Related papers: On the equivalence of different presentations of T…
A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with…
This paper concerns the explicit treatment of substitutions in the lambda calculus. One of its contributions is the simplification and rationalization of the suspension calculus that embodies such a treatment. The earlier version of this…
The transformer is a neural network component that can be used to learn useful representations of sequences or sets of data-points. The transformer has driven recent advances in natural language processing, computer vision, and…
We formally define algorithmic capture of combinatorial tasks as the ability of a transformer to extrapolate to arbitrary task sizes with controllable error and logarithmic sample adaptation, providing a sharp scaling criterion for…
We study Milner's lambda-calculus with partial substitutions. Particularly, we show confluence on terms and metaterms, preservation of \b{eta}-strong normalisation and characterisation of strongly normalisable terms via an intersection…
One takes advantage of some basic properties of every homotopic $\lambda$-model (e.g.\ extensional Kan complex) to explore the higher $\beta\eta$-conversions, which would correspond to proofs of equality between terms of a theory of…
Terms in the lambda-calculus can be represented as planar trees decorated with symbols for abstraction and application, and having variables as leaves. In this paper, we concentrate on the branches of such trees, rather than on the trees…
The aim of this work is to characterize three fundamental normalization proprieties in lambda-calculus trough the Taylor expansion of $ \lambda$-terms. The general proof strategy consists in stating the dependence of ordinary reduction…
In a world of proliferating data, the ability to rapidly summarize text is growing in importance. Automatic summarization of text can be thought of as a sequence to sequence problem. Another area of natural language processing that solves a…
We give a formal treatment of simple type theories, such as the simply-typed $\lambda$-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional…
The amount of text data available online is increasing at a very fast pace hence text summarization has become essential. Most of the modern recommender and text classification systems require going through a huge amount of data. Manually…
The algebraic $\lambda$-calculus is an extension of the ordinary $\lambda$-calculus with linear combinations of terms. We establish that two ordinary $\lambda$-terms are equivalent in the algebraic $\lambda$-calculus iff they are…
Recently, there has been growing interest in the ability of Transformer-based models to produce meaningful embeddings of text with several applications, such as text similarity. Despite significant progress in the field, the explanations…
Abstraction is a desirable capability for deep learning models, which means to induce abstract concepts from concrete instances and flexibly apply them beyond the learning context. At the same time, there is a lack of clear understanding…
The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin-Barnes integral. The fundamental idea is to transform these integral…
Tensor completion is a natural higher-order generalization of matrix completion where the goal is to recover a low-rank tensor from sparse observations of its entries. Existing algorithms are either heuristic without provable guarantees,…
There is still a lot of confusion about "optimal" sharing in the lambda calculus, and its actual efficiency. In this article, we shall try to clarify some of these issues.
In terms of signal samples, we propose and justify a new rank reduced multi-term transform, abbreviated as MTT, which, under certain conditions, may provide better-associated accuracy than that of known optimal rank reduced transforms. The…
The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…
The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the…