Not so fast - you might have precise and efficient functions that do things like basic arithmetic. What you might not have is a model that can reason mathematic

Not so fast - you might have precise and efficient functions that do things like basic arithmetic. What you might not have is a model that can reason mathematically. You need a model to do things like basic arithmetic functions so that semantic and arbitrary relations get encoded in the weights of a network. You see this type of glitch crop up in tokenizing schemes in large language models. If you attempt working with character level reasoning or output construction, it will often fail. Trying to get ChatGPT 4 to output a sentence, and then that sentence backwards, or every other word spelled backwards, is almost impossible. If you instead prompt the model to produce an answer with a delimiter between every character, like #, also to replace spaces, it can resolve the problems much more often than with standard punctuation and spaces. The idea applies to abstractions that aren't only individual tokens, but specific concepts and ideas that in turn serve as atomic components of higher abstractions. In order to use those concepts successfully, the model has to be able to encode the thing and its relationships effectively in the context of whatever else it learns. For a given architecture, you could do the work and manually create the encoding scheme for something like arithmetic, and it could probably be very efficient and effective. What you miss is the potential for fuzzy overlaps in the long tail that only come about through the imperfect, bespoke encodings learned in the context of your chosen optimizer.