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One of the major open problems in automata and logic is the following: is there an algorithm which inputs a regular tree language and decides if the language can be defined in first-order logic? The goal of this paper is to present this…
We present an algorithm, called the Offset Tree, for learning to make decisions in situations where the payoff of only one choice is observed, rather than all choices. The algorithm reduces this setting to binary classification, allowing…
Finite-state tree automata are a well studied formalism for representing term languages. This paper studies the problem of determining the regularity of the set of instances of a finite set of terms with variables, where each variable is…
We define a notion of randomness for individual and collections of formal languages based on automatic martingales acting on sequences of words from some underlying domain. An automatic martingale bets if the incoming word belongs to the…
The issue of estimating residual variance in regression models has experienced relatively little attention in the machine learning community. However, the estimate is of primary interest in many practical applications, e.g. as a primary…
We define an algorithm k which takes a connected graph G on a totally ordered vertex set and returns an increasing tree R (which is not necessarily a subtree of G). We characterize the set of graphs G such that k(G)=R. Because this set has…
We study the class of quasi-alphabetic relations, i.e., tree transformations defined by tree bimorphisms with two quasi-alphabetic tree homomorphisms and a regular tree language. We present a canonical representation of these relations; as…
When trained on language data, do transformers learn some arbitrary computation that utilizes the full capacity of the architecture or do they learn a simpler, tree-like computation, hypothesized to underlie compositional meaning systems…
This paper presents a decidable characterization of tree languages that can be defined by a boolean combination of Sigma_1 sentences. This is a tree extension of the Simon theorem, which says that a string language can be defined by a…
Let $T$ be a weighted tree. The weight of a subtree $T_1$ of $T$ is defined as the product of weights of vertices and edges of $T_1$. We obtain a linear-time algorithm to count the sum of weights of subtrees of $T$. As applications, we…
Traditional language models treat language as a finite state automaton on a probability space over words. This is a very strong assumption when modeling something inherently complex such as language. In this paper, we challenge this by…
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more…
Most of modern neural machine translation (NMT) models are based on an encoder-decoder framework with an attention mechanism. While they perform well on standard datasets, they can have trouble in translation of long inputs that are rare or…
This note presents a method of interpreting the tree adjoining languages as the natural third step in a hierarchy that starts with the regular and the context-free languages. The central notion in this account is that of a higher-order…
We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and…
The focus of this paper is the analysis of real-time systems with recursion, through the development of good theoretical techniques which are implementable. Time is modeled using clock variables, and recursion using stacks. Our technique…
We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic…
Tree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, R\'acz, and Rashtchian used combinatorial…
Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if the tree contains a node n levels from the root.…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen randomly proportionally to its weight. Under some assumptions on the sequence of…