Scott McCallum

James H. Davenport, Matthew England, Scott McCallum

This paper presents two enhancements to cylindrical algebraic decomposition (CAD) based quantifier elimination (QE) for cases in which multiple equational constraints are present in the given input formula $ϕ^*$. The first enhancement provides more detail in the output when there is a conceptual partition of the set of variables of $ϕ^*$ into parameters and unknowns. In such cases, we describe how to partition the parameter space so that: (1) in each open set of the partition the number $ν$ of associated unknowns is a finite constant or is infinite; and (2) for each such open set for which $ν$ is finite, an expression for the unknowns in terms of the parameters is provided. The second enhancement is an efficiency gain achievable in certain situations. Indeed, when certain conditions are met, the second CAD equational projection step can be reduced more significantly than is supported by the prior existing theory. Relevant theorems and worked examples for both enhancements are provided. Application areas include approximation theory, cuspidal manipulator classification, and biological/chemical systems.

Gregory D. Baker, Scott McCallum, Dirk Pattinson

Many real-world machine learning problems involve inherently hierarchical data, yet traditional approaches rely on Euclidean metrics that fail to capture the discrete, branching nature of hierarchical relationships. We present a theoretical foundation for machine learning in p-adic metric spaces, which naturally respect hierarchical structure. Our main result proves that an n-dimensional plane minimizing the p-adic sum of distances to points in a dataset must pass through at least n + 1 of those points -- a striking contrast to Euclidean regression that highlights how p-adic metrics better align with the discrete nature of hierarchical data. As a corollary, a polynomial of degree n constructed to minimise the p-adic sum of residuals will pass through at least n + 1 points. As a further corollary, a polynomial of degree n approximating a higher degree polynomial at a finite number of points will yield a difference polynomial that has distinct rational roots. We demonstrate the practical significance of this result through two applications in natural language processing: analyzing hierarchical taxonomies and modeling grammatical morphology. These results suggest that p-adic metrics may be fundamental to properly handling hierarchical data structures in machine learning. In hierarchical data, interpolation between points often makes less sense than selecting actual observed points as representatives.