Checking extracted rules in Neural Networks

In this paper we investigate formal verification of extracted rules for Neural Networks under a complexity theoretic point of view. A rule is a global property or a pattern concerning a large portion of the input space of a network. These rules are algorithmically extracted from networks in an effort to better understand their inner way of working. Here, three problems will be in the focus: Does a given set of rules apply to a given network? Is a given set of rules consistent or do the rules contradict themselves? Is a given set of rules exhaustive in the sense that for every input the output is determined? Finding algorithms that extract such rules out of networks has been investigated over the last 30 years, however, to the author's current knowledge, no attempt in verification was made until now. A lot of attempts of extracting rules use heuristics involving randomness and over-approximation, so it might be beneficial to know whether knowledge obtained in that way can actually be trusted. We investigate the above questions for neural networks with ReLU-activation as well as for Boolean networks, each for several types of rules. We demonstrate how these problems can be reduced to each other and show that most of them are co-NP-complete.