Related papers: Nested Recursions, Simultaneous Parameters and Tre…
A family of nested recurrence relations $a(n+1) = n - a^{(m)}(n) + a^{(m+1)}(n)$, parameterized by an integer $m \ge 1$ with initial condition $a(1)=1$, is studied. We prove that $a(n)=n-h(n)$ is the unique solution satisfying this…
This paper studies the performances of BERT combined with tree structure in short sentence ranking task. In retrieval-based question answering system, we retrieve the most similar question of the query question by ranking all the questions…
We introduce a general recursive method to construct continuum random trees (CRTs) from independent copies of a random string of beads, that is, any random interval equipped with a random discrete probability measure, and from related…
The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration…
We show a new simple algorithm that solves the model-checking problem for recursion schemes: check whether the tree generated by a given higher-order recursion scheme is accepted by a given alternating parity automaton. The algorithm…
Tree kernels are fundamental tools that have been leveraged in many applications, particularly those based on machine learning for Natural Language Processing tasks. In this paper, we devise a parallel implementation of the sequential…
Net-trees are a general purpose data structure for metric data that have been used to solve a wide range of algorithmic problems. We give a simple randomized algorithm to construct net-trees on doubling metrics using $O(n\log n)$ time in…
We explore a physical model of ordered sums of integers as trains of rods. The trains for a fixed, possibly infinite, set of rod lengths naturally correspond to nodes in a tree; relations among finite linear recursions encoded in the…
For any real-valued $k > 1$, we consider the tree rooted at 0, where each positive integer $n$ has parent $\lfloor\frac{n}{k}\rfloor$. The average number of children per node is $k$, thus this definition gives a natural way to extend…
Given an ensemble of randomized regression trees, it is possible to restructure them as a collection of multilayered neural networks with particular connection weights. Following this principle, we reformulate the random forest method of…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on reasoning frameworks for path expressions where node…
We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant $k$, we construct linear decision trees that solve the $k$-SUM problem on $n$ elements…
In order to speed-up classification models when facing a large number of categories, one usual approach consists in organizing the categories in a particular structure, this structure being then used as a way to speed-up the prediction…
Complex reasoning problems are most clearly and easily specified using logical rules, but require recursive rules with aggregation such as count and sum for practical applications. Unfortunately, the meaning of such rules has been a…
Empirically, neural networks that attempt to learn programs from data have exhibited poor generalizability. Moreover, it has traditionally been difficult to reason about the behavior of these models beyond a certain level of input…
We introduce the concept of Random Sequential Renormalization (RSR) for arbitrary networks. RSR is a graph renormalization procedure that locally aggregates nodes to produce a coarse grained network. It is analogous to the (quasi-)parallel…
We study ranked enumeration of join-query results according to very general orders defined by selective dioids. Our main contribution is a framework for ranked enumeration over a class of dynamic programming problems that generalizes…
The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is…
This paper studies the "explanation problem" for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A whose…
This paper considers the enumeration of ternary trees (i.e. rooted ordered trees in which each vertex has 0 or 3 children) avoiding a contiguous ternary tree pattern. We begin by finding recurrence relations for several simple tree…