Polynomial Threshold Functions of Bounded Tree-Width: Some Explainability and Complexity Aspects

The tree-width of a multivariate polynomial is the tree-width of the hypergraph with hyperedges corresponding to its terms. Multivariate polynomials of bounded tree-width have been studied by Makowsky and Meer as a new sparsity condition that allows for polynomial solvability of problems which are intractable in general. We consider a variation on this theme for Boolean variables. A representation of a Boolean function as the sign of a polynomial is called a polynomial threshold representation. We discuss Boolean functions representable as polynomial threshold functions of bounded tree-width and present two applications to Bayesian network classifiers, a probabilistic graphical model. Both applications are in Explainable Artificial Intelligence (XAI), the research area dealing with the black-box nature of many recent machine learning models. We also give a separation result between the representational power of positive and general polynomial threshold functions.

Interpreting Deepcode, a learned feedback code

Deep learning methods have recently been used to construct non-linear codes for the additive white Gaussian noise (AWGN) channel with feedback. However, there is limited understanding of how these black-box-like codes with many learned parameters use feedback. This study aims to uncover the fundamental principles underlying the first deep-learned feedback code, known as Deepcode, which is based on an RNN architecture. Our interpretable model based on Deepcode is built by analyzing the influence length of inputs and approximating the non-linear dynamics of the original black-box RNN encoder. Numerical experiments demonstrate that our interpretable model -- which includes both an encoder and a decoder -- achieves comparable performance to Deepcode while offering an interpretation of how it employs feedback for error correction.

Non-characterizability of belief revision: an application of finite model theory

A formal framework is given for the characterizability of a class of belief revision operators, defined using minimization over a class of partial preorders, by postulates. It is shown that for partial orders characterizability implies a definability property of the class of partial orders in monadic second-order logic. Based on a non-definability result for a class of partial orders, an example is given of a non-characterizable class of revision operators. This appears to be the first non-characterizability result in belief revision.