Related papers: Automatic discovery of structural rules of permuta…
We show that all permutations in $S_n$ can be generated by affine unicritical polynomials. We use the $\operatorname{PGL}$ group structure to compute the cycle structure of permutations with low Carlitz rank. The tree structure of the group…
A generate and test algorithm is described which parses a surface form into one or more lexical entries using linearly ordered phonological rules. This algorithm avoids the exponential expansion of search space which a naive parsing…
Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with…
We review and extend previous results on coincidence of mesh patterns. We introduce the notion of a force on a permutation pattern and apply it to the coincidence classification of mesh patterns, completing the classification up to size…
We present a constructive recognition algorithm to decide whether a given black-box group is isomorphic to an alternating or a symmetric group without prior knowledge of the degree. This eliminates the major gap in known algorithms, as they…
Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a…
Transformation-based learning has been successfully employed to solve many natural language processing problems. It has many positive features, but one drawback is that it does not provide estimates of class membership probabilities. In…
In recent years, the interest in interpretable classification models has grown. One of the proposed ways to improve the interpretability of a rule-based classification model is to use sets (unordered collections) of rules, instead of lists…
We introduce a use of the \(M\)-cover (or \(M\)-layer) transform for machine learning. The method replicates a model \(M\) times, but instead of coupling the copies through parameter averaging or an explicit attractive force, as in…
Complex networks provide a powerful mathematical representation of complex systems in nature and society. To understand complex networks, it is crucial to explore their internal structures, also called structural regularities. The task of…
Transformers have recently been shown to be capable of reliably performing logical reasoning over facts and rules expressed in natural language, but abductive reasoning - inference to the best explanation of an unexpected observation - has…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
Despite its importance, choosing the structural form of the kernel in nonparametric regression remains a black art. We define a space of kernel structures which are built compositionally by adding and multiplying a small number of base…
We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise…
This paper addresses the general problem of modelling and learning rank data with ties. We propose a probabilistic generative model, that models the process as permutations over partitions. This results in super-exponential combinatorial…
Any permutation has a disjoint cycle decomposition and concept generates an equivalence class on the symmetry group called the cycle-type. The main focus of this work is on permutations of restricted cycle-types, with particular emphasis on…
We demonstrate how by using a reinforcement learning algorithm, the deep cross-entropy method, one can find explicit constructions and counterexamples to several open conjectures in extremal combinatorics and graph theory. Amongst the…
Rewriting is a formalism widely used in computer science and mathematical logic. The classical formalism has been extended, in the context of functional languages, with an order over the rules and, in the context of rewrite based languages,…
We here introduce a novel classification approach adopted from the nonlinear model identification framework, which jointly addresses the feature selection and classifier design tasks. The classifier is constructed as a polynomial expansion…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…