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A class of relative importance measures based on orthogonalization and reallocation, ORMs, has been found to effectively approximate the General Dominance index (GD). In particular, Johnson's Relative Weight (RW) has been deemed the most…
In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from $\RT$, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five"…
Despite the superior performance of Large Reasoning Models (LRMs), their reasoning behaviors are often counterintuitive, leading to suboptimal reasoning capabilities. To theoretically formalize the desired reasoning behaviors, this paper…
Continuity is one of the most central notions in mathematics, physics, and computer science. An interesting associated topic is decompositions of continuity, where continuity is shown to be equivalent to the combination of two or more weak…
In his book, John Stillwell wrote "finding the exact strength of the Brouwer invariance theorems seems to me one of the most interesting open problems in reverse mathematics." In this article, we solve Stillwell's problem by showing that…
This paper presents a reverse mathematical analysis of several forms of the sorites paradox. We first illustrate how traditional formulations are reliant on H\"older's Representation Theorem for ordered Archimedean groups. While this is…
This paper falls within the general program of investigating the proof theoretic strength (in terms of reverse mathematics) of combinatorial principals which follow from versions of Ramsey's theorem. We examine two statements in graph…
Reasoning in large language models has long been a central research focus, and recent studies employing reinforcement learning (RL) have introduced diverse methods that yield substantial performance gains with minimal or even no external…
Reversible computing is motivated by both pragmatic and foundational considerations arising from a variety of disciplines. We take a particular path through the development of reversible computation, emphasizing compositional reversible…
Argument mining (AM) is the process of automatically extracting arguments, their components and/or relations amongst arguments and components from text. As the number of platforms supporting online debate increases, the need for AM becomes…
We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a $\subseteq$-maximal…
Neural reasoners such as Tiny Recursive Models (TRMs) solve complex problems by combining neural backbones with specialized inference schemes. Such inference schemes have been a central component of stochastic reasoning systems, where…
We deal with the random combinatorial structures called assemblies. By weakening the logarithmic condition which assures regularity of the number of components of a given order, we extend the notion of logarithmic assemblies. Using the…
Despite the success of test-time scaling, Large Reasoning Models (LRMs) frequently encounter repetitive loops that lead to computational waste and inference failure. In this paper, we identify a distinct failure mode termed Circular…
We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$…
A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications. This has both numerical…
In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…
We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original…
Current large language models (LLMs) primarily rely on linear sequence generation and massive parameter counts, yet they severely struggle with complex algorithmic reasoning. While recent reasoning architectures, such as the Hierarchical…
In the era of Large Language Models (LLMs), alignment has emerged as a fundamental yet challenging problem in the pursuit of more reliable, controllable, and capable machine intelligence. The recent success of reasoning models and…