?

   Symbolic system feedback is often non-unique and ambiguous. We design methods that explicitly account for multiple plausible explanations and learn from them efficiently C?.

3. How do we speed up reasoning?

   Logical reasoning problems are generally NP-hard and cannot always be solved quickly. We use learning-guided methods for fast, reliable approximation P?.

Our goal is to combine theory, algorithms, and computation to build robust hybrid AI systems for real-world use.