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The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the…
In 1998 Hoholdt, van Lint and Pellikaan introduced the concept of a ``weight function'' defined on a F_q-algebra and used it to construct linear codes, obtaining among them the algebraic-geometric (AG) codes supported on one point. Later it…
Recent advances in large language models (LLMs) have demonstrated impressive capabilities in formal theorem proving, particularly on contest-based mathematical benchmarks like the IMO. However, these contests do not reflect the depth,…
Listwise rerankers based on large language models (LLM) are the zero-shot state-of-the-art. However, current works in this direction all depend on the GPT models, making it a single point of failure in scientific reproducibility. Moreover,…
This short expository paper outlines applications of computer algebra to the implication problem of conditional independence for Gaussian random variables. We touch on certificates for validity and invalidity of inference rules from the…
We tackle the problem of deciding whether two probabilistic programs are equivalent in Probabilistic NetKAT, a formal language for specifying and reasoning about the behavior of packet-switched networks. We show that the problem is…
This paper introduces a SAT-based technique that calculates a compact and complete symmetry-break for finite model finding, with the focus on structures with a single binary operation (magmas). Classes of algebraic structures are typically…
We provide an extension of concurrent Kleene algebras to account for probabilistic properties. The algebra yields a unified framework containing nondeterminism, concurrency and probability and is sound with respect to the set of…
We introduce a new cross-validation method based on an equicorrelated Gaussian randomization scheme. Our method is well-suited for problems where sample splitting is infeasible, either because the data violate the assumption of independent…
Large language models (LLMs) achieve strong results on knowledge graph question answering (KGQA), but most benchmarks assume complete knowledge graphs (KGs) where direct supporting triples exist. This reduces evaluation to shallow retrieval…
Knowledge graphs offer a structured representation of real-world entities and their relationships, enabling a wide range of applications from information retrieval to automated reasoning. In this paper, we conduct a systematic comparison…
The principal goal of Group Testing (GT) is to identify a small subset of "defective" items from a large population, by grouping items into as few test pools as possible. The test outcome of a pool is positive if it contains at least one…
This paper discusses the no-cloning theorem in a logico-algebraic approach. In this approach, an orthoalgebra is considered as a general structure for propositions in a physical theory. We proved that an orthoalgebra admits cloning…
Adaptive methods are extremely popular in machine learning as they make learning rate tuning less expensive. This paper introduces a novel optimization algorithm named KATE, which presents a scale-invariant adaptation of the well-known…
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…
We investigate the equational theory for Kleene algebra terms with variable complements and constant complements -- (language) complement where it applies only to variables or constants -- w.r.t. languages. While the equational theory…
We present Alias Refinement Types (ART), a new approach to the verification of correctness properties of linked data structures. While there are many techniques for checking that a heap-manipulating program adheres to its specification,…
The categoricity spectrum of a class of structures is the collection of cardinals in which the class has a single model up to isomorphism. Assuming that cardinal exponentiation is injective (a weakening of the generalized continuum…
Conformal inference, cross-validation+, and the jackknife+ are hold-out methods that can be combined with virtually any machine learning algorithm to construct prediction sets with guaranteed marginal coverage. In this paper, we develop…
We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational…