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In years past, GOFAI (Good Old Fashioned AI) was heavily based on "rules" and symbolic computation based on rules. Unfortunately, that approach ran into stumbling blocks, and the world moved heavily towards statistical/probabilistic approaches leading to the current wave of interest in "machine learning".

It seems though, that the symbolic/rule-based approach probably still has application. So, could one "learn" rules using a probabilistic rule induction method, and then layer symbolic computation on top? If so, how could the whole process be made truly two-way, so that something "learned" from processing rules, can be fed back into how the system learns rules?

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Sure! This is a somewhat hot area right now.

There are lots of ways to do it.

Probably the main line of research is with Bayesian Networks (1980's) and Casual Networks (1990's). These are basically rule-based systems for reasoning probabilistically. They rely on a user-designed model, which corresponds well to rules (e.g. when blood pressure is high, then heart attack rates are elevated), but provide a robust way to reason about uncertainties in the presence of these rules. Contrast this with a pure learning approach, like a decision tree or a neural network, which tends to rely less on rules. Modern research in this area focuses on learning the structure of the network (which corresponds to the learning probabilistic rules) from data.

While it's possible to learn rules from data and then do symbolic reasoning atop them using other techniques (e.g. rule induction), this approach runs into the same problems that plague the learning of the structure of Bayesian networks: when is correlation causation? Causal Diagrams are the only good tool for answering this question, but my impression is that inferring their structure automatically is still an open question.

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