Related papers: Specializing Wide Aronszajn Trees without Adding R…
An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow…
In this paper, we extend the notion of (word) derivatives and partial derivatives due to (respectively) Brzozowski and Antimirov to tree derivatives using already known inductive formulae of quotients. We define a new family of extended…
We introduce a new geometric tool for analyzing groups of finite automata. To each finite automaton we associate a square complex. The square complex is covered by a product of two trees iff the automaton is bi-reversible. Using this method…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…
We advance the recently proposed neuro-symbolic Differentiable Tree Machine, which learns tree operations using a combination of transformers and Tensor Product Representations. We investigate the architecture and propose two key…
We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root…
We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove…
We investigate active learning by pairwise similarity over the leaves of trees originating from hierarchical clustering procedures. In the realizable setting, we provide a full characterization of the number of queries needed to achieve…
In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily…
The empirical success of deep learning is often attributed to deep networks' ability to exploit hierarchical structure in data, constructing increasingly complex features across layers. Yet despite substantial progress in deep learning…
A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time…
We present several results that rely on arguments involving the combinatorics of "bushy trees". These include the fact that there are arbitrarily slow-growing diagonally noncomputable (DNC) functions that compute no Kurtz random real, as…
We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for…
We study effective randomness-preserving transformations of path-incompressible trees. Some path-incompressible trees with infinitely many paths do not compute perfect path-random trees with computable oracle-use. Sparse perfect…
We give an amalgamation construction of free multiple trees with a strongly transitive automorphism group. The construction shows that any partial codistance function on a tuple of finite trees can be extended to yield multiple trees.
We analyze the trade-off between model complexity and accuracy for random forests by breaking the trees up into individual classification rules and selecting a subset of them. We show experimentally that already a few rules are sufficient…
A well-known conjecture of Orlov asks whether the existence of a full exceptional collection implies rationality of the underlying variety. We prove this conjecture for arithmetic toric varieties over general fields. We also investigate a…
A set of real $n$th roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.
Reverse search is a convenient method for enumerating structured objects, that can be used both to address theoretical issues and to solve data mining problems. This method has already been successfully developed to handle unordered trees.…
We study stochastic processes that generate non-growing complex networks without self-loops and multiple edges (simple graphs). The work concentrates on understanding and formulation of constraints which keep the rewiring stochastic…