arXiv subjects

Lucius T. Schoenbaum

Publications and source records attributed to Lucius T. Schoenbaum.

5 recordsLinked to original sources

arXiv2021

A Tapered Floating Point Extension for the Redundant Signed Radix 2 System Using the Canonical Recoding

A tapered floating point encoding is proposed which uses the redundant signed radix 2 system and is based on the canonical recoding. By making use of ternary technology, the encoding has a dynamic range exceeding that of the recently-proposed Posit number system and the IEEE 754-1985 Standard for Floating Point Arithmetic (IEEE-754-1985), and precision equal to or better than that of the IEEE-754-1985 system and the recently proposed Posit system when equal input sizes are compared. In addition, the encoding is capable of supporting several proposed extensions, including extensions to integers, boolean values, complex numbers, higher number systems, low-dimensional vectors, and system artifacts such as machine instructions. A detailed analytic comparison is provided between the proposed encoding, the IEEE-754-1985 system, and the recently proposed Posit number system.

cs.AR↗

arXiv2018

Towards Theory and Applications of Generalized Categories to Areas of Type Theory and Categorical Logic

Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to higher category theory, we undertake a detailed study of a new mathematical abstraction, the generalized category. It is a partially defined monoid equipped with endomorphism maps defining sources and targets on arbitrary elements, possibly allowing a proximal behavior with respect to composition. We first present a formal introduction to the theory of generalized categories. We describe functors, equivalences, natural transformations, adjoints, and limits in the generalized setting. Next we indicate how the theory of monads extends to generalized categories. Next, we present a variant of the calculus of deductive systems developed by Lambek, and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed categories and exponential-preserving morphisms that leverages the theory of generalized categories. Next, we develop elementary topos theory in the generalized setting of ideal toposes, utilizing the formalism developed for the Curry-Howard-Lambek theorem. In particular, we prove that ideal toposes possess the same Heyting algebra structure and squares of adjoints that ordinary toposes do. Finally, we develop generalized sheaves, and show that such categories form ideal toposes. We extend Lawvere and Tierney's theorem relating $j$-sheaves and sheaves in the sense of Grothendieck to the generalized setting.

math.CT↗

arXiv2018

Simulations to Analyze Cellular Voting Systems for Side Effects of Democratic Redistricting

Motivated by the problem of partisan gerrymandering, we introduce an electoral system for a representative democracy called democratic cellular voting, designed to make modern packing and cracking strategies irrelevant by allowing districts to be influenced directly by voters through elections. We introduce an example of a democratic cellular voting system, called CV0, that is suitable for dynamic modelling. We develop a modification of the theory of discrete Markov chains using the algebraic structure of the semiring $[0,\infty]$, which is used as a space of correlation coefficients. We use this to measure voter preferences and model representatives, voters, and districts in computationally feasible models with a guarantee of long-term stability.

physics.soc-ph↗

arXiv2017

Foundations of a Recent Extension of Category Theory and Topos Theory

This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural transformations, adjoints, and limits in a generalized setting, giving a concise outline of these frequently arising constructions.

math.CT↗

arXiv2010

On the Syntax of Logic and Set Theory

We introduce an extension of the propositional calculus to include abstracts of predicates and quantifiers, employing a single rule along with a novel comprehension schema and a principle of extensionality, which are substituted for the Bernays postulates for quantifiers and the comprehension schemata of ZF and other set theories. We prove that it is consistent in any finite Boolean subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti, and others, and discuss the relationship of the system to other set theoretic systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order quantification and abstraction, and then very briefly discuss the application of one of these methods to areas of mathematics outside of logic.

math.LO↗