Related papers: Kleene's Two Kinds of Recursion

First and second order corrections for the scattering of different types of particles by a weak gravitational field, treated as an external field, are calculated. These computations indicate a violation of the Equivalence Principle: to…

Different constructions in the recursion theory use the so-called priority arguments. A general scheme was suggested by A.~Lachlan. Based on his work, we define the notion of a priority-closed class of requirements. Then, for a specific…

We define two classes of functions, called regular (respectively, first-order) list functions, which manipulate objects such as lists, lists of lists, pairs of lists, lists of pairs of lists, etc. The definition is in the style of regular…

We provide an amendment to the first theorem of "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design" by Manchester & Slotine in the form of an additional technical condition required to show…

Open sets are central to mathematics, especially analysis and topology, in ways few notions are. In most, if not all, computational approaches to mathematics, open sets are only studied indirectly via their 'codes' or 'representations'. In…

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions…

Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $\mathbf{LK}$, several other proofs have been proposed. Even though the…

The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this…

We provide a new proof of a important theorem in the Lagrangian formalism about necessary and sufficient conditions for a second-order variational system of equations to follow from a first-order Lagrangian.

First-order methods (FOMs) have recently been applied and analyzed for solving problems with complicated functional constraints. Existing works show that FOMs for functional constrained problems have lower-order convergence rates than those…

This paper proposes a novel formulation for reinforcement learning (RL) with large language models, explaining why and under what conditions the true sequence-level reward can be optimized via a surrogate token-level objective in policy…

Using a regularization with the properties of dimensional regularization, higher order local consistency conditions on one loop anomalies and divergent counterterms are given. They are derived without any a priori assumption on the form of…

Classically, the time complexity of a first-order method is estimated by its number of gradient computations. In this paper, we study a more refined complexity by taking into account the `lingering' of gradients: once a gradient is computed…

IRT models are being increasingly used worldwide for test construction and scoring. The study examines the practical implications of estimating individual scores in a paper-and-pencil high-stakes test using 2PL and 3PL models, specifically…

This work is devoted to the study of the existence and sign of Green's functions for first order linear problems with constant coefficients and initial (one point) conditions. We first prove a result on the existence of solutions of $n$-th…

In this paper we use the Recursion Theorem to show the existence of various infinite sequences and sets. Our main result is that there is an increasing sequence e_0, e_1, e_2 .. such that W_{e_n}={e_{n+1}} for every n. Similarly, we prove…

Large language models (LLMs) are typically trained by reinforcement learning (RL) with verifiable rewards (RLVR) and supervised fine-tuning (SFT) on reasoning traces to improve their reasoning abilities. However, how these methods shape…

Pronk's theorem on bicategories of fractions is applied, in almost all cases in the literature, to 2-categories of geometrically presentable stacks on a 1-site. We give an proof that subsumes all previous such results and which is purely…

Bilevel reinforcement learning (RL), which features intertwined two-level problems, has attracted growing interest recently. The inherent non-convexity of the lower-level RL problem is, however, to be an impediment to developing bilevel…

The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction…