Symbol Correctness in Deep Neural Networks Containing Symbolic Layers

We identify puzzles (Yang et al., 2020; Li et al., 2023b; Wang et al., and formalize an intuitive, high-level principle 2023; Topan et al., 2021) and performing commonsense that can guide the design and analysis of NS-reasoning about images (Yang et al., 2020; Li et al., 2023a). DNNs: symbol correctness, the correctness of the intermediate symbols predicted by the neural When a NS-DNN forward pass reaches the boundary between layers with respect to a (generally unknown) a neural layer and a symbolic layer, the neural layer's ground-truth symbolic representation of the input real-valued predictions are instantiated as symbols for the data. We demonstrate that symbol correctness is subsequent symbolic layer to operate over. Because training a necessary property for NS-DNN explainability is end-to-end, the neural layers need to learn a mapping from and transfer learning (despite being in general impossible raw input data to these intermediate symbols without any to train for). Moreover, we show that supervision of the symbols. For example, in visual addition, the framework of symbol correctness provides a a pair of handwritten digits is labeled with the mathematical precise way to reason and communicate about sum of the digits, not the individual summands.