Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1705.06822

The Cayley-Dickson Construction in ACL2

John Cowles·Ruben Gamboa
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Cayley-Dickson Construction is a generalization of the familiar construction of the complex numbers from pairs of real numbers. The complex numbers can be viewed as two-dimensional vectors equipped with a multiplication. The construction can be used to construct, not only the two-dimensional Complex Numbers, but also the four-dimensional Quaternions and the eight-dimensional Octonions. Each of these vector spaces has a vector multiplication, v_1*v_2, that satisfies: 1. Each nonzero vector has a multiplicative inverse. 2. For the Euclidean length of a vector |v|, |v_1 * v_2| = |v_1| |v2|. Real numbers can also be viewed as (one-dimensional) vectors with the above two properties. ACL2(r) is used to explore this question: Given a vector space, equipped with a multiplication, satisfying the Euclidean length condition 2, given above. Make pairs of vectors into "new" vectors with a multiplication. When do the newly constructed vectors also satisfy condition 2?

Explore related subjects

cs.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

John Cowles, Ruben Gamboa. 2017-05-03. The Cayley-Dickson Construction in ACL2. https://doi.org/10.4204/eptcs.249.2

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Deciding Predicate Logical Theories of Real-Valued Functions

The notion of a real-valued function is central to mathematics, computer science, and many other scientific fields. Despite this importance, there are hardly any positive results on decision procedures for predicate logical theories that reason about real-valued functions. This paper defines a first-order predicate language for reasoning about multi-dimensional smooth real-valued functions and their derivatives, and demonstrates that - despite the obvious undecidability barriers - certain positive decidability results for such a language are indeed possible.

cs.LO↗
arXiv2026

Structural Liveness of Conservative Petri Nets

We show that the EXPSPACE-hardness result for structural liveness of Petri nets [Jancar and Purser, 2019] holds even for a simple subclass of conservative nets. As our main result, we prove that for structurally live conservative nets, the values of the minimal live markings are at most doubly exponential in the size of the net. This implies the EXPSPACE-completeness of structural liveness for conservative Petri nets. The result also applies to structurally bounded Petri nets, whereas the complexity of the general case remains open. As a proof ingredient of independent interest, we present an extension of known results on the bounds of minimal integer solutions to Boolean combinations of linear equalities, inequalities, and divisibility constraints.

cs.LO↗
arXiv2026

Verifying Numerical Methods with Isabelle/HOL

Modern machine learning pipelines and ODE solvers are built on numerical algorithms. Reliable numerical methods are thus a prerequisite for trustworthy machine learning and cyber-physical systems. We evaluate a framework designed for verifying imperative programs and the Isabelle proof assistant as tools for proving the total correctness of four numerical algorithms: the bisection method, the fixed-point method, the perceptron, and the gradient descent algorithm. Our verifications required subtle extensions and generalisations to Isabelle's version of Taylor's theorem and higher-order derivatives. Finally, we reflect on the framework's automation, friendly syntax, and on further requirements to turn it into a verification tool for numerical methods.

cs.LO↗