Sandia National Laboratories (SNL) requested a measure of possible human error for each state verification method for a safety mechanism performed at partnering production agencies. A team of three human factors individuals were tasked with conducting observations during site visits of both production agencies in order to complete a Human Reliability Analysis (HRA). A HRA will be used because it provides both qualitative and quantitative reports of human error. This report is the first phase of that effort, which will describe the methods which occur at one of the production agencies.
Despite their noted potential in polynomial-system solving, there are few concrete examples of Newton-Okounkov bodies arising from applications. Accordingly, in this paper, we introduce a new application of Newton-Okounkov body theory to the study of chemical reaction networks and compute several examples. An important invariant of a chemical reaction network is its maximum number of positive steady states Here, we introduce a new upper bound on this number, namely the ‘Newton-Okounkov body bound’ of a chemical reaction network. Through explicit examples, we show that the Newton-Okounkov body bound of a network gives a good upper bound on its maximum number of positive steady states. We also compare this Newton-Okounkov body bound to a related upper bound, namely the mixed volume of a chemical reaction network, and find that it often achieves better bounds.
In this work, we introduce a family of novel activation functions for deep neural networks that approximate n-ary, or n-argument, probabilistic logic. Logic has long been used to encode complex relationships between claims that are either true or false. Thus, these activation functions provide a step towards models that can efficiently encode information. Unfortunately, typical feedforward networks with elementwise activation functions cannot capture certain relationships succinctly, such as the exclusive disjunction (p xor q) and conditioned disjunction (if c then p else q). Our n-ary activation functions address this challenge by approximating belief functions (probabilistic Boolean logic) with logit representations of probability and experiments demonstrate the ability to learn arbitrary logical ground truths in a single layer. Further, by representing belief tables using a basis that associates the number of nonzero parameters with the effective arity of each belief function, we forge a concrete relationship between logical complexity and sparsity, thus opening new optimization approaches to suppress logical complexity during training. We provide a computationally efficient PyTorch implementation and test our activation functions against other logic-approximating activation functions on both traditional machine learning tasks as well as reproducing known logical relationships.
