Related papers: On Rational Recursive Sequences
Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding recurrence relations is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements…
We study the growth behaviour of rational linear recurrence sequences. We show that for low-order sequences, divergence is decidable in polynomial time. We also exhibit a polynomial-time algorithm which takes as input a divergent rational…
We consider estimation procedures which are recursive in the sense that each successive estimator is obtained from the previous one by a simple adjustment. We propose a wide class of recursive estimation procedures for the general…
Finding whether a linear-constraint loop has a linear ranking function is an important key to understanding the loop behavior, proving its termination and establishing iteration bounds. If no preconditions are provided, the decision problem…
Relational concept analysis (RCA) is an extension of formal concept analysis allowing to deal with several related contexts simultaneously. It has been designed for learning description logic theories from data and used within various…
Sequences of rational powers \left( \xi\left( \frac{p}{q} \right)^{n} \right)_{n\ge 0}, especially in the case \frac{p}{q}=\frac{3}{2}, have a connection with many important combinatorics and number theory problems as for example Syracuse,…
We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are…
We define the syntax and reduction relation of a recursively typed lambda calculus with a parallel case-function (a parallel conditional). The reduction is shown to be confluent. We interpret the recursive types as information systems in a…
Complex functions have multiple uses in various fields of study, so analyze their characteristics it is of extensive interest to other sciences. This work begins with a particular class of rational functions of a complex variable; over this…
D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and…
Pre-trained language models achieves high performance on machine reading comprehension (MRC) tasks but the results are hard to explain. An appealing approach to make models explainable is to provide rationales for its decision. To…
This paper presents the Relational Machine Calculus (RMC): a simple, foundational model of first-order relational programming. The RMC originates from the Functional Machine Calculus (FMC), which generalizes the lambda-calculus and its…
We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…
In this article one builds a class of recursive sets, one establishes properties of these sets, and one proposes applications.
Circular (or cyclic) proofs have received increasing attention in recent years, and have been proposed as an alternative setting for studying (co)inductive reasoning. In particular, now several type systems based on circular reasoning have…
Polynomially-recursive sequences generally have a periodic behavior mod $m$. In this paper, we analyze the period mod $m$ of a second order polynomially-recursive sequence. The problem originally comes from an enumeration of avoiding…
Given a full column rank matrix $A \in \mathbb{R}^{m\times n}$ ($m\geq n$), we consider a special class of linear systems of the form $A^\top Ax=A^\top b+c$ with $x, c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. The occurrence of $c$ in…
We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone…
We study a quadruple of interrelated subexponential subsystems of arithmetic WKL$_0^-$, RCA$^-_0$, I$\Delta_0$, and $\Delta$RA$_1$, which complement the similarly related quadruple WKL$_0$, RCA$_0$, I$\Sigma_1$, and PRA studied by Simpson,…
Recursive reasoning models such as Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM) show that small, weight-shared networks can solve compute-heavy and NP puzzles by iteratively refining latent states, but their training…